Open and Closed-loop Model Identification and Validation Hernan Guidi ©

Open and Closed-loop Model Identification and Validation Hernan Guidi ©
Open and Closed-loop Model
Identification and Validation
Hernan Guidi
© University of Pretoria
Open and Closed-loop Model Identification
and Validation
by
Hernan Guidi
A dissertation submitted in partial fulfillment
of the requirements for the degree
Master of Engineering (Control Engineering)
in the
Department of Chemical Engineering
Faculty of Engineering, the Built Environment and Information
Technology
University of Pretoria
Pretoria
5th December 2008
© University of Pretoria
Open and Closed-loop Model Identification and
Validation
Author:
Date
Supervisor:
Department:
Degree:
Hernan Guidi
5th December 2008
Professor P. L. de Vaal
Department of Chemical Engineering
University of Pretoria
Master of Engineering (Control Engineering)
Synopsis
Closed-loop system identification and validation are important components in dynamic
system modelling. In this dissertation, a comprehensive literature survey is compiled on
system identification with a specific focus on closed-loop system identification and issues
of identification experiment design and model validation. This is followed by simulated
experiments on known linear and non-linear systems and experiments on a pilot scale distillation column. The aim of these experiments is to study several sensitivities between
identification experiment variables and the consequent accuracy of identified models and
discrimination capacity of validation sets given open and closed-loop conditions. The
identified model structure was limited to an ARX structure and the parameter estimation method to the prediction error method.
The identification and validation experiments provided the following findings regarding the effects of different feedback conditions:
• Models obtained from open-loop experiments produced the most accurate responses
when approximating the linear system. When approximating the non-linear system,
models obtained from closed-loop experiments were found to produce the most
accurate responses.
• Validation sets obtained from open-loop experiments were found to be most effective
in discriminating between models approximating the linear system while the same
may be said of validation sets obtained from closed-loop experiments for the nonlinear system.
These finding were mostly attributed to the condition that open-loop experiments produce
more informative data than closed-loop experiments given no constraints are imposed on
system outputs. In the case that system output constraints are imposed, closed-loop
experiments produce the more informative data of the two. In identifying the non-linear
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system and the distillation column it was established that defining a clear output range,
and consequently a region of dynamics to be identified, is very important when identifying
linear approximations of non-linear systems. Thus, since closed-loop experiments produce
more informative data given output constraints, the closed-loop experiments were more
effective on the non-liner systems.
Assessment into other identification experiment variables revealed the following:
• Pseudo-random binary signals were the most persistently exciting signals as they
were most consistent in producing models with accurate responses.
• Dither signals with frequency characteristics based on the system’s dominant dynamics produced models with more accurate responses.
• Setpoint changes were found to be very important in maximising the generation of
informative data for closed-loop experiments
Studying the literature surveyed and the results obtained from the identification and
validation experiments it is recommended that, when identifying linear models approximating a linear system and validating such models, open-loop experiments should be used
to produce data for identification and cross-validation. When identifying linear approximations of a non-linear system, defining a clear output range and region of dynamics is
essential and should be coupled with closed-loop experiments to generate data for identification and cross-validation.
Keywords: Closed-loop system identification; LTI approximations; Cross-validation;
Prediction error method; Identification experiment design, ARX
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Acknowledgements
I would like to give my sincere thanks to Professor Philip de Vaal for his guidance and
support on not only this dissertation but through out my academic career.
A mis padres y a mi hermano. Me siento eternamente endeudado y les agradezco de
mi corazn por todo su apoyo incesante y sin condiciones que ha sido inestimable para mi.
Los quiero mucho.
”Joy in looking and comprehending is nature’s most beautiful gift” – Albert Einstein
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CONTENTS
1 Introduction
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Philosophy and Purpose Behind Modelling and System Identification
1.2.1 Models to Satisfy Scientific Curiosity . . . . . . . . . . . . . . . .
1.2.2 Modelling for Diagnosis of Faults and Inadequacies . . . . . . . .
1.2.3 Models for Simulation . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Models for Prediction and Control . . . . . . . . . . . . . . . . . .
1.3 Model Structures and Model Types . . . . . . . . . . . . . . . . . . . . .
1.4 System Identification Approach and Concepts . . . . . . . . . . . . . . .
1.5 Closed-Loop System Identification . . . . . . . . . . . . . . . . . . . . . .
1.6 Identification Experiment Design . . . . . . . . . . . . . . . . . . . . . .
1.7 Model Validation and Discrimination . . . . . . . . . . . . . . . . . . . .
1.8 Linear Approximations of Non-linear systems . . . . . . . . . . . . . . .
1.9 Identification and Control . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Scope and Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . .
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2 System Identification Theory
2.1 Background and History . . . . . . . . . . . . . .
2.2 Linear System Representation and Fundamentals
2.2.1 Linear Response Modelling . . . . . . . . .
2.2.2 Transforms and Transfer Functions . . . .
2.2.3 Prediction and Parameterisation . . . . . .
2.2.4 A Class of Model Structures . . . . . . . .
2.3 Non-Linear Time-Varying System Identification .
2.3.1 Non-Linearity and System Memory . . . .
2.3.2 Identification Approaches and Methods for
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Non-linear
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4 Model Validation Theory
4.1 General Linear Model Validation . . . . . . . . . . . . . . . . . . . . . .
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Non-Parametric - Frequency Domain Methods . . . . . . . . .
2.4.1 Co-Correlation, Cross-Correlation and Spectral Density
2.4.2 The Empirical Transfer-function Estimate . . . . . . .
Parametric Estimation Methods . . . . . . . . . . . . . . . . .
2.5.1 Principles Behind Parameter Estimation . . . . . . . .
2.5.2 The Prediction Error Estimation Framework . . . . . .
2.5.3 Other Common Estimation Frameworks . . . . . . . .
Convergence and Asymptotic Properties . . . . . . . . . . . .
2.6.1 Convergence and Identifiability . . . . . . . . . . . . .
2.6.2 Asymptotic Distribution . . . . . . . . . . . . . . . . .
Multivariable System Identification . . . . . . . . . . . . . . .
2.7.1 Notation and System Description . . . . . . . . . . . .
2.7.2 Parameterisation and Estimation Methods . . . . . . .
2.7.3 Convergence and Asymptotic Properties . . . . . . . .
Model Validation . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . .
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3 Closed-Loop System Identification Theory
3.1 Basic Concepts, Issues and Method Classifications . . . . . . . . . .
3.1.1 Closed-Loop System Description, Notation and Conventions
3.1.2 Issues in Closed-Loop Identification . . . . . . . . . . . . . .
3.1.3 Problematic Identification Methods . . . . . . . . . . . . . .
3.1.4 Classification of Closed-Loop Identification Approaches . . .
3.2 Closed-Loop Identification in the Prediction Error Framework . . .
3.2.1 The Direct Method . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 The Indirect Method . . . . . . . . . . . . . . . . . . . . . .
3.2.3 The Joint Input-Output Method . . . . . . . . . . . . . . .
3.2.4 Overview on Closed-Loop Approaches . . . . . . . . . . . . .
3.3 Multivariable identification and Controller Specific Effects . . . . .
3.3.1 Multivariable Closed-Loop Identification . . . . . . . . . . .
3.3.2 Linear Decentralised Controllers . . . . . . . . . . . . . . . .
3.3.3 Non-linear Controllers and MPC . . . . . . . . . . . . . . .
3.4 Non-Linear Closed-Loop System Identification . . . . . . . . . . . .
3.4.1 The Youla-Kucera Parameter Approach . . . . . . . . . . . .
3.4.2 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.2
4.3
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4.1.1 Statistical Tools, Hypothesis Testing and Confidence Intervals
4.1.2 Model-comparison Based Validation . . . . . . . . . . . . . . .
4.1.3 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Simulation and Consistent Model Input-Output Behaviour . .
4.1.5 Residual Analysis . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.6 Model Uncertainty Considerations . . . . . . . . . . . . . . . .
Linear Time Invariant Approximations of Non-linear Systems . . . . .
4.2.1 LTI Aprroximation Sensitivity to System Non-Linearity . . . .
4.2.2 Validation of LTI Approximations of Non-linear Systems . . .
Multivariable Model Validation . . . . . . . . . . . . . . . . . . . . .
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Experimental Design Theory
5.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . .
5.2 Informative Experiments . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Informative Data . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Asymptotic Covariance and Information Matrices . . . .
5.3 Input Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Step Signals . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Gaussian White Noise . . . . . . . . . . . . . . . . . . .
5.3.3 Pseudorandom Sequences . . . . . . . . . . . . . . . . .
5.4 Other Design Variables . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Model Structure and Order Selection . . . . . . . . . . .
5.4.2 Bias Considerations and Sampling Intervals . . . . . . .
5.5 Closed-Loop Experiments . . . . . . . . . . . . . . . . . . . . .
5.6 Multivariable Inputs . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Data Preparation . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7.1 High-frequency Disturbances . . . . . . . . . . . . . . . .
5.7.2 Bursts and Outliers . . . . . . . . . . . . . . . . . . . . .
5.7.3 Slow Disturbances: Offset . . . . . . . . . . . . . . . . .
5.7.4 Slow Disturbances: Drift, Trends and Seasonal Variations
5.8 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . .
6 Identification of Simulated Systems
6.1 Investigative Approach and Model Description .
6.1.1 Investigative Approach . . . . . . . . . .
6.1.2 Linear Model - System A . . . . . . . . .
6.1.3 Non-linear Model - System B . . . . . .
6.2 Experimental Method : Framework and Design
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6.3
6.4
6.5
6.2.1 General Identification Method . . . . . . . . . . . . . . . . . . . .
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6.2.2 Model Structure and Order Selection . . . . . . . . . . . . . . . .
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6.2.3 Identification Experiment Conditions . . . . . . . . . . . . . . . .
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6.2.4 Validation Techniques and Experiment Conditions . . . . . . . . . 106
6.2.5 General Execution and Software . . . . . . . . . . . . . . . . . . . 110
Identification and Validation Results for System A - The Linear System . 111
6.3.1 Model Variance and Uncertainty . . . . . . . . . . . . . . . . . . . 112
6.3.2 Model Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3.3 Simulation and Prediction . . . . . . . . . . . . . . . . . . . . . . 116
6.3.4 Frequency Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3.5 Residual Correlation Analysis . . . . . . . . . . . . . . . . . . . . 129
6.3.6 Results Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Identification and Validation Results for System B - The Non-Linear System138
6.4.1 Model Variance and Uncertainty . . . . . . . . . . . . . . . . . . . 139
6.4.2 Simulation and Prediction . . . . . . . . . . . . . . . . . . . . . . 141
6.4.3 Frequency Content Analysis . . . . . . . . . . . . . . . . . . . . . 148
6.4.4 Residual Correlation Analysis . . . . . . . . . . . . . . . . . . . . 150
6.4.5 Results Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5.1 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5.2 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5.3 Other Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7 Identification of a Distillation Column
7.1 Investigative Approach and System Description . . . . . . . . . . . . .
7.1.1 Investigative Approach . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Description of the Pilot Scale Distillation Column . . . . . . . .
7.2 Experimental Method : Framework and Design . . . . . . . . . . . . .
7.2.1 System and Identification Problem Definition and Assumptions
7.2.2 Identification Framework and Validation Methods . . . . . . . .
7.2.3 Model Structure and Order Selection . . . . . . . . . . . . . . .
7.2.4 Identification and Validation Experiment Conditions . . . . . .
7.2.5 General Execution . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Identification and Validation Results . . . . . . . . . . . . . . . . . . .
7.3.1 Simulation and Prediction . . . . . . . . . . . . . . . . . . . . .
7.3.2 Frequency Content Analyses . . . . . . . . . . . . . . . . . . . .
7.3.3 Residual Correlation Analysis . . . . . . . . . . . . . . . . . . .
7.3.4 Results Overview . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A ’Real’ Identification Problem . . . . . . . . . . . . . . . . . . .
Experimental Conditions . . . . . . . . . . . . . . . . . . . . . . .
Validation Techniques . . . . . . . . . . . . . . . . . . . . . . . .
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8 Discussion and Conclusions
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8.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.1.1 Sensitivity to Noise and Model Uncertainty . . . . . . . . . . . . . 203
8.1.2 Closed-loop Vs. Open-Loop . . . . . . . . . . . . . . . . . . . . . 204
8.1.3 Linear Approximation of Non-linear Systems . . . . . . . . . . . . 205
8.1.4 Identification and Control . . . . . . . . . . . . . . . . . . . . . . 205
8.1.5 Simulation Vs. Experimentation and Beyond . . . . . . . . . . . . 206
8.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
8.2.1 Model Accuracy Sensitivity to Identification Experiment Conditions 207
8.2.2 Validation Set Bias and Discrimination Sensitivity to Validation
Experiment Conditions . . . . . . . . . . . . . . . . . . . . . . . . 209
8.2.3 Validation Techniques . . . . . . . . . . . . . . . . . . . . . . . . 209
8.2.4 Linear Approximation of Non-linear Dynamics . . . . . . . . . . . 210
8.2.5 Identification and Validation of a real System . . . . . . . . . . . 210
8.2.6 General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 211
9 Recommendations and Further Investigation
9.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Further Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.1 Identification of Linear Approximations of Non-linear Systems
9.2.2 Identification of the Pilot Scale Distillation Column . . . . . .
9.2.3 MPCI and Other Recommendations . . . . . . . . . . . . . . .
A Software and Identified Models
A.1 Description of Software Used and Compiled . . . . .
A.1.1 Signal Generation . . . . . . . . . . . . . . . .
A.1.2 Identification and Validation Data Generation
A.1.3 Model Generation . . . . . . . . . . . . . . . .
A.1.4 Model Validation . . . . . . . . . . . . . . . .
A.2 Model Parameters of Identified Models . . . . . . . .
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LIST OF FIGURES
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1.2
Flow diagram illustrating identification and validation logic sequence . .
Comparison of closed and open-loop configurations . . . . . . . . . . . .
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Possible Selection of Input Disturbances . . . . . . . . . . . . . . . . .
Linear System Response . . . . . . . . . . . . . . . . . . . . . . . . . .
Open-loop Transfer Function Representation . . . . . . . . . . . . . . .
General Structure Transfer Function Representation . . . . . . . . . . .
(a) Hammerstein Model (b) Wiener Model . . . . . . . . . . . . . . . .
Auto-correlation Function and Power Spectrum of a white noise signal .
Auto-correlation Function and Power Spectrum of a sinusoidal signal .
Possible Selection of Input Disturbances . . . . . . . . . . . . . . . . .
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Closed-loop system representation . . . . . . . . . . . . . . . . . . . . . .
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Model Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Residual plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a) Acceptable residual plot (b) - (d) Unacceptable residual plots
Output-Error Model Structure . . . . . . . . . . . . . . . . . . . .
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A white Gaussian signal . . . . . . . . . . . . . . . . . . . . . . . . . . .
A pseudorandom binary signal . . . . . . . . . . . . . . . . . . . . . . . .
A pseudorandom multilevel signal . . . . . . . . . . . . . . . . . . . . . .
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General distillation column . . . . . . . . . . . . . . . . . . . . . . . . .
Open-loop step response of linear system - system A . . . . . . . . . .
Closed-loop response to setpoint changes for linear system - system A .
Response to 2 and 5 percent step disturbances for the non-linear system
system B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Higher order correlation test for non-linear dynamics . . . . . . . . . .
91
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6.5
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6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.23
6.24
6.25
6.26
6.27
6.28
6.29
6.30
Closed-loop response to setpoint changes for non-linear system - system B 97
Open and closed-loop output excitation of system A . . . . . . . . . . . . 105
Illustration of data cross-validation in the context of data generation and
model identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Parameter and response variance due to random noise realisations . . . . 113
Monte Carlo Simulation for case CL 6 . . . . . . . . . . . . . . . . . . . . 115
Average output response standard deviations due to parameter uncertainty 115
Percentage fit values for simulation validation against open-loop validation
sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Percentage fit values for simulation validation against closed-loop validation sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Percentage fit simulations validated against ValOL 1 . . . . . . . . . . . 118
Model simulation revealing closed-loop identification sensitivity to setpoint
direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Frequency response comparing experimental cases of different identification signal frequency characteristics . . . . . . . . . . . . . . . . . . . . . 123
Frequency response comparing submodel responses to removal of identification signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Frequency responses of cases with indistinguishable simulation fits against
ValOL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Frequency analysis of simulation responses validated against ValOL 1 . . 128
Frequency analysis of simulation responses validated against ValCL 3 . . 128
Illustration of sensitivity of correlation tests to validation data for OL cases130
Illustration of sensitivity of correlation tests to validation data for CL cases 131
Input-Residual correlations for OL 3 correlated against ValOL 1 and ValCL3133
Input-Residual correlations for CL 4 correlated against ValOL 1 and ValCL3134
Parameter variance of non-linear system estimates due to random noise
realisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Standard deviations of output responses found through Monte Carlo simulations showing model uncertainty . . . . . . . . . . . . . . . . . . . . . 140
Simulation percentage fit values validated against open-loop validation
data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Simulation percentage fit values validated against closed-loop validation
data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Comparison between input signals generated under open and closed-loop
conditions for normal and larger disturbance ranges . . . . . . . . . . . . 144
Simulation validated against open-loop step response validation data set
NValOL 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
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6.31 Illustration of the characteristic problem in linear model approximation of
a non-linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.32 Comparison of input signal frequency content between normal input range
and larger input range validation sets. . . . . . . . . . . . . . . . . . . .
6.33 Comparison of output signal frequency content between normal input range
and larger input range validation sets. . . . . . . . . . . . . . . . . . . .
6.34 Residual correlation results for case NCL 8 using NValCL 4 and NValCL
5 as validation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.35 Residual correlation results for case NOL 6 using NValOL 1 and NValOL
2 as validation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.36 Input-Residual correlation between u1 and 1 , validated against closed-loop
data, for some of the cases showing similar results . . . . . . . . . . . . .
6.37 Input-Residual correlations between u1 and 1 , validated against closedloop data, for the cases showing results different correlations to the general
trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.38 Input-Residual correlation between u1 and 1 , validated against open-loop
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.39 Input-Residual correlations between u1 and 1 , validated against open-loop
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.40 Input-Residual correlations for case NOL 5 showing no common trend
between correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.41 Input-Residual correlations for case NOL 6 showing common trends between correlations with between residuals and the same inputs . . . . . .
6.42 Higher order correlation: Ru2 2 , of a select group of cases validated against
closed-loop data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.43 Higher order correlation: Ru2 , of a select group of cases validated against
closed-loop data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.44 Higher order correlation: Ru2 , of a select group of cases validated against
open-loop data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Detailed illustration of the pilot scale distillation column . . . . . . . . .
Diagram showing piping and Instrumentation of the pilot scale distillation
column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simplified illustration of relevant control loops . . . . . . . . . . . . . . .
Steam presssure control and manipulation . . . . . . . . . . . . . . . . .
Open-loop step response . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distillation column setpoint tracking . . . . . . . . . . . . . . . . . . . .
Input disturbance for open-loop distillation column identification experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
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145
149
149
152
152
154
154
155
156
157
157
158
159
159
169
171
172
174
175
175
179
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
7.18
7.19
7.20
7.21
7.22
(a)-(b) u2 and y2 values generated from closed-loop identification experiment DCL 2. (c)-(d) u2 and y2 values generated from open-loop identification experiment DOL 1 . . . . . . . . . . . . . . . . . . . . . . . . . .
Percentage fit values when validating against open-loop data sets . . . .
Percentage fit values when validating against closed-loop data sets . . . .
Validation simulations against set DVCL 1 . . . . . . . . . . . . . . . .
Validation simulations against set DVCL 3 . . . . . . . . . . . . . . . . .
Experimentation with and without dither signals . . . . . . . . . . . . .
Validation predictions against the closed-loop validation sets . . . . . . .
Frequency analysis of simulation validation for DOL 1 against DVCL 1 .
Residual auto-correlation for a case DOL 1 against validation sets DVCL
1, DVCL 3 and DVOL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
Residual auto-correlation for a case DCL 1 against validation sets DVCL
1, DVCL 3 and DVOL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
Residual auto-correlation for a cases DCL 1, DCL 2 and DOL 1 against
validation set DVOL 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Input-residual correaltions for DCL 1 . . . . . . . . . . . . . . . . . . . .
Input-residual correlations for DOL 1 . . . . . . . . . . . . . . . . . . . .
Higher order correlation profile, Ru2 , for DOL 1 against DVCL 1 . . . .
Higher order correlation profile, Ru2 , for DOL 1 against DVOL 1 . . . .
A.1 Illustration of the fundamental software components and the associated
matlab m-files. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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183
185
185
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188
189
190
191
193
193
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195
196
197
198
216
LIST OF TABLES
2.1
Common Model Structures . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.1
Significance of higher order correlation results . . . . . . . . . . . . . . .
72
6.1
6.2
6.3
6.4
6.5
Model parameters for the linear ARX model representing system A . . .
93
Model parameters for non-linear ARX model representing system B . . .
95
Open-loop experimental conditions for identification of the linear model . 101
Closed-loop experimental conditions for identification of the linear model 101
Closed-loop experimental conditions used for identification of the nonlinear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Open-loop experimental conditions for identification of the non-linear model103
Conditions for generation of validation data for linear model . . . . . . . 109
Conditions for generation of validation data for non-linear model . . . . . 109
6.6
6.7
6.8
7.1
7.2
7.3
7.4
7.5
7.6
Distillation column principle variables and normal operation ranges . . .
Steady state values used as initial states of experiments . . . . . . . . . .
Open-Loop cases used to generated data for identification of the distillation
column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Closed-Loop cases used to generated data for identification of the distillation column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Open-Loop cases used to generated data for validation of the distillation
column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Closed-Loop cases used to generated data for validation of the distillation
column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
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173
177
180
180
181
181
NOMENCLATURE
Roman Symbols
ū
Steady State Input
ȳ
Steady State Output
a, b, c, d
Regression Coefficients
Dc
The set into which θ̂N converges
E
Expectation
e
White Noise
fe (·)
Probability Density Function of Noise
G
System Transfer Function
g
Unit Impulse Response
G(q, θ)
Parameterised System Transfer Function
G0,OE
OE-LTI-SOE transfer function
G0
True System Transfer Function
H
Noise Transfer Function
P
Probability
Pθ
Asymptotic Covariance Matrix of θ
q
Time Shift Operator
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RSW
Cross-Correlation Function between Signals S and W
RS
Auto-Correlation Function of Signal S
s
Laplace Operator
S0
True Sensitivity Function
t
Time
TN
Total Experiment Time
u
System Input
v
System Disturbance Variable
VN
Criterion function to be minimised
W
Weighting function
y
System Output
ZN
Data Set
Acronyms
ACF
Auto-Correlation Function
AIC
Akeike’s Information Criterion
ARMA
Auto-Regressive with Moving Average
ARMAX
Auto-Regression with Moving Average Exogenous
ARX
Auto-Regressive with External Input
CCF
Cross-Correlation Function
ETFE
Empirical Transfer Function Estimate
ETFE
Empirical Transfer-function Estimate
FIR
Finite Impulse Response
FPE
Akeike’s Final Prediction Error
IID
Independant and Identically Distributed
IV
Instrumental Variable
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LS
Least-Squares
LTI
Linear Time Invariant
MA
Moving Average
ML
Maximum Likelihood
MLE
Maximum Likelihood Estimator
NARX
Non-linear auto-regression with external inputs
OE-LTI-SOE
Output Error LTI Second Order Equivalent
PDF
Probability Distribution Function
PEM
Prediction Error Method
PRBS
Pseudorandom Binary Sequence
PRBS
Pseudorandom Multilevel Sequence
RBS
Random Binary Signal
SNR
Signal-to-Noise Ratio
Greek Symbols
Prediction Error
F
Filtered Prediction Error
λ0
Variance of e(t)
ω
Frequency
ΦS
Power Spectrum of Signal S
ΦSW
Cross-Power Spectrum between signal S and W
ψ
The gradient of ŷ with respect the θ
σ
Signal Magnitude or Step Size
τ
Operator Index
θ
Parameter Vector
θ0
Parameter vector reflecting true system
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ϕ(t)
Regression Vector
ϕ(t, θ)
Pseudo-Linear Regression Vector
ζ(t)
Instrumental Vector
Subscripts
0
The True System
Prediction Error
k
Operator Index
N
Discrete Data Index
t
Time
u
System Input
v
System Disturbance
ML
Maximum Likelihood
Superscripts
cl
Closed Loop
m
Measured Value
T
Matrix or Vector Transpose
IV
Instrumental Variable
LS
Least-Squares
Others Symbols
R̄u
Covariance matrix of signal u
V̄N
Limit of the Criterion Function
`(·)
Prediction Error Norm
θ̂
Estimated Parameter Vector
ˆ
Ĝ( N )(ejωt )
Empirical Transfer Function Estimate
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θ̂N
Estimate of θ0 based on data set Z N
ŷ(t | θ)
Output prediction based on parameterised model
ŷ(t | t − 1)
Prediction of output y at time t based on past data
F
Fourier Transform
L
Laplace Transform
M
Model Structure
M(θ̂N )
Model member out of model set defined by θ estimate
S
The True System
Cov(x)
Covariance Matrix of vector x
Rd
Euclidian d-dimensional space
w.p. 1
With Probability 1
xix
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CHAPTER 1
Introduction
This chapter presents the introduction to this work. A concise presentation of the
problem statement and investigative approach is made followed by some introductory concepts and theory that further detail the investigative direction taken. The
last section of this chapter presents the structure of this dissertation by briefly
indicating what the subsequent chapters entail and contain.
1.1
Introduction
System identification deals with the problem of building mathematical models of dynamical systems from observed data. The generation of such observed data is dependent on
the identification experiment (Ljung, 1987). Given the system to be identified normally
operates as part of a closed-loop configuration, there are various experimental approaches
that may be used to generate identification data. An obvious approach is an open-loop
experiment where controller feedback loops are opened, the system is disturbed, and
data accumulated. Alternatively the identification experiment may be conducted under
closed-loop conditions where the controllers maintain their normal operating condition
and disturbances are introduced via setpoint changes and/or superimposed input signals.
Literature does indicate open-loop experiments to allow for more persistent excitation
of the system and generation of more informative data (Goodwin & Payne, 1977). The
question at hand however is what are the advantages and disadvantages of each approach
and how do they manifest themselves in the quality of the identified model. From this, the
importance of model validation is revealed since it is through validation that models are
discriminated and measures of accuracy are determined. Furthermore, there are several
other experiment design variables, namely those concerned with the characteristics of
disturbance signals. It is not so well understood how the sensitivities between these
1
© University of Pretoria
CHAPTER 1. INTRODUCTION
2
experimental variables and model accuracy change given different feedback conditions.
The question of how these sensitivities change further extends itself when considering the
effects of approximating a non-linear system with a linear model.
The objective of this work may thus be given by the following points:
• Compile an extensive literature survey on system identification approaches with a
specific focus on closed-loop system identification, model validation techniques and
identification experiment design.
• Through the use of known linear and non-linear models, simulate open and closedloop system identification and validation experiments to obtain and validate linear
approximations of these known models. From this, obtain an understanding with
regard to the sensitivities of the key identification variables, specifically the disturbance signal characteristics, and how they change given a non-linear system.
• Specifically concentrate on cross-validation approaches and assess how the different
experimental conditions used to generate validation data affect the discrimination
and bias of validation sets.
• Use a pilot scale distillation column to obtain an understanding regarding pragmatic
issues of implementing identification and validation techniques on a real system that
would not necessarily arise in simulated experiments.
While the literature survey will attempt to cover most of the pertinent methods and
techniques, the identification efforts through simulated experiments and those performed
on the distillation column will use the prediction error method as the identification
method and the direct approach for the closed-loop identification experiments. Furthermore, the model type and structure used will be limited to parametric ARX structures.
These choices will be explained and justified in the sections and chapters to come.
1.2
The Philosophy and Purpose Behind Modelling
and System Identification
It is appropriate at this point to mention a maxim accredited to statistician George Box:
”All models are wrong, but some are useful” (Box, 1979). From this one may gather
that pursuing the perfect model to perfectly represent the system is baseless and that,
defining the purpose for a model is the first step in obtaining a useful one. Dynamical
models have found application in areas as diverse as engineering and the hard sciences,
economics, medicine and the life sciences, ecology and agriculture (Norton, 1986). The
same few basic purposes underlie identification in all these fields.
© University of Pretoria
CHAPTER 1. INTRODUCTION
3
The next few sections further present some of the key model applications and purposes
for system identification.
1.2.1
Models to Satisfy Scientific Curiosity
Norton (1986) states that a principal characteristic of science is its use of mathematical
models to extract the essentials from complicated evidence and to quantify the implications. This is the essence of scientific modelling, to increase an understanding of some
mechanism by finding the connections between observations relating to it.
1.2.2
Modelling for Diagnosis of Faults and Inadequacies
A great benefit of system identification is its ability to uncover shortcomings and anomalies. In this application models are used in the disclosure of unexpected or atypical
behaviour of a system. It is generally understood that modelling for fault detection may
require efforts that are typically different to those required for other purposes. The main
reason for this is that, in order to assure a robust diagnostic system which is aptly able
to isolate faults, modelling of abnormal regions of operation is essential (Frank, 2000)
1.2.3
Models for Simulation
Mathematical models allow for the possibility to explore situations which in actuality
are hazardous, difficult or expensive to set up. Aircraft and space vehicle simulators are
good examples of this. Through such applications of models valuable insight may be
obtained with regard to the dynamics of the system being studied. Model accuracy and
comprehensiveness are essential for such purposes (Norton, 1986).
1.2.4
Models for Prediction and Control
The desire to predict is a common and powerful motive for dynamical modelling. From
a strictly pragmatic point of view, a prediction model should be judged solely on the
accuracy of its predictions. Norton (1986) states that the plausibility and simplicity of
the prediction model and its power to give insight are all incidental. Section 1.9 further
elaborates on the issues of system identification and control.
1.3
Model Structures and Model Types
Provided a purpose for modelling, the next most definitive issue is to find a suitable
model type and structure that would adequately represent the system keeping in mind
its intended purpose. Ljung (1987) states a basic rule in estimation, namely that, one
© University of Pretoria
CHAPTER 1. INTRODUCTION
4
should not estimate what is already known. In other words, one should utilise prior
knowledge and physical insight about the system when selecting the model structure
and type to be used. Literature customarily distinguishes between three types of models
based on the prior knowledge at hand. These are as follows (Sjöberg et al.):
White Box Models : This is the case when a model is perfectly known and has been
possible to construct it from prior knowledge and physical insight.
Grey Box Models : This is the case when some physical insight is available, but several
parameters remain to be determined from observed data. It is useful to consider
two sub-cases
• Physical Modelling: A model structure can be built on physical grounds, which
has a certain number of parameters to be estimated from data. This could for
example be a state space model of given order and structure.
• Semi-physical Modelling: Physical insight is used to suggest certain non-linear
combinations of measured data signal. These new signals are then subjected
to structures of black box character.
Black Box Models : No physical insight is available or used, but the chosen model
structure belongs to families that are known to have good flexibility and have been
successful in the past.
From these categories it is established that the type of model to be used in an identification task greatly depends on the information at hand. On the one end of the scale
are white box models, which are used when the physical laws governing the behaviour of
the system are known. On the other end are black box models where no physical insight
on the system is known.
It has been found that despite the quite simplistic nature of many black-box models,
they are frequently very efficient for modelling dynamical systems. In addition to this,
such models typically require less engineering time to construct than white-box models
(Forssell, 1999).
An additional model characteristic that is of importance is whether it is defined as
linear or non-linear. While it is understood that most systems in ”real-life” to some
degree are non-linear, linear models are used more in practice. It is true that they
provide idealised representations of the ”real-life” non-linear systems, even so, they are
justified in that they usually provide good results in many cases (Ljung, 1987).
More details on different models and modelling approaches are given in chapter 2
© University of Pretoria
CHAPTER 1. INTRODUCTION
1.4
5
System Identification Approach and Concepts
As has been mentioned earlier, system identification deals with the construction of models
from data. The components that characterise the procedure of such model constructions
are now further introduced, with figure 1.1 showing the procedural interactions between
the key components (Norton, 1986):
Problem Formulation : This initial step in identifying a system is to first establish all
that is necessary in order to determine the type of identification effort. That is,
whether a priori knowledge is available and what the purpose of the model will be.
Experimental Design : This section is perhaps the most important. Here the system
to be identified is defined in terms of the inputs, outputs and assumptions. Typically
the system to be identified is part of a larger system, it is thus necessary to make
assumptions and define which components will be identified. From these definitions
and assumptions, an experiment is designed. The key design variables are typically
the characteristics of the disturbance signal and whether the experiment is done
under open or closed-loop conditions (further discussed in section 1.5).
Data Preparation : Before the data obtained from the identification experiment may
be used to generate a model, it needs to be prepared. This preparation involves
cleaning the data of drifts, outliers and trends.
Estimation : This identification component involves the mapping of information from
the experimental data set to the model coefficients. This typically involves optimisation routines of functions with norms that are to be maximised or minimised.
Model Validation : This section is essential in determining the success of any identification efforts. While section 4.1 details the different approaches to model validation,
it is simply understood as the determination of whether the model is ”good enough”
for its intended use.
More detail regarding each of these identification components is provided in the subsequent chapters.
1.5
Closed-Loop System Identification
A central focus point of this work is closed-loop system identification. Closed-loop identification results when the identification experiment is performed under closed-loop conditions, this is, where a signal feedback mechanism is incorporated into the system so as
to control a variable via manipulation of another. Figure 1.2 illustrates the closed and
© University of Pretoria
CHAPTER 1. INTRODUCTION
6
PROBLEM FORMULATION
Yes
Purpose and scope of model?
What effort is justified?
Revise
Enough effort,
time available?
Record Data
DATA PREPARATION
No
Stop
Check data informally for errors,
anomalies, missing values,
effects of disturbances, drift
Yes
What is known of system,
environment, what models exist?
Select and tidy up data
Existing model
adequate?
Yes
Stop
ESTIMATION
No
Yes
Estimate structural parameters
of model, estimate model
coefficients
Data available?
No
EXPERIMENT DESIGN
Yes
VALIDATION
Choose inputs to perturb (if any),
outputs to observe, identification
method
Estimates credible?
Structure adequate?
Good fit to observations?
Revise
Try
again?
No
Yes
Examine actuators, instruments,
access, operating constraints
Apply model for intended
purpose
Choose type ad parameters of
perturbation, design sampling
shedule
Performance
adequate?
No
Yes
Satisfactory
experiment possible?
No
Document model
Stop
Revise
Stop
Figure 1.1: Flow diagram illustrating identification and validation logic sequence
r
+
-
C
u
G
y
Closed-loop system
u
G
y
Open-loop system
Figure 1.2: Comparison of closed and open-loop configurations
© University of Pretoria
No
Stop
CHAPTER 1. INTRODUCTION
7
open-loop configurations next to each other with u being the system input, y the system
output, r the reference or setpoint signal, C the controller and G represents the system.
Due to possible unstable behaviour of a plant, required safety and/or efficiency of
operation, many industrial processes need to be identified under closed-loop conditions.
This is especially the case when attempting to identify biological and economic systems
that inherently contain feedback loops that can not be removed or opened.
As it will be demonstrated in chapter 3, closed-loop system identification has some
advantages and disadvantages. The primary advantages may be given as (Forssell &
Ljung, 1999):
• Due to the feedback loop being closed, the disturbance to the system is reduced.
• Allows for better identification of the system given output constraints.
• Allows for better identification of control relevant dynamics.
the disadvantages may be stated as follows (Forssell, 1999):
• The feedback loop causes correlations between the input signals and the system
noise which results in the failure of many identification methods. Additionally
several model structures become more difficult to use specifically the non-parametric
structures (Ljung, 1987).
• By definition a closed-loop system contains a controller which has a purpose of
reducing system disturbances. This reduces the excitation of the system and thus
the information content in the data used to identify models from.
Further details of these advantages and disadvantages will be given in the theoretical
chapters that follow. Of specific interest will be how these characteristics extend to linear
estimations of non-linear systems.
1.6
Identification Experiment Design
The primary principle behind identification experiment design is that the choice of experimental conditions has a significant effect on the achievable accuracy of the identified
model. The following are the primary identification experiment design variables (Goodwin & Payne, 1977):
1. Inputs signal characteristics.
2. Total experimental time.
3. Total number of samples.
© University of Pretoria
CHAPTER 1. INTRODUCTION
8
4. Sampling rate
The objective behind optimising these variables is to obtain the most informative data
of the system with the model objective in mind while keeping within specified constraints.
The primary variable of those mentioned is undoubtedly the input signal. This is the
signal which disturbs the system in a specific manner so as to obtain informative data.
There are several characteristics of the signal that may be specified so as to maximise
the persistent excitation of the system. These are typically the signal type, amplitude
constraints, frequency characteristics and power constraints.
It is important to note that the success of accurately identifying a system depends
on the generation of informative data. For open-loop experiments the only criteria for
informative data is that the input signal be sufficiently exciting. For the closed-loop
experiments this is not sufficient (Forssell, 1999).
1.7
Model Validation and Discrimination
Model validation can loosely be defined as the determination of whether a model is good
enough for its intended use. While there are several statistical tools available that validate models based on confidence intervals and hypothesis testing, this work will focus on
cross-validation approaches. Cross-validation implies the condition where the identified
model response is simulated using different measured input data to that used for identification and the output is compared to the measured output (Forssell, 1999). From these
simulation results, the differences between the model output and the measured output,
defined residuals, may be analysed in the time domain and in the frequency domain.
Further more the residuals may be tested for correlations to allow for further insight into
the extent of model inaccuracy.
An important aspect of this work will be to test the validation results. That is, to
determine how accurate are the validation results and whether certain validation techniques are more biased than others. This effectively means assessing the ability and
conditions by which validation sets discriminate between models. Since cross-validation
relies on experimental data, the experimental conditions under which the validation data
are generated undoubtedly should affect the validation results. Determining validation
results sensitivity to validation data and the conditions they are generated under will be
an essential component of this work.
1.8
Linear Approximations of Non-linear systems
A central investigative aspect of this work will be identification and validation of linear approximations of non-linear systems. This delves into assessing what experimental
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CHAPTER 1. INTRODUCTION
9
characteristics improve linear approximations of non-linear systems. Additionally, model
validation takes on some different challenges to the typical validation of linear models of
non-linear systems. It becomes important to determine to what extent does the linear
system approximate the non-linear system.
In studying linear estimation of non-linear systems convergence theory, Enqvist et al.
(2007) states that one may establish the ”best” linear representation of a non-linear
system and assess the factors that affect the convergence of linear estimations towards
this best linear representation. So the question becomes, can one determine what is
the best linear approximation of the non-linear system from experimental data and how
does one validate a linear model in view of this best linear approximation? While this
issue of establishing the best possible linear approximation and determining how close
linear models are to this best approximation is not directly addressed in this work, it is
of relevance and discussed in the validation theory presented and assessment of model
validations.
1.9
Identification and Control
Gevers (1986) proclaims that few people would object to the assertion that control engineering has been the key component in the development of identification theory. This may
be specifically attributed to the fast development and implementation of model predictive controllers (MPC) and the extension of such controllers to be adaptive. This implies
that models are to be continuously generated, validated and implemented in such controllers with the objective of optimising controller performance. This while maintaining
the system under closed-loop conditions.
As stated earlier, one of the advantages of closed-loop system identification is its
ability to identify better models for control. Zhu & Butoyi (2002) attributes this to two
reasons:
1. The identification experiment efforts in information generation may be concentrated
over strictly defined output ranges due to feedback control.
2. Closed-loop experiments have been shown to be particularly accurate at identifying
frequency regions that are relevant for control.
3. Closed-loop experiments have been prominent features in identifying regions where
uncertainty is not tolerated. Reduced model uncertainty is essential in producing
robust model based control.
These points indicating the central role closed-loop system identification plays in
identification for control alone justify the importance of studying closed-loop identification
and validation.
© University of Pretoria
CHAPTER 1. INTRODUCTION
1.10
10
Scope and Outline of Dissertation
This document comprises of four theoretical chapters, two experimental chapters, one
discussions and conclusions chapter and a recommendations chapter.
It is noted that a relatively detailed literature survey of relevant system identification
theory is one of the primary outcomes of this work. The theoretical chapters may be
described as follows:
Chapter 2 - System Identification Theory : Presents the fundamental theory and
background of system representation and system identification. While a general
basis is established that covers most of the different approaches, the later sections
presenting the asymptotic and convergence theories focus more on the parametric
model structures.
Chapter 3 - Closed-loop System Identification Theory : Here the closed-loop system identification problem is presented along with the key concepts and approaches.
As with the general system identification theory, theory focuses on parametric approaches.
Chapter 4 - Model Validation Theory : This chapter presents the fundamentals behind model validation, specifically residual analysis. Additionally the chapter delves
into convergence theory of linear approximations of non-linear systems and how
these approximations of such systems may be validated.
Chapter 5 - Experimental Design Theory : Presents the theory regarding designing identification experiments. Basic theory is presented regarding the conditions
for informative data and how the different design variables affect the experiment
outcomes.
The chapters following these are those concerned with the simulation and experimentation work. The approach taken was such that simulation efforts in system identification
and validation were conducted before attempting to identify a pilot scale distillation column. These chapters may be further detailed as follows:
Chapter 6 - Identification of Simulated models : This chapter presents simulated
identification and validation efforts with a purpose of gaining insight into sensitivity relationships between experimental design variables and identification and
validation results. Two systems were simulated, one via a linear ARX model and
the other via a non-linear ARX model. Several identification experiments were designed for both and models were identified from all the data sets generated by these
experiments. In a similar fashion several different experimental designs were used
to generate validation sets. All the identified models were validated against all the
validation sets for both the linear and non-linear systems.
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CHAPTER 1. INTRODUCTION
11
Chapter 7 - Identification of a Distillation Column : This chapter presents efforts
in identifying and validating models of a pilot scale distillation column. Findings
from the previous chapter were used to design identification and validation experiments under the knowledge that the column is highly non-linear.
Chapter 8 - Discussion and Conclusions : Presents discussions and conclusions based
on the simulations and distillation column identification and validation efforts and
results.
Chapter 9 - Recommendations and Further Investigation : This chapter presents
recommendations regarding general approaches to system identification and validation based on the study of literature and experimental findings presented in this
work. Additionally, suggestions are made on how to further work in the identification of the pilot scale distillation column.
© University of Pretoria
CHAPTER 2
System Identification Theory
This chapter stresses the theoretical background and principles defining system
identification. The general fundamentals, methods and approaches to representing
linear and non-linear systems are surveyed. The later sections in this chapter focus
on parametric estimation methods. Specific attention is given to the prediction
error method and the convergence and asymptotic properties of this method.
Theoretical background chapters 3 and 4, on closed loop system identification
and model validation respectively, grow upon this chapter.
2.1
Background and History
Up until the 1950’s most of control design relied on Bode, Nyquist and Nichols charts
or step response analyses. These methods where mostly limited to SISO systems. In
the early 1960’s Kalman introduced the state-space representation and through this
established state-space based optimal filtering and optimal control theory with Linear
Quadratic optimal control as a cornerstone for model-based control design (Gevers, 2006).
It was on the heels of this introduction and rise of model-based control that system
identification developed. Literature shows how two landmark papers, Åström et al. (1965)
and Ho & Kalman (1965), gave birth to the two primary streams of research in system
identification in the mid 1960’s. Ho & Kalman (1965) gave the first solutions to the
state-space realisation theory. This went on to establish stochastic realisation and later
to subspace identification. The contribution by Åström et al. (1965) was such that it laid
the foundations for the highly successful Prediction Error Identification framework. These
two mainstream identification techniques, subspace and prediction error, still dominate
the identification field today.
For a more elaborate extension of the advances made in the field of system identifi12
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
13
cation over the more recent years and the origin of these advances, the reader is referred
to literature. Specifically (Raol & Sinha, 2000) and Gevers (2006).
While there have been, and will continue to be, many sources and contributions to the
ever growing compendium of System Identification techniques, present day contributions
made by Lennart Ljung must be commended. The book ”System Identification: Theory
for the user”, Ljung (1987), is said to have had a major impact on the engineering
community of system identifiers and has established system identification as a design
problem that plays a central role in many engineering problems. Much of the literature
presented in the forthcoming sections will be from this text.
2.2
Linear System Representation and Fundamentals
Input
A very well understood and common approach to identification is to introduce known
input disturbances to the system and record the system’s response. From knowledge of
the input and the type of response it yields, system dynamics may be extracted and
represented in an appropriate form. For such an identification approach typical inputs
are the impulse, unit pulse, step and sinusoidal-waves with figures 2.1 and 2.2 illustrating
(Norton, 1986: 24). Figure 2.2 illustrates the important connotation of a linear system
and assumption that facilitates the extraction of a system’s dynamics from a response to
an input. That is, a feature in the dynamics of a linear system is the additivity of output
responses to separate inputs.
0
Impulse
Pulse
Step
Wave
Time
Figure 2.1: Possible Selection of Input Disturbances
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
Linear System
u(t)
14
y(t)
Total Response
Input
Figure 2.2: Linear System Response
2.2.1
Linear Response Modelling
Impulse, Pulse and Step Responses
Common types of identification approaches based on recorded system responses are those
of the impulse and unit pulse response. An impulse, which is a pulse covering an infinitesimally short time, ∆τ , may be used to obtain working knowledge of the system. In other
words, the total response of a system may be determined by summing the contributions
from infinitesimally short sections of the input. With the impulse being precisely such an
input, the total response of a system may be given via the convolution or superposition
integral (Norton, 1986: 24),
Z
∞
g(τ )u(t − τ )dτ
y(t) =
(2.1)
0
where g is the response to the impulse. A unit-pulse response can be used in the same
manner. The advantage this approach has over the impulse response is that in practise it
is easier to implement a unit pulse than an infinitesimally short pulse. This superposition
of an isolated response may also be represented in discrete time form,
yt =
∞
X
gk ut−k
(2.2)
k=0
this is a useful representation as typically recorded data is obtained in discretised form.
A system representation in the form of a convolution sum based on a step response is
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
15
also obtained by extension of these concepts.
Sinusoidal-Wave Responses
Although a larger component of this project is centred on discrete-time approaches it is
important to note the sine-wave response based system representation. Once again the
convolution integral is used to represent the system response to a sine-wave. Algebraically
this is stated as the response to ejωt , applied from −∞,
Z
∞
y(t) =
g(t − τ )ejωτ dτ
(2.3)
−∞
This form of representation is central to frequency domain based system representations.
This is further discussed in the next section.
2.2.2
Transforms and Transfer Functions
Fourier and Laplace transforms form the basis of classical control design. The Laplace
and Fourier transform definitions are recalled as (Norton, 1986: 30),
∞
Z
f (t)e−st dt
L [f (t)] ≡ F (s) ,
0
Z
F [f (jω)] ≡ F (jω) ,
(2.4)
∞
f (t)e−jωt dt
(2.5)
0
and have the major advantage in that through the transform they simplify the inputoutput convolution relation given in equations (2.1) and (2.3) to a multiplication of
rational transforms,
Y (s) = G(s)U (s)
Y (jω) = G(jω)U (jω)
(2.6)
(2.7)
G(s) being the Laplace transform of the impulse response g(t) and G(jω) being the
frequency transform of the impulse response. This makes solving convolution equations
much easier as the appropriate transform inversion of the transform solution provides the
time domain solution (Norton, 1986: 35). All these characteristics make using a transfer
function very practical in terms of quantitative descriptions a system.
At this point it is convenient to introduce the basic notation that will be used from here
on to represent the essential components that describe a system. Recalling the discretised
form of the convolution integral, equation (2.2), this may be given in shorthand notation
by the introduction of the time shift operator q,
qu(t) = u(t + 1),
q −1 u(t) = u(t − 1)
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
16
this allows equation (2.2) to be represented in the following form:
"
y(t) =
∞
X
#
g(k)q −k u(t)
(2.8)
k=1
A complete linear model, illustrated in figure 2.3, may thus be defined as follows
(Ljung, 1987: 69):
y(t) = G(q)u(t) + v(t),
v(t) = H(q)e(t)
fe (·), the PDF of e
with
G(q) =
∞
X
g(k)q
−k
,
H(q) = 1 +
k=1
∞
X
h(k)q −k
(2.9)
(2.10)
k=1
G(q) is the transfer function of the linear system subject to an input u(t) resulting
in the output response y(t). The system disturbance variable is represented by v(t),
which is further defined through the noise model H(q). H(q) is assumed monic, i.e.
P
−k
, and inversely stable, i.e. (H(q))−1 is stable. Furthermore, e(t) is
H(q) = ∞
k=0 h(k)q
an unpredictable white noise signal with zero mean, a variance of λ0 and a probability
density function (PDF) of fe (·). This probability density function allows for a description
of the probability that the unpredictable values of e(t) fall with in certain ranges, this is
given by:
Z
b
P (a ≤ e < b) =
fe (x)dx
a
This representation of a system in a discrete form is advantageous since this is the typical
data acquisition mode (Ljung, 1987: 14).
2.2.3
Prediction and Parameterisation
Given the description expressed by equation (2.9) and the assumption that y(s) and u(s)
are known for s ≤ t − 1, since
v(s) = y(s) − G(q)u(s)
(2.11)
v(s) is also known for s ≤ t − 1, this means that it is possible to obtain a prediction for
y(t) = G(q)u(t) + v(t)
given by the following one-step-ahead prediction expression (Ljung, 1987: 56):
ŷ(t|t − 1) = H −1 (q)G(q)u(t) + 1 − H −1 (q) y(t)
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(2.12)
CHAPTER 2. SYSTEM IDENTIFICATION THEORY
17
e
H
v
u
+
G
y
+
Figure 2.3: Open-loop Transfer Function Representation
With some further manipulation of equation (2.12) the predictor may be extended to a
k-step-ahead predictor (Ljung, 1987: 57),
ŷ(t|t − k) = Wk (q)G(q)u(t) + [1 − Wk (q)] y(t)
with
Wk (q) = H̄(q)H
−1
(q),
H̄k (q) =
k−1
X
h(l)q −l
(2.13)
(2.14)
l=0
From equation (2.9) and (2.12) it is found that the prediction error y(t) − ŷ(t|t − 1)is
given by
y(t) − ŷ(t|t − 1) = −H −1 (q)G(q)u(t) + H −1 (q)y(t) = e(t)
(2.15)
This means that variable e(t) represents the component of output y(t) that cannot be
predicted from past data.
From this we can establish that through representation of the system transfer function,
G(q), and disturbance transfer function, H(q), we have a working model of the system.
However, representing these transfer functions by enumerating the infinite sequences contained in equation (2.10) would render the task of prediction through equation (2.12) very
impractical. Instead, one chooses to work with structures that permit the specification
of G(q) and H(q) in terms of a finite set of numerical values.
This representation of a system through a finite number of numerical values, or coefficients, has a very important consequence. Quite often it is not possible to determine these
coefficients a priori from knowledge or some physical interpretation of the mechanisms
that govern the dynamics of a system (Ljung, 1987: 70). This means that the determination of these values enter the system description as parameters to be determined by
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
18
some estimation procedure. These parameters are denoted by the vector θ, and thus we
have a system description given as
y(t) = G(q, θ)u(t) + v(t),
v(t) = H(q, θ)e(t)
fe (x, θ), the PDF of e
(2.16)
with fe being the probability density function (PDF) of e which is required for the
specification of the modelled disturbance. The parameter vector θ ranges over a subset
of Rd , where d is the dimension of θ (Ljung, 1987: 70):
θ ∈ DM ⊂ R d
At this point it is important to note that equation (2.16) is no longer a model but a set of
black-box models defined through the range of θ. It is through the estimation procedure
that the vector θ is obtained and the model defined from the set. Using equation (2.12)
we can compute the one-step-ahead prediction for equation (2.16).
2.2.4
A Class of Model Structures
Linear Regression
Ljung (1987: 71) states that the most immediate way of parametrizing G(q) and H(q)
is to represent them as rational functions with parameters entering as the coefficients of
the numerator and denominator. This parameterized representation of the relationship
between the input and output may be described as a linear difference equation (Ljung,
1987: 71):
y(t) + a1 y(t − 1) + . . . + ana y(t − na )
= b1 u(t − 1) + . . . + bnb u(t − nb ) + e(t)
(2.17)
where the choice of coefficients a and b and model order, na and nb , define the system
representation. The adjustable parameters may be represented as
θ = [a1 a2 . . . ana b1 . . . bnb ]T
(2.18)
Such a linear difference equation is written in the transfer function form expressed in
equation (2.16) through the use of the time shift operators as follows:
A(q) = 1 + a1 q −1 + . . . + ana q −na ,
B(q) = b1 q −1 + . . . + bnb q −nb
(2.19)
It must be noted that in some cases the dynamics from u(t) to y(t) contains a delay of
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
19
nk samples. This means that some of the leading coefficients of B(q) would be zero, thus
according to Ljung (1987):
B(q) = bnk q −nk + bnk +1 q −nk −1 + . . . + bnbk +nb −1 q −nk −nb +1
(2.20)
Applying this transfer function expression described in equations (2.17) and (2.19) to
equation (2.16) allows G(q, θ) and H(q, θ) to be defined as
G(q, θ) =
B(q)
,
A(q)
H(q, θ) =
1
A(q)
(2.21)
This model is called an Auto-Regressive with External Input (ARX) model. If the case
were such that na in equation (2.19) was zero, the model would be come a Finite Impulse
Response (FIR) model.
The prediction function given in equation (2.12) can now be given as follows:
ŷ(t|θ) = B(q)u(t) + [1 − A(q)] y(t)
(2.22)
This may be re-written as a linear regression (Ljung, 1987: 72),
ŷ(t|θ) = θT ϕ(t) = ϕ(t)T θ
(2.23)
ϕ(t) = [−y(t − 1) . . . − y(t − na ) u(t − 1) . . . u(t − nb )]T
(2.24)
with
This shows the predictor to be the scalar product between a known data vector ϕ(t), also
known as the regression vector, and the parameter vector θ.
A General Family of Model Structures
Consider the following generalised model structure (Forssell, 1999):
A(q)y(t) =
C(q)
B(q)
u(t) +
e(t),
F (q)
D(q)
(2.25)
where C(q) and D(q) are polynomials similar to A(q) and B(q) defined by equation
2.19. The structure is also depicted in figure 2.4. The advantage of this type of system
description is that there is more freedom in describing the properties of the disturbance
term. The introduction of polynomial C(q) does precisely this as it introduces flexibility
in describing the equation error as a moving average (MA) of white noise.
Table 2.1 shows the different structures and their names based on how one defines the
model by use of the different combinations of polynomials. Of particular value is the ARMAX structure. This structure has become a standard tool in control and econometrics
© University of Pretoria
CHAPTER 2. SYSTEM IDENTIFICATION THEORY
20
e
C
—
D
v
u
+
B
—
F
+
1
—
A
y
Figure 2.4: General Structure Transfer Function Representation
Polynomials Used in eq (2.25) Name of Model Structure
B
FIR (finite impulse response)
AB
ARX
ABC
ARMAX
AC
ARMA
ABD
ARARX
ABCD
ARARMAX
BF
OE (output error)
BFCD
BJ (box-jenkins)
Table 2.1: Common Model Structures
(Ljung, 1987: 78)
for both system description and control design. A version with an enforced integration
in the system description is the ARIMAX model which has become useful to describe
systems with slow disturbances (Box & Jenkins, 1970).
It is noted that for model structures other than ARX and FIR, the regression is not
strictly linear. For example, given an ARMAX structure, the regression vector becomes:
ϕ(t, θ) = [−y(t − 1) . . . − y(t − na ) u(t − 1) . . . u(t − nb ) (t − 1, θ) . . . (t − nc , θ)]T (2.26)
with the resulting predictor being
ŷ(t|θ) = ϕT (t, θ)θ
(2.27)
which is a pseudolinear regression, due to the non-linear effect of θ on the vector ϕ(t, θ)
(Ljung, 1987: 74). Additionally, literature occasionally refers to the FIR structures as
non-parametric due to fact that FIR structures require much more parameters to describe
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
21
the same dynamic behaviour .
2.3
Non-Linear Time-Varying System Identification
While non-linear system identification methods are not the primary focus of this work
some key approaches and concepts are worth discussing as they provide insight into the
non-linear framework which is relevant.
2.3.1
Non-Linearity and System Memory
Non-Linearity
Norton (1986: 246) explains the case of non-linearity very appropriately in stating that
in contrast to the tight constraints on system behaviour imposed by linearity and timeinvariance, non-linear behaviour is vastly diverse and complex. We can thus expect
non-linear system identification to be more difficult
A system L is said to be linear if the following properties hold true for the system:
Additive Property
Homogenous Property
L[u1 + u2 ] = L[u1 ] + L[u2 ]
L[cu] = cL[u]
(2.28)
(2.29)
If a system or model does not satisfy the properties given by equations (2.28) and (2.29)
it is non-linear (Bendat, 1991: 2).
In the case of a linear system, the system output, or response, to a random Gaussian
input sequence will have a Gaussian probability density function, while non-linear systems
will tend to have non-Gaussian outputs in response to Gaussian inputs (Bendat, 1991:
3).
It is understood that most of the dynamics that models try to approximate are nonlinear, that is, most of ”real-life” is non-linear.
System Memory
At this point it is appropriate to define system memory. System memory is a characteristic that is used in defining some of the common approaches to non-linear system
identification. The memory of a system refers to the effect that past inputs have on the
present outputs of a system and may be classified as follows (Bendat, 1991: 3):
• Infinite Memory,
• Finite Memory,
• Zero Memory.
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
22
A system with infinite memory has an output that is influenced by all past inputs.
For most systems the inputs exerted on the system have negligible effects after a certain
time, such a system has a fading memory or a finite memory. The system defined as
having zero memory is a system that is not affected by the past inputs. The output of
such a system may be predicted from the current input alone.
2.3.2
Identification Approaches and Methods for Non-linear Systems
While research in the field of non-linear system identification has yielded many useful
methods and approaches, a few of the more common and established are now briefly
introduced.
Linear Approximation
It is common practice to approximate non-linear systems via linear models as they are
typically simpler and are usually accurate enough for the intended application (Enqvist
et al., 2007). In addition to this, or in assertion of this, it is noted that while most
industrial processes are non-linear, some processes are sufficiently linear around a given
operating point. It is through this approach that the fields of non-linear dynamics detection and validation of linear approximations has established themselves. The former
being a tool used to ascertain the extent of non-linearities that govern the system allowing
one to determine whether or not a linear approximation would suffice. The latter being a
tool that establishes whether a linear approximation is a valid enough approximation of
the non-linear system. These fields are further discussed in section 4.2 and are important
topics in this work.
Volterra-Series Models
As introduced in section 2.2.1, the response of a system may be represented through the
superposition of a recorded response to a known input which may be given in the form of
an input-output convolution. The non-linear generalisation of this convolution equation
is the Volterra series (Norton, 1986: 247),
Z
∞
Z
∞
Z
∞
g1 (τ )u(t − τ )dτ +
y(t) =
0
g2 (τ1 , τ2 )u(t − τ1 )u(t − τ2 )dτ1 dτ2 + . . .
0
Z
+
∞
Z
...
0
0
∞
gr (τ1 , τ2 . . . , τr )
0
r
Y
u(t − τi )dτi + ...∞
(2.30)
i=1
The Volterra series is generally seen as a higher order extension of the linear impulse
response model where the gr coefficients are called Volterra kernels. While its use and
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
23
application in non-linear identification is extensive, it does have drawbacks. The primary
drawback of this approach is its impracticality and computational intensity of an nth
order constrained Volterra series when n > 2. Thus in most applications reduced order
Volterra series are used with some of the coefficients constrained to zero (Norton, 1986:
247).
Norton (1986) mentions that there are complications in calculating the Kernel coefficients. There is, however, extensive literature on the different estimation methods that
could be used to obtain such coefficients. Schetzen (1980) surveys some of these methods
with a focus on correlation approaches.
Block-Oriented Models
Block-oriented models are represented by a combination of static non-linearities and
linear dynamic components. A non-linearity is defined as static when it has zero memory
dynamics, that is, it does not depend on past input values (Ljung, 1987: 132).
This means that a system that exhibits memoryless non-linearities may be represented
as cascades of linear dynamic and non-linear instantaneous sub-systems. Figure 2.5 illustrates two possible block-oriented models of this sort. Literature shows that physical
insight into the system does help in identifying the different blocks but there are alternative methods that explore the property of separability via cross-correlation functions
that allow one to determine f (u(t)) (Norton, 1986: 248).
u(t)
(a)
N
f(u(t)
Linear System
Static nonlinearity
u(t)
(b)
H
H
y(t)
f(u(t)
Linear System
N
y(t)
Static nonlinearity
Figure 2.5: (a) Hammerstein Model (b) Wiener Model
Two common or well known extensions of the block-oriented models are the Hammerstein and Weiner models with the former illustrated above the latter in figure 2.5 .
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
24
The figure shows how each is a combination of a zero memory non-linear block, N , and
a linear dynamic element with a steady state gain of 1, represented by H. While the
two different models contain the same blocks the difference in order has the significance
in that their steady state behaviour will be the same but their dynamic behaviour is
different (Pearson & Pottmann, 2000).
Regression Models
Ljung (1987) states how, if one were to recall the linear regression model structure defined
by equation (2.23), the regressor vector, ϕ(t), is a vector of the lagged input and output
values. How this regressor vector forms or is defined is however immaterial, that is, ϕ(t)
need not be consecutive lagged outputs and inputs, eg.
ϕ(t) = [u(t − 1), u(t − 5)]
All that matters is that it is a known quantity at time t. Thus one could write
ŷ(t|θ) = θ1 ϕ1 (ut , y t−1 ) + . . . + θd ϕd (ut , y t−1 ) = ϕT (t)θ
(2.31)
with arbitrary functions, ϕi , of past data. The structure of equation (2.31) could be
regarded as a finite-dimensional parametrisation of a general, unknown, non-linear model
(Ljung, 1987: 130). The key is to determine the functions ϕi (ut , y t−1 ) and this usually
requires physical insight into the system, however, determination in terms of input-output
data only is also possible (Sjöberg et al.).
This approach extends itself to very large sets of possible regression options which
may ultimately be described through
ŷ(t|θ) = g(ϕ(t), θ)
(2.32)
where g is a non-linear function that serves to map the regressor space to the output
space. This translates the regression problem into two partial problems for dynamical
systems (Sjöberg et al.):
1. How to choose the regression vector ϕ(t) from past inputs and outputs
2. How to choose the non-linear mapping function g(ϕ).
While there are several possible ways to define the regression vector, those defined
in section 2.2.4, summarised in table 2.1, are commonly used. In the case of the ARX
structure, under the non-linear form given by equation (2.32), it becomes a non-linear
auto-regression with external inputs (NARX).
There are other choices for regressors, typically based on physical insight of the system
(van den Hof & Boker, 1994).
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
25
Defining the non-linear mapping, g(ϕ, θ), is typically approached by defining the mapping as parameterised function expansions (Sjöberg et al.):
g(ϕ, θ) =
X
αk κ(βk (ϕ − γk ))
(2.33)
where κ is a non-linear function, αk , βk and γk are parameters of the non-linear estimator.
The non-linear function κ is some times termed the mother basis function. There are
various forms and structures of this function that allow for certain approaches to solving
the parameters defining the function. Sjöberg et al. details some of these structures of
which the more common ones are:
• Wavelet Networks
• Kernel Estimators
• Sigmoid Neural Networks
2.4
Non-Parametric - Frequency Domain Methods
This section will explore some of the different identification techniques available that
do not explicitly employ a finite-dimensional parameter vector in the search of a best
description of the system.
2.4.1
Co-Correlation, Cross-Correlation and Spectral Density
Definitions
Before establishing some of the frequency domain based identification approaches found
in literature it is essential to define some of the fundamental signal analysis tools. Frequency analysis is all about the analysis of signals in terms of their frequency content
and/or contribution. A signal frequency analysis tool that is widely used throughout
identification theory is the power spectrum - or spectral density. In order to define the
spectral density of a signal, or cross-spectral density between two signals, it is necessary
to define the auto-correlation function (ACF) - also termed the covariance - of a signal,
S, and cross-correlation function (CCF) - also known as the cross-covariance - between
signals S and W . These are respectively given as (Ljung, 1987: 27-28):
RS (τ ) = Ē[s(t)s(t − τ )]
(2.34)
RSW (τ ) = Ē[s(t)s(t − τ )]
(2.35)
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
with
26
N
1 X
Ē[f (t)] = lim
E[f (t)]
N →∞ N
t=1
The notation E[f (t)] is the mathematical expectation of f (t) and may be interpreted as
the mean of a signal if the signal were a stationary stochastic signal. That is, E[f (t)] is
analogous to the mean in that if f (t) were a deterministic signal the mean, f¯, would be
determined by the limit of the normalised sum,
N
1 X
¯
f (t)
f = lim
N →∞ N
t=1
Broch (1990) explains how the correlation function and power spectrum reveal certain
characteristic properties of the signals or processes they are applied to, but goes on to
mention how such characteristics are not easily interpreted. Nevertheless, Broch (1990:
33) explains the auto-correlation function of signal S, RS , as the description of how
a particular instantaneous amplitude value of the signal S, at time t, depends upon
previously occurring instantaneous amplitude values at time t − τ . While the crosscorrelation function may be interpreted as the dependence between the magnitude value
of a signal S at an arbitrary instant of time, t, and the magnitude value of the observed
signal, W , at time t − τ . Equations (2.34) and (2.35) reflect this mathematically.
With the correlation functions now defined this allows for the definition of the spectral
density, or power spectrum, of a signal S as the Fourier transform of the auto-correlation
function, RS , given in discretised form by
ΦS (ω) =
∞
X
RS (τ )e−iτ ω
(2.36)
τ =−∞
while the cross-power spectrum, or cross-spectral density, between signals S and W is
similarly defined as the Fourier transform of the cross-correlation function (CCF), RSW ,
and given by
∞
X
ΦSW (ω) =
RSW (τ )e−iτ ω
(2.37)
τ =−∞
Note that by definition of the inverse Fourier transform we have
1
Ēs = RS (0) =
2π
2
Z
π
ΦS (ω)dω
(2.38)
−π
The relationship between the auto-correlation and spectral density is known as the
Wiener-Khintchine relation which is a special case of Parseval’s theorem (Rayner, 1971).
It must be noted at this point that the correlation function and power spectrum may
be used to qualitatively identify characteristics within signals and determine relationships
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
27
amongst signals. Figures 2.6 and 2.7 illustrate this well. Each illustrates the autocorrelation (b), power spectrum (c), of the the corresponding signal (a).
(a) − Signal
Amplitude
1
0.5
0
Power/Frequency (dB/Hz)
Sample Auto−Correlation
0
0.5
1
1.5
2
2.5
3
Time (s)
(b) − Sample Autocorrelation Function
3.5
4
4.5
5
80
90
100
40
45
50
1
0.5
0
−0.5
0
10
20
30
40
50
60
Lag
(c) − Power Spectral Density
70
5
0
−5
−10
−15
−20
0
5
10
15
20
25
Frequency (Hz)
30
35
Figure 2.6: Auto-correlation Function and Power Spectrum of a white noise signal
The signal in figure 2.6(a) is a signal of white noise. Broch (1990) states that if one
were to consider an ”ideal” stationary random signal the auto-correlation function would
consist of an infinitely narrow impulse at τ = 0 as all instantaneous signal magnitudes are
independent of each other. This is reflected in figure 2.6(b) for the noisy signal. In terms
of the power spectrum of white noise, literature points out that the profile should be
flat as there is no signal frequency that contributes substantially more than the another,
figure 2.6(c) reflects this.
In contrast, the signal assessed in figure 2.7 and given in 2.7(a) is composed of two
sinusoidal functions superimposed on top of each other. The auto-correlation function
is effective in recognising the governing relationship within the signal while the power
spectrum shows two peaks at frequencies corresponding to each sinusoidal function. The
peak magnitudes are proportional to the corresponding sine functions contribution to the
signal in terms of functions amplitude magnitude.
These applications of frequency domain analysis tools will be further emphasised in
chapter 4.1 when theory concerning model validation methods is introduced.
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
28
(a) − Signal
Amplitude
5
0
−5
Power/Frequency (dB/Hz)
Sample Auto−Correlation
0
2
4
6
20
40
60
8
10
12
Time (s)
(b) − Sample Autocorrelation Function
14
16
18
20
80
140
160
180
200
1
0.5
0
−0.5
−1
0
100
120
Lag
(c) − Power Spectral Density
40
20
0
−20
0
2
4
6
Frequency (Hz)
8
10
12
Figure 2.7: Auto-correlation Function and Power Spectrum of a sinusoidal signal
Spectral Density Transformations
Let signal S be defined by the following linear system:
s(t) = G(q)w(t)
Ljung (1987: 43) shows that
Φ(ω)S = |G(ejω )|2 Φw
Φ(ω)SW = G(ejω )Φw
The multivariable extension of this is given in section 2.7
2.4.2
The Empirical Transfer-function Estimate
In section 2.2.2 it was shown that through use of the Fourier transform one can transform a
system description into the frequency domain. Extending this transformation, in discrete
form, to output and input values of a system we have (Ljung, 1987: 146):
PN
−jωt
YN (ω)
ˆ
jω
t=1 y(t)e
ĜN (e ) ,
= PN
−jωt
UN (ω)
t=1 u(t)e
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
29
This estimate is usually referred of as the Empirical Transfer-function Estimate (ETFE),
and is considered the basic building block for most of the existing non-parametric frequency function estimates (Ljung, 1987).
It is well understood that the ETFE is a very crude estimate of the true transfer
frequency function, G0 (eiω ) (Stenman et al., 2000). This is mostly due to the fact that
the observed output, y(t), can be corrupted by measurement noise, v(t), which propagates
into the ETFE via the Fourier transform. However, it is noted that typically for a fixed
input u(t) and sufficiently large N , the ETFE allows for an unbiased estimate of the
true transfer frequency function. This condition is clouded by the fact that the variance
associated with the ETFE does not decay to zero as N increases. Instead it tends to the
noise-to-input-signal ratio at the frequency ω.
As a result of these conditions, and the understanding that in some cases the estimate
YN (ω)
is undefined for some frequency ω such that UN (ω) = 0, the ETFE is not very
UN (ω)
smooth. This is problematic as the true transfer frequency function, G0 (eiω ), is a smooth
continuous function of ω. For this reason much of the literature presented on the ETFE
focuses on the smoothing out of the ETFE.
Amongst the approaches established are simple but not so effective solutions like assuming that the values of the ETFE can be estimated via a local averaging procedure
which assumes that neighbouring ETFE values are related. This is typically done with
some weighting function (Stenman et al., 2000). More elaborate successful approaches
to smoothing out the ETFE are available. Stenman et al. (2000) suggests an adaptive parametric approach while Ljung (1987: 153) discusses established Blackman-Tukey
(Blackman & Tukey, 1958) methods that use spectral estimates.
2.5
2.5.1
Parametric Estimation Methods
Principles Behind Parameter Estimation
In section 2.2.4 a parameterised family of model structures was introduced where a parameter vector, θ, was used to define the model with in a model structure. This section
introduces the different methods that literature presents to obtaining a parameter vector
estimate θ̂ and the governing principles.
Given the information or data set,
Z N = [y(1), u(1), y(2), u(2), . . . , y(N ), u(N )]
(2.40)
the objective, or problem to resolve, is how to use this information to select a vector θ̂N ,
and hence the proper model member, M(θ̂N ), amongst the possible sets of models, M∗ .
Strictly speaking one wants to determine a mapping from the data set, Z N , to DM , the
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
30
set of θ̂N values in the model structure M (Ljung, 1987: 170):
ZN → θ̂N ∈ DM
(2.41)
This forms the very essence of the identification problem in terms of parametric models
and will be addressed next.
2.5.2
The Prediction Error Estimation Framework
In the prediction error framework one uses the difference between the predictor and the
data set to optimise the estimate. Additionally there are parameters or functions in the
form of norms and filters within the optimisation procedure that allow a ”truer fit” in
the face of unrepresentative data. This does essentially mean that these norms and filters
are variable functions or scalars that could be optimised with respect to minimising the
prediction error.
It is further noted that the prediction error framework is amongst the broader of
parameter estimation families and can be applied to quite arbitrary model parameterisations (Ljung, 2002). For these reasons and more, most of the work presented here
will be relative to the prediction error estimation method being the primary estimation
candidate.
The Prediction Error
At this point it is convenient recall section 2.2.3 and to further introduce the prediction
error between the measured output, y(t), and the predicted output, ŷ(t|θ),
(t, θ) = y(t) − ŷ(t|θ)
(2.42)
Together with the data set Z N , containing the system’s measured response y(t) and
inputs u(t), the prediction errors can be computed for t = 1, 2, 3, . . . N .
The prediction error sequence defined by equation (2.42) can be seen as a vector. The
size of this vector may be measured using any norm, be it quadratic or non-quadratic.
It is through the minimisation of the size of the prediction error sequence, i.e. the norm,
that defines the procedure for determining the estimate for a vector θ that best represents
the system at hand. Thus, let the prediction error be filtered through a stable linear filter
L(q) (Ljung, 1987: 171):
F (t, θ) = L(q)(t, θ),
1≤t≤N
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
31
applying a norm:
N
1 X
VN (θ, Z ) =
`(F (t, θ))
N t=1
N
(2.44)
where `(· ) is the norm usually given by a positive scalar-valued norm function. The
standard choice is the quadratic norm,
1
`((t, θ)) = 2F (t, θ)
2
(2.45)
Recalling and rearranging equation (2.15), and implementing some linear, monic, parameterised prefilter, L(q, θ) we have
(t, θ) = L(q, θ)H −1 (q, θ)(y(t) − G(q, θ)u(t))
(2.46)
Thus the effect of the prefilter can be included in the noise model and L(q, θ) = 1 can be
assumed without loss of generality.
The estimate θ̂N is then defined by the minimisation of equation (2.44),
θ̂N = θ̂N (Z N ) = arg min VN (θ, Z N )
θ∈DM
(2.47)
The mapping given by (2.41) is thus consolidated by equations (2.43) through (2.47).
The general term prediction-error identification method (PEM) is defined as the family
of approaches that correspond to this set of equations (Ljung, 1987: 171).
Numerical Issues and the Least-Square Estimate
The actual calculation of the minimising argument presented by equation (2.47) can be
complex with substantial computations, and possibly a complicated search over a function
with several local minima. Literature shows the typical numerical search is carried out
using the damped Gauss-Newton method (Ljung, 2002) apart from this other frequently
used methods for solving this optimisation problem are documented in Evans & Fischl
(1973) and Steiglitz & Mcbride (1965). There are however exceptions to the case.
Given the case where the model structure used to estimate y(t) is a linear regression
ŷ(t|θ) = ϕ(t)T θ
a unique feature is established via the choice of a quadratic norm, equation (2.45), in that
a least-squares (LS) criterion is ensued (Forssell, 1999). This means that equation (2.47)
can be minimised analytically, forgoing the necessity for a complex numerical search,
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
32
provided the inverse exists (Ljung, 1987: 176),
#−1
N
N
X
1 X
1
N
T
= arg min VN (θ, Z ) =
ϕ(t)ϕ (t)
ϕ(t)y(t)
θ∈DM
N t=1
N t=1
"
LS
θ̂N
(2.48)
This is the well known Least-Squares Estimate.
2.5.3
Other Common Estimation Frameworks
Instrumental Variable
Analogous to the least-squares estimate, Forssell (1999) states that through the standard instrumental variable method, the parameter vector estimate, θ̂N , may be obtained
through
#−1
"
N
N
X
1 X
1
T
IV
ζ(t)ϕ (t)
ζ(t)y(t)
(2.49)
θ̂N =
N t=1
N t=1
where ζ(t) is known as the instrumental variable vector. It must be noted that the
least-squares method is obtained if ζ(t) = ϕ(t). The elements of vector ζ(t) are termed
instruments and are usually obtained from past inputs by linear filtering and can be
conceptually written as
ζ(t) = ζ(t, ut−1 )
(2.50)
IV
to tend to θ0 at large values of N , the instrumental vector and hence the
For θN
elements that form the vector should allow for the following conditions (Forssell, 1999):
Ēζ(t)ϕT (t) be nonsignular
(2.51)
Ēζ(t)e(t) = 0
(2.52)
In loose terms, the instrument vector should be well correlated with lagged inputs and
outputs but uncorrelated with the noise sequence. The natural choice for an instrument
vector that satisfies these conditions would be a vector similar to the linear regression in
equation (2.17) so as to secure condition (2.51) but different in that it is not influenced
by e(t) (Ljung, 1987: 193),
ζ(t) = K(q)[−x(t − 1) − x(t − 2) . . . − x(t − na )u(t − 2) . . . u(t − nb )]T
(2.53)
where K is a linear filter and x(t) is generated from the input through a linear system
N (q)x(t) = M (q)u(t)
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
33
with
N (q) = 1 + n1 q −1 + · · · + nnn q −nn
M (q) = m0 + m1 q −1 + · · · + mnm q −nm
This further shows how ζ(t) can be generated from past inputs.
Maximum Likelihood
In contrast to the prediction error approach previously mentioned, the maximum likelihood method is to some extent very similar. The difference being that the ML (Maximum
likelihood) method appeals to statistical arguments for the estimation of θ (Forssell, 1999).
More specifically it uses the probability density functions of the observed variable y N ,
given by fy (θ; xN ) such that
N
Z
P (y ∈ A) =
fy (θ; xN )dxN
xN ∈A
The ML estimation function is such that probability, P , that the observed value, y N ,
is equal to the estimated value, y∗N is maximised. This is said to be proportional to
fy (θ; xN ) and thus the ML parameter estimation seeks
θ̂M L (y∗N ) = arg max fy (θ; y∗ N )
(2.54)
this function is called the maximum likelihood estimator (MLE) and is further discussed
in Ljung (1987)
2.6
Convergence and Asymptotic Properties
It has been previously established that the central nature of the identification process is
the mapping from the data set, Z N , to the estimation vector, θ̂N . The previous section
introduced some common mapping or estimation approaches with the most pertinent
method being that of the prediction error framework.
This section continues by characterising the convergence and asymptotic properties
of the prediction error estimation method.
It is noted that the property of convergence bias is discussed in section 3.2 since this
property is best discussed after introducing the closed-loop identification condition.
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
2.6.1
34
Convergence and Identifiability
Informative Data Set
Identifiability of model structures is concerned with whether different parameter vectors
may describe the same model in the model set M∗ . This is directly related to whether
the data set Z N allows for the distinction between different models in the model set. If a
data set does allow for this distinction then the data set is said to be informative enough
(Ljung, 1987).
This concept of informative data is related to the concept of persistent signal excitation. Ljung (1987) details the definition of an input signal that is persistently excited. He
states that such a signal allows for the identifiability of the model structure that uniquely
describes the data set if, and only if, the input is persistently excited to a sufficiently
high order. More of this concept is addressed in chapter 5.2.
A Frequency Domain Description of Convergence
Consider the prediction error method with a quadratic norm (Forssell, 1999),
N
1 X1 2
(t, θ)
VN (θ, Z ) =
N t=1 2
(2.55)
1
VN (θ, Z N ) → V̄ (θ) = Ē 2 (t, θ) w.p. 1 as N → ∞
2
(2.56)
θ̂N → Dc = arg min V̄ (θ) w.p. 1 as N → ∞
(2.57)
N
under moderate conditions
and (Ljung, 1978)
θ∈DM
with w.p. signifying with probability, Dc signifying the model set into which θ̂N converges
and V̄N (θ) the convergence limit of VN (θ, Z N ). This result is quite general and can be
applied to other norms and non-linear time varying systems. It states that the estimate
will converge to the best possible approximation of the system that is available in the
model set (Ljung, 1987: 216).
Recalling Parseval’s theorem, the relationship between auto-correlations and spectral
density given by
Z π
1
2
Ēs =
ΦS (ω)dω
2π −π
applying this to equation (2.56) we have
1
V̄ (θ) =
2π
Z
π
−π
1
Φ (ω)dω
2
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
35
where Φ (ω) is the power spectrum of the prediction errors. Given the true system is
described by,
y(t) = G0 (q)u(t) + v0 (t)
(2.59)
this is substituted into the prediction error equation previously given by equation (2.15)
where a linear estimation is used,
(t, θ) = H −1 (q, θ) [y(t) − G(q, θ)u(t)]
(2.60)
= H −1 (q, θ) [(G0 (q) − G(q, θ))u(t) + v0 (t)]
This allows for the definition of Φ ,
Φ (ω, θ) =
|G0 (ejω ) − G(ejω , θ)|2 Φu (ω) + Φv (ω)
|H(ejω , θ)|2
(2.61)
which may now be used in the convergence limit definition,
1
V̄ (θ) =
4π
Z
π
Φu (ω)
|G0 (e ) − G(e , θ)|
dω +
|H(ejω , θ)|2
−π
jω
jω
2
Z
π
−π
Φv (ω)
dω
|H(ejω , θ)|2
(2.62)
This definition of the convergence limit allows for the specification of the estimation
vector in equation (2.57) (Ljung, 2002),
θ̂N → Dc
Z π
1
Φu (ω)
dω
= arg min
|G0 (ejω ) − G(ejω , θ)|2
θ∈DM 4π −π
|H(ejω , θ)|2
Z π
1
Φv (ω)
+
dω w.p. 1 as N → ∞
4π −π |H(ejω , θ)|2
(2.63)
Φu
|H(ejω ,θ)|2
is termed the quadratic frequency norm (Ljung, 1987: 227). It is interesting to
consider this norm as a frequency weighting function in that:
• If H(ejω , θ) has primarily high frequency components, a larger weight is put on low
frequency misfits between G0 and G(θ).
• If Φu (ω) has more high frequency components than low, a larger weight is put on
high frequency misfits between G0 and G(θ).
• The ratio between Φu (ω) and H(ejω , θ) is known as the model signal-to-noise ratio.
It is through these points that it is understood that the extent of signal excitation
through the input u(t) and definition of the noise model play important roles in the
estimate convergence. This being said, equation (2.63) shows that if the true system, S,
is represented in the model set, M∗ , the estimate will converge towards the true system.
It must be noted that this stated property of convergence for a Least-Squares Prediction Error Method (PEM) approach is bound to the condition that N tends to ∞. This
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
36
is obviously strictly theoretical as all applications are bound to finite limitations in N
Ljung (1976).
2.6.2
Asymptotic Distribution
A General Expression for Asymptotic Variance
With convergence covered in the previous section it is now appropriate to present the
asymptotic properties of the estimation approach specific to the PEM. Consideration
of the asymptotic distribution deals with the rate at which the convergence limit is
approached (Ljung, 2002).
Let it be assumed that the true model of the system is defined within the model set.
The basic outline of this property is such that if the prediction error, (t, θ), is Gaussian
√
with variance λ0 , then the distribution of the random vector N (θ̂N − θ∗ ) converges to a
normal distribution with zero mean. Ljung (1987: 242) states that the covariance matrix
of θˆN , Cov θ̂N , is related to the asymptotic covariance matrix Pθ of θ through
Cov θ̂N ∼
1
Pθ
N
(2.64)
where Pθ is directly given by
Pθ = λ0 R−1
R = Ēψ(t, θ∗ )ψ T (t, θ∗ )
(2.65)
and
λ0 = Ē2 (t, θ∗ )
d
d
ψ(t) = − (t|θ)|θ=θ∗ =
ŷ(t|θ)|θ=θ∗
dθ
dθ
(2.66)
(2.67)
Ljung (1987: 243) explains that equation (2.65) has a natural interpretation. As
equation (2.67) shows, ψ, is the gradient of ŷ with respect the the estimation parameter,
that is, ψ shows how much the estimation error, , changes as the parameter vector
estimation, θ̂, changes. This is also termed the sensitivity derivative. Additionally it is
noted that Ēψ(t)ψ T (t) is by definition the covariance of ψ. Therefore equation (2.65)
naturally defines the asymptotic covariance of the estimate as being proportional to the
inverse of the covariance matrix of the sensitivity derivative.
Ljung (1995) further elaborates that the variability of the model expressed through
the covariance matrix is an indication of the model uncertainty. It is through model
uncertainty that an understanding of how model parameters will differ if the estimation
procedure is repeated if another data set, Z N (with the same input sequences as the
original data set), was used.
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
37
Asymptotic Black-box Variance
While the previous section shows an analysis of the covariance of the parameter vector,
it is sometimes useful to study the covariance of the transfer function itself.
Ljung (1985) shows that when given a black box parameterisation of linear models,
the transfer function estimates, G(ejω , θ̂N ) and H(ejω , θ̂N ), will, as N and model order n
(i.e. the dimensionality of θN ) tend to infinity, have a normal distribution with covariance
matrix
"
#
"
#−1
G(ejω , θ̂N )
Φu Φue
n
Cov
≈ Φv (ω)
(2.68)
N
H(ejω , θ̂N )
Φeu λ0
Here Φue is the cross power spectrum between input u and noise source e. This expression of the covariance matrix of the transfer function estimates is a general expression
applicable to both open and closed loop conditions.
Further discussion on this will be done in section 3.2.1
2.7
Multivariable System Identification
So far the theory presented has been of systems with a scalar input and a scalar output.
This section deals with the extension or necessary adaptations of such theory to systems
with p outputs and m inputs. Such a system is termed multivariable and is depicted in
figure 2.8. Ljung (1987: 35) states that the work required in this adaptation may be
broken into two parts:
1. The easy part: Mostly notation changes, keeping track of transposes and that
certain scalars become matrices and might not commute.
2. The difficult part: Multivariable models have richer internal structures, this has
consequences that result in non-trivial parameterisations and estimations.
u1
y1
u2
y2
u3
y3
G
um
yp
Figure 2.8: Possible Selection of Input Disturbances
Note that multiple input and single output (MISO) systems do not have the problems
stated in part 2 (Ljung, 1987: 35).
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
2.7.1
38
Notation and System Description
Representing a multivariable system is done by collecting the p output components and
m input components into respective p- and m- dimensioned column vectors, y(t) and
u(t) respectively. Let the disturbance e(t) also be represented as a p-dimensional column
vector and we have,


y1
 . 
. 
y(t) = 
 . 
yp


u1
 . 
. 
u(t) = 
 . 
um


e1
 . 
. 
e(t) = 
 . 
ep
This allows for the basic multivariable system to be represented in a similar manner as
the scalar system description (Ljung, 1987: 37):
y(t) = G(q)u(t) + H(q)e(t)
In this case G(q) is a transfer function matrix of dimension p × m and H(q) has the
dimension p × p. The sequence e(t) is a sequence of p-dimensional vectors with zero mean
values and covariance matrix Ee(t)eT (t).
This translates into a system description that is formulated by subsystem transfer
functions,


G11 (q) G12 (q) · · · G1m (q)


 G21 (q) G22 (q) · · · G2m (q) 


(2.69)
G(q) = 
..
..
..
..

.
.
.
.


Gp1 (q) Gp2 (q) · · · Gpm (q)
Where each subsystem in G(q) is denoted with the indices, i and j, allowing for the
scalar transfer function from the input number j to the output number i to be denoted
as Gij (q).
The adaptation of the system description for a multivariable system extends to definitions of multivariable correlations and power spectrum transformation. That is
RS (τ ) = Ē[s(t)sT (t − τ )]
RSW (τ ) = Ē[s(t)wT (t − τ )]
with RS and RSW now being matrices and s(t) and w(t) being column vectors.
The definitions of the power spectra remain unchanged while the manner in which
they are transformed given a multivariable system
s(t) = G(q)w(t)
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CHAPTER 2. SYSTEM IDENTIFICATION THEORY
39
becomes
Φ(ω)S = G(ejω )Φw GT (−ejω )
2.7.2
Parameterisation and Estimation Methods
Parametrisation
Representing a multivariable system as a parametric model structure is done in a similar
way to the mostly straightforward scalar counterparts. In terms of a linear regression
ARX structure, equations (2.17) through (2.19), the regression is no longer scalar polynomials but matrix polynomials (Ljung, 1987: 80),
y(t) + A1 y(t − 1) + . . . + Ana y(t − na )
= B1 u(t − 1) + . . . + Bnb u(t − nb ) + e(t)
(2.70)
using the time shift operator
A(q) = I + A1 q −1 + . . . + Ana q −na , and
(2.71)
B(q) = B1 q −1 + . . . + Bnb q −nb
where Ai are p × p matrices and Bi are p × m matrices.
This allows for multivariable parametric rational transfer functions of q,
G(q, θ) = A−1 (q)B(q),
H(q, θ) = A−1
(2.72)
The derivation of the parameter vector, θ, and representation of the multivariable linear
regression equation is a fairly subtle issue and is slightly more complex than what these
given representations initially connote. This is primarily as the multivariable condition
results in a higher degree or order specification. That is, the choice of structure used for
representation requires the specification of the matrix fraction description.
Specifically, this entails determining how each subsystem’s regression order is defined.
If one was to consider the coefficient matrix A(q), given in equation (2.71), and expand
for a system with p outputs and m inputs the following is obtained (Ljung, 1987: 116):



A(q) = 



A11 (q) A12 (q) . . . A1p (q)

A21 (q) A22 (q) . . . A2p (q) 
 = A(0) + A(1) z −1 + . . . + A(na) z −na
..
..
..
..

.
.
.
.

Ap1 (q) Ap2 (q) . . . App (q)
(2.73)
with subsystem entries as follows:
(0)
(1)
(2)
(naij ) −naij
Aij (q) = aij + aij z −1 + aij z −2 + . . . + aij
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z
(2.74)
CHAPTER 2. SYSTEM IDENTIFICATION THEORY
40
This means that the order of the multivariable ARX system representation is done by
specifying the orders naij and nbij (i.e. p(p + m) integers). This results in a staggering
amount for possible model structures that allow for different parameterisations.
Certain parameterisations allow for completely independent subsystems which means
that each subsystem is parameterised as if it were a SISO system. Other parameterisations
result in MIMO system representations that are compiled by partial models that are
constructed from all the inputs and one output, i.e. multiple MISO models. Zhu &
Butoyi (2002) shows that these types of parameterisation do typically result in bias models
as they do not completely incorporate variable interactions especially in ill-conditioned
systems. The most suitable MIMO system representation results from parameterisations
where all inputs and all outputs are parameterised and estimated simultaneously (Ljung
(1987: 80-81, 115) and Johnson & Wichern (2007: 360)).
In any case, an immediate analogy of the scalar parameter vector is possible. Suppose
all the matrix entries in equation (2.70) are included in θ, we may now define the [na ·
p + nb · m] × p matrix as
θ = [A1 A2 · · · Ana B1 · · · Bnb ]T
(2.75)
and the [na · p + nb · m] dimension column vector


−y(t − 1)


..


.




 −y(t − na ) 


ϕ=

u(t
−
1)




..


.


u(t − nb )
(2.76)
y(t) = θ T ϕ(t) + e(t)
(2.77)
allowing for
It is noted that not all structure specifications allow for this representation to be used.
Parameter Estimation
The multivariable counterpart of the prediction error frame work is very similar to those
represented by equations (2.43) through (2.47) (Ljung, 1987: 175). The quadratic norm
and the prediction error method once again lend themselves to the often-used leastsquares estimation method given by
1
`() = T Λ−1 2
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(2.78)
CHAPTER 2. SYSTEM IDENTIFICATION THEORY
41
and
f (t, θ) = L(q, θ)(t, θ)
N
1 X T
N
VN (θ, Z ) =
(t, θ)Λ−1 F (t, θ)
N t=1 F
θ̂ N = arg min VN (θ, Z N )
θ∈DM
(2.79)
(2.80)
(2.81)
with Λ being a symmetric, positive definite p×p weighting matrix and L once again being
a monic filter that can be incorporated into H(q, θ), i.e. L = I (Johnson & Wichern,
2007: 364).
2.7.3
Convergence and Asymptotic Properties
Ljung (1987: 223) states that the convergence and asymptotic results for multivariable
systems are entirely analogous to the scalar case with the only major difference being
notation changes.
2.8
Model Validation
Model validation should form the final stage of any identification procedure (Billings &
Voon, 1986). While the parameter estimation procedure picks the ”best” model within
the chosen structure, determination of the quality of the model is essential. The different
quantitative measures of model quality and under which circumstances certain definitions
are used, is the central aspect of model validation.
The literature presented in this work will focus on two primary aspects of model
validation. The first being the various techniques and approaches and the second being
the necessity for higher order validation techniques in the case of linear estimates of nonlinear systems. The chapter of Model validation (Chapter 4.1) presents these topics in
detail.
2.9
Chapter Summary
In overview of this chapter it must be noted that the focus began on general modelling
and system representation approaches and gradually moved towards parametric model
structures. Through the surveying of literature it was found that the simple structure of
such models to lend themselves to facilitated estimation procedures. Parameter estimation convergence and asymptotic theory showed the importance of system excitation via
system inputs and how the correct definition of noise models and pre-filters may serve as
frequency weighting functions.
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CHAPTER 3
Closed-Loop System Identification Theory
Chapter 2 presented the basis and fundamental theory of system identification.
This chapter extends this theory to the condition where the data set, ZN , used to
generate the parameter vector, θ̂N , is obtained under closed-loop conditions. The
problems and issues concerned with closed-loop system identification are introduced along with the three primary approaches to successfully identifying a model
from closed-loop data. Once again the convergence and asymptotic properties
are presented relative to the prediction error framework along an assessment into
the role controller complexity has in system identification.
3.1
Basic Concepts, Issues and Method Classifications
3.1.1
Closed-Loop System Description, Notation and Conventions
A closed-loop system, represented by figure 3.1, is a system that is under feedback control.
In general, this work will assume that the controller is a linear controller with a mechanism
that is not always known. Recalling the representation of the true system (denoted by
the 0 subscript),
y(t) = G0 (q)u(t) + v0 (t)
= G0 (q)u(t) + H0 (q)e(t)
42
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
43
e
H
v
r1
u
+
+
G
+
y
+
-
C
+
r2
Figure 3.1: Closed-loop system representation
defining the input signal as
u(t) = r1 (t) + C(q)(r2 (t) − y(t))
(3.1)
where r1 (t) is regarded as the dither or excitation signal while r2 (t) is the set-point
(Forssell & Ljung, 1997). For purposes of this work it is sufficient to consider these as
the reference signal defined by
r(t) = r1 (t) + C(q)r2 (t)
(3.2)
where r(t) is assumed independent of the noise e(t). The input signal may now be
redefined by
u(t) = r(t) − C(q)y(t)
(3.3)
Allowing for the closed-loop system representation through
y(t) = G0 (q)S0 (q)r(t) + S0 (q)H0 (q)e(t)
(3.4)
with S0 (q) termed the sensitivity function and defined by
S0 (q) =
1
1 + C(q)G0 (q)
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(3.5)
CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
44
We may further define closed loop transfer functions as
Gcl
0 (q) = G0 (q)S0 (q)
H0cl (q) = S0 (q)H0 (q)
this allows equation (3.4) to be re-written,
cl
y(t) = Gcl
0 (q)r(t) + v (t)
cl
= Gcl
0 (q)u(t) + H0 (q)e(t)
(3.6)
with the input further defined as
u(t) = S0 (q)r(t) − C(q)v cl (t)
(3.7)
From the above equation it is noted that the input signal has two components, one sourced
in the reference signal and the other in the system disturbance. It is thus convenient to
establish the power spectrum for the input
Φu (ω) = |S0 (ejω )|2 Φr (ω) + |C(ejω )|2 |S0 (ejω )|2 Φv (ω)
(3.8)
where Φr (ω) and Φv (ω) are the power spectrums for the reference signal and the disturbance signal respectively. It follows that it is possible to denote
Φru (ω) = |S0 (ejω )|2 Φr (ω)
Φeu (ω) = |C(ejω )|2 |S0 (ejω )|2 Φv (ω)
(3.9)
(3.10)
with Φru (ω) and Φeu (ω) interpreted as the part of the input spectrum that originates from
the reference signal r and noise e respectively.
3.1.2
Issues in Closed-Loop Identification
The Input-Noise Correlation
The primary problem or difficulty to overcome in closed-loop system identification is the
correlation between unmeasurable noise and the input. Considering figure 3.1 it becomes
clear that if the feedback controller, C(q), is not zero, the input and the noise will be
correlated. This correlation typically results in a biased estimate (Forssell, 1999).
This condition of a biased estimate due to input-noise correlation is revealed if we
were to consider a true system representation given by,
y(t) = B(q)u(t) + v(t) = b1 u(t − 1) + . . . + bn (t − n) + v(t)
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
45
which may be re-written as a linear regression,
y(t) = ϕT (t)θ0 + v(t)
with
θ0 = [b1 . . . Bn ]T
and ϕ(t) = [u(t − 1) . . . u(t − n)]T
From the above we note that the chosen model structure, M, contains the true system,
S, that is, S ∈ M. Additionally θ0 is the true parameter vector. Thus, given N data
points, the least-squares estimate may be solved for analytically using equation (2.48)
(Forssell, 1997),
#−1
N
N
1 X
1 X
T
=
ϕ(t)ϕ (t)
ϕ(t)y(t)
N t=1
N t=1
#−1
"
N
N
1 X
1 X
T
ϕ(t)ϕ (t)
ϕ(t)y(t)
= θ0 +
N t=1
N t=1
"
LS
θ̂N
(3.11)
It is known that, under mild conditions, θ̂N → θ∗ w.p. 1 (Forssell, 1997), thus
−1
θ∗ = lim E θ̂N = θ0 + Ēϕ(t)ϕT (t)
Ēϕ(t)v(t)
N →∞
(3.12)
From this it is quite clear that if there is an input-noise correlation, Ēϕ(t)v(t) 6= 0, θ∗ will
result in a biased estimate. In closed loop systems this is the case, and this typically why
many open-loop identification methods fail when applied to closed-loop data (Forssell,
1997).
Signal Excitation and Information
An underlying assumption in system identification is that the data set used, Z N , is
sufficiently rich so as to allow conditions that make it possible to uniquely determine
the system parameters. This is a much studied subject in closed-loop identification and
open-loop identification (Anderson & Gevers, 1982)(Gustavsson et al., 1977).
In section 2.6.1 the concept of persistent signal excitation allowing for informative data
was established for the open-loop case. For the closed-loop system the condition is not
the same. Ljung (1987: 365) and Forssell (1999) show that the condition of informative
data in a closed-loop system is not guaranteed by sufficient signal excitation. Gustavsson
et al. (1977) goes on to say that this is natural as one of the purposes of the feedback
controller is to minimise the output variance and hence the consequent minimisation of
output information content.
The methods or aspects concerned with guarantying informative conditions, and hence
identifiability, will be discussed in section 3.2 and again in section 5.2. This will be done
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
46
relative to the different methods commonly available for closed-loop system identification.
3.1.3
Problematic Identification Methods
Due to the issues mentioned in the previous section, specifically the problem on inputnoise correlation, most of the well-known identification methods, which work well when
used with open-loop data, fail when applied directly to input-output data obtained from
a system operated under feedback (Forssell, 1999).
A few of the better known identification methods that typically fail are now mentioned
together with a brief explanation as to why they fail. In some cases references to literature
sources that investigate adaptations to the respective method allowing for identification
under closed-loop conditions are made.
Instrumental Variable (IV) Methods
Under open-loop conditions the typical approach taken when using the IV method is to let
the instruments defining the regression vector be filtered delayed versions of the input.
This results in biased estimates due to the conditions, given by equations (2.51) and
(2.52), required for consistent identification not being met as a result of the input-noise
correlation (Forssell, 1999).
Söderström et al. (1987) indicates that through constructing instruments from the
reference signal which are uncorrelated to noise or choosing delayed versions of the regressor with large delays, the conditions for unbiased closed-loop identification can be
met. It must be said that it is not completely straightforward to apply the IV method to
closed loop conditions.
Subspace Method
The problem with subspace method is very much similar to that of the IV method. Here
the problem is finding a suitable multiplication matrix, which is usually obtained from
delayed inputs, which is uncorrelated to noise allowing for the elimination of the noise
term in the Hankel Matrix equation.
More detail on this problem is given in Forssell (1999), while alternative solutions
or adaptations to the method that overcome the input-noise correlation problem are
surveyed in Lin et al. (2005) and Ljung & McKelvey (1996).
Correlation and Spectral Analysis Methods
The identification principle behind these non-parametric methods is based on time domain
correlations (covariance) or frequency contributions (power spectrum) between the input
and the output data. These methods obviously encounter problems when presented with
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
47
closed-loop data due to the input-noise correlation. Forssell (1999) further details this
problem while possible solutions have been proposed by Akaike (1967)
3.1.4
Classification of Closed-Loop Identification Approaches
The classification of approaches is based on the distinction of the following assumptions
when dealing with the feedback (Forssell & Ljung, 1999):
1. Assume no knowledge of the nature of the feedback and use only measurements of
the system input,
2. Assume the feedback to be known and use possible reference signals and set-points
3. Assume the regulator the be unknown but of a certain structure
In assumptions (a) and (b) it is common to assume that the controller is linear. However,
if the controller were non-linear the same principles would apply with the price that
the estimation procedure becomes more involved. Unless otherwise stated most of the
theory presented here will assume a linear controller. It is however noted that most
controllers in industry are non-linear. This is as no controller has a simple form, most
have various delimiters, anti-windup functions and other components that would instill
even in a strictly linear controller structure some type of non-linearity (Forssell & Ljung,
1999).
From these assumptions the closed-loop identification methods correspondingly fall
into approach groups as follows (Forssell & Ljung, 1999):
1. The Direct Approach : Ignore the feedback and identify the system using only
in input and output measurements
2. The Indirect Approach : Identify the closed-loop transfer function and determine the open-loop parameters using knowledge of the (linear) controller.
3. The Joint Input-Output Approach : Regard the input and output jointly as
the output from the system driven by some reference or set-point signal and noise.
Then use a method to determine open loop parameters from this augmented system.
3.2
Closed-Loop Identification in the Prediction Error Framework
As the previous section established, most non-parametric system identification methods
do not directly work when using closed-loop data (Ljung, 1987). Literature also repeatedly indicates the prediction error method (PEM) as the best approach under closed-loop
conditions (Forssell & Ljung, 1999).
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
48
Ljung (2002) states that this is mostly due to the PEM being a very robust method,
applicable to broad data conditions and parameterisation structures. Huang & Shah
(1997) additionally supports the PEM approach primarily based on its ability to achieve
a minimum asymptotic variance near to that achieved under open-loop conditions.
These different approaches, and their respective properties, will be further detailed in
the next sections. The properties concerning the approach of directly applying the PEM
to closed-loop data, in accordance with literature, will firstly be established followed by
those of the Indirect and Joint Input-Output methods. This detailing of the convergence
and asymptotic properties of these closed-loop approaches complement theory presented
for the general identification condition discussed in section 2.6. For a complete and
more detailed elaboration of the convergence, asymptotic and covariance properties of
the different approaches see Forssell & Ljung (1997) and Ljung (1985).
3.2.1
The Direct Method
As pointed out in the previous section, the direct approach ignores the feedback and
directly uses the input and output measurements. This entails an identification procedure
that coincides with the open loop approach. Thus, the identification methodology detailed
in section 2.5.2 is directly applied.
Literature consistently indicates that this direct identification approach should be seen
as the natural approach to closed-loop data analysis (Forssell, 1999) (Forssell & Ljung,
1997). The main reasons given for this are (Forssell & Ljung, 1997):
• It requires no knowledge of the regulator or character of the feedback and works
regardless of the complexity of the regulator.
• No special algorithms or software are required.
• Consistency and optimal accuracy is obtained if the model structure contains the
true system (including the noise properties).
There are two primary drawbacks with this direct approach (Forssell & Ljung, 1997):
• Unlike open-loop identification, an accurate noise model, Ĥ(q, θ), is required.
• The second problem is a consequence of the first. When a simple model is sought
that should approximate the system dynamics, the noise model deviation from true
noise characteristics will introduce bias.
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
49
Convergence and Identifiability
Let us recall equation (2.60),
(t, θ) = H −1 (q, θ) [y(t) − G(q, θ)u(t)]
= H −1 (q, θ) [(G0 (q) − G(q, θ))u(t) + (H0 (q) − H(q, θ))e(t)] + e(t) (3.13)
and further re-arrange this equation such that
(t, θ) = H −1 (q, θ)T̃ T (q, θ)χ0 (t) + e(t)
(3.14)
T̃ (q, θ) = T0 (q) − T (q, θ)
(3.15)
with
and
"
T (q, θ) =
G(q, θ)
H(q, θ)
"
#
T0 (q) =
G0 (q)
H0 (q)
#
"
χ0 (t) =
u(t)
e(t)
#
(3.16)
Assuming that G(q, θ)u(t) and G0 (q)u(t) depend only on e(s) for s < t and that H(q, θ)
(and H0 (q)) is monic, then the last term in equation (3.13) is independent of the rest and
in reflection of equations (2.60) through (2.63) we have (Forssell, 1999),
Φ (ω) =
1
|H(ejω , θ)|2
T̃ T (ejω , θ)Φχ0 (ω)T̃ (e−jω , θ) + λ0
with
"
Φχ0 (ω) =
Φu Φue
Φeu λ0
(3.17)
#
(3.18)
this, w.p 1 as N → ∞, translates into,
1
θ̂N → Dc = arg min
θ∈DM 4π
Z
π
T̃ T (ejω )Φχ0 (ω)T̃ (e−jω )
−π
1
|H(ejω , θ)|2
dω
(3.19)
In equation (3.19) we thus have an understanding that the prediction error estimate
T (q, θ̂N ) will converge to the true transfer function T0 (q) if the parameterisation is flexible
enough so that S ∈ M and if the structure T̃ T (ejω , θ) does not lie in the left null space of
Φχ0 (ω) (Forssell, 1999). This latter condition of T̃ T (ejω , θ) not lying in the left null space
of Φχ0 (ω) is directly related to an informative enough data set. An informative data set
is achieved when the matrix Φχ0 (ω) is positive definite for almost all frequencies.
It is important to note that equation (3.19) is equation (2.63) factorised differently
in order to allow for a better interpretation of the convergence properties under closed
loop conditions. Note that under open-loop conditions the off-diagonal values in equation
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
50
(3.18) will be zero.
Consider the factorisation
#"
#
"
#"
1
0
1 Φue (ω)/λ0
Φru (ω) 0
Φχ0 (ω) =
0
λ0
Φeu (ω)/λ0 1
0
1
(3.20)
It follows that Φχ0 (ω) is positive definite for almost all frequencies if Φru (ω) is positive
for almost all frequencies. Recalling equation (3.9), stating that Φru (ω) is part of the
input power spectrum that originates from the reference signal r and, under a linear
feedback law, is defined as Φru (ω) = |S0 (jω)|2 Φr (jω). Since S0 (ejω ) can only have finitely
many zeros, the informative data condition becomes Φr (ω) > 0 for almost all frequencies.
In other words, the reference signal, r(t), should be persistently exited for closed loop
identifiability (Forssell, 1999).
Bias Distribution
Inserting the alternative factorisation of Φχ0 (ω),
"
Φχ0 (ω) =
1
0
Φeu (ω)/Φu (ω) 1
#"
Φu (ω)
0
r
0
Φe (ω)
#"
1 Φue (ω)/Φu (ω)
0
1
#
(3.21)
into equation (3.19), gives the following characterisation of the limit estimate w.p. 1 as N →
∞ (Forssell, 1999):
θ̂N → Dc
1
= arg min
θ∈DM 4π
Z
π
h
|G0 (ejω ) + B(ejω , θ) − G(ejω , θ)|2 Φu (ω)
−π
i
+ |H0 (ejω ) − H(ejω , θ)|2 Φre (ω)
1
|H(ejω , θ)|2
dω
(3.22)
where
B(ejω , θ) = (H0 (ejω ) − H(ejω , θ))Φeu (ω)/Φu (ω)
(3.23)
B is termed the Bias Pull and is an indicator of the bias inclination in the system estimate.
It follows that this bias inclination will be small over frequency ranges where either the
noise model is good, i.e. (|H0 (ejω ) − H(ejω , θ)|) is small, or the input power spectrum,
Φu , dominates the cross power spectrum, Φeu (ω). It is also noted that the bias term,
B(ejω , θ) is always identically zero in open-loop since Φeu (ω) is zero.
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
51
Asymptotic Properties
Recalling the definition of the black-box transfer function covariance matrix given by
equation (2.68) and the definition given by equation (3.18) we have,
"
Cov
G(ejω , θ̂N )
H(ejω , θ̂N )
#
≈
n
Φv (ω)Φ−1
χ0
N
This expression indicates that the asymptotic covariance of the transfer function is given
by the noise-to-signal ratio (v, noise; χ0 , signal) multiplied by the model order, n, to
number of data, N , ratio.
Additionally this expression holds regardless of the nature of the feedback. Under
open-loop conditions the off diagonal values of Φχ0 are zero and we have (Ljung, 1987:
251)
Cov G(ejω , θ̂N ) ≈
n Φv (ω)
N Φu (ω)
(3.24)
Cov H(ejω , θ̂N ) ≈
n
|H0 (ejω )|2
N
(3.25)
The above formulations reflect the points discussed in section 2.6.1. It is clear that
the signal-to-noise ratio and hence the extent of input signal excitation have a large role
to play in model estimation.
Under closed-loop conditions equation (3.24) becomes (Forssell & Ljung, 1997)
Cov G(ejω , θ̂N ) ≈
n Φv (ω)
N Φru (ω)
(3.26)
This illustrates precisely the closed-loop informative data condition for accurate estimation in contrast to the open-loop condition. Here it is required that the input signal
spectral component that originates in the reference signal r be excited.
3.2.2
The Indirect Method
Given the structure and characteristics of the feedback controller, C(q), are known, together with the reference signal, the indirect approach may be used. This approach may
be broken down into two components (Forssell & Ljung, 1997):
1. Identify the closed-loop system from the reference signal r to the output y defined
by Gcl
0.
2. Compute the open-loop model, ĜN , from the closed-loop model using knowledge of
the regulator.
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The advantage this method has is that any identification technique may be used since
estimating the closed-loop system from measurements of r and y is an open-loop problem.
This means that methods such as spectral analysis, IV and subspace methods may be
used (Forssell & Ljung, 1997).
In the prediction error method, which allows for arbitrary parameterisations, it is
natural to parameterise the model of the closed-loop system in terms of the open-loop
model (Forssell, 1999),
y(t) =
G(q, θ)
r(t) + H∗ (q)e(t)
1 + C(q)G(q, θ)
(3.27)
Here a fixed noise model H∗ is used which is common in this approach. This is mostly
due to the fact that since the identification problem has become an open-loop one, there
is no loss in consistency if a fixed noise model is chosen to shape the bias distribution.
Through this parameterisation, computing the open loop estimate is avoided and the
open-loop model is delivered directly making the second step superfluous.
An alternative parameterisation approach is to use the so called dual Youla parameterisation that parameterises all systems that are stabilised by a regulator C(q). See
Vidyasagar (1985)
Bias Distribution
Given the system description in equation (3.27) the following may be established (Forssell,
1999):
2
jω
jω
1
G0 (e ) − G(e , θ) |S0 (ejω )|2 Φr (ω)
dω
θ̂N → Dc = arg min
θ∈DM 4π −π 1 + C(ejω )G(ejω , θ) |H∗ (ejω )|2
Z
π
(3.28)
This indicates that the indirect method can give consistent estimates of G0 (ejω ) provided S ∈ M even with a fixed noise model. Unlike the direct approach it is difficult to
directly quantify the bias. The estimation will however be a compromise between making
G(ejω , θ) close to G0 and making the model sensitivity function, 1/(1 + C(ejω )G(ejω , θ)),
small. There is thus a bias-pull towards a transfer function that gives small sensitivity for the given regulator (controller) C. Literature (Gevers, 1986) shows this to be
advantageous if the model is to be used for control.
Asymptotic Properties
From equation (3.27) the asymptotic variance of Gcl (ejω , θ̂N ) is
Cov Gcl (ejω , θ̂N ) ≈
n |S0 (ejω )|2 Φv (ω)
N
Φr (ω)
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
53
regardless of the noise model H∗ (Forssell & Ljung, 1997). Forssell & Ljung (1997) goes
on to verify that from the above expression the following is obtained:
Cov G(ejω , θ̂N ) ≈
n Φv (ω)
N Φru (ω)
(3.30)
which is the same asymptotic variance obtained under the direct approach
3.2.3
The Joint Input-Output Method
The third of the primary approaches used in closed-loop system identification is the so
called joint input-output approach. As mentioned before the primary assumption here
is that the input is generated using a regulator of a known form. Exact knowledge of
the regulator is not required. Note this is an advantage over the indirect method where
complete knowledge of the regulator is necessary.
There have been many adaptations and variations of this approach. The three most
discussed in literature are the Coprime, Two-Stage and Projection methods. It is noted
however that each of these methods do fall under a unifying framework as Forssell &
Ljung (1999) shows for the multivariable case.
Of these three joint input-output approaches, focus will be put on the projection
method. This is due to the fact that the projection method’s characteristic dissimilarities
from the other methods are what allow it to be more applicable to systems with arbitrary,
non-linear feedback mechanisms.
The projection method consists of the following two steps (Forssell & Ljung, 2000):
1. Estimate the parameters sk in the non-causal FIR model
u(t) = S(q)r(t) + e(t) =
M2
X
sk r(t − k) + e(t)
k=−M1
where M1 and M2 are chosen so large that any correlation between u(t) and r(s)
for t − s ∈
/ [−M1 , . . . , M2 ] can be ignored, and simulate the signal û(t) = Ŝ(q)r(t)
2. Identify the open-loop system using a model of the kind
y(t) = G(q, θ)û(t) + H∗ e(t)
The first step in this approach may be viewed as a least squares projection of u
onto r, hence the name the projection method. Conceptually it is best understood as
an identification procedure where the system input and the output maybe be considered
as outputs of a system driven by the reference input r(t) and unmeasured noise v(t).
Knowledge of the system and the controller is recovered from this joint model.
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The two-stage method can essentially be explained with the same two steps. What
distinguishes the projection method from the two-stage method is that a non-causal
model is used in the first step instead of a causal model. In the two-stage method causal
high-order FIR or ARX models in are implemented in the first step (Forssell & Ljung,
2000).
The properties of the projection method will be discussed in the next sections. For
further detail on the Coprime, Two-stage and Projection methods see van den Hof et al.
(1995), van den Hof & Schrama (1993) and Forssell & Ljung (2000) respectively.
Bias Distribution
Restating that for the projection method we have the following definitions,
y(t) = G(q, θ)û(t) + H∗ e(t)
û(t) = Ŝ(q)r(t)
ũ(t) = u(t) − û(t)
(3.31)
this results, w.p. 1 as N → ∞, in
1
θ̂N → Dc = arg min
θ∈DM 4π
Z
π
|G0 (ejω ) + B̃(ejω , θ) − G(ejω , θ)|2
−π
Φu (ω)
dω
|H∗ (ejω )|2
(3.32)
where
B̃(ejω ) = G0 Φũû (ω)Φ−1
û (ω)
(3.33)
This holds true for both the two-stage and the projection methods for arbitrary feedback
mechanisms (Forssell, 1999).
Equation (3.33) shows that Φũû (ω) needs to be zero (i.e. ũ and û be uncorrelated)
for consistency to prevail for both the two-stage and the projection methods. By construction the projection method asymptotically achieves this regardless of the nature of
the feedback . This condition which is more applicable to the projection method is due
to the non-causal model used in the first step of the approach (Forssell & Ljung, 2000).
Asymptotic Properties
(Forssell & Ljung, 1997) shows that the transfer function asymptotic variance may be
given by
n Φv (ω)
Cov G(ejω , θ̂N ) ≈
N Φru (ω)
which is the same as that established for the direct and indirect approaches. This hold
for the projection, two-stage and coprime adaptations of the approach.
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3.2.4
55
Overview on Closed-Loop Approaches
The following overview is compiled with reference to Forssell & Ljung (1999), Forssell &
Ljung (1997) and Forssell (1999):
The Direct Approach
• If S ∈ M and the noise model contains the true noise model, then consistency and
optimal accuracy will be achieved regardless of the feedback.
• Requires a correct noise model to avoid incurring bias in the system transfer function.
• Does not require any knowledge of the feedback mechanism or controller.
The Indirect Approach
• Requires perfect knowledge of the controller but can be applied with fixed noise
models and still allow for consistency.
• Allows for other open-loop identification approaches to be applied that would otherwise fail under closed-loop conditions.
• Does typically give the best results in terms of identification for control.
The Joint Input-Output Approach
• Gives consistent estimations of the open-loop system regardless of the noise model
used.
• With exception of the projection method, only knowledge or specification on the
controller is that it has a certain linear structure.
• The projection method is capable of consistent identification under non-linear controller structures.
Other Comments
• Given a linear feedback mechanism, as the model order tends to infinity the asymptotic variance expressions are the same for all methods .
• Given a finite model order, the indirect approach generally gives the worst accuracy.
• The improved accuracy achieved by the direct approach can be attributed to the
fact that a constrained noise model that contains the true noise dynamics is used.
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3.3
56
Multivariable identification and Controller Specific Effects
Most literature concerned with the fundamentals of closed-loop system identification
assumes that the system is scalar and may be accurately represented through a linear
model and is controlled by a linear controller. It must be said however that the majority
of systems are to some extent non-linear and multivariable and in some instances are
controlled by non-linear controllers.
3.3.1
Multivariable Closed-Loop Identification
In section 2.7 it was mentioned the extension of system identification methods from
those applicable to scalar systems to those applicable to multivariable systems, is mostly
a task or notation changes and some slight non-trivialities in the parameterisation and
estimation approaches. Under the condition where the data set, Z N , is closed-loop data,
the same can generally be said.
Forssell & Ljung (1999) produced a survey paper which presents the entire closed-loop
identification problem and all its aspects concerning the prediction error framework, parameter estimation and asymptotic properties for the multivariable condition. The paper
affirms the understanding that all methods presented for the SISO case are analogously
applicable to the MIMO case with a few exceptions that are mostly concerned with the
properties and type of controller used.
The next sections introduce some aspects on controller specific effects on identification.
This will allow for some insight into the reasoning behind the theory presented in section
3.2 and additionally illustrate some of the multivariable closed-loop identification issues .
3.3.2
Linear Decentralised Controllers
Sections 3.1.2 and 3.2 showed how introducing the feedback can reduce the generation
of informative data via. The extent of this can be controller specific. The tuning of the
controller determines the strength of the ”feedback path”. A very detuned controller
means the control actions will be sluggish resulting in the ”forward path” dynamics
dominating the data (Doma et al., 1996). This implies that the controller parameters do
have effects on identifiability.
3.3.3
Non-linear Controllers and MPC
As stated in section 3.2, Forssell (1999) discussed how the indirect and joint inputoutput approaches become very involved and complex in the face of non-linear controllers
with the possible exception of the projection method. The problem with this is that
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57
many multivariable systems are dynamically complex and are consequently controlled
by centralised or model-based controllers which are non-linear. Furthermore, in some
cases even the linear PID controllers exhibit some non-linear traits due to the necessary
delimiters and anti-windup functions (Forssell & Ljung, 1999) (note that the extent of
this is dependant on input saturation which is dependent on controller tuning). Thus it
is relatively important to have an understanding of the effect of a non-linear controller
on the closed-loop identification process.
In the previous section it was mentioned that controller tuning parameters have effects
on the extent of identifiability. van den Hof (1998) takes this further and establishes that
switching controller parameters contributes to providing informative data that allows
for consistent identification. In essence a variable parameter controller is a non-linear
controller. This means that the introduction of a non-linear controller allows for the data
to be more informative while additionally complicating the identification methodology.
This is with exception of the direct approach as this approach is completely independent
of the feedback and the type of controller used.
From this we gather that the direct approach is the most suitable for complex, nonlinear controllers. This is directly applicable to model predictive control (MPC) which
is very much a non-linear controller since each control action is computed based on an
optimisation routine every iteration (Zhu & Butoyi, 2002).
Special attention has been given in literature to closed-loop identification for MPC
or on a system using MPC. Zhu & Butoyi (2002) and Zhu (1998) have concentrated
on a method termed the asymptotic method, which uses automated multivariable tests
to obtain specified desired input signal power spectra. This and other approaches have
also been extended into recursive online model predictive control identification where the
model is iteratively updated (Shouche et al., 1998)(Shouche et al., 2002).
3.4
Non-Linear Closed-Loop System Identification
While the focus of this work is primarily on linear estimation methods, it is of value
to address some common non-linear estimation methods found in literature that have
been applied to closed-loop conditions. It is noted however, that non-linear identification
approaches that do not rely on physical insight are rare and this is more so the case when
using closed-loop data. The more established method of the Youla-Kucera approach will
briefly be introduced followed by some other general approaches.
3.4.1
The Youla-Kucera Parameter Approach
This approach is related to the Dual-Yuala approach (the indirect method in section
3.2) where specific tailor-made parameterisations are used. It was initially investigated
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58
in Hansen (1989) and further investigated in a non-linear framework in Desgupta &
Anderson (1996). Fujimoto et al. (2001) explains this approach as one which relies on the
ability to parameterise the unknown system in terms of a known nominal model and the
known controller along with the so-called Youla-Kucera parameter (Vidyasagar, 1985)
associated with the system through the use of coprime factorisations. This turns the
identification problem into an open-loop one.
While the given description of the approach is brief it must be noted that this approach
has been extensively studied in literature. Additionally there are many adaptations
of this approach to study non-linear controllers (Linard et al., 1999) and kernel form
representations (Fujimoto et al., 2001).
3.4.2
Other Approaches
While the method identified in the previous section does seem to be the more widely
covered non-linear closed-loop system identification method in literature, many other
approaches have been established. In section 2.3.2 the Block-Oriented models were introduced as an approach to modelling non-linear systems. There has been some effort in
applying this to the closed loop data, in particular Eskinat et al. (1991). In this particular instance it was shown that the non-linear Hammerstein Block-Oriented model did
generally better represent the non-linear system than a linear model.
Again recalling a non-linear modelling method introduced in section 2.3.2, the VolterraSeries has been successfully applied to closed loop-data. Zheng & Zafiriou (1994) does
this with a control-relevant approach. A Volterra-series model obtained from closed-loop
data was combined with an uncertainy model and implemented in a high performance
model based controller.
Neural Networks models trained on closed-loop data, have also been used successfully
(Atsushi & Goro, 2001). Although this approach is far from the typical parametric or
convolution approaches discussed in this work, its success illustrates the extent of diverse
approaches and continuous effort and justification in furthering the field of closed-loop
identification.
3.5
Chapter Summary
This chapter presented the closed-loop system identification problem as being one where
feedback causes a lesser degree of information content in identification data and inputnoise correlations cause bias estimations. Three approaches to closed-loop system identification were presented:
• The direct approach.
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CHAPTER 3. CLOSED-LOOP SYSTEM IDENTIFICATION THEORY
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• The indirect approach.
• The join input-output approach.
with a comparative overview of these approaches given in section 3.2.4. Of these three
approaches the direct approach was found to be most suitable due to its simple application
and its ability to readily identify a system regardless of the controller complexity. It was
however found that in order to reduce model inaccuracy and bias using this method, it
is important to persistently excite the system and assure an accurate noise model.
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CHAPTER 4
Model Validation Theory
This chapter deals with the final component in system identification, model validation. Several essential validation techniques are presented, with a specific emphasis on linear approximations of non-linear systems and the validation of such
approximations. The last sections further study non-linear dynamics detection
and the determination of the extent of unmodelled non-linear dynamics via higher
order residual correlation analysis.
4.1
General Linear Model Validation
Several methods have been developed for linear model validation. Sections 4.1.1 through
4.1.5, that follow, present the more common approaches to model validation or gaining
insight into the validity of a model.
4.1.1
Statistical Tools, Hypothesis Testing and Confidence Intervals
There are several chapters in statistical texts dedicated to defining validation criteria,
confidence intervals and hypothesis testing, specifically for linear regression models.
Literature shows hypothesis testing to have wide applicability in modelling, specifically parametric modelling. Ljung (1987: 422) illustrates the use of statistical hypothesis
testing for comparing model structures and rejecting model structures and parameterisations based on defined confidence intervals. Montgomery et al. (2001: 319) presents a
section on hypothesis testing in linear regressions where tests for regression significance
and tests on individual regression coefficients are presented. The former allowing for
verification of the linear relationship between the true response variable, y(q), and the
60
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CHAPTER 4. MODEL VALIDATION THEORY
61
regression coefficients, the latter allowing for an investigation into the potential value
of each of the regression coefficients in the model. This is very useful in determining
the order of the model as it essentially statistically investigates the value in reducing or
increasing the model order.
There are several other statistically based goodness of fit testing methods. An example
that has found wide application in system identification is the Chi-squared (χ2 ) goodness
of fit test based on the Chi-squared distribution. An overview on this method is found
in Cressie & Read (1989).
These tests together with the many well developed procedures to obtain confidence
intervals in coefficients themselves, allow for a very practical understanding of the goodness of fit of the model and give insight into the uncertainty of the model accuracy
(Montgomery et al., 2001: 326).
4.1.2
Model-comparison Based Validation
This validation approach involves applying statistical tests that generate quantitative
values that may be used to compare models pairwise and to select the best model with
the minimum or maximum statistical value. A commonly used criterion that falls under
this approach is Akaike’s Final Prediction Error (FPE) and the closely related Information
Theoretic Criterion (AIC). Once several possible models of a system have been computed,
they may be compared using these criteria. According to Akaike’s theory, the most
accurate model has the smallest FPE and AIC (Ljung, 1995).
Akaike’s FPE is defined as follows (Ljung, 1987: 420):
FPE =
1 + dM /N
1 − dM /N
VN (θ̂N , Z N )
(4.1)
with
dM = dim θM
signifying the number of estimation parameters. For a quadratic norm this becomes
FPE =
1 + dM /N
1 − dM /N
1
· 2 (t, θ̂N )
2
(4.2)
This criterion may be understood as a reflection of the prediction-error variance obtained, on average, between the model defined by θ̂N , and a data set other than that used
in the identification procedure.
Akaike’s AIC is presented as (Ljung, 1995):
AIC = log VN (θ̂N , Z N ) +
2dM
N
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CHAPTER 4. MODEL VALIDATION THEORY
For dM << N
62
2dM
AIC = log VN (θ̂N , Z ) + 1 +
N
N
(4.4)
this stems out of general identification theory closely related to the maximum-likelihood
approach, the advantage being that this criterion is applicable to all model structures.
Ljung (1987: 189) shows how through Akaike’s AIC, a form of mean log-likelihood identification criterion is obtained and that minimisation of this criterion is a reflection of the
goodness of fit.
It is noted however, that AIC and FPE comparison criteria are mostly geared towards
model structure comparisons than comparisons between models of the same structure.
4.1.3
Model Reduction
A procedure or test that indicates whether the estimate is a simple and appropriate
description of the true system is to reduce is model order reduction. Ljung (1987: 426)
states that if the model order can be reduced without affecting the input-output properties
very much, then the original model was unnecessarily complex.
4.1.4
Simulation and Consistent Model Input-Output Behaviour
While it is good practise to compare system and simulated responses to common inputs,
Ljung (1987: 425) states that it is essential to compare system responses and model
responses in Bode Plots. Especially useful is the comparison between spectral analysis
estimates, such as those rooted in smoothed out frequency domain Empirical TransferFunction Estimates, and Bode plots as they are formed from different underlying assumptions. This Bode plot comparison approach is especially useful when comparing different
models obtained by PEM (Prediction Error Method) with different structures. This typically does allow for insight into whether essential dynamic features of the system have
been captured in a specific model structure.
Additionally, while most validation and model comparison approaches investigate the
accuracy of the model’s ability to simulate, it is also important to assess the model’s prediction capacity, specially if the model is to be used for control purposes. The primary
difference between simulation and prediction is that the former uses previously calculated
outputs generated by the model to generate a next value while the latter uses previously
measured (from the data set) and previously calculated values to generate the next value
(Ljung, 1995). From the model validation point of view, the difference is that simulation typically emphasises low frequency behaviour while predictions emphasise mid-range
frequency behaviour.
Validation through simulation is typically termed cross-validation. This is as the
model is validated against an independent data set of measurements containing system
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inputs and outputs of the true system. The model is disturbed (simulated) using the
measured inputs and response is compared against the measured outputs. Figure 4.1
illustrates.
4.1.5
Residual Analysis
Residual (estimation errors) analysis is a very well developed field of model diagnosis and
validation founded mostly in the 1960’s by Anscombe & Tukey (1963) and is very much
a strong component of cross-validation. An enormous amount has been written about
this field and many text books have been published providing its different applications
and adaptations, an example is Cook & Weisberg (1982). As Box & Jenkins (1970)
states, studying the difference between the model output and the output of the true
systems - residual analysis - allows for the study of the existence and nature of model
inadequacies, thus its place in model validation. Figure 4.1 illustrates. Specifically in the
Model
G(q,θ)
u(t)
y(t,θ)
+
є(t)
-
G0(q)
y0(t)
True
System
Figure 4.1: Model Residual
case of regression models, there are many well known methods of residual analysis where
statistical tools are applied to extract information about the model and ascertain as to
how good a representation of the true system it really is.
In this work concise literature on two approaches will be presented. The first is
standard residual plotting and the second is residual correlation analysis. Section 4.2.2
discusses an extension of the second approach.
Standard Residual Plotting
Standard residual plots are those in which the residuals, N (t, θ), are plotted against the
estimated values, ŷN (t, θ), or other functions of ϕ(t) (Cook & Weisberg, 1982). Figure
4.2 is an example of such a plot. Draper & Smith (1998: 63) reveals that a plot of
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²N (t, θ)
CHAPTER 4. MODEL VALIDATION THEORY
64
0
ŷN (t, θ)
Figure 4.2: Residual plot
residuals from a satisfactory or correctly specified model would resemble a horizontal
band of haphazard points such as figure 4.3a. In contrast to this there are many possible
unsatisfactory plots that indicate nonconstant error variances of which the most typical
forms are given in figures 4.3b through 4.3d. It is noted that figure 4.2 resembles the
residual plot indicated in figure 4.3d revealing an unsatisfactory model. Anscombe (1973)
makes the interesting argument in motivation for residual plotting in that through the
indication of nonconstant error variances, unmodelled non-linearites are implied.
The haphazardness or randomness of data presented in residual plots that reflect a
correctly specified model does have reason behind it. Recalling equation (2.15)
y(t) − ŷ(t|t − 1) = −H −1 (q)G(q)u(t) + H −1 (q)y(t) = e(t)
and the concept implied that the prediction error represents the component of the output,
y(t), that cannot be predicted from past data and is formulated by e(t), an unpredictable
white noise signal. It is thus understood that for an accurate model the residuals should
be white - or random - and fit the assumption that states:
(t) ≈ e(t)
(4.5)
Draper & Smith (1998) is referred to as a further source for more detailed literature
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CHAPTER 4. MODEL VALIDATION THEORY
65
(a)
(b)
(c)
(d)
Figure 4.3: (a) Acceptable residual plot (b) - (d) Unacceptable residual plots
on residual plots, specifically on gathering model diagnostic interpretations from plots.
Residual Correlations
The analysis of correlations amongst estimation residuals (estimation errors) and between
the estimation residuals and the system inputs are commonly used linear validation approaches. This analysis may be considered as a direct quantification of the concept behind
residual plotting and consists of the following two tests (Ljung, 1995):
The Whiteness Test : This test is based on the condition that the residuals, or the
prediction errors between the model and the system, of a good model should be
independent of each other and of past data. Therefore a good model has residuals
that are uncorrelated, i.e confirmation of the condition assumed through equation
(4.5). The whiteness test is thus obtained through the residual auto-correlation
(Ljung, 1987: 428),
R (τ ) = Ē[(t)(t − τ )]
(4.6)
The Independence Test : Is a measure of the correlation between the residuals and
the corresponding inputs. A good model has residuals uncorrelated to past inputs.
A peak in the residual-input correlation function at lag τk indicates that the output
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66
y(t) that originates from input u(t − τk ) is not properly described in the model.
Thus the model has unmodelled dynamics. This correlation is given by the crosscorrelation function as follows (Ljung, 1987: 429):
Ru (τ ) = Ē[(t)u(t − τ )]
(4.7)
Through the assertion of these two tests the following conditions for model adequacy
are formed (Billings & Voon, 1986):
R (τ ) = δ(τ )
Ru (τ ) = 0 ∀τ
)
(4.8)
where δ(τ ) is the auto-correlation function of a white signal, as discussed in section 2.4.1
(see figure figure 2.6(b)), which shows no correlations patterns but an infinitely narrow
impulse at τ = 0,
(
1,
τ =0
δ(τ ) =
(4.9)
0,
otherwise
Box & Jenkins (1970) further notes that if the system model is correct but the noise
model is incorrect, the residual auto-correlation function shows marked correlation patterns such that R (τ ) 6= δ(τ ) but they will be uncorrelated with the input such that
Ru (τ ) = 0 ∀τ . Alternatively, if both the noise model and the system model are inadequate, both the residual auto-correlation and residual-input cross-correlation will show
correlation patterns.
4.1.6
Model Uncertainty Considerations
An important consideration in modelling is the extent of uncertainty in the model. There
are many texts that delve into the details of accounting for model uncertainty in control,
specifically where model based controllers are implemented. Skogestad & Postlethwaite
(1997: 255) details that the various sources of model uncertainty may be grouped as
follows:
Parametric Uncertainty : Here the structure of the model is known, but some of the
parameters are uncertain.
Neglected and Unmodelled Dynamics Uncertainty : Here model uncertainty is rooted
in unmodelled dynamics.
Skogestad & Postlethwaite (1997: 255) further details the different manners in which
these model uncertainties may be represented in the control system description.
In terms of the identification - validation procedure, there have been efforts to incorporate uncertainty. Hjalmarsson (1993) presented work where considerations on the effects
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of incorporating model uncertainty and ill-defined model terms in the model validation
process where studied. Furthermore as sections 2.6.2 and 3.2 revealed, the description of
model uncertainty through the covariance of the estimations is an established approach
to viewing mode uncertainty information (Ljung, 1995).
4.2
Linear Time Invariant Approximations of Nonlinear Systems
In this section we look linear model validation under the condition where the dynamics
of the system being modelled might be non-linear. This introduces issues such as being
able to determine whether the residual is due to bad identification rooted in unfavourable
model parameterisation or structure selection or due to unmodelled non-linearities. Additionally if a system is non-linear it is useful to determine how well the linear approximation fits the linear components of the non-linear system and under what conditions
do the non-linearities distort this linear approximation.
4.2.1
LTI Aprroximation Sensitivity to System Non-Linearity
The LTI Second Order Equivalent
LTI approximations of non-linear systems have been studied in many frameworks. Related material on these different frameworks can be found in Schoukens et al. (2003),
Pintelon & Schoukens (2001) and Mäkilä & Partington (2003). This work focus on
framework presented by Enqvist & Ljung (2004), where under the PEM the input and
output signals are assumed to have the following properties:
1. The input u(t) and output y(t) are real-valued stationary stochastic processes with
E[u(t)] = E[y(t)] = 0.
2. There exists K > 0 and α, 0 < α < 1 such that the second order moments, Ru (τ ),
Ryu (τ ) and Ry (τ ) satisfy
|Ru (τ )| < Kα|τ |
∀τ ∈ Z
|Ryu (τ )| < Kα|τ |
∀τ ∈ Z
|Ry (τ )| < Kα|τ |
∀τ ∈ Z
3. The z-spectrum, Φu (z) (the z-transform of Ru (τ )) has a canonical spectral factorisation
Φu (z) = L(z)ru L(z −1 )
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where L(z) and 1/L(z) are causal transfer functions, L(+∞) = 1 and ru is a
positive constant. This implies the power spectrums, or spectral densities, Φu (ejω )
and Φyu (ejω ), are well defined for all ω ∈ [−π, π].
In addition to these assumptions it is implied that a OE (output-error) structure is used,
i.e. H(q, θ) = 1, which is illustrated in figure 4.4.
e
u
+
G
y
+
Figure 4.4: Output-Error Model Structure
These assumptions and the afore mentioned structure selection allow for the definition
of the Output Error LTI Second Order Equivalent (OE-LTI-SOE), as the cuasal, stable,
LTI approximation of a non-linear system that minimises the mean-square error, E[(y(t)−
G(q)u(t))2 ]. i.e. a convergence theorem is presented such that
G0,OE (q) = arg min E[(y(t) − G(q)u(t))2 ]
G∈Y
where
Φyu (ejω )
G0,OE (e ) =
Φu (ejω )
jω
(4.10)
(4.11)
causal
given a non-linear system. Note Y denotes the set of all stable and causal LTI models
(Enqvist & Ljung, 2004). Further elaboration and proof for equation (4.11) is given in
Ljung (1987).
From this the following convergence features are noted:
• Φyu /Φu may correspond to a non-causal function even if the true system is causal,
it is thus essential to assure the causal part is taken. The condition of non-causality
is realised under feedback conditions.
√
• THE OE-LTI-SOE will be typically approached with the rate 1/ N and the path
taken to the limit will depend on the input.
Properties of OE-LTI-SOE for Slightly Non-Linear Systems
With the OE-LTI-SOE being defined as the linear, time-invariant model to which the
least-squares optimisation converges given a non-linear system, its properties are extremely pertinent in understanding linear approximations of non-linear systems and the
key factors involved.
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Enqvist & Ljung (2004) shows that a slightly non-linear system may be represented
as a function of the linear component and the non-linear component as follows:
y(t) = Gl (q)u(t) + αyn (t) + c(t)
(4.12)
where Gl (q)u(t) is the linear part and αyn is the non-linear component defined as a
non-linear function of u(t). The term α defines the size of the non-linear component of
the system. Given the assumption that the input signal is generated by filtering white,
possibly non-Gaussian, noise e(t) through a minimum phase filter,
u(t) = Lm (q)e(t)
the z-transform form of equation (4.11) may be expressed as
G0,OE (z) =
Φyu (z)
Φye (z)
=
Φu (z)
Lm (z)Re (0)
(4.13)
where
Re (0) = E[(e)2 ]
Defining the minimum phase input filter by the constant c, (eg.
cq −1 )2 for 0 < c < 1) we have:
G0,OE (z, α, c) =
Φyu (z, α, c)
Φu (z, c)
= Gl (z) + α
Φyn e (z, c)
Lm (z, c)Re (0)
Lm (q, c) = (1 −
(4.14)
(4.15)
This description of the OE-LTI-SOE is very useful in that it allows for the an investigation
of the effects of the input signal used and the extent of non-linearity of the system. Enqvist
& Ljung (2004) concluded the following points through such an investigation:
• The larger α is, the size of the non-linear component of the system, the further
OE-LTI-SOE will be from the linear component, Gl (q), of the system. Small values
of α allowed for an OE-LTI-SOE near to Gl (q)
• Regardless of how linear or not the system is, there is always a bounded input signal
such that its OE-LTI-SOE is far from Gl (ejω ) for ω = 0.
• The distribution of the input signal components affects the results significantly.
While this is not the case for linear approximation of linear systems, when approximating a non-linear system a non-Gaussian input signal distribution can result in
the OE-LTI-SOE being far form the linear component of the system.
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• Gaussian input distributions reduced the sensitivity of the OE-LTI-SOE to nonlinaerities.
It is clear from the above that Enqvist & Ljung (2004) reveals the importance of
input signal design in system identification. This importance of using the correct input
signal to curve the effects of non-linearities on the linear estimate together with condition of informative data generation through persistent input signal excitation has had
much attention in literature. Chapter 5 introduces some aspects presented in literature
concerning input signal design in the context of experimental design theory.
4.2.2
Validation of LTI Approximations of Non-linear Systems
Categorisation of Non-linearity Detection Tests
The following is a categorisation of the different Non-linear detection tests as per Enqvist
et al. (2007):
1. Methods based on data from more than one experiment.
2. Methods based on comparisons with an estimated non-linear model.
3. Methods based on the assumption that the true system is a second order linear
system
4. Methods based on the assumption that the mean-square error optimal linear approximation of the system is noncausal
5. Methods based on second order moments of the input and output signals.
6. Methods for detection of non-linearities in time series.
7. Methods based on the use of special multi-sine input signals.
8. Methods based on higher order moments, either in time or in frequency domain.
For a more extensive and detailed survey of methods for nonlinearity detection in identification see Haber & Keviczky (1999).
Enqvist et al. (2007) states that categories 1 and 2 are most probably the most
common approaches to non-linear detection and linear model validation. However, these
methods are not ideal for the task as they have limited applicability in that category 1
assumes the input signals are unconstrained and category 2 requires a non-linear model
before hand.
Categories 3 and 4 are very much pertinent to the OE-LTI-SOE tool described in the
previous section. Enqvist et al. (2007) mentions that while these methods are useful in
particular instances, they are not applicable to the standard non-linear system. A Linear
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system or the linear component of a system can have a higher order than two, this means
that these methods would indicate false non-linearities in such condiations. Additionally
causality is generally implied through these approaches which is not always the case,
specifically if feedback is used.
The use of second order moments of input and output data in the 5th category,
detailed in section 4.1.5, has problems distinguishing unmodelled non-linearities from
measurement noise. Billings & Voon (1986) found that auto-correlation functions and
cross-correlation functions of residuals does not detect all unmodelled non-linear dynamics
making it relatively ineffective.
Category 6 and 7 approaches are labelled as only being able to detect non-linearities
in some special cases. This means they cannot by considered as a general all purpose
approach to non-linear dynamics detection.
Enqvist et al. (2007) and Billings & Zhu (1994) do however reveal that category 5
methods, correlation-based model approaches, have an advantage compared with modelcomparison based methods due to the fact that it is possible to diagnose directly if an
identified model is adequate or not without testing all the possible model sets. These
authors further mention that it is possible to increase the diagnosis power of this method
by considering higher order moments of the data. This approach is categorised as category
8.
Linear models are said to be able to describe the second order moments in a data set
with input and output measurements but in general not the higher order moments. This is
the underlying principle behind the fact that linear validation techniques fail in the face on
non-linear dynamics even if a linear approximation is attempted. This is additionally the
driving factor behind increasing the diagnosis power of correlation approaches in category
8. Higher order spectral densities or correlation functions are said to be the better of the
non-linear detection/linear model validation approaches. Furthermore, an advantage with
these methods is that they give only statistically significant model inadequacies. This
makes it easy to control the risk of false detections. The section that follows, further
details higher order correlation tests.
Higher Order Correlation Test
In section 4.1.5 it was established that correlation functions, Ru (τ ) and R (τ ), could
be used to determine whether a linear model has unmodelled linear dynamics. These
correlation functions may be extended to higher order correlation functions that allow
for the determination of whether a system is linear or non-linear. Billings & Voon (1986)
shows that if,
Ryy2 (τ ) = Ē[y(t − τ )y 2 (t)] = 0
∀τ
(4.16)
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Ru2 2 (τ )
Ru2 (τ )
Ru (τ )
Significance
=0
=0
=0
System model is adequate and unbiased
6= 0
=0
=0
Inadequate noise model, even or odd terms
missing
6= 0
6= 0
6= 0
System model is inadequate or System and
noise model is inadequate
6= 0
6= 0
=0
Even powered unmodelled non-linearity in
system model and possibility of inadequate
noise model
6= 0
=0
6= 0
Odd powered unmodelled non-linearity in
system model and possibility of inadequate
noise model
Table 4.1: Significance of higher order correlation results
where y(t) is the system’s response to a gaussian signal or any other signal with zero
third-order moments of u(t), then the system that produced y(t) is linear.
In addition to this, there have been several extensions of these higher order correlation
functions that allow for the determination of whether a linear estimation has unmodelled
non-linear dynamics. Of the many contributions to this field of study, Billings & Zhu
(1994) and Enqvist et al. (2007) have been found to be the most accessible. This work
will focus on the approach presented by Billings & Voon (1986) for its pragmatic form.
Enqvist & Ljung (2004) states that correlation tests may be used to look for dependencies between u(t)2 and (t)2 , u(t)2 and (t) in addition to the already introduced
pair, u(t) and (t). From these correlations, the following conditions are presented that
indicate the model to be unbiased and adequate are:
Ru2 2 (τ ) = 0 ∀τ
Ru2 (τ ) = 0 ∀τ
Ru (τ ) = 0 ∀τ



(4.17)


with table 4.1 summarising the significance of the different combinations of these conditions. The interpretations of these correlation results show that these test to not only
be model adequacy tests but also tests that give insight into the type of unmodelled
non-linear dynamics. This allows for further model discrimination.
These higher order correlation functions are straight forward extensions of the corre-
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lation functions defined in section 2.4.1,
Ru2 2 (τ ) = Ē[u2 (t)2 (t − τ )]
Ru2 (τ ) = Ē[u2 (t)(t − τ )]
Ru (τ ) = Ē[u(t)(t − τ )]
(4.18)
Billings & Voon (1986) illustrates through simulations the importance of using these
higher order correlation techniques for model validation. The lower order validation techniques have been found to give false model validations that would otherwise be detected
by these higher order correlations functions.
4.3
Multivariable Model Validation
It is understood that with the scope of this work primarily focusing on regression modelling, the model validation problem becomes one regression analysis. Multivariable regression analysis is a well developed field and deals the the fact that for a system with
p outputs and m inputs there can be up to p × m regressions to fit. This can make the
extension of single variable validation techniques more complex or less effective (Draper
& Smith, 1998).
The extension of most validation techniques to the multivariate condition is mostly a
matter of notation changes. In the case of the higher order correlation functions, along
with the notation change of input and output variables to column vectors, the residual
variable becomes


1
 . 
. 
(t) = 
 . 
p
allowing for the correlation matrix function. In the case of Ru2 (τ ) (Billings & Zhu, 1995),



Ru2 (τ ) = Ē[u (t) (t − τ )] = 


2
T
Ru21 1 (τ )
Ru22 1 (τ )
..
.
Ru21 2 (τ ) · · ·
Ru22 2 (τ ) · · ·
..
...
.
Ru21 p (τ )
Ru22 p (τ )
..
.






(4.19)
Ru2m 1 (τ ) Ru2m 2 (τ ) · · · Ru2m p (τ )
With respect to the conditions for model adequacy inferred on these correlation matrices, they are direct extensions of the scalar conditions expressed by equation (4.17)
but in matrix form.
Billings & Zhu (1995) shows how the application of linear validation tests, given
by equation (4.8), to a multivariable system with p outputs and m inputs, essentially
translates into p × m + p × p + 2 local tests.
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In efforts to reduce this number, Billings & Zhu (1995) further introduces a global
test formed from the local auto-correlations among all the submodel residuals and crosscorrelations among all the inputs and submodel residuals using,
Rη (τ ) = Ē[η(t)η(t − τ )]
(4.20)
Rβη (τ ) = Ē[β(t)η(t − τ )]
(4.21)
η(t) = 1 (t) + 2 (t) + . . . + p (t)
(4.22)
β(t) = u1 (t) + u2 (t) + . . . + up (t)
(4.23)
where
with the model adequacy conditions for Rη and Rβη respectively being those of the scalar
R and Ru .
4.4
Chapter Summary
In overview of this chapter, several model validation techniques were presented with
a specific concentration on cross-validation based residual analysis techniques. Residual
correlation tests were shown to be able to determine whether the identified noise model or
the dynamics model are the source of inaccuracy. In addition to this, literature revealed
higher order residual correlation tests to improve the efforts of model discrimination,
specifically when validating linear estimates of non-linear systems. A study of convergence
theory of linear approximation of non-linear systems showed that certain characteristics
do facilitate the identification of the best linear fit of a non-linear system. It was revealed
that Gaussian input signals are most favourable in producing informative data for linear
approximation generations.
© University of Pretoria
CHAPTER 5
Experimental Design Theory
In the preceding chapters, specifically chapters 2 and 3, an overview of open and
closed-loop system identification techniques was presented. In this chapter the
pragmatic nature of system identification is addressed through the presentation
of the theory behind experimental design. The concepts of informative data and
signal excitation are explored, with the later sections focusing on input signal
design, other key experimental variables and data preparation.
5.1
General Considerations
With the concepts of persistent excitation and informative data already introduced in
chapters 2 and 3, it is at this stage appropriate to study the extension of these principles into practical experimental implications. These implications are realised through
experimental design variables that together form an identification experiment or test.
Before the primary design variables are refined it is important to properly characterise
the experimental problem. This involves clarifying and defining which variables are to
be considered as measured system outputs and which variables are system inputs. Ljung
(1987) stresses an important point in that in a system there are signals that rightly are
to be considered as inputs (in the sense that they have an effect on the system), even
though it is not possible or feasible to manipulate them. If such variables are measured it
is still highly desirable to include them among the measured input signals even though,
from an operational point of view, they are considered as measurable disturbances.
Furthermore, it is important to understand the purpose of the model being identified
in the experiment. This being said, the experimental conditions need to resemble the
situation for which the model is to be used, specifically when approximating a system
that might be non-linear with linear model (Norton, 1986).
75
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Let all the design variables associated with the experiment be denoted by X . These
variables, in practise, are limited by the conditions incurred by the system and the operation of such system. As much of this work has detailed, many instances require
closed-loop system operation and determining whether this is required/necessary or not
must be done early in the identification procedure as there are implied constraints on the
design variables. Goodwin & Payne (1977) lists the typical constraints as follows:
1. Amplitude constraints on inputs, outputs or internal variables.
2. Power constraints on inputs, outputs or internal variables.
3. Total time available for the experiment.
4. Total number of samples that can be taken or analysed.
5. Maximum sample rate.
6. Availability of transducers and filters.
7. Availability of hardware and software for analysis.
5.2
Informative Experiments
Creating conditions for informative experimental data and optimum input signals is a
matter that may be divided into two areas of consideration. One is concerned with
higher order information of u(t) such as Rue (τ ) and Φu (τ ) and the other is the shape of
the input signal. Aspects concerned with the former are studied through the correlation
matrix, covariance matrix and information matrix.
5.2.1
Informative Data
In section 2.6.1 it was mentioned that for an experimental data set to be informative
enough to allow for model discrimination, the input signal used needs to be persistently
excited to a sufficiently high order. Ljung (1987: 365) proves that this excitation order,
of a quasi-stationary signal, may be defined by the correlation (also termed covariance)
matrix such that


Ru (0)
Ru (0)
· · · Ru (n − 1)


 Ru (1)
Ru (0)
· · · Ru (n − 2) 

(5.1)
R̄u (n) = 
..
..
..
..


.
.
.
.


Ru (n − 1) Ru (n − 2) · · ·
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Ru (0)
CHAPTER 5. EXPERIMENTAL DESIGN THEORY
77
where the input u(t) is persistently exciting of order n, if and only if R̄u (n) is nonsingular.
This is a direct translation of the definition of persistent excitation where a signal is
persistently excited of order n, if Φu (ω) is different from zero at at least n points in the
interval −π < ω ≤ π.
It is at this point important to recall that persistent excitation of a signal is not sufficient to guarantee informative data from a closed-loop system (see section 3.1.2). Section
3.2 details the different conditions pertinent to the different closed-loop identification approaches that result in informative data. In addition to this section 3.3.3 established that
literature indicates non-linear controllers or the shifting between controller parameters
as important aspects of creating informative data from closed-loop experiments.
5.2.2
Asymptotic Covariance and Information Matrices
Ljung (1987: 369) states that the requirement of informative experiments still leaves considerable freedom in input signal optimisation. The pursuit for the optimum input signal
requires a better objective function or optimisation problem. Much literature presents
the minimisation of the asymptotic covariance matrix as an optimisation function,
min α(Pθ (X ))
X ∈X
(5.2)
where α(P ) is a scalar measure of how large the asymptotic covariance matrix Pθ is and
X is the set of admissible designs as per the constraints on the design variables. This
covariance matrix function is further used to set the condition where if Pθ satisfies
Pθ ≥ M −1
(5.3)
then the covariance matrix is that of an unbiased estimation of G0 (q). M is termed the
Fisher information matrix, Ljung (1987: 370) shows that this information matrix may
be defined as
N
1 X
MN =
·
E[ψ(t, θ0 )ψ T (t, θ0 )]
(5.4)
κ0 t=1
with the average information matrix per sample, M̄ , being
N
1
1 X
Ē[ψ(t, θ0 )ψ T (t, θ0 )]
MN =
N →∞ N
κ0 t=1
M̄ (X ) = lim
where
1
=
κ0
Z
∞
−∞
(5.5)
0
[fe (x)]2
dx
fe (x)
It is noted that if e(t) were a Gaussian innovation, κ0 = λ0
This condition, expressed by equation (5.3) and termed the Cramer-Rao lower bound
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inequality, is not only a sufficient condition for unbiased estimates but also for estimation
efficiency. From this, the idea to choose X so that the scalar transformations of the
asymptotic covariance matrix Pθ (X ) and the information matrix M̄ (X ) are minimised
and maximised respectively, is established.
There has been much analysis of these matrices for both the open and closed-loop
conditions. From these analyses, information is extracted concerning input properties.
Ljung (1987: 371) indicates that a frequency analysis on the information matrix shows
how at frequencies where there is large parametric variance (i.e. large covariance matrix)
input power spectra should be more prominent to allow for informative data and the
consequent larger information matrix. Goodwin & Payne (1977) presented a study on
how constraints on input and output signal energy content - power spectra - affect the
information content and consequently the relative performance between open-loop and
closed-loop experiments. It was revealed that under constraints on input energy, openloop experimentation is preferable. However, under output energy constraints closed-loop
experimentation is preferable.
5.3
Input Design
Until now second order properties (information matrix, covariance matrix, power spectrum) and their characteristics with respect desirable input sequences have been discussed. At this point it is appropriate to present some typical input sequence types that
are used to generate data for system identification along with the signal-to-noise ratio
(SNR) characteristic. It must however be noted that while these are generic signals, the
functions and properties mentioned in the previous section, specifically the information
matrix, are shown in literature as tools to design the optimal input signal for a specific
system.
5.3.1
Step Signals
Step inputs and other deterministic type signals (see figure 2.1) are widely used in the
identification of linear models. Step signals may be defined as follows (Söderstrom &
Stoica, 1989):
(
0,
t<0
u(t) =
(5.6)
σ,
t≥0
where σ is the the step size. Step signals do allow for informative data about the system
regarding system gains, dead time and dominant time constant, however, revelation of
non-linear dynamics via such a signal does depend on the size of the step. This means
under constrained input conditions, step signals are not the most practical (Sung & Lee,
2003).
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
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This is reflected through the covaraince function (Ru (n)) in that given a step size of
σ, Ru (n) = σ 2 for all τ . Thus the covariance matrix (or correlation matrix) R̄u (n) is
non-singular if and only if n = 1, which means a step input is persistently exciting of
order 1. Such a low order of excitation means that the consistent estimation of more
than one parameter in a noisy environment cannot be achieved with a single step input.
(Söderstrom & Stoica, 1989).
5.3.2
Gaussian White Noise
Gaussian white noise signals are white noise signals with Gaussian distributions. These
signals are understood to have interesting consequences when the system being identified
is non-linear. As was discussed in section 4.2.1, if a signal is non-Gaussian then it is
likely that a linear approximation of a non-linear system will deviate from the best linear
approximation of this system. That is, Gaussian signals disturb the system so as to
allow for data that is more likely to provide better linear estimates. On the other hand,
Gaussian signals are understood as being poor at identifying non-linearities as the signal
operates about the origin and hence the full operating range is not used (Nowak & Veen,
1994).
This means that Gaussian signals are favourable for linear modelling, even if the
system is non-linear, and non-Gaussian signals are favourable for non-linear modelling.
Figure 5.1 illustrates a white Gaussian signal. The covariance matrix of a Gaussian white
noise sequence of variance λ0 is R̄u (n) = λ0 In with In being an identity matrix of size n.
Since this matrix is invertible for all n, Gaussian white noise sequences are persistently
exciting of all orders (Söderstrom & Stoica, 1989: 121).
5.3.3
Pseudorandom Sequences
Binary
In the presence of constraints, the use of bounded (or amplitude constrained) signals that
have desirable second order properties that allow for informative data is important. In
satisfying this condition it is often better to let the input signal be binary:
u(t) = u1
or u2
where u1 and u2 are permissible input levels.
Two binary signal forms have been widely used throughout identification experiments.
One is the random binary signal (RBS) and the other is the pseudorandom binary signal
(PRBS). A pseudorandom binary signal is typically generated by a random sequence of
length P s and repeating this sequence as many times as required, as figure 5.2 illustrates
(Söderstrom & Stoica, 1989). These sequences are deterministic signals that are designed
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
80
Signal Amplitude
White Gaussian Signal
0
Time (s)
Figure 5.1: A white Gaussian signal
to have the same covariance function, independent of signal strength, as white noise, see
figure 5.2(b) (horizontal lines are 95 percent confidence intervals), it can be shown that
a covariance function for a PRBS of period P s is given by
(
Ru (τ ) =
σ2,
2
− Pσ s ,
τ = 0, ±P s, ±2P s . . .
otherwise
(5.7)
since Ru (τ ) = Ru (1) for all τ 6= 0, ±P s, ±2P s . . ., the covariance matrix for a PRBS is
given by


2
2
σ 2 − Pσ s · · · − Pσ s
 σ2
2 
 − P s σ 2 · · · − Pσ s 
R̄u = 
(5.8)
..
.. 
..
 ..

.
.
. 
 .
2
2
− Pσ s − Pσ s · · ·
σ2
(Söderstrom & Stoica, 1989) additionally reveal that optimum time between each
sequence switch is about 20% of the largest system response time and that the best
length of signal perturbation period is several times the largest time constant.
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
81
Signal Amplitude
(a) − Pseudorandom Binary Signal
0
Time (s)
Sample Auto−Correlation
(b) − Sample Autocorrelation Function of (a)
1
0.5
0
−0.5
0
10
20
30
40
Lag
50
60
70
80
Figure 5.2: A pseudorandom binary signal
Multilevel
Pseudorandnom binary sequences have been found to be persistently exciting for linear
systems, when used on non-linear systems, the extent of persistent excitation is less (Toker
& Emara-Shabaik, 2004). Pseudorandom Multilevel Sequences (PRMLS), illustrated in
figure 5.3, are similar to PRBSs except that the signal magnitudes randomly vary between
binary signal magnitudes. The second order and higher order properties of PRMLSs
approach those of white noise. Figure 5.3(b) illustrates the covariance function, compared
to that of the PBRS in figure 5.2(b), it is noted that the 95 percent confidence intervals
are not completely satisfied. Toker & Emara-Shabaik (2004) does however show that
through determination of the number of different levels used and the signal switching
probability, the closeness of the properties of PRMLSs to white noise can be improved.
These signals have been found to be more persistently exciting for non-linear systems
than PRBSs.
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
82
Signal Amplitude
(a) − Pseudorandom Multilevel Signal
0
Time (s)
Sample Auto−Correlation
(b) − Sample Autocorrelation Function of (a)
1
0.5
0
−0.5
0
10
20
30
40
Lag
50
60
70
80
Figure 5.3: A pseudorandom multilevel signal
5.4
5.4.1
Other Design Variables
Model Structure and Order Selection
Generally there are two approaches to model structure and order selection. The first is
based on analysing preliminary data for indications of the true structure of the system.
The second is a comparative approach where different structures are used and compared
through validation techniques. The former, also termed data-aided model structure selection, appears to be an underdeveloped field. However, the importance of a priori
knowledge of the system is always useful in facilitating the determination of which structure is most representative (Ljung, 1987: 413).
An exception to this underdevelopment of data-aided selection approaches is model
order estimation. There have been several developments in different techniques to systematically select model orders. Literature specifically does this with emphasis on pole-zero
cancellation under conditions of over-specifying model orders. The following are categories for order selection methods (Ljung, 1987: 413):
1. Examination of spectral analysis estimates of the transfer function.
2. Testing the rank in sample covariance matrices.
3. Correlating variables
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
83
4. Examining the information matrix.
5.4.2
Bias Considerations and Sampling Intervals
Recalling section 2.6.1, specifically equation (2.63), and assuming a monic or fixed noise
model we have,
1
θ̂N → Dc = arg min
θ∈DM 4π
Z
π
|G0 (ejω ) − G(ejω , θ)|2 Q(ω, θ)dω
(5.9)
−π
where
Q(ω, θ) =
Φu (ω)
|H(ejω , θ)|2
As was previously established, Q may be considered as a weighting function that would
allow for the correction of model bias by selecting the following (Ljung, 1987: 351):
• Input Spectrum, Φu (ω)
• Noise model, H1 (q)
• Prefilter, L(q)
• Prediction horizon, k
where H1 (q) is a component of the noise model that is usually fixed to give the noise
model a frequency weighting.
The sampling interval is an additional variable that also contributes to the bias of
the model. Before this model bias due to sampling interval is presented, it is valuable to
introduce the role of the sampling interval.
Due to the discretised nature of data-acquisition systems there is an inevitable loss
of data associated with sampling intervals. This makes it important to select a sampling
interval, T , that reduces or makes information loss insignificant (Norton, 1986).
If the total experiment time, 0 ≤ t ≤ TN , is limited and the acquisition of data
within this time is costless, it is clearly advantageous from an theoretical point of view
to sample as fast as possible. However, there are aspects that are undesirable when
doing so. Firstly, creating a model based on data obtained using a high sample rate
can become computationally intensive, secondly, model fitting may concentrate on the
high-frequency band. The former problem is becoming less and less relevant with ever
increasing availability of computing power. The latter problem may be dealt with by
determining the best sampling interval relative to the bandwidth over which the concerned
dynamics needs to be captured. This however is not the typical approach. This problem
is usually dealt with by prefiltering the data so as to redistribute the bias (Ljung, 1987:
378).
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
84
(Ljung, 1987) shows where the T -dependence enters in parameter convergence theory
through equation 5.9. As sample interval, T , tends to zero, the frequency range over which
the fit is made in equation (5.9) increases. However, system dynamics are normally such
that G0 (ejω ) − G(ejω , θ) is well damped for high frequencies so that the contribution from
higher values of ω in equation (5.9) will be insignificant even if the input is has a broad
bandwidth.
An important exception is the case where the noise model is coupled with the system
model. Such is the case of the ARX structure where H(ejωT ) = 1/A(ejωT ). Thus the
product
|G0 (ejω ) − G(ejω , θ)|2
|H(ejωT , θ)|2
does not tend to zero as ω increases and the fit in equation (5.9) is pushed into very high
frequency bands as T decreases. This means small sample intervals are undesirable as in
effect it results in putting a higher weighting on higher frequency prediction errors and in
turn providing a model that is biased to higher frequency dynamics (Ljung, 1987: 383).
This effect can be countered in two ways, firstly the effects in the bias may be counteracted by a pre-filter that essentially functions as a frequency counter-weight to that
created by the short sampling interval. Secondly the prediction horizon in the prediction
error calculation may be increased.
5.5
Closed-Loop Experiments
In a similar fashion as open-loop experiments, covariance matrices and information matrices may be derived for the closed-loop case where the reference signal is perturbed and
optimised. Goodwin & Payne (1977) does make it clear that the only condition where
closed-loop experimentation allows for larger information matrices is under output energy
constraints. Additionally it is established that white setpoint perturbations of variance
λ are the optimal inputs for experiments with feedback where the λ may be derived from
an information matrix analysis.
In addition to this, as illustrated in figure 3.1 and explained in section 3.1.1, besides
the setpoint signal, r2 , there is the additional signal r1 termed the dither or excitation
signal. Perturbing the system via the dither signal is typically the preferred approach as
it disturbs the system directly via the input and not through the controller. This means
that the perturbation is superimposed on the controller efforts to maintain the setpoint.
This implies to some extent the identification process is occurring in the background of
the control system (Norton, 1986).
Furthermore, as has already been mentioned, designing a closed-loop experiment that
is formed by sub-experiments where the feedback law is varied improves the information
content of the experiment.
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
5.6
85
Multivariable Inputs
There are essentially three basic approaches to multivariable testing and the signals involved (McLellan, 2005),
• Perturb inputs sequentially and estimate SISO models (SISO)
• Perturb all inputs simultaneously and estimate models for a given output (MISO)
• Perturb all inputs simultaneously and estmate models for all outputs simultaneously
(MIMO)
Each has advantages and disadvantages. It may generally be said that while the SISO
and MISO approaches do allow for easy identification of the true structure and dynamics
between individual variables, the loss of information due to not accounting for all the
variable interactions results in a poorer model in comparison the MIMO approach (Zhu,
1998).
This makes automatic simultaneous multivariable signal perturbation the best option
for multivariable system identification. In terms of the signals used to induce such perturbations, all those previously mentioned and all their advantages and disadvantages apply
to the multivariable case. McLellan (2005) goes further to suggest that cross-correlated
input signals should be used.
5.7
Data Preparation
After data has been collected via the identification experiment, it is likely that the data
is not ready for immediate use in identification algorithms. Once data collection is completed it is typical to plot the data to inspect it for deficiencies that could cause problems
later in the identification procedure. The following are several typical deficiencies that
should be attended to (Ljung, 1987: 386):
1. High-frequency disturbances in the data record, above the frequencies of interest to
the system dynamics.
2. Occasional bursts and outliers.
3. Drift and offset, low-frequency disturbances, possibly of periodic character.
This chapter presents methods and approaches on how to polish the data and avoid
problems later in the identification procedure due to these deficiencies.
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
5.7.1
86
High-frequency Disturbances
As mentioned earlier when discussing sampling intervals, if a sampling interval is too
small the resulting data set could contain higher energy components at bandwidths that
are not of concern. A typical solution to treating data that has this characteristic is to
re-sample this original data set, at a different sampling rate thus effectively increasing
the sampling interval (Ljung, 1987: 386).
5.7.2
Bursts and Outliers
Isolated erroneous or highly disturbed values of measured data may have substantial
effects on the resulting estimate. Protecting the estimate from such bad data is thus very
necessary. A robust identification procedure, specifically a robust identification norm,
does accomplish this. See Ljung (1987: 396) for a more elaborate discussion on the
selection of a robust norm. It is however noted that bad data is typically easy to detect
and remove or replace with some type of interpolation (Ljung, 1987: 387).
5.7.3
Slow Disturbances: Offset
Low-frequency disturbances, offsets, trends, drift and periodic (seasonal) variations are
typically common in data and usually stem from external sources that we may or may
not prefer to include in the system model. There are basically two different approaches
to deal with these issues (Ljung, 1987: 387):
1. Removing the disturbances by explicit pre-treatment of data.
2. Letting the noise model take care of the disturbances.
The first approach involves removing trends and offsets by direct subtraction while the
second relies on accurate noise models. These two approaches will now be discussed with
respect to the offset problem.
The offset problem is best understood as the result of estimation procedure constraint
in modelling the static behaviour of a system. This means that it is typically necessary
to specify how the estimation algorithm is going to treat the initial state of data.
Consider the standard linear model:
A(q)y(t) = B(q)u(t) + v(t)
(5.10)
This model describes both the dynamics properties of the system and the static relationships between the steady state input, ū, and the steady state output, ȳ, given by
A(1)ȳ = B(1)ū
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
87
In practice, the raw input-output data used for identification are the measurements,
um (t) and y m (t). These measurements are recorded in physical units, the values of which
may be quite arbitrary. This being said, the constraint on equation (5.10), and the
estimation algorithm, to satisfy the fitting of the arbitrary values in equation (5.11) is
unnecessary. There are five ways to resolve this problem (Ljung, 1987: 387):
Let u(t) and y(t) be deviations from equilibrium: The natural approach is to determine the steady state values at the desired operating point and define :
y(t) = y m (t) − ȳ
(5.12)
u(t) = um (t) − ū
(5.13)
as the deviations from the steady state. This approach emphasises the physical
values in the dynamic models as linearisations around an equilibrium. Furthermore
this approach allows for the definition of the arbitrary physical values while not
imposing a constraint on the estimation procedure.
Subtract sample means: This approach is closely related to the first approach. Here
we define
N
N
1 X m
1 X m
y (t),
ū =
u (t)
(5.14)
ȳ =
N t=1
N t=1
and use equation (5.12). If an input is used that varies around ū, and the resulting
output varies around ȳ, the system is likely to be near an equilibrium point.
Estimate the offset explicitly: One could also model the system using variables in
the original physical units and add a constant that corrects the offsets,
A(q)y m (t) = B(q)um (t) + α + v(t)
(5.15)
Comparing equation(5.10) to equation (5.12) we find that α corresponds to A(1)ȳ −
B(1)ū. The value α is then included in the parameter vector θ and estimated from
data.
Using a noise model with integration In equation (5.15) the constant α could be
viewed as a constant disturbance,
α
δ(t)
1 − q −1
where δ(t) is the unit pulse at time zero. The resulting model is thus
y m (t) =
B(q) m
1
u (t) +
w(t)
−1
A(q)
(1 − q )A(q)
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CHAPTER 5. EXPERIMENTAL DESIGN THEORY
88
where w(t) is the combined noise source αδ(t) + v(t) − v(t − 1). The offset α can
thus be described by changing the noise model from 1/A(q) to 1/[(1 − q −1 )A(q)].
this is equivalient to prefiltering the data through the filter L(q) = 1 − q −1 , that is,
differencing the data (Ljung, 1987: 388):
yFm (t) = L(q)y m (t) = y m (t) − y m (t − 1)
m
m
m
um
F (t) = L(q)u (t) = u (t) − u (t − 1)
(5.17)
Extending the noise model: It is noted that the mode described in equation (5.16)
becomes a special case of the model described by equation (5.10) if the polynomial
orders of A and B are increased by 1. A common factor can then be included in
A and B which implies a higher-order model, when applied to the raw measured
data, y m and um , will converge to a model like in equation (5.16).
Of these several approaches to dealing with offset, the first one is recommended. In
conditions where a steady state condition in the experiment in not feasible, then the
second approach of explicitly estimating the offset is recommended (Ljung, 1987: 389).
It is important to note that in the case of non-linear model fitting it is generally better
not to remove offsets.
5.7.4
Slow Disturbances: Drift, Trends and Seasonal Variations
Coping with the other slow disturbances such as drifts, trends and seasonal variations,
are very much analogous to the approaches taken for the offset slow disturbance. Drifts
and trends may be interpreted as time varying steady states, or time varying means. High
pass filters have also been used to compensate for these conditions. As for the seasonal
variations, Box & Jenkins (1970) have developed several useful techniques.
5.8
Chapter Summary
This chapter presented a concise account of system identification experiment design and
the involved variables. The more important variable was found to be the disturbance
signal used to persistently excite the system so as to generate informative data. Step
signals were found to be least informative while the PRBS and the White Gaussian signals
were found to be most exciting with the White Gaussian signal being more favourable for
non-linear information extraction. The importance of bias manipulation via input signal
properties and noise model frequency weighting was established. The data sampling
rate was found to be of particular value in manipulating the frequency weighting of the
estimation procedures especially when using an ARX model structure.
© University of Pretoria
CHAPTER 6
Identification of Simulated Systems
This chapter is the first of two chapters establishing the investigative experimental
work done. In this first chapter, the experimental work presents itself through
simulation. That is, open and closed-loop experiments are conducted on known
mathematical models through the simulation of system disturbances (inputs) and
the consequent responses (ouputs). The data generated from these experiments
is then used for model identification and validation. The identification framework
and validation methods used is to be defined over the next few chapters together
with the objectives of these experiments. It is important to note that the results
obtained from these experiments will form the basis of the experimental design
efforts for the identification of the pilot scale distillation column. This is presented
in the next chapter.
6.1
6.1.1
Investigative Approach and Model Description
Investigative Approach
As stated in chapter 1, the primary experimental objectives of this work are to:
• Investigate, within a specific framework, how well different identification and validation experiment designs perform.
• Specifically assess the effects of disturbance signal characteristics and feedback conditions (open/closed-loop).
• Focus the validation assessment on cross-validation methods and study how the
different experimental conditions under which validation data is generated affect
89
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
90
the validation results. A specific focus is again made on the effects of disturbance
signals and feedback conditions (open/closed-loop).
• Investigate the effect of different identification and validation experiment conditions
on the linear approximation of non-linear systems.
This chapter pursues these objectives through the use of known true systems given
by known mathematical models where system disturbances and responses are simulated.
These true systems are those to be identified. The direct objectives of this chapter may
thus be further refined as follows:
1. To obtain suitable linear and non-linear multivariate mathematical models which
will be the true known systems that need to be identified. The linear and non-linear
models used to represent the true systems are from here on referred to as system A
and system B respectively.
2. To design a broad range of open and closed-loop experiments to be conducted on
system A and system B to generate data for identification of these known systems
and validation of the identified models.
3. To generate linear approximations of system A and system B using the data generated during the identification experiments.
4. To validate the linear approximations of each system using different validation data
sets and techniques.
5. To assess the differences between the validation results obtained from the different
validation techniques.
With the investigative approach now generally defined, it is at this point important to
reiterate that identification and validation methods and approaches are generally defined
by the purpose of identification, be it simulation or control (prediction). The experiments
and the techniques used were carried out in such a way so as to yield information for both
perspectives of identification. It is however important to recall that the ultimate model
validation test with respect to control purposes is the extent of successful implementation
of such a model in a model based controller. Since this is not done the conclusions reached
in this work are more suited to identification for simulation.
The next sections introduce the linear and non-linear systems to be identified. That
is, the mathematical models used to represent system A and system B respectively. Furthermore the identification and validation framework used is defined.
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
6.1.2
91
Linear Model - System A
System Description
The mathematical model used to represent the linear system to be identified was obtained
from work by Nooraii et al. (1999). In this work the authors investigated uncertainty
characterisation and robustness of a pilot scale distillation column (illustrated in figure
6.1) separating water and ethanol. Their work consequently lead to the generation of the
higher order parametric ARX model to be used in this work. Nooraii et al. (1999) details
that a flow-sheeting package, SPEEDUP, was used together with Matlab’s identification
toolbox to identify the model.
x6
C
x4
L
x2
D
TP
x1
F
x5
R
S
TB
x3
B
Figure 6.1: General distillation column
Referring to figure 6.1, the authors modelled the distillation column such that the
multivariable ARX structure was selected with two outputs and two inputs. The outputs,
© University of Pretoria
CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
92
y1 and y2 , being the temperature of plate 12 and 5 respectively (TP and TB on the figure)
with the inputs, u1 and u2 , being the reboiler duty (energy transfered to R by S) and the
reflux flow rate (L) respectively. The authors scaled the model to be a deviation model.
For experimental consistency the normal operating range of both model inputs was taken
to be −0.5 to 0.5 while the output range for both open and closed-loop simulations was
taken as −0.5 to 0.5.
The model generated was documented as reflecting the column’s high levels of variable
interaction and ill-conditioned nature. Figure 6.2 illustrates the open-loop step responses
of the column while table 6.1 contains the parameters defining the model. In addition to
this a system pole-zero analysis revealed that the system does contain poles out side the
unit circle on the z-plane (Nooraii et al., 1999). This implies an inherent instability in
the system.
Open-loop y response to unit step in u
1
Open-loop y response to unit step in u
1
1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
Time (s)
20
25
0.02
0.015
0.01
0.005
0
0
30
Open-loop y response to unit step in u
2
10
15
Time (s)
20
25
30
2
2
∆ Plate 5 Temperature ( C)
8
10
o
o
∆ Plate 5 Temperature ( C)
5
Open-loop y response to unit step in u
1
12
8
6
4
2
0
0
2
0.025
o
∆ Plate 12 Temperature ( C)
o
∆ Plate 12 Temperature ( C)
0.7
5
10
15
Time (s)
20
25
30
6
4
2
0
0
5
10
15
Time (s)
20
25
30
Figure 6.2: Open-loop step response of linear system - system A
System Control
The input variables, u1 and u2 , were directly manipulated to control the model’s outputs
under closed-loop conditions. While it is evident that the best controller, in the face of
such conditions, would be a model predictive controller, a PI controller was used. This
was mostly due to its easier application and the fact that, while not the best, the PI
controller was found to provide adequate control.
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
93
Parameter Values ×104
a11
a12
a21
a22
Coefficient
A
A1
A2
A3
A4
A5
A6
A7
A8
10000
0
40.2
-3.26
113.5 -24.8
62.4 -0.305
14.8
0
1.33
0
0.068
0
24.6
0
0
446.8
205.9
53.05
8.63
0.83
0.037
18.5
10000
-583
-57
-8.63
-0.56
-0.018
0
0
B
b11
b12
b21
b22
B1
B2
B3
B4
B5
B6
B7
B8
0
135
1765
2720
1332
301.3
11.26
1
0
1
16.65
8.015
0
0
0
0
0
7309.3
58936
25766
3544
423.3
16.04
1
0
7714
44790
154.57
840
100
0
0
Table 6.1: Model parameters for the linear ARX model representing system A
Closed-loop response of y to setpoint change
1
o
∆ Plate 12 Temperature ( C)
0.2
0.1
0
-0.1
-0.2
-0.3
0
Output
Setpoint
50
100
150
200
250
Time (s)
300
350
400
450
500
Closed-loop response of y to setpoint change
2
o
∆ Plate 5 Temperature ( C)
3
Output
Setpoint
2
1
0
-1
-2
0
50
100
150
200
250
Time (s)
300
350
400
450
500
Figure 6.3: Closed-loop response to setpoint changes for linear system - system A
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
94
A controllability analysis conducted by Nooraii et al. (1999) revealed that the system
has substantial levels of interaction and is ill-conditioned. The step responses given in
figure 6.2 reflect the ill-conditioned state of the system in that the steady state gains are
much larger for output y2 . It was found that the best variable pairing for the decentralised
controller is; y1 -u1 and y2 -u2 . The controller parameters were found by trail and error.
Figure 6.3 shows the closed-loop response to setpoint changes.
6.1.3
Non-linear Model - System B
System Description
In a similar fashion as the linear system, the non-linear system was obtained from efforts
by Chien (1996) to identify a non-linear model of a distillation column separating water
and ethanol. The column was characterised as a high purity column and being highly
non-linear.
Through open-loop experimentation on the distillation column, Chien (1996) used
disturbance and disturbance response data to identify the non-linear approximation of
the column. The model obtained from these efforts and used in this work is thus a
multivariable non-linear ARX model (NARX) defined by:.
y1 (1 + k) = C1 + C2 y1 (k) + C3 u1 (k) + C4 u2 (k) + C5 y1 (k)u1 (k)
+C6 y1 (k)u2 (k) + C7 y12 (k)u1 (k) + C8 y13 (k)
+C9 y12 (k − 1)u1 (k) + C10 y12 (k − 3)u2 (k) + C11 y12 (k − 2)
(6.1)
y2 (1 + k) = D1 + D2 y2 (k) + D3 u1 (k) + D4 u2 (k) + D5 y1 (k)u1 (k)
+D6 y1 (k)u2 (k) + D7 y22 (k)u2 (k) + D8 y2 (k)u22 (k)
+D9 y2 (k − 1)u22 (k)
(6.2)
and
with the parameters given in table 6.2. The outputs, y1 and y2 , were modelled as the
temperatures on tray 21 and 7 respectively (TP and TB on the figure 6.1), while the
inputs, u1 and u2 , are the reflux flow rate and the boilup rate respectively (L and R).
The model was scaled to be a model of deviation from a specific steady state.
In assessing the model and its extent of non-linearity, the size of the regions over which
non-linear dynamics were small enough to be negligible were quickly identified. Figure
6.4 shows the response of the system to a 2 percent and 5 percent change in the inputs.
If the system were linear there would be a direct proportionality between the change
in input and the change in output. This is clearly not the case. While the changes in
response to input u1 do not so clearly indicate this deviation from direct proportionality
in response changes, the differences in responses to magnitude changes in u2 do. This is
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
Output 1 (y1 )
Coefficient Parameter Values
C1
0
C2
0.98
C3
-0.27
C4
0.21
C5
-1.91
C6
1.44
C7
0
C8
0
C9
5.96
C10
-3.71
C11
-0.11
95
Output 2 (y2 )
Coefficient Parameter Values
D1
0
D2
0.99
D3
-0.32
D4
0.31
D5
0.76
D6
-1.02
D7
-1.22
D8
0
D9
-12.1
Table 6.2: Model parameters for non-linear ARX model representing system B
Open-loop y response to step in u
1
Open-loop y response to step in u
1
1
2% step
5% step
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
0
100
200
Time (s)
300
0.3
0.25
0.2
0.15
0.1
Open-loop y response to step in u
2
100
200
Time (s)
300
400
Open-loop y response to step in u
1
2
2
0.2
2% step
5% step
-0.5
-1
-1.5
100
200
Time (s)
300
400
o
∆ Plate 7 Temperature ( C)
o
∆ Plate 7 Temperature ( C)
2% step
5% step
0.05
0
0
400
0
-2
0
2
0.35
o
∆ Plate 21 Temperature ( C)
o
∆ Plate 21 Temperature ( C)
0
2% step
5% step
0.15
0.1
0.05
0
0
100
200
Time (s)
300
400
Figure 6.4: Response to 2 and 5 percent step disturbances for the non-linear system - system
B
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
96
especially the case for the y2 response, one notes that the response to a 5 % input step
presents an inverse response that is not present at in the response to a 2 % input step.
This dynamic behaviour may be accounted for by the final term in equation (6.2).
This term is an exponential function of input u2 , compounding this is coefficient D9 ,
which is negative and the largest of all coefficients.
In addition to this, while further perusing an understanding of the dynamics of the
system, particularly the extent of the non-linearity, the higher order correlation test for
non-linear dynamics defined by equation 4.16 was used. The result is given in figure 6.5
together with the result obtained for the linear model given white Guassian input signals.
This clearly shows a deviation from linear dynamics.
Higher order correlation of inear system
0.6
0.5
0.3
R
yy
2
0.4
0.2
0.1
0
-0.1
-400
-300
-200
-100
0
100
200
300
400
200
300
400
Lag (s)
Higher order correlation of non-linear system
1
0.8
0.4
R
yy
2
0.6
0.2
0
-0.2
-0.4
-400
-300
-200
-100
0
100
Lag (s)
Figure 6.5: Higher order correlation test for non-linear dynamics
From this understanding of the effect of input signal magnitude on the types of dynamics exhibited and the regions and boundaries over which the system may be expected to
exhibit smaller or greater extents of non-linear dynamics, it was made possible to design
experiments to study the effect of input signal properties on linear model identification
and validation of the non-linear system.
For experimental consistency the normal operating range of both model inputs was
taken to be −0.03 to 0.03 while the normal output range for both outputs was taken
as −0.03 to 0.03. In addition to these normal operating ranges over which most of the
simulations took place, larger ranges were also chosen for investigative purposes, these
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
97
are; −0.06 to 0.06 and −0.05 to 0.05 for the input and output respectively.
System Control
The same approach was taken for the closed-loop simulation of the non-linear model
(system B) as the linear model (system A). The closed-loop system was controlled by a
decentralised controller in the form of a PI controller which was tuned by trial and error.
An extensive study into controller performance given different variable pairing showed
the best pairing configuration to be y1 -u1 and y2 -u2 . Figure 6.6 shows the closed-loop
response of the system.
Closed-loop response of y to setpoint change
1
o
∆ Plate 21 Temperature ( C)
0.01
0
-0.01
-0.02
-0.03
-0.04
0
Output
Setpoint
100
200
300
Time (s)
400
500
600
Closed-loop response of y to setpoint change
2
o
∆ Plate 7 Temperature ( C)
0.06
Output
Setpoint
0.04
0.02
0
-0.02
-0.04
0
100
200
300
Time (s)
400
500
600
Figure 6.6: Closed-loop response to setpoint changes for non-linear system - system B
It Is noted that there is a substantial amount of controller interaction, this and the
non-linear nature of the system resulted in a de-tuned controller being the only controller
(not including advanced controllers) that could adequately control the system without
introducing instability under different variations in setpoint changes.
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6.2
6.2.1
98
Experimental Method : Framework and Design
General Identification Method
The identification method used to identify system A and system B was that of the prediction error method (PEM) together with an unfiltered, unweighted quadratic norm making
the identification approach a least squares estimate defined by equation (2.48).
When applied to closed-loop data this was done via the direct approach. As established in section 3.2, the direct approach is the more robust closed-loop identification
method, specifically since its adaptation to complex or non-linear controllers is the least
involved. This, together with the consideration that modern-day controllers are ever increasing in complexity and tending towards non-linear forms such as MPC controllers,
justifies the value in its investigation.
It is understood that prior knowledge of a system’s controllability problems or unique
characteristics, such as being ill-conditioned, should be incorporated into the identification method through proper weighting functions or norm adjustments. These adjustments
are forgone in this work, however, the consequences and effects will be discussed in later
sections.
6.2.2
Model Structure and Order Selection
It is understood that for an approximation of a system to converge to the true system,
the model structure must contain the true system. It is further understood that there is
value in studying identification cases where it is unknown whether the true structure is
contained in the model structure or not. This understanding and the fact that the focus
of this work is on other identification variables, lead to a selection method for model
structure and order that was based on reducing complexity and calculation considerations.
As one may deduce from the choice of identification method, parametric model structures were used, specifically the multivariable auto regression with external input (ARX)
structure, given by equations (2.17) through (2.24).
Although the ARX structure has the advantages of being compact and conducive to
simpler parameter estimation routines, there is a disadvantage in terms of its ability to
independently model the system noise from the system dynamics. Since the ARX model
structure is defined by
1
B(q)
u(t) +
e(t)
(6.3)
y(t) =
A(q)
A(q)
it is clear that the disturbance model, 1/A(q), is directly coupled to the dynamics model,
B(q)/A(q). This is specifically problematic for closed-loop identification since it has
greater demand on an accurate noise model, due to input - noise correlations, in order to
minimise model bias.
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In terms of order selection and delay considerations, while the true orders (and structure) of both system A and system B are known, reduced orders were typically used for
the estimates. While several orders were briefly assessed in an effort to obtain an understanding on structure selection and its effects, the selection of model order is not dealt
with in depth in this work. The selection of the orders used for the estimates of both
the linear system (system A) and the non-linear system (system B) was done based on a
brief simulation fit analysis. The final order selections were two for system A estimates
and five for system B estimates. In terms of the delay considerations, the known system
delays were used and incorporated into the identification procedure.
Thus, to reiterate, the model structure used to approximate both system A and B is
defined in equation (6.3), with its expanded form given as
y(t) + A1 y(t − 1) + . . . + Ana y(t − na )
= B1 u(t − 1) + . . . + Bnb u(t − nb ) + e(t)
(6.4)
where
"
Ai (q) =
"
Bi (q) =
ai11 ai12
ai21 ai22
#
bi11 bi12
bi21 bi22
#
(6.5)
(6.6)
with na = 2 and nb = 2 for the system A estimates while for the system B estimates
na = 5 and nb = 5. Note that with both system A and B being multivariate systems of
size 2 × 2, the correct order expression, in the case of the system A approximation, is
"
ni =
2 2
2 2
#
however, since the same order is chosen for all submodels, the scalar representation is
used to represent the multivariate representation.
6.2.3
Identification Experiment Conditions
It is at this point appropriate to state that the two primary identification experiment
design variables are the feedback condition, whether the experiment is operated under
open or closed-loop conditions, and the signal used to disturb the system in order to
generate informative data for identification and validation purposes.
Recalling figures 2.3 and 3.1, the open and closed-loop block diagrams respectively.
In the case of the open-loop experiments, these disturbance signals were implemented via
the input signal, u(t), while for the closed-loop case the disturbance signal entered via
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100
the dither signal, r1 (t), with the setpoint signal, r2 (t), being values of typical operation.
It is important to note that the open-loop experiments were designed by specifying the
input signal magnitude ranges, thus open-loop experiments were operated under input
constraints. The closed-loop experiments on the other hand were designed by specifying
output signal magnitude ranges (setpoint ranges), thus they were operated under output
constraints.
The following types of disturbance signals were used:
• Step
• White Gaussian
• PRBS
• PRMLS
with several variations of each signal in terms of switching probabilities for the pseudorandom signals. Since both the linear (system A) and non-linear (system B) systems have
two inputs and two outputs, two sets of disturbance signals are required to disturb the
system. For the white Gaussian, PRBS and PRMLS signals, this was done by creating
two signals that are maximally shifted from each other.
In addition to varying the disturbance signal characteristics the following conditions
were implemented so as to create different experimental conditions, these are mostly
relevant to the closed-loop experiments:
Control Parameter Changes : Controller parameters were varied for some experiments so as to introduce a slightly non-linear controller.
Input Constraints : Constraints were implemented on the system inputs for some experiments.
Disturbance Signal Implementation : For some experiments no dither signal disturbances were implemented and for others no setpoint changes were made.
The different identification experiment cases given by the different disturbance signals,
setpoint variations and controller settings used on system A (the linear system) under
open and closed-loop conditions are given in table 6.3 and 6.4 respectively. While those
used for identification of system B (the non-linear system) are similarly given in tables
6.6 and 6.5. Note that the case tags were created by firstly describing whether the
experiment was done under open or closed-loop conditions (OL or CL) followed by the
experiment number. The same is done for all experiments, with the only difference being
that experiments concerned with system B (tables 6.5 and 6.6) are supplemented with
the prefix ”N” for the purpose of distinction, ”N” signifying non-linear.
© University of Pretoria
CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
Case
Tag
Input Signal (u(t))
Signal Type
Range
B
OL 1 White Gaussian
[-0.5, 0.5]
1
OL 2 PRBS
[-0.5, 0.5]
1
OL 3 PRBS
[-0.5, 0.5]
2
OL 4 PRMLS
[-0.5, 0.5]
2
[0, 0.5]
-
OL 5 Step
Table 6.3: Open-loop experimental conditions for identification of the linear model
Case
Tag
Dither Signal (r1 (t))
Signal Type
Range
B
SNR = 2.5
Setpoint Signal (r2 (t))
SP Values [SP1 ],[SP2 ]
CL 1
White Gaussian
[-0.0125, 0.0125]
1
[0.3, -0.1],[-0.5,0.4]
CL 2
PRBS
[-0.0125, 0.0125]
1
[0.3, -0.1],[-0.5, 0.4]
CL 3
PRBS
[-0.0125, 0.0125]
2
[0.3, -0.1],[-0.5, 0.4]
CL 4
PRBS NL
[-0.0125, 0.0125]
2
[0.3, -0.1],[-0.5, 0.4]
CL 5
PRBS CNL
[-0.0125, 0.0125]
2
[0.3, -0.1],[-0.5,0 .4]
CL 6
PRBS no ∆ SP
[-0.0125, 0.0125]
2
[0, 0],[0, 0]
CL 7
PRMLS
[-0.0125, 0.0125]
2
[0.3, -0.1],[-0.5, 0.4]
CL 8
PRMLS sync ∆ SP
[-0.0125, 0.0125]
2
[0.3, -0.1],[0.5,-0.4]
CL 9
No Dither Signal
-
-
[0.3, -0.1],[-0.5, 0.4]
Table 6.4: Closed-loop experimental conditions for identification of the linear model
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101
CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
Case
Tag
Dither Signal (r1 (t))
Signal Type
Range
B
SNR = 2.5
102
Setpoint Signal (r2 (t))
SP Values [SP1 ],[SP2 ]
NCL 1
White Gaussian
[-0.00125, 0.00125]
1
[-0.02, 0],[0.03, -0.01]
NCL 2
PRBS
[-0.00125, 0.00125]
1
[-0.02, 0],[0.03, -0.01]
NCL 3
PRBS
[-0.00125, 0.00125]
10
[-0.02, 0],[0.03, -0.01]
NCL 4
PRBS L
[-0.00125, 0.00125]
10
[-0.04, 0],[0.035, -0.05]
NCL 5
PRBS NL
[-0.00125, 0.00125]
10
[-0.02, 0],[0.03, -0.01]
NCL 6
PRBS CNL
[-0.00125, 0.00125]
10
[-0.02, 0],[0.03, -0.01]
NCL 7
PRBS no ∆ SP
[-0.00125, 0.00125]
10
[0, 0],[0, 0]
NCL 8
PRMLS
[-0.00125, 0.00125]
10
[-0.02, 0],[0.03 ,-0.01]
NCL 9
No Dither Signal
-
-
[-0.02, 0],[0.03, -0.01]
SNR = 4
NCL 10 White Gaussian
[-0.002, 0.002]
1
[-0.02, 0],[0.03, -0.01]
NCL 11 PRBS
[-0.002, 0.002]
1
[-0.02, 0],[0.03, -0.01]
NCL 12 PRBS
[-0.002, 0.002]
10
[-0.02, 0],[0.03, -0.01]
NCL 13 PRBS L
[-0.002, 0.002]
10
[-0.04, 0],[0.035, -0.05]
NCL 14 PRBS NL
[-0.002, 0.002]
10
[-0.02, 0],[0.03, -0.01]
NCL 15 PRBS CNL
[-0.002, 0.002]
10
[-0.02, 0],[0.03, -0.01]
NCL 16 PRBS no ∆ SP
[-0.002, 0.002]
10
[0, 0],[0, 0]
NCL 17 PRMLS
[-0.002, 0.002]
10
[-0.02, 0],[0.03 ,-0.01]
Table 6.5: Closed-loop experimental conditions used for identification of the non-linear model
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
Case
Tag
Input Signal (u(t))
Signal Type
Range
103
B
NOL 1 White Gaussian
[-0.03,0.03]
1
NOL 2 PRBS
[-0.03,0.03]
1
NOL 3 PRBS
[-0.03,0.03]
10
NOL 4 PRBS L
[-0.06,0.06]
10
NOL 5 PRMLS
[-0.03,0.03]
10
NOL 6 Step
[-0.03,0.03]
-
Table 6.6: Open-loop experimental conditions for identification of the non-linear model
The remainder of this section further elaborates on how the identification experiments
were designed and clarifies the content of the tables detailing the experiments.
The following can be said regarding the identification experiment conditions:
• Values of all variables are initially zero, thus all the input and disturbance signal
magnitudes are specified as deviation values
• All inputs of both the linear (system A) and non-linear (system B) multivariate
systems were disturbed simultaneously, thus making the experiments multivariable
experiments.
• The amplitude of the White Gaussian signal superimposed on the output signals
used to simulate output noise was taken as 1% of the full normal operating range
of each system.
• The same noise realisation was used for all the identification cases unless otherwise
stated (i.e. where sensitivity to different noise realisations was investigated - this is
discussed in the following sections)
• The amplitude of the dither signal (r1 ) superimposed on the system input for closedloop experiments was taken as 2.5% the normal operating range in the case of system
A. For system B two dither signal amplitudes were used, 2.5% and 4%. Figures
6.7(a) and 6.7(b) illustrate the differences in output excitation between the open
and closed-loop condition on the linear system. The effect of the dither signal is
additionally illustrated.
• With the exception of the step signal which is only defined by the range, all other
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104
signals are defined by the type, range and average time of constant signal interval
(B), given in seconds.
• Signals with B = 1 are white signals, the other signals have B values selected as
30% of the largest time constant. Theory dictates this approach to selecting B
values as optimal for generating response data reflecting the dominant dynamics of
the system (Norton, 1986).
• The identification experiment duration was taken as several times the system’s
closed-loop response in order to allow for several setpoint changes.
• Two setpoint changes were made to both controlled variables during closed-loop
experiments. As mentioned earlier, the initial setpoint is zero. The changes made
to each setpoint were done in different directions and a different times.
• In the case of system A the setpoint changes for output y1 and y2 were made at
100 and 300 seconds and 200 and 400 seconds respectively. In the case of system B
the setpoint changes for output y1 and y2 were made at 100 and 350 seconds and
200 and 450 seconds respectively. Figures 6.6 and 6.3 respectively illustrate these
changes for system A and system B.
The following further elaborates on the information presented in tables 6.3, 6.4, 6.6
and 6.5:
Range : Specifies the range of the input and dither signals used in the open and closedloop experiments respectively. Since both input u1 and u2 are similarly scaled for
both system A and B, the range of these input (u(t)) and the dither (r1 (t)) signals
are only given once e.g. [-0.5, 0.5] specifies the lower and upper bound of both
signal, u1 and u2 .
SP Values : Specifies the changes in setpoint values made for each output. [SP1 ] are
the setpoint changes made to setpoint 1, [SP2 ] are the setpoint changes made to
setpoint 2.
NL : Implies a non-linear controller, which was achieved by varying the controller parameters between four sets of parameters throughout the closed-loop experiment
period.
CNL : Implies the same NL condition mentioned above but with the input signals constrained between specified limits.
no ∆SP : is the condition where no setpoint changes were incurred.
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105
Open-loop response of y for linear system with noise and dither
1
0.4
0.3
o
∆ Plate 12 Temperature ( C)
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
50
100
150
200
250
Time (s)
300
350
400
450
500
(a) Open-loop output excitation of system A using PRMLS input signals and superimposed noise
Closed-loop response of y for linear system with noise
1
o
∆ Plate 12 Temperature ( C)
0.3
0.2
0.1
0
-0.1
-0.2
Output
Setpoint
-0.3
-0.4
0
50
100
150
200
250
Time (s)
300
350
400
450
500
Closed-loop response of y for linear system with noise and dither
1
o
∆ Plate 12 Temperature ( C)
0.3
0.2
0.1
0
-0.1
-0.2
Output
Setpoint
-0.3
-0.4
0
50
100
150
200
250
Time (s)
300
350
400
450
500
(b) Closed-loop output excitation of system A indicating the effect of introducing noise and PRBS dither
signals
Figure 6.7: Open and closed-loop output excitation of system A
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106
sync ∆SP : is the condition where setpoint changes were made on both controlled variables at the same time and in the same direction.
No Dither Signal : is the case where no dither signal was used, only setpoint changes.
L : For the experiments on system B, the indicator L, was assigned to label experiments
which operated over a larger range than the typical experiments. This larger range
was selected so as to assure operation in regions of more pronounced non-linear
dynamics. The normal identification range was selected to be closer to the lower
extents of non-linearity. It is important to note that when referring to a larger range
for the open-loop experiment, this is specified by larger input signal magnitudes,
while for the closed-loop experiment this is achieved by larger setpoint changes.
Thus 14 (5 open-loop and 9 closed-loop) experimental conditions for system A (the
linear system) and the 23 (6 open loop and 17 closed-loop) for system B (the non-linear
system) were used to generate 14 and 23 models for each system respectively.
As will be revealed in the discussion of the simulation results in section 6.3, the process
of identification and model generation was repeated 50 times for each case for both system
A and system B. The primary purpose of this was to obtain an understanding of the effects
of the random realisations of noise on the models generated by specific signals. From this
parameter variances and basic validation result variances were obtained.
It is however important to note that while several models where generated due to the
different noise realisations and a basic assessment was made into each of these models,
the models obtained by the first realisation of noise were taken as the primary models
for investigation.
6.2.4
Validation Techniques and Experiment Conditions
The validation of a model may be considered as the essence of the identification procedure
since it determines whether the model generated is useful or not. As established in section
4.1, there are several techniques available for model validation. The following techniques
were used to assess the validity and accuracy of the models generated (further detail on
the execution of these techniques is presented in the following section):
• Model Variance and Uncertainty
• Percentage Fit it for simulations and predictions
• Frequency Content Analysis
• Residual Correlations
• Higher Order Residual Correlations
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In addition to these, two other techniques were implemented. These are model pole-zero
stability and frequency response analysis (Bode plot). Both of these techniques were only
applied in validation efforts of the linear system (system A).
From the list of validation methods being used it must be noted that most of them
are cross-validation techniques. These techniques involve validating the identified model
against a validation data set. Figure 6.8 illustrates the process of cross-validation in the
context of generating data and identifying models.
Validation data generation
experiment
Data
Generation
Identification data generation
experiment
Experiment conditions
•
Open/closed loop
•
Disturbance signal
•
Sample rate
Validation Data Set
Identification Data Set
Input Data
Output Data
Input Data
Output Data
Parameter Estimation
•
•
Model
Estimation
Model structure
Map Data
u(t)
G(q,θ)
y(t)
G(q,θ)
y(t)
Validation Data Set
Input Data
Output Data
Input
Data
Model
Cross-Validation
Crossvalidation
Output
Data
Cross-Validation Techniques
•
•
•
Response fit
Frequency component fit
Residual Correlation
Figure 6.8: Illustration of data cross-validation in the context of data generation and model
identification
Thus, given the primary focus on cross-validation techniques, attention was given to
generating several validation data sets from different experimental conditions in order to
obtain an understanding into the effects of experimental conditions on the accuracy of
cross-validation techniques. The primary experiment variables studied in terms of their
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108
effects were disturbance signal characteristics and feedback conditions.
Thus several validation data sets were generated from experiments using PRBS and
step signals under both open and closed-loop conditions on both systems A and B. In
generating validation data for the non-linear system, system B, data was generated over
normal and larger operating ranges.
Tables 6.7 and 6.8 show the different experimental conditions used to create different
validation data sets for the linear and non-linear system experiments. In a similar fashion
as the experimental condition used to generate identification data, these are given experimental case tags. The same rules apply in that ”N” is used to signify the experiment
was done on system B, ”OL” and ”CL” indicate open-loop and closed-loop conditions
respectively and following this is the experiment number. In addition to this is the phrase
”Val” signifying the experiment is for generating validation data.
The following can be said about the generation of data for validation:
• Even though some validation data sets were generated from step disturbance experiments, due to the fact that research into literature clearly showed the low information content of step response data, more emphasis was put into assessing validation
results from data generated by PRBS disturbance signals.
• The setpoint changes made during the closed-loop experiments were slightly different to those implemented for the identification experiments, however, in a similar
fashion as identification experiments, setpoints were changed twice for each validation experiment.
• The average constant signal interval values (B) of the disturbance signals used to
generate validation data were slightly larger than those of the disturbance signals
used to generate data for identification. However, they were not so much larger so
as to generate data with frequency characteristics outside of the frequency band of
interest for modelling.
• Noise was not always superimposed on the output data of the validation data sets.
This is clarified later.
• Just as with the identification data generation experiments for system B, two experimental regions were created for validation data generation. One over a smaller
region of operation, and another over a larger region of operation. This was done
in hope of creating sets of validation data with different dynamic contents in terms
of non-linear dynamics.
From the tables describing the experimental conditions used to generate validation
data (tables 6.7 and 6.8), it can be said that 3 validation data sets (2 open-loop and 1
© University of Pretoria
CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
Open-Loop Validation Data
Input Signal (u(t))
Signal Type
Range
Case
Tag
ValOL 1 PRBS
ValCL 3
B
[-0.5, 0.5]
2.5
[0, 0.5]
-
ValOL 2 Step
Case
Tag
109
Closed-Loop Validation Data
Dither Signal (r1 (t))
Setpoint (r2 (t))
Signal Type
Range
B
SP Values [SP1 ],[SP2 ]
PRBS
[-0.0125, 0.0125]
2.5
[0.3, 0.5, -0.1],[0.5, -0.4]
Table 6.7: Conditions for generation of validation data for linear model
Case
Tag
Open-Loop Validation Data
Input Signal (u(t))
Signal Type
Range
B
NValOL 1 PRBS
[-0.02, 0.02]
5
NValOL 2 PRBS L
[-0.05, 0.05]
5
[0, 0.02]
-
NValOL 3 Step
Case
Tag
Closed-Loop Validation Data
Dither Signal (r1 (t))
Setpoint (r2 (t))
Signal Type
Range
B
SP Values [SP1 ],[SP2 ]
NValCL 4
PRBS
[-0.00125, 0.00125]
5
[0.02, 0, -0.01],[-0.01, 0.01]
NValCL 5
PRBS L
[-0.00125, 0.00125]
5
[0.05, 0, 0.04],[0.04, -0.055]
Table 6.8: Conditions for generation of validation data for non-linear model
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
110
closed) were generated for system A and 5 validation data sets (3 open-loop and 2 closedloop) were generated for system B. Thus in terms of summarising the cross-validation
efforts, 14 identified models were cross-validated against 3 validation sets for system A,
for sytem B, 23 models were cross-validated against 5 validation sets.
Recalling that all the identified models have ARX structures given by equation (6.3),
1
which may be broken down into a dynamics model, B(q)
, and a noise model, A(q)
. It is
A(q)
at this point appropriate to note that most of the cross-validation techniques validated
only the dynamics model. That is, no noise signals were superimposed on the validation
data outputs allowing for direct validation of the identified dynamics model against the
true known system. The only validation techniques that assess the validity of the noise
model together with the dynamics model are the residual correlation assessments.
It is acknowledged that the topic of model stability as a validation technique is important, specifically when modelling for control. This validation tool is not used to great
depth in this work and is limited to the pole-zero stability comparisons between linear
system, system A, and the identified linear approximations of the system.
Is is noted that model variance and uncertainty play very large roles in assessing
how well a model represents a system and that ultimately, knowledge of this uncertainty
should be included in all validation assessments. The approach taken by this work was
such that instead of imposing variance and uncertainty measures on all the validation
techniques and results, the variance and uncertainty measures were directly assessed on
their own.
6.2.5
General Execution and Software
The generation of system disturbance signals and the simulated responses of both systems
A and B, together with the identification and validation of models from data comprising
of these disturbances and responses was all done through Matlab. The regression models
representing systems A and B were simulated in discrete form, the controllers were thus
discrete controllers in the velocity form. The following can be said about the functions
used and generated and how they were executed:
• The function idinput was used to create all the disturbance signals except for the
PRMLS, a function was written to generate this signal based on the idinput function.
• The function ARX was used to generate multivariable ARX models with specified
orders and delays given input and output data.
• The compare function was used to create simulations and predictions to generate
percentage fits between the identified model responses and validation data sets.
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Matlab’s percentage fit is defined as follows:
|y − ŷ|
percentage fit = 100 × 1 −
|y − ȳ|
(6.7)
where y is the ’measured’ (simulated) output, ŷ is the output generated by the
identified model given the input used to generate the ’measured’ output. ȳ is the
mean of y.
• The fft function, a discrete Fourier transformation function, was used to convert
discrete time data to discrete frequency data. This, combined with the compare
function, allowed for the assessment of the cross-validation of the identified model
response fits in the frequency domain.
• The function frqresp was used to extract frequency response data from linear parametric models.
• The simsd function was used to assess each identified model’s covariance matrix
and the consequent indication of model uncertainty. This is achieved through Matlab’s Monte Carlo simulation engine which expresses the model’s covariance matrix
(obtained from the model parameter estimation procedure) as simulation response
variance.
• The function pzmap was used to determine the poles and zeros of models and
consequently their stability.
• For residual auto-correlations and input-residual cross-correlations the corr and
autocorr functions where used respectively.
• A function was created called crossco for the higher order correlations and crosscorrelations. This function was based on the higher order correlation functions
found in the Neural Network based system identification toolbox (Norgaard, 2000).
See Appendix A for a complete description of the functions and programs used.
6.3
Identification and Validation Results for System
A - The Linear System
This section presents and discusses the results of the identification of models approximating system A and the validation of such models. Note that all the parameters of
each identified model are presented in appendix A.2. Each of the 14 identified models (5
generated from open-loop data, 9 generated from closed-loop data) and the experimental
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conditions that generated the respective data used to identify them will be referred to
by the concerned experimental case label. That is, the case labels given in table 6.3 for
the models obtained from open-loop experiments and table 6.4 for the models obtained
from closed-loop. Note however that a brief mention of the conditions characterising the
experiment used to generate the model or validation set being discussed is done where
necessary and relevant to the point being made.
Section 6.3.1 presents the assessment of variance and uncertainty of the identified
models while section 6.3.2 presents and assessment of the identified model stability. These
model validity assessments are not cross-validation techniques, they thus do not use the
validation data sets generated. However, in assessing the variance and uncertainty of the
identified models it was necessary to invoke responses. Normal PRBS signals confined to
the normal input ranges were used as inputs to produce such responses.
Sections 6.3.3 through 6.3.5 primarily focus on the validation results and findings
based on cross-validation techniques as presented in figure 6.8. These techniques are based
on comparing simulated responses from the identified models to responses from the true
known system, system A. This is done using the validation data sets generated. These
are given in table 6.7, recall that there are two sets generated under open-loop conditions
(ValOL 1 and 2) and one under closed-loop conditions (ValCL 3). Each validation set
contains input and output data. The inputs are used to disturb the identified model
so as to allow for a simulated output response which is to be compared to the response
contained in the validation data set. Thus effectively comparing the simulated response
accuracy of the identified models.
6.3.1
Model Variance and Uncertainty
The assessment of model variance and uncertainty was done on two levels. The first
being model parameter sensitivity to random noise realisations in identification data.
The second being parameter uncertainty as per the covariance matrix obtained from the
parameter estimation procedure expressed via simulated response variances (Monte Carlo
simulations).
Variance Due to Sensitivities to Random Noise Realisations
The investigation into experimental design sensitivity to noise realisations was done by
repeating all the identification and consequent model estimation experiments 50 times.
The number of repetitions was selected based on the observation that using fewer than
50 repetitions did not produce consistent results. The execution conditions were such
that all variables, besides the random realisation of the White Guassian noise superimposed on the output signals, were kept constant. The resulting model parameters of the
models identified from these repeated experiments were then compared to each other to
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Average parameter variance due to noise realisation
OL 1
OL 2
OL 3
OL 4
OL 5
CL 1
CL 2
CL 3
CL 4
CL 5
CL 6
CL 7
CL 8
CL 9
0.06
variance ( σ 2)
0.05
0.04
0.03
0.02
0.01
0
Identification Case
(a) Parameter variance due to random noise realisations
Simulated responses for output y revealing effect of variance due to noise sensitivity for OL 5
1
o
∆ Plate 12 Temperature ( C)
0.4
0.2
Variance = 0.0649
0
-0.2
-0.4
0
10
20
30
40
50
Time (s)
60
70
80
90
100
Simulated responses for output y revealing effect of variance due to noise sensitivity for CL 5
1
o
∆ Plate 12 Temperature ( C)
0.4
0.2
Variance = 0.000674
0
-0.2
-0.4
0
10
20
30
40
50
Time (s)
60
70
80
90
100
(b) Model variance shown through simulation response variance
Figure 6.9: Parameter and response variance due to random noise realisations
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assessed for variance. The average variance of all the parameters was used as the variance
measurement. Figure 6.9(a) illustrates the findings.
It is clear from this figure that case OL 5, defined as an open-loop step disturbance
experiment, results in a significantly larger parameter variance due to random noise
realisations. This could reflect the experiment’s inability to produce informative data
which consequently resulted in the estimation procedure modelling the random noise
realisations. While the significance of the differences in parameter variance measurements
between the other cases might be questionable, since they are relatively small, it must
be noted that, of all the other experimental cases, CL 9 was found to have the largest
variance. Note that CL 9 is the closed-loop identification experiment where no dither
signal was used.
It is interesting to note that while there is significant distinction when comparing the
parameter variance due to noise experienced by case OL 5 to that of the other cases, such
distinction is not observed in the output responses. This is indicated in figure 6.9(b).
Here the 50 responses of output y1 to the same input (PRBS), generated by each of
the 50 models identified from experiments OL 5 and CL 5 given the 50 different noise
realisations, are plotted together. Note that cases OL 5 and CL 5 are the cases with the
largest and smallest parameter variances respectively, as per figure 6.9(a). The difference
in observable variance in response y1 is relatively insignificant. The same was found for
output y2 . This does imply that while the parameter variance due to noise sensitivity in
case OL 5 is clearly larger than the others, it is still an insignificant amount of variance.
Model Uncertainty
The response variance of each identified model due to parameter uncertainty was measured by standard deviations obtained from Monte Carlo simulations. Using the covariance matrices generated during the estimation procedure of each identified model, the
simulations produced 50 random response simulations to a PRBS input signal reflecting
the parameter uncertainties. From these simulated responses, standard deviations at each
sample point (i.e. at each time measure) were obtained. Figure 6.10(a) illustrates the
50 response simulations of output y1 for the model identified from experiment CL 9 and
figure 6.10(b) shows those of output y2 for the same case. Figure 6.11 shows the averaged
response standard deviation values over time of each identified model.
From these averaged response standard deviations, it is evident that there is no experimental case that produced a model with significantly larger extents of uncertainty
than the others. No clear trend may be observed besides that the model identified from
the open-loop step response experiments, OL 5, has the lowest average response standard
deviations due to parameter uncertainty for both outputs.
The results for output y1 are particularly less distinguishable in comparison to those
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(a) Simulated responses for output y revealing effect parameter uncertainty for case CL 9
1
o
∆ Plate 12 Temperature ( C)
0.6
0.4
Average std deviation = 0.0288
0.2
0
-0.2
-0.4
0
10
20
30
40
50
Time (s)
60
70
80
90
100
(b) Simulated responses for output y revealing effect parameter uncertainty for case CL 9
2
o
∆ Plate 5 Temperature ( C)
10
5
Average std deviation = 0.262
0
-5
-10
0
10
20
30
40
50
Time (s)
60
70
80
90
100
Figure 6.10: Monte Carlo Simulation for case CL 6
Output standard deviation for y
1
Standard Deviation ( σ)
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
Identification Case
Standard Deviation ( σ)
Output standard deviation for y
2
0.5
0.4
0.3
0.2
0.1
0
Identification Case
OL 1
OL 2
OL 3
OL 4
OL 5
CL 1
CL 2
CL 3
CL 4
CL 5
CL 6
CL 7
CL 8
CL 9
OL 1
OL 2
OL 3
OL 4
OL 5
CL 1
CL 2
CL 3
CL 4
CL 5
CL 6
CL 7
CL 8
CL 9
Figure 6.11: Average output response standard deviations due to parameter uncertainty
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of y2 . The standard deviation values for the models obtained from open-loop experiments
(the OL cases) show larger values for output y2 relative to those of models obtained from
closed-loop experiments. Looking at the results for output y1 , it may however be observed
that the models generated by closed-loop, cases CL 7-9, revealed slightly larger extents
of model uncertainty than most of the other closed-loop cases. Cases CL 7-9 being the
experiments defined by PRMLS dither signals (CL 7-8) and the lack of a dither signal
(CL 9).
Note that the standard deviation values for output y2 are larger than those of y1 . This
may be justified by the larger system gains between the system inputs and output y2 .
6.3.2
Model Stability
The brief assessment made into model stability was based on the understanding that the
true system is characterised as containing unstable poles.
An assessment into the system poles and zeros of each identified model revealed that
the model obtained from case OL 5, the open-loop step response experiment, was the
only model with no pole situated outside the unit circle on the z-plane. This implies
that, unlike the true system which had several poles outside the unit circle making it
an unstable system, the model obtained from case OL 5 is stable. This does reflect the
experimental design’s inability to obtain informative data on the system and furthermore
implies that its incorporation into a model based controller will not only result in poor
control but in an unstable controller.
All the other experiments generated models which had unstable poles. However, as
mentioned earlier, since the structure of the identified models is different to that of true
system, the unstable poles were not located in the same regions as those of the true
system. Given the multivariable nature of the system, this implies that the identified
models will be unstable in different directions in comparison to the true system.
6.3.3
Simulation and Prediction
This section introduces the first and primary component of the cross-validation techniques. Here percentage fit values are used to validate the accuracy of each identified
model’s simulated response in comparison to the true system’s response given by the
validation set. This is done for all 14 models against all 3 validation sets. Since all the
models were generated under different experimental conditions and all the validation data
sets were generated under different experimental conditions, the cross-validation results
will allow for insight into experimental effects on model accuracy and validation results.
Figures 6.12 and 6.13 show the response percentage fit values for each model validated against the open and closed-loop validation sets respectively. Figure 6.14 shows
an example of simulated responses using validation set ValOL 1, which is characterised
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
(a) Response Percentage Fit Against ValOL 1 and ValOL 2 for y
1
100
Percentage Fit
80
60
40
20
0
Validated against ValOL 1
Validated against ValOL 2
(b) Response Percentage Fit Against ValOL 1 and ValOL 2 for y
2
100
Percentage Fit
80
60
40
20
0
Validated against ValOL 1
Validated against ValOL 2
117
OL 1
OL 2
OL 3
OL 4
OL 5
CL 1
CL 2
CL 3
CL 4
CL 5
CL 6
CL 7
CL 8
CL 9
OL 1
OL 2
OL 3
OL 4
OL 5
CL 1
CL 2
CL 3
CL 4
CL 5
CL 6
CL 7
CL 8
CL 9
Figure 6.12: Percentage fit values for simulation validation against open-loop validation sets
(a) Response Percentage Fit Against ValCL 3 for y
1
100
Percentage Fit
80
60
40
20
0
Validated against ValCL 3
(b) Response Percentage Fit Against ValCL 3 for y
100
Percentage Fit
80
60
40
20
0
Validated against ValCL 3
2
OL 1
OL 2
OL 3
OL 4
OL 5
CL 1
CL 2
CL 3
CL 4
CL 5
CL 6
CL 7
CL 8
CL 9
OL 1
OL 2
OL 3
OL 4
OL 5
CL 1
CL 2
CL 3
CL 4
CL 5
CL 6
CL 7
CL 8
CL 9
Figure 6.13: Percentage fit values for simulation validation against closed-loop validation sets
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by open-loop experimental conditions and PRBS disturbance signals (see table 6.7 for
more details on the validation set). It is noted that figure 6.14(a) plots the response of
models obtained from open-loop experiments while figure 6.14(b) plots those obtained
from closed-loop experiments.
o
∆ Plate 12 Temperature ( C)
(a) Response fit for models obtained under open-loop experiments for y
1
0.3
0.2
ValOL 1 (100%)
OL 1: 18.3 %
OL 2: 4.76 %
OL 3: 53.2 %
OL 4: 65 %
OL 5: 24.7 %
0.1
0
-0.1
-0.2
-0.3
-0.4
0
10
20
30
40
50
Time (s)
60
70
80
o
∆ Plate 12 Temperature ( C)
(b) Response fit for models obtained under closed-loop experiments for y
90
100
1
0.3
ValOL 1 (100 %)
CL 1: 52.6 %
CL 2: 49.8 %
CL 3: 52.9 %
CL 4: 53.3 %
CL 5: 49.7 %
CL 6: 52.3 %
CL 7: 45.6 %
CL 8: 39.7 %
CL 9: 46.6 %
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
10
20
30
40
50
Time (s)
60
70
80
90
100
Figure 6.14: Percentage fit simulations validated against ValOL 1
Assessment of Validation Sets - Model Discrimination and Bias
Looking at the percentage fit values in figures 6.12 and 6.13, it may be said that the
percentage fit values generally indicate worse fits for the identified models when using
validation sets obtained from open-loop experiments (ValOL 1 and 2). Furthermore the
models obtained from open-loop experiments (OL 1 -5) did very poorly when validated
against closed-loop validation data while the models obtained under closed-loop conditions (CL 1 -9) all showed very large percentage fit values. It is additionally noted that
when validating the models against validation set ValOL 2, obtained from an open-loop
step disturbance experiment, the difference in percentage fit values between output y1
and y2 for each model was more significant than when using the other validation sets.
This lack of consistent results between the two outputs made it difficult to determine
any clear trend regarding how the different models performed relative to each other when
validated against ValOL 2.
Further analysing the differences in percentage fit values when using ValOL 2 in
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comparison to those when using the other validation sets, it is noted that validation set
ValOL 2 is the only set that produced a relatively large percentage fit value for case OL
5. Case OL 5 referring to the model identified from the experimental case OL 5 defined
as an open-loop experiment where the disturbance signals were step disturbances (i.e.
ValOL 2 and case OL 5 are both rooted in open-loop step disturbance experiments). In
fact, it may be said that the model produced by experiment OL 5 produced the most
accurate response when validated against ValOL 2, while it was amongst the worst when
validated against ValOL 1 and ValCL 3. On the other hand the model produced by case
CL 6, characterised by a closed-loop experiment with no setpoint changes, performed
very poorly when validated against ValOL 2. In fact, it was the worst performing case
of all the closed-loop cases against ValOL 2 by a significant amount. However, when
validated against the other validation data sets, this distinction was not found.
Observing the differences in percentage fits when using the validation data sets obtained under open-loop conditions to those obtained under closed-loop conditions, it is
clear that the closed-loop validation data set, produced by case ValOL 3, did not produce
informative enough data to allow for the discrimination between the models generated
by the closed-loop cases. On the other hand, validation set ValOL 2, which was expected
to be very uninformative since it was generated from step disturbances, did a better job
than the closed-loop validation set ValCL 3 at discriminating between the models obtained from these closed-loop experiments. However, validation set ValOL 2 found the
model obtained from the least informative disturbance signal (the step disturbance, i.e.
OL 5) to be the most accurate since it gave the best fit.
From these findings it is clear that the closed-loop validation set is bias towards models
generated from closed-loop experiments while the open-loop step disturbance validation
set is especially bias towards models that where obtained from similar open-loop step
disturbance experiments and generally bias towards experimental data that contain the
dynamics revealed by step disturbances and low frequency setpoint changes.
Another way of interpreting this is that these validation sets were highly prejudice
against models obtained from experimental conditions different to their own.
The validation data set produced by PRBS under open-loop conditions, ValOL 1,
allowed for the best discrimination between models and revealed the most consistent
trends amongst the two outputs.
Assessment of Identified Models - Identification Experiment Condition Sensitivities
The following observations and comments are made regarding the response percentage
fit results of the different models obtained from the different experiment conditions when
validated against open-loop validation set ValOL 1, which was found to be least biased
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and most discriminating (see figure: 6.12 - ValOL 1 results)
• It is observed that the only difference between cases OL 2 and OL 3 is that OL 3 had
a larger input signal average constant interval value, B (which may infer frequency
characteristics). This larger value of B for case OL 3, was selected considering
disturbance signal characteristics that more closely resemble the dominant dynamic
characteristics of system A. The much larger percentage fit value for case OL 3
does indicate the effect of designing the disturbance signal in accordance with the
dominant dynamics of the system being identified. However, this larger value of
B used in case OL 3, relative to OL 2, is also nearer to that used to generate the
validation set, thus the improved response fit might not necessarily be due to better
system identification but to the validation set’s bias towards the model obtained
from disturbance signals with B values nearer to its own.
• Interestingly enough, the model produced by case OL 5 (defined by step disturbance
input signals), showed better validation results than the models obtained from OL
1 and OL 2. Recalling that cases OL 1 and OL 2 are defined as experiments that
used higher frequency PRBS input signals (smaller B values) than that used to generate validation set ValOL 1. This could imply that input signals that are not very
persistently exciting produce models with better cross validation results than models produced by persistently exciting signals with characteristics that significantly
differ form those used to generate the validation data.
• It is clear that of all the open-loop cases, case OL 4, which is characterised as using
a PRML input signal, produced the model that best fits the validation set ValOL
1.
• With respect to the closed-loop cases, it is clear that even when validated against
the validation set thought to be most informative (ValOL 1), there is little to
distinguish and discriminate between them. This may be accounted for by the controller’s efforts in dampening the effects of the excitation signal used to identify the
system. This would thus reduce the effective differences between the identification
experiment designs.
• An alternative interpretation of this lack of response fit distinction among the models obtained from closed-loop experiments is that the dither signals were not as effective as expected. When comparing the results of the models obtainted from cases
CL 1-6 there is little significant difference in simulation results. This implies that
the differences in dither signal characteristics might have done little to differently
excite the system.
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• The differences between results of the models obtained from experiments CL 3, 6
and 9 are of significant interest and address the previous point made. All three are
defined as closed-loop experiments with case CL 3 using a PRBS dither signal and
two setpoint changes to disturb the system. Case CL 6 is further characterised as
having the same dither signal as CL 3 but no setpoint changes. While case CL 9 is
further defined as having the same setpoint changes as CL 3 but no dither signal.
Of the three models obtained from these cases, that produced by case CL 3 had the
largest percentage fit values with those of case CL 6 closely following. The model
generated by case CL 9 clearly had the lower percentage fit values of the 3. From
this one may deduce that the excitation in output caused by servo tracking seems
to contribute little since its elimination did little to worsen the fit, while the effect
of not exciting the system via the dither signal was more significant. This implies
that the dither signals where valuable in producing informative data.
• It is interesting to note that, while the previous observation claims the effect on
identification due to excitation from setpoint change is minimal, the difference between the percentage fit values of cases CL 7 and 8 states otherwise. With the only
difference in experimental design between these 2 cases being setpoint direction and
time of setpoint change, it is important to note that, even though the percentage
fit is significantly different only for y1 , this difference does imply that setpoint direction and controller interaction play important roles in multivariable closed-loop
system identification. Figure 6.15 shows the difference in identifed model response
caused by setpoint direction changes in the identification experiments.
Model response variance due to setpoint directon differences for y
1
CL 7: 45.61 %
CL 8: 39.68 %
0.3
o
∆ Plate 12 Temperature ( C)
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
10
20
30
40
50
60
70
Time (s)
Figure 6.15: Model simulation revealing closed-loop identification sensitivity to setpoint direction
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• Observing that there is little difference in percentage fit values between the two
identification cases defined by non-linear and constrained non-linear controller conditions, CL 6 and 7 respectively, and those of cases CL 1-5, this indicates that the
effect of varying controller parameters and constrained inputs was not significant.
In addition to the observations made, it must be noted that there was no significant
difference found between the validation results obtained from simulation and those obtained from prediction. This does suggest that the results obtained may be used to infer
indications of model validity in terms of control.
6.3.4
Frequency Analyses
The approach taken in using the frequency domain to analyse the identified models and
the validation data was done on two fronts. The first concerns comparing the identified
models obtained by the different experimental cases with the true known system in the
frequency domain. This involves comparing the frequency responses (bode plots) of the
identified model to that of the known mathematical model which represents system A.
The second involves the same cross-validation technique as the previous section, but
in the frequency domain. That is, the simulated response of the identified model to the
input signals from a validation set, is compared against output response contained in
that validation set but in the frequency domain. This is done by taking the validation
set response data and the identified model’s simulated response data and representing
them in the discrete frequency domain. This is done through the use of discrete Fourier
transforms and allows for the frequency content of the simulated response fits against each
validation set to be compared. Furthermore, the assessment of the frequency content of
the response data in the validation sets can be compared to the frequency response of the
true system. This allows for an investigation into how well the validation set represents
the true system.
Frequency Response Analysis of Identified Models
The analyses of the different frequency responses of the identified models did allow for a
clearer understanding of the effects of different identification experiment designs on the
identified model’s accuracy.
Figure 6.16 illustrates the effect of changing the frequency characteristic of the disturbance signals used to generate information from system A for identification. With
figure 6.16(a) being the frequency responses of models identified from open-loop experiments OL 2 and OL 3 and 6.16(b) being those of closed-loop experiments CL 2 and CL
3. The only difference between the experiment conditions used to generate to models
that produce the two responses in each plot ((a) and (b)) being the average constant
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signal time interval (B) of the disturbance signal used. Cases OL 2 and CL 2 have B
values of 1 which characterises their disturbance signals as white signals, while OL 3 and
CL 3 have B values of 2 which is more representative of the dominant time constant of
system A. It is observed that increasing the B value (effectively decreasing the frequency)
of the input signal used in the open-loop identification experiments produced a model
with a more accurate frequency response. Figure 6.16(a) shows this clearly in that the
model identified from experiment OL 3 has a frequency response that much better fits
the frequency response of the true system, system A.
(a) Effect of disturbance frequency on open-loop experiment results for y -u
1
0
1
10
System A
OL 2 (B=1)
OL 3 (B=2)
Amplitude
-1
10
-2
10
-3
10
-2
10
-1
0
10
1
10
10
Frequency (rad/s)
(b) Effect of disturbance frequency on closed-loop experiment results for y -u
1
0
1
10
Amplitude
-1
10
System A
CL 2 (B=1)
CL 3 (B=2)
-2
10
-3
10
-2
10
-1
0
10
10
1
10
Frequency (rad/s)
Figure 6.16: Frequency response comparing experimental cases of different identification signal frequency characteristics
Figure 6.16(b) shows the frequency response fits of the two models identified from
closed-loop experiments. The same is seen here in that increasing the dither signal’s
average constant signal time, B, from 1 to 2, produces a model with a slightly improved
the frequency fit. This improvement is however less significant than that for the openloop experiment. This may be accounted for by the closed-loop controllers efforts to
reduce dither signal’s disturbance effects and the smaller magnitude of the dither signal
in comparison to the open-loop disturbance signal.
This finding proves to be very valuable. In the previous section, assessing simulated
response fits showed that this increase in B value does provide a better fit of the validation data. It was however uncertain whether this was due to the validation set’s bias
towards a frequency characteristic (B value) nearer to that of its own, or due to an inher-
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ently better representation of the true system. These results establish that the better fit
was due to B being more representative of the dominant dynamics of the system being
identified, not due to the validation set’s bias towards a model obtained from similar experimental conditions. This additionally attests to validation set ValOL 1 being a good
representation of the true system.
While most of the findings in analysing the frequency responses of the identified models
did reflect the response simulation findings in the previous section, some interesting cases
did reveal themselves.
Figure 6.17 compares the frequency responses of models obtained from closed-loop
experiments CL 3, 6 and 9. As was done in the previous section, this compares the model
obtained from case CL 3 defined as an experiment that used both setpoint changes and
dither signal disturbances to excite the system with that from case CL 6, that used only
dither signal disturbances, and case CL 9, that used only setpoint changes to disturb the
system. The figure further reveals how the frequency responses between output y1 and
input u1 (figure (a)) differ from that between y2 and u2 (figure (b)). Specifically it is noted
how not using certain disturbances signals affected the accuracy of the different output
responses of the models in different ways. The response between y1 and u1 shows that
experimentation without setpoint changes, case CL 6, produced a submodel with reduced
accuracy at lower frequencies, while removing the dither signal (case CL 9) revealed
little effect. The frequency response between y2 and u2 shows that there is no effect in
not changing setpoints while the lack of dither signal in the closed-loop identification
experiment caused a larger submodel mismatch at higher frequencies.
Figure 6.18 shows the frequency responses of both outputs to both inputs for the
models identified from cases CL 3, CL 4, CL 5, OL 3 and OL 4. All of these models
showed little distinction between each other when validated against ValOL 1 in terms of
response fit accuracy in the previous section. That is, validation of simulated response
fits against validation set ValOL 1, which is expected to be the most informative set,
did little to distinguish between the models generated by these cases. Analysing figure
6.18 it can be said that frequency response analysis does not discriminate between the
closed-loop cases since the differences in responses are not significant enough.
However, the responses of the models identified from open-loop experiments were significantly different to those of the models obtained from closed-loop experiments. The
open-loop cases were found to be less accurate at lower frequencies for most of the responses. These findings better reflect the response fit results when validated against the
closed-loop validation set, ValCL 3, than against the open-loop validation set, ValOL 1.
It is noted (in figure 6.18) that case OL 4 has a smaller deviation from the true model
frequency response in comparison to OL 3. This discrimination is found in the response
fit results when validated against the closed-loop validation set, ValCL 3 (in figure 6.13),
but not when validated against the open-loop validation set, ValOL 1 (in figure 6.12).
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(a) Effect of setpoint change and dither signal on frequency response for y -u
1
0
1
10
System A
CL 3
CL 6
CL 9
-1
Amplitude
10
-2
10
-3
10
-2
10
-1
0
10
1
10
10
Frequency (rad/s)
(b) Effect of setpoint change and dither signal on frequency response for y -u
2
2
System A
CL 3
CL 6
CL 9
0.7
Amplitude
10
0.5
10
0.3
10
0.1
10
-2
-1
10
0
10
1
10
10
Frequency (rad/s)
Figure 6.17: Frequency response comparing submodel responses to removal of identification
signals
(b) Frequency response for y -u
(a) Frequency response for y -u
1
0
1
10
Amplitude
Amplitude
10
-1
10
-2
10
-2
10
-1
0
10
10
1
10
10
Frequency (rad/s)
10
10
1
0
-1
-2
-3
-2
2
-1
10
(c) Frequency response for y -u
2
2
0
1
10
10
Frequency (rad/s)
10
(d) Frequency response for y -u
1
2
2
10
0.7
Amplitude
Amplitude
10
1
10
0.5
10
0.3
10
0.1
10
0
10
-2
10
-1
0
10
10
Frequency (rad/s)
System A
1
-2
10
CL 3
10
CL 4
CL 5
-1
0
10
10
Frequency (rad/s)
OL 3
OL 4
1
10
Figure 6.18: Frequency responses of cases with indistinguishable simulation fits against ValOL
1
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Once again these findings suggest that the inability of the closed-loop validation set
(ValCL 3) to discriminate between models obtained from closed-loop experiments through
response fit validation (see figure 6.13) may not have been due to bias towards models
obtained from experiments with the same feedback condition. This is since the frequency
response analysis is not a cross-validation technique in that it is dependent only on
the characteristics of the models themselves, independent of validation sets. Yet the
frequency response results given in figure 6.18 more closely resemble the cross-validation
results when using the closed-loop validation set than any other in that there is little to
distinguish between the responses of the models obtained from closed-loop experiments.
This suggests the closed-loop validation data set may more accurately reflects the true
system.
Further analysing the frequency responses of the models obtained from open-loop
experiments in figure 6.18 (cases OL 3 and 4), it is observed that there is a general
mismatch between OL 3 and OL 4 at the lower frequencies with OL 3 producing the
worse fit. The only difference between the two experiments is the type of input signal
used. OL 3 used PRBS disturbances signals while OL 4 used PRMLS. Also, 6.18(b)
shows how the open-loop cases generated models with better fits at higher frequencies
than the closed-loop cases.
This better fit at higher frequencies by the models identified from open-loop experiments, while not absolutely evident in all the responses, might be accounted for by the
fact that the identified models all have ARX structures, which do not adequately allow
for independent modelling of noise and dynamics. This is an established problem of the
ARX structure as has been explained in section 5.4.2. This issue of dynamic model corruption due to noise is expected to be more of a problem for the closed-loop experiments
as the feedback condition further increases this inability to independently model noise
and dynamics. This might justify the better high frequency fit for model obtained from
open-loop experiments.
Frequency Content of Response Fits
Through the previous assessment into each identified model’s frequency response and how
they compare to the true system’s frequency response, an understanding was obtained regarding the true accuracy of the different models over certain frequencies without the use
of validation data. Following this, it is appropriate to assess how the different validation
sets reveal the model accuracies and how this compares to the true accuracy assessment
done via the frequency response. Thus in essence, while the previous assessment presented model accuracy based on inherent model characteristics independent of validation
sets, here the presented accuracy of the identified model is dependent on the validation
set. As explained earlier, this is done by analysing response fits between the identified
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model and the validation data set via discrete frequency representation and serves the
purpose of testing the credibility of the validation sets.
The assessment is done against validation sets ValOL 1 and ValCL 3 (the sets generated by PRBS under open and closed-loop conditions respectively) for cases: OL 2, OL
3, OL 4 and CL 3, Cl 4, CL 5. This is as these cases yielded the most information and
discussion on the simulated response fits and model frequency response analyses (figure
6.18).
The comparisons of frequency component fits between each of the models obtained
from these cases and validation set ValOL 1 for output y1 are shown on figure 6.19. From
comparison with the frequency response results show in figure 6.18, it is deduced that the
validation data set ValOL 1 does to some extent allow for accurate validation and model
discrimination that reflects the differences between the identified models and true model.
Figure 6.19 shows how responses from models obtained from closed-loop experiments
(figures 6.19(d),(e) and (f)) validated against ValOL 1 closely fit the validation data
at lower frequencies while slightly drifting at the higher frequencies. In comparison,
the frequency content analysis of the responses from models obtained from open-loop
experiments (figures 6.19(a),(b) and (c)) have worse fits at the lower frequencies. This is
exactly what was found in the frequency response analysis in figure 6.18.
This may once again be explained by the fact that the closed-loop experiments used
setpoint changes and dithers signal disturbances to generate information on the system,
whereas the open-loop experiments only used input disturbances. The condition of using
low frequency setpoint changes may explain the better fit at lower frequencies for the
models generated from closed-loop data.
Looking at figure 6.19 it is observed that at the higher frequencies the signals become
noisy. This is thought to be since the response data of the validation set (ValOL 1)
and the simulated model responses may not have sufficient and accurate information
at those frequencies since they were not sufficiently disturbed at such frequencies. It
is however strange to find that the model obtained from case OL 2 in figure 6.19(a),
produced a response trend that is much smoother at these high frequencies than the
others, even though it was disturbed by the same signal. What distinguishes this model
from all the other models being assessed is that the experiment used to generate data
for its identification is characterised by an input disturbance signal with a much smaller
constant signal interval value (B). This means the PRBS input signal disturbed the system
at a higher frequency. Thus this model should be more accurate at higher frequencies,
however, this accuracy was not expected to manifested its self since the validation set
did not disturb the system at such frequencies and thus was not expected to produce
consistent information at such frequencies.
Figure 6.20 shows the frequency component representations of the same models as
6.19 (those obtained from experiment cases OL 2, OL 3, OL 4 and CL 3, Cl 4, CL
© University of Pretoria
CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
(a)
Response Fit Frequency Analysis for Case OL 2
0
(d) Response Fit Frequency Analysis for Case CL 3
0
-2
ValOL 1 (100 %)
OL 2: 4.76 %
-1
10
10
Amplitude
Amplitude
10
10
-1
10
ValOL 1 (100 %)
OL 3: 53.2 %
10
-2
10
ValOL 1 (100 %)
CL 4: 53.3 %
0
-1
10
10
Frequency (rad/s)
(c) Response Fit Frequency Analysis for Case OL 4
0
10
Frequency (rad/s)
(f) Response Fit Frequency Analysis for Case CL 5
0
0
10
Amplitude
10
Amplitude
0
10
Frequency (rad/s)
0(e) Response Fit Frequency Analysis for Case CL 4
Amplitude
Amplitude
-2
-1
ValOL 1 (100 %)
CL 3: 52.9 %
0
10
10
-2
10
10
Frequency (rad/s)
(b)
Response Fit Frequency Analysis for Case OL 3
0
10
128
-2
10
ValOL 1 (100 %)
OL 4: 65 %
-1
10
-2
10
ValOL 1 (100 %)
CL 5: 49.7 %
0
-1
10
10
Frequency (rad/s)
0
10
Frequency (rad/s)
Figure 6.19: Frequency analysis of simulation responses validated against ValOL 1
(a) Response Fit Frequency Analysis for Case OL 2
(d) Response Fit Frequency Analysis for Case CL 3
0
0
ValCL 3 (100 %)
OL 2: -57 %
-2
10
-2
10
-1
10
10
Amplitude
Amplitude
10
-2
10
0
-2
10
10
Frequency (rad/s)
(b) Response Fit Frequency Analysis for Case OL 3
0
10
0
ValCL 3 (100 %)
OL 3: 11 %
-2
10
-2
10
-1
10
10
Amplitude
Amplitude
-1
10
Frequency (rad/s)
(e) Response Fit Frequency Analysis for Case CL 4
0
10
ValCL 3 (100 %)
CL 4: 98.4 %
-2
10
0
-2
10
10
Frequency (rad/s)
(c) Response Fit Frequency Analysis for Case OL 4
-1
10
0
10
Frequency (rad/s)
(d) Response Fit Frequency Analysis for Case CL 5
0
0
ValCL 3 (100 %)
OL 4: 59 %
-2
10
-2
10
-1
10
0
10
10
Amplitude
10
Amplitude
ValCL 3 (100 %)
CL 3: 98.4 %
ValCL 3 (100 %)
CL 5: 98.2 %
-2
10
-2
10
Frequency (rad/s)
-1
10
0
10
Frequency (rad/s)
Figure 6.20: Frequency analysis of simulation responses validated against ValCL 3
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
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5) but validated against ValCL 3, the data set obtained under closed-loop conditions.
The first thing that is noticed is the earlier decay in amplitude in comparison to the
frequency analysis against ValOL 1 (figure 6.19). With the frequency profile having
larger magnitudes persisting at low frequencies of small bandwidths and small magnitudes
with larger persistence at the higher frequencies, this reveals the closed-loop identification
condition. Where the large disturbances are implemented by setpoint changes and at very
low frequencies leaving small magnitude dither signals to extract most of the information
at higher frequencies.
Recalling a previous observation made from figure 6.18(b) where the models identified
from open-loop experiments (OL 3 and OL 4) produced slightly improved frequency
response fits at the higher frequencies than the closed-loop cases. That is, the models
obtained from closed-loop experiments were found to have a high frequency respons misfit
when compared to the true system. Looking at figures 6.19 and 6.20, this misfit at higher
frequencies of the models obtained from closed-loop experiments, characterised by the rise
in magnitude where the validation set falls, is made more evident when using validation
set ValCL 3 (figures 6.20(d),(e) and (f)) than ValOL 1 (figures 6.19(d),(e) and (f)). This
means that the closed-loop validation set, ValCL 3 did a better job at revealing these
high frequency inaccuracies of the models obtained from closed-loop experiments than
the validation set obtained from the open-loop experiment.
This finding reveals an interesting conditions. Previous assessments suggested that the
models produced by closed-loop experiments are less accurate at the higher frequencies
than those obtained from open-loop experiments. While this last finding reveals that
the validation set obtained from closed-loop experiments better discriminates at higher
frequencies, which implies more accurate information, than the validation set obtained
from open-loop conditions.
6.3.5
Residual Correlation Analysis
Both the whiteness and independence correlation tests were used to assess the model residuals. With the residuals being generated by cross-validation, the same cross-validation
approach is used here as in the simulation and prediction section. The inputs used to
generate responses from the identified models were those of validation sets ValOL 1 and
ValCL 3 (the open and closed-loop experiments conditions respectively with both using
PRBS signals). ValOL 2 (the open-loop step response experiment case) was not used
since its inability to persistently excite the system did not yield informative enough data
for effective correlation analyses. In essence this section conveys the findings from the implementation of correlation tests on the residuals generated when assessing the response
simulation percentage fits in section 6.3.3.
The important difference, however, is that the simulations that provided the responses
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used to assess percentage fit values where done without the noise model. This was done
so as to compare the identified dynamics model and the true system (system A). These
residual correlation tests are done on simulated responses of identified models include
the noise model. Noise was accordingly also superimposed on the validation output data.
Thus, as was established in section 4.1.5, if the identified model was to perfectly represent
system A, the residuals would only be white random realisations due to the difference
between realisations of noise used in simulating responses from the identified model to
that superimposed on the validation data set output.
The reader is referred to section 4.1.5 for the residual correlation result interpretation
rules and is reminded that 1 and 2 signify the residuals for output 1 and 2 respectively.
Whiteness Test Results
The whiteness test, assessed by residual auto-correlation plots, of the models generated
by open-loop experiments gave some debatable results. It was found that the results
heavily depended on the validation data by which they were disturbed by and simulated
against. Figure 6.21(a) shows the residual correlations for the model identified from case
OL 2 when simulated against validation set ValOL 1 (the open-loop set) while figure
6.21(b) shows the result against ValCL 3 (the closed-loop set).
(b) Auto-Correlation of Residual ε for case OL 2 against ValOL 1
2
Sample Auto-Correlation
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-200
-150
-100
-50
0
Lag
50
100
150
200
150
200
(b) Auto-Correlation of Residual ε for case OL 2 against ValCL 3
2
Sample Auto-Correlation
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-200
-150
-100
-50
0
Lag
50
100
Figure 6.21: Illustration of sensitivity of correlation tests to validation data for OL cases
The correlation against ValOL 1 does not show perfect whiteness, however, in com-
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131
parison to the correlation against ValCL 3 these correlations are relatively insignificant.
Interpretation of this correlation dependence on validation set suggests that the closedloop validation set, ValCL 3, allows for the detection of unmodelled dynamics where
ValOL 1 did not. This trend is seen throughout the models obtained from open-loop
experiments for both residuals, 1 and 2 .
If the simulated response percentage fit results were a reflection of these correlation
results, then all the open-loop cases would have generated models that performed worse
when validated against ValCL 3 than when validated against ValOL 1. The fact is that
to some extent this is the case, recalling figures 6.12 and 6.13, one observes that all the
models obtained from open-loop experiments (the OL cases) showed worse percentage fit
values when validated against ValCL 3 than against ValOL 1.
(b) Auto-Correlation of Residual ε for case CL 4 against ValOL 1
1
Sample Auto-Correlation
1
0.5
0
-0.5
-1
-200
-150
-100
-50
0
Lag
50
100
150
200
150
200
(b) Auto-Correlation of Residual ε for case CL 4 against ValCL 3
1
Sample Auto-Correlation
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-200
-150
-100
-50
0
Lag
50
100
Figure 6.22: Illustration of sensitivity of correlation tests to validation data for CL cases
The residual correlation analyses of the models generated from closed-loop experiments showed much more consistency and independence of the validation set against
which they were validated. All the correlation plots resembled those shown in figure 6.22
illustrating the results for case CL 4. The results indicate very little significant correlation regardless of the validation set against which they were validated and suggest little
differences in indications of unmodelled dynamics. This is contrary to simulated response
percentage fit results for the models obtained from closed-loop experiments (CL cases) as
they implied large extents of umodelled dynamics when validated against ValOL 1 and
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132
almost none when validated against ValCL 3.
In terms of the differences in residual correlations between the two outputs, the same
trends were shown for both, that is, there where no differences in correlation results
between 1 and 2 .
Independence Test Results
The results on residual independence tests reflected those of the whiteness test. The
models generated from open-loop data showed larger sensitivities to the validation set
used in comparison to those generated from closed-loop experiments. Figure 6.23(a)
shows how the model obtained from case OL 3 reveals little residual dependence on input
signals when validated against the open-loop validation set, ValOL 1. In contrast to this,
figure 6.23(b), shows a much larger extent of input dependence of the same model when
validated against the closed-loop validation set, ValCL 3. This larger dependence implies
a larger component of the model that wrongly fits the true system. This was found for
all the models obtained from open-loop experiments.
Almost all the models obtained from closed-loop experiments showed little residual
dependence to input signals regardless of the validation set used. Only case CL 6 showed
larger dependencies that suggested model dynamics that wrongly describe the true system, the rest of the closed-loop cases showed input-residual correlations that closely
resembled that of case CL 4 shown by figure 6.24(a) validated against the open-loop
validation set, ValOL 1, and figure 6.24(b) against the closed-loop validation set, ValCL
3.
It must be noted that case CL 6, the only closed-loop case that generated a model that
revealed some significant input-residual correlation, is characterised as the closed-loop
experiment where no setpoint changes where implemented. This means only the dither
signals were used to excite the system, this resembles the open-loop experiments but
with lower magnitude disturbance signals and controller action reducing the disturbance
effects. From this one may gather that the signal structure changes caused by the setpoint
changes are beneficial in assuring independence between inputs and residuals.
In trying to interpret the differences in input-residual correlations obtained when
using closed-loop validation sets to those when using open-loop validations sets, it is
important to recall that the input-residual correlation results validated against closedloop experiments are not to be completely trusted. Closed-loop experiments generate
data with definite correlations between the inputs and output signals due to the feedback
loop. This undoubtedly means there is a correlation between the input and the residual
values when using closed-loop data. From this, it is implied that these correlations cannot
purely signify incorrectly modelled dynamics.
© University of Pretoria
CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
(a) Input-Residual Cross-Correlation for u -ε
1
(c) Input-Residual Cross-Correlation for u -ε
1
1
2
0.2
Sample Cross-Correlation
Sample Cross-Correlation
0.2
0
-0.2
-0.4
-0.6
-0.8
-200
-100
0
Lag
100
(b) Input-Residual Cross-Correlation for u -ε
2
0
-0.2
-0.4
-0.6
-0.8
-200
200
-100
0
Lag
100
200
(d) Input-Residual Cross-Correlation for u -ε
1
2
2
0.3
Sample Cross-Correlation
0.3
Sample Cross-Correlation
133
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-200
-100
0
Lag
100
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-200
200
-100
0
Lag
100
200
(a) Input-Residual correlations correlated against ValOL 1 for OL 3
(a) Input-Residual Cross-Correlation for u -ε
1
(c) Input-Residual Cross-Correlation for u -ε
1
1
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-200
-100
0
Lag
100
(b) Input-Residual Cross-Correlation for u -ε
2
0.4
0.2
0
-100
0
Lag
100
200
(d) Input-Residual Cross-Correlation for u -ε
1
2
2
Sample Cross-Correlation
0.2
1
0.8
0.6
0.4
0.2
0
-0.2
-200
0.6
-0.2
-200
200
1.2
Sample Cross-Correlation
2
0.8
Sample Cross-Correlation
Sample Cross-Correlation
0.4
-100
0
Lag
100
200
0
-0.2
-0.4
-0.6
-0.8
-200
-100
0
Lag
100
200
(b) Input-Residual correlations correlated against ValCL 3 for OL 3
Figure 6.23: Input-Residual correlations for OL 3 correlated against ValOL 1 and ValCL3
© University of Pretoria
CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
(a) Input-Residual Cross-Correlation for u -ε
1
(c) Input-Residual Cross-Correlation for u -ε
1
1
2
0.4
Sample Cross-Correlation
Sample Cross-Correlation
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-200
-100
0
Lag
100
(b) Input-Residual Cross-Correlation for u -ε
2
0.2
0
-0.2
-0.4
-200
200
-100
0
Lag
100
200
(d) Input-Residual Cross-Correlation for u -ε
1
2
2
0.3
Sample Cross-Correlation
0.4
Sample Cross-Correlation
134
0.2
0
-0.2
-0.4
-200
-100
0
Lag
100
0.2
0.1
0
-0.1
-0.2
-200
200
-100
0
Lag
100
200
(a) Input-Residual correlations correlated against ValOL 1 for CL 4
(a) Input-Residual Cross-Correlation for u -ε
1
(c) Input-Residual Cross-Correlation for u -ε
1
1
0.05
0
-0.05
-0.1
-200
-100
0
Lag
100
(b) Input-Residual Cross-Correlation for u -ε
2
0.05
0
-0.05
-100
0
Lag
100
200
(d) Input-Residual Cross-Correlation for u -ε
1
2
2
Sample Cross-Correlation
0.1
0.05
0
-0.05
-0.1
-200
0.1
-0.1
-200
200
0.1
Sample Cross-Correlation
2
0.15
Sample Cross-Correlation
Sample Cross-Correlation
0.1
-100
0
Lag
100
200
0.05
0
-0.05
-0.1
-0.15
-200
-100
0
Lag
100
200
(b) Input-Residual correlations correlated against ValCL 3 for CL 4
Figure 6.24: Input-Residual correlations for CL 4 correlated against ValOL 1 and ValCL3
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6.3.6
135
Results Overview
The identification and validation results for system A, the linear system, allowed for some
interesting findings. These may be summarised as follows:
Model Variance, Uncertainty and Stability
Variance due to Noise Sensitivity:
• The information generated from the open-loop step disturbance experiment (OL 5)
was found to be most sensitive to different noise realisations. This implies a low
capacity to generate dynamics information by a step disturbance.
Model Uncertainty:
• The model obtained from the open-loop step disturbance experiment (OL 5) was
found to have the smallest measure of model uncertainty. This was however not by
a significant amount.
Stability:
• The model obtained from the open-loop step disturbance experiment (OL 5) was
found to be the only model that did not have a pole outside the unit circle on the
z-plane as the true system did.
Simulation and Prediction
Bias and Discrimination:
• The worst response fits from the identified models were obtained when validated
against the open-loop validation ValOL 1 (characterised by PRBS disturbance signals).
• The response fit results when validated against the closed-loop validation set, ValCL
3, suggested the validation set was biased towards models obtained from closed-loop
experiments.
• The response fit results when validated against the open-loop step disturbance validation, ValOL 2, suggested that validation set ValOL 2 was biased towards models
obtained from experiments with similar signal disturbances i.e. step disturbances
and setpoint changes.
• The above point was emphasised by the fact that validation set ValOL 2, was the
only set to reveal a terrible fit for the model obtained from closed-loop experiments
with no setpoint changes.
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Experimental Condition Sensitivity:
• It was established that models generated by experiments with disturbance signal
frequency characteristics chosen in consideration of the dominant time constant of
system A had better fits. These disturbance signal frequency characteristics were
also closer to that of the validation set being used. This meant it was not possible
to distinguish whether the better fit was due to better inherent model accuracy or
validation set bias.
• The PRMLS disturbance signal (closely followed by the PRBS) proved to produce
the model with the most accurate responses from those that were generated from
open-loop experiments while the PRBS proved to be most consistent for the closedloop experiments.
• The models obtained from closed-loop experiments were found to be less sensitive
to changes in experimental design variables. This was thought to be due to the
controller dampening the effects of disturbances signals.
• The previous point means that given disturbance signals that were relatively similar
in terms of persistent excitation, the open-loop experiments produced models with
more accurate responses. However, given poorly designed disturbance signals that
were not very persistently exciting, the open-loop experiments were affected more
in that they generated models that produced less accurate responses compared to
those generated from closed-loop experiments.
• When validated against the open-loop PRBS disturbed validation set, it was revealed that models identified from experiments that did not use dither signal disturbances did worse than those identified from experiments that did not use setpoint
changes.
• The direction and timing of setpoint changes made during identification experiments
were found to have effects on the identified model. It was found that setpoint
changes in opposite direction and at different times were most informative.
Frequency Analysis
Frequency Response Analysis:
• Frequency response analysis allowed for more insight into the simulate response fits
results.
• It was found that the better simulated response fits of model obtained from experiments disturbed by signals with frequency characteristics chosen in consideration
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137
of the dominant time constant of system A was in fact not due to validation set
bias.
• It was shown that not using a dither signal disturbance can reduce model accuracy at higher frequencies and that implementing setpoint changes can drastically
improve model accuracy at low frequencies.
• It was shown that experiment variable changes can have different effects on the
response accuracy of different outputs of the same model.
• Assessing how the frequency response of each identified model varied relative to
each other and the response of the true system, system A, it was revealed that the
results indicated by the frequency responses best resemble the response fits obtained
when validated against the closed-loop validation set.
• Frequency response analyses suggested that the models obtained from open-loop
experiments have better model accuracy at higher frequencies. This was thought
to reflect the condition of dynamic model corruption by noise which closed-loop
models are more susceptible to.
Frequency Content of Response fits:
• Analysing the simulated response fits in the frequency domain and comparing the
finding to the frequency response analysis it was found that both the open-loop
PRBS disturbed (ValOL 1) and the closed-loop (ValCL 3) validation sets both did
well in representing the true system in terms of information content.
• However, the validation set obtained from closed-loop conditions was found to better
represent the true system at higher frequencies.
Residual Correlation
Whiteness Test:
• The whiteness test reflected much of the findings made from the simulated response
fit analyses.
• The closed-loop validation set was found to better discriminate between the models
obtained from open-loop experiments sets than the open-loop validation set.
• The whiteness tests revealed that the models obtained from closed-loop experiments
have less unmodelled dynamics.
Independence Test:
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• All the models obtained from open-loop experiments showed little residual-input
correlations when validated against the open-loop validation set but large correlations against the closed-loop validation set.
• All but one of the models obtained from closed-loop experiments showed little
residual-input correlation regardless of the validation set used.
• The model obtained from experiment CL 6, characterised as a closed-loop experiment with not setpoint changes, was the only one obtained from closed-loop experiments that showed significant residual-input correlations. This was the case against
both open and closed-loop validation sets. This result clearly revealed the value of
setpoint changes in closed-loop experiments in assuring model residuals were not
correlated to inputs.
• Emphasis was placed on the use of closed-loop validation data in assessing inputresidual correlations. Closed-loop data inherently contain correlations between the
inputs and outputs via the feedback, thus introducing correlations between the
input and residuals that don’t necessarily imply incorrectly modelled dynamics.
• Thus, disregarding the input-residual correlation results obtained when validated
against closed-loop validation sets, the only model that showed input-residual correlation was that of CL 6, obtained from the closed-loop experiment that did not
implement setpoint changes.
6.4
Identification and Validation Results for System
B - The Non-Linear System
This section presents and discusses the results of the identification of linear models approximating the non-linear system, system B, and the validation of such models. Note
that all the parameters of each identified model are presented in appendix A.2 Each of
the 23 identified models (6 generated from open-loop experiments and 17 from closedloop experiments) are first assessed for variance sensitivities and model uncertainty in
section 6.4.1 in a similar fashion as was done for the models approximating system A
(section 6.3.1). Following this the identified models are cross-validated against the 5
validation sets (3 obtained from open-loop experiments and 2 obtained from closed-loop
experiments) generated from system B. The validation approach is similar to that of the
previous section in that cross-validation techniques (see figure 6.8) are primarily used.
It is at this point appropriate to recall tables 6.6 and 6.5 for the open and closedloop experimental conditions respectively used to identify models from. Table 6.8 details
the experimental conditions used to generate validation data. It is important to note
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the case tags of each identification experiment and validation experiment as they will be
continuously referred to in this section. As with the previous section, a brief mention of
the conditions characterising the experiment used to generate the model or validation set
being discussed is done where necessary and relevant to the point being made.
6.4.1
Model Variance and Uncertainty
The assessment of sensitivity to noise and the model uncertainties of the models identified
from each identification experiment in efforts to approximate system B, was done in the
same manner as that of the linear system, system A. That is, 50 repetitions of each
identification experiment was done to generate 50 different noise realisation and assess
the effects thereof. Also, Monte Carlo simulations were used to generate 50 simulated
responses reflecting the response variance due to parameter uncertainty.
Variance Due to Sensitivities to Random Noise Realisations
Average parameter variance due to noise realisation
NOL 1
NOL 2
NOL 3
NOL 4
NOL 5
NOL 6
NCL 1
NCL 2
NCL 3
NCL 4
NCL 5
NCL 6
NCL 7
NCL 8
NCL 9
NCL 10
NCL 11
NCL 12
NCL 13
NCL 14
NCL 15
NCL 16
NCL 17
0.025
variance ( σ 2)
0.02
0.015
0.01
0.005
0
Identification Case
Figure 6.25: Parameter variance of non-linear system estimates due to random noise realisations
Figure 6.25 shows the resulting average parameter variances of the models identified
for the respective identification experiments given the 50 repetitions of each. It is interesting to note that the two cases that yielded the largest parameter variances due to noise
sensitivities were cases NCL 7 and NCL 16, these are both closed-loop experiments with
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no setpoint changes. While the variance magnitudes of these two cases are significantly
larger than the others, they are also significantly different to each other. That is, case
NCL 16 which is precisely the same as NCL 7 but with a larger signal-to-noise ratio has
the smaller parameter variance due to noise sensitivity.
Furthermore, all the other closed-loop experiments showed the same variance while
the open-loop experiments showed no trend or allowed for any logical deductions with case
NOL 2, characterised as a white PRBS disturbance signal, having the largest variance.
It must however be noted that even though the variances experienced through cases
NCL 7 and NCL 16 might be considered significantly larger than those of the other cases,
none of the variance magnitudes were large enough to effectively show significant variance
in simulated responses to PRBS input disturbances.
Model Uncertainty
The average standard deviations obtained via Monte Carlo simulations reflecting the
model uncertainties are shown in figure 6.26. Once again, as with the parameter variance
due to noise sensitivities, the closed-loop cases with no setpoint changes (NCL 7 and
NCL 16) gave the largest average simulated response standard deviations. This implies
that the model generated from these experimental conditions had the largest extents of
model uncertainty.
Output standard deviation for y
1
Standard Deviation ( σ)
0.025
0.02
0.015
0.01
0.005
0
Identification Case
Output standard deviation for y
2
Standard Deviation ( σ)
0.06
0.05
0.04
0.03
0.02
0.01
NOL 1
NOL 2
NOL 3
NOL 4
NOL 5
NOL 6
NCL 1
NCL 2
NCL 3
NCL 4
NCL 5
NCL 6
NCL 7
NCL 8
NCL 9
NCL 10
NCL 11
NCL 12
NCL 13
NCL 14
NCL 15
NCL 16
NCL 17
0
Identification Case
Figure 6.26: Standard deviations of output responses found through Monte Carlo simulations
showing model uncertainty
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The trends for the different cases are very much the same for both output y1 and y2 .
The model uncertainty does however seem to be greater in components associated with
output y2 . This is most likely due to fact that the larger system gains are between both
inputs and output y2 .
6.4.2
Simulation and Prediction
General Observations
Figure 6.27 shows the percentage fit values for the identified model responses when validated against the open-loop validation data sets (NValOL 1, NValOL 2, NValOL 3).
Comparing the results between the 3 open-loop validation sets it is quite evident that
they vary immensely between the different sets. Figure 6.28 shows the percentage fits
for the identified model responses when validated against the closed-loop sets (NValCL
4 and NValCL 5). These results do not show a clear trend when comparing the results
between the two validation sets. This is mostly on account of the difference in fit values
between y1 and y2 for NValCL 5.
The response fit results validated against the closed-loop sets do however show greater
variance within the sets. That is, it is evident that the closed-loop validation sets allowed
for greater discrimination between the identified model responses than the open-loop
validation sets.
Bias and Discrimination by PRBS Disturbance Validation Sets - Disturbance
Signal Magnitude Sensitivities
Looking at the results validated against the open-loop data sets, figure 6.27, the drastic
reduction in percentage fit values between validation set NValOL 1 and NValOL 2 introduces the characteristic problem behind estimating a non-linear system with a linear
model. With the only difference between the two validation sets being that set NValOL 2
was generated using larger disturbance signals than NValOL 1, it may be said that these
larger system disturbances reveal the non-linear nature of the system to a greater extent
and consequently the linear models, identified from smaller disturbance magnitudes and
their responses, performed worse.
This is further emphasised when looking at the results of the models identified from
experiments that used larger input signals similar to those used to generate validation
set NValOL 2. These cases, NOL 4, NCL 4 and NCL 13, are characterised as the dips
in the percentage fit profile validated against NValOL 1 and the peaks when validated
against NValOL 2. This is more evident for the y2 fits.
As with the open-loop validation sets just discussed, the only difference between the
two closed-loop validation sets, NValCL 4 and NValCL 5, is that set NValCL 5 was allowed
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(a) Response Percentage Fit Against Open Loop Validation Data : NValOL 1, NValOL 2, NValOL 3 for y
1
100
Percentage Fit
80
60
40
20
0
Validated against NValOL 1
Validated against NValOL 2
Validated against NValOL 3
(b) Response Percentage Fit Against Open Loop Validation Data : NValOL 1, NValOL 2, NValOL 3 for y
100
Percentage Fit
80
60
40
20
0
Validated against NValOL 1
Validated against NValOL 2
2
NOL 1
NOL 2
NOL 3
NOL 4
NOL 5
NOL 6
NCL 1
NCL 2
NCL 3
NCL 4
NCL 5
NCL 6
NCL 7
NCL 8
NCL 9
NCL 10
NCL 11
NCL 12
NCL 13
NCL 14
NCL 15
NCL 16
NCL 17
Validated against NValOL 3
Figure 6.27: Simulation percentage fit values validated against open-loop validation data sets
(a) Response Percentage Fit Against Closed Loop Validation Data : NValCL 4, NValCL 5 for y
1
100
Percentage Fit
80
60
40
20
0
Validated against NValCL 4
Validated against NValCL 5
(b) Response Percentage Fit Against Closed Loop Validation Data : NValCL 4, NValCL 5 for y
2
100
Percentage Fit
80
60
40
20
0
Validated against NValCL 4
NOL 1
NOL 2
NOL 3
NOL 4
NOL 5
NOL 6
NCL 1
NCL 2
NCL 3
NCL 4
NCL 5
NCL 6
NCL 7
NCL 8
NCL 9
NCL 10
NCL 11
NCL 12
NCL 13
NCL 14
NCL 15
NCL 16
NCL 17
Validated against NValCL 5
Figure 6.28: Simulation percentage fit values validated against closed-loop validation data set
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larger setpoint changes and consequently operated over a larger output range. Figure 6.28
shows the response percentage fits of each identified model validated against these two
closed-loop validation sets. The effect of varying the magnitude of the disturbance signals,
and hence the dynamic region over which data was generated, on the experiments used
to generate to closed-loop validation sets was not as evident as when using the open-loop
validation sets. As mentioned earlier, this may be attributed to inconsistencies in results
between the results for output y1 and y2 . Looking at the results for output y1 (figure
6.28(a)) it is noted that when validating the models against the closed-loop validation set
generated by larger disturbances signals (NValCL 5), the change in response fit, relative
to the results validated against NValCL 4, is not evident for most models. In contrast to
this the results for output y2 (figure 6.28(b)) do more clearly indicate that most of the
models do have significantly worse fits when validated against NValCL 5 in comparison
to results when validated against NValCL 4.
This is very interesting. This condition that the accuracy of the response fits for
ouput y2 , in comparison to y1 , are more sensitive to the magnitude of the disturbance
signals and output ranges used in generating models and validation data sets affirms
the larger extent of non-linear dynamics associated with output y2 . This is a definitive
characteristic of the system and was discussed in section 6.1.3.
It must however be said that differences between response fit results validated against
NValOL 1 and NValOL 2 to those between NValCL 4 and NValCL 5 are not entirely
comparable. This is since the difference in dynamic region, over which the data was generated, between NValOL 1 and NValOL 2, was specified by differences in the magnitudes
of the input signals disturbing the system (since they were open-loop experiments). In
the case of the closed-loop experiments, the difference in dynamic region was specified
by setpoint ranges, thus the experimental range was defined by the output range. This
means that the effect of increasing the disturbance signal magnitudes in the open-loop
experiments is not directly comparable to that of increasing the magnitude of disturbance
signals of the closed-loop experiments since the former is done via the input signal and
the latter is done via the setpoint signal. However, looking at figure 6.29, which compares
the input signals used to generate the open-loop validation sets, NValOL 1 and 2 (figure 6.29(a)), and the input signals used by the controllers in generating the closed-loop
validation sets, NValCL 4 and 5 (figure 6.29(b)), the ranges are very similar.
Bias and Discrimination by Step Disturbance Validation Sets - Disturbance
Signal Magnitude Sensitivities
The results generated by validation set NValOL 3 were extremely interesting. Looking
at the results when using validation set NValOL 3 (figure 6.27), characterised as the
open-loop step disturbance experiment, all the identified models performed very poorly.
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(a) Input signals used to generate open-loop validation data for NValOL 1 and NValOL 2 (u )
-1
∆ Reflux Flow Rate (mol min )
1
0.05
NValOL 1
NValOL 2
0
-0.05
100
120
140
160
180
200
Time (s)
220
240
260
280
300
(b) Controller induced input signals used to generate closed-loop validation data for NValCL 4 and NValCL 5 (u )
-1
∆ Reflux Flow Rate (mol min )
1
0.05
0
NValCL 4
NValCL 5
-0.05
100
150
200
250
300
350
Time (s)
400
450
500
550
600
Figure 6.29: Comparison between input signals generated under open and closed-loop conditions for normal and larger disturbance ranges
The exception to this statement being the fits produced by the models generated from
experiments NCL 4 and NCL 13 for output y1 and NCL 4 for y2 . It is interesting to note
that these experiments are those characterised as having larger disturbance signals.
This greater discrimination between models based on the region over which they
were identified, and consequently the exposure to non-linear dynamics, proved to make
validation set NValOL 3 very informative and revealing in terms of allowing insight into
the non-linear system identification problem. This is specifically the case for response fits
for output y2 (figure 6.27(b)). Output y2 is of is of particular interest as it exhibits the
larger sensitivity to input magnitude changes. Section 6.1.3, in describing the dynamics of
the true system - system B, characterised the dynamic behaviour of output y2 as having an
inverse response given sufficiently large disturbances. Looking at the simulated responses
for output y2 of the identified models, figure 6.30, when validated against validation set
NValOL 3, it is evident that most of the identification experiments generated models
that did not identify the inverse response. This does indicate that the identification
experiments designed to disturb system B so as to generate data for identification, did
not all force the inverse response.
The experiments that did allow for the identification of the inverse response were NOL
3, NOL 6, NCL 4 and NCL 6 with NCL 4 having the best fit. Note how NCL 4 is the
only experiment of the 4 characterised by larger disturbance signal ranges. Additionally,
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∆ Plate 7 Temperature ( C)
(a) Response fit for models obtained from open-loop experiments for y
145
2
0
NValOL 3 (100 %)
NOL 1: -1776 %
NOL 2: -821 %
NOL 3: -119 %
NOL 4: -293 %
NOL 5: -1817 %
NOL 6: -59.9 %
-0.01
-0.02
-0.03
0
50
-3
o
∆ Plate 7 Temperature ( C)
4
x 10
100
150
200
250
Time (s)
300
350
400
450
NValOL 3 (100 %)
NCL 1: -216 %
NCL 2: -125 %
NCL 3: -230 %
NCL 4: 59.9 %
NCL 5: -264 %
NCL 6: -159 %
NCL 7: -587 %
NCL 8: -243 %
NCL 9: -194 %
NCL 10: -64 %
NCL 11: -247 %
NCL 12: -331 %
NCL 13: -49 %
NCL 14: -398 %
NCL 15: -281 %
NCL 16: -580 %
NCL 17: -221 %
(b) Response fit for models obtained from closed-loop experiments for y
2
2
0
-2
-4
-6
-8
-10
-12
0
50
100
150
200
250
Time (s)
300
350
400
450
Figure 6.30: Simulation validated against open-loop step response validation data set NValOL
3
(a) Simulation validation against true model y response to 2 % input disturbance
-3
o
∆ Plate 7 Temperature ( C)
4
2
x 10
2
0
NValOL 3 (100 %)
NCL 4: 59.9 %
NCL 5: -264 %
-2
-4
-6
0
x 10
50
100
150
200
250
Time (s)
300
350
400
450
(b) Simulation validation against true model y response to 1 % input disturbance
-3
2
o
∆ Plate 7 Temperature ( C)
1
0
NValOL 3mod (100 %)
NCL 4: -621 %
NCL 5: 52.3 %
-1
-2
-3
-4
0
50
100
150
200
250
Time (s)
300
350
400
450
Figure 6.31: Illustration of the characteristic problem in linear model approximation of a nonlinear system
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looking at tables 6.6 and 6.8 one notes the experimental conditions of NOL 6 are the same
open-loop step disturbance conditions as those used to generate validation set NValOL
3 but with a larger step disturbance magnitude. Figure 6.30(a) shows how the model
generated by experiment NOL 6 produced a more pronounced inverse response than the
validation set response. Recalling section 6.1.3, it was established that the larger the step
disturbance, the more pronounced the inverse response in output y2 for system B. Since
experiment NOL 6 used a larger step disturbance magnitude than that used in validation
set NValOL 3, the inverse response used to identify the model was more pronounced,
thus the result in figure 6.30(a).
Figure 6.31(a) shows the same simulated responses from the models obtained from
experiments NCL 4 and NCL 5 given in figure 6.30, validated against the same NValOL 3,
isolated from the others for better visibility. Here it is clearly seen that the experimental
conditions of NCL 4 allowed for the identification of the inverse response, while those of
NCL 5 did not. Below figure 6.31(a), figure 6.31(b) shows the simulated responses generated by the same identified models but validated against a modified version of validation
set NValOL 3, NValOL 3mod. The only difference being that the while NValOL 3 was
generated using a 2% step disturbance magnitude, NValOL 3mod was generated using
a 1% step disturbance magnitude. It is noted that the reduction in step magnitude of
the signal disturbing the true non-linear system, system B, produced a much less pronounced inverse response. More importantly, it is noted that the linear models identified
from experiments NCL 4 and NCL 5 maintain the same dynamic characteristics in their
responses (as per definition of a linear system). The model generated from NCL 4 performed very well when validated against NValOL 3 but poorly against NValOL 3mod
and vice-versa for NCL 5. .
This shows how a poor model may accurately fit a validation set response if that
validation set is not appropriately chosen and obtained from experiments appropriately
designed so as to generate informative data. Through this the importance of generating
validation data, and hence designing experiments, based on the defined dynamic region
over which the identified model must satisfy its purpose is emphasised, especially when
approximating a non-linear system with a linear model
Other Experiment Condition Sensitivities
In analysing the sensitivities of the identified models to experimental variables other than
disturbance signal magnitudes, it must be said that the open-loop validation sets showed
little ability in discriminating based on the variances of such variables. The closed-loop
validation sets however seemed to be able to better discriminate between models based
on the effects of such variances on these model. This is clearly seen when comparing the
response fit profiles validated against the open-loop validation set NValOL 1 in figure6.27
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to the profile when validated against the closed-loop validation set NValCL 4 in figure
6.28. The larger variance in response fit results when validated against the closed-loop
validation set is clearly evident.
This implies the closed-loop validation set better discriminated between the models
identified from different experimental conditions. Thus the following observations regarding the identification experiment variable sensitivities are predominantly made relative to
the response fits validated against the closed-loop validation set NValCL 4 (figure 6.28):
• Looking at the experimental conditions of identification experiments NCL 10 - NCL
16, it is noted that they are respectively precisely the same as those of NCL 1 8 but with larger dither signal magnitudes, effectively giving these experiments
larger signal-to-noise ratios. The effect of increasing the signal-to-noise ratio only
presented itself in the models identified from the experiments where no setpoint
changes were incurred. That is, the difference between experiments NCL 7 and NCL
16. Experiment NCL 16, different to NCL 7 only in the larger signal-to-noise ratio,
generated a model with significantly better response fits when validated against the
closed-loop validation set NValCL 4 in comparison to NCL 7. This further reveals
the importance of setpoint changes in closed-loop identification experiments since
it alleviates the strain off the dither signal in exciting the system. That is, without
setpoint changes, dither signals need to be much more persistently exciting to allow
for the same level of informative data generation.
• With respect to the different types of disturbance signals and their effect on the
accuracy of the identifed model. The White Gaussian signals were expected have
greater capacity to persistently excite non-linear systems (see section 5.3.2), this
was not the case as the model identified from the open-loop experiment which
used white Gaussian disturbance signals, case NOL 1, performed worst of all the
modes when validated against the closed-loop validation set. Looking at the other
models obtained from open-loop experiments, no clear trend regarding the model’s
performance relative to the disturbance signal may be made since there was little
consistency in results between the fits for output y1 and y2 .
• In terms of the models generated from closed-loop experiments and their sensitivity
to the disturbance signals used, the relevant experiments are those of NCL 3 and
NCL 8. Comparing the response fits of the models generated from experiment NCL
3, which used a PRBS disturbance signal, to that of NCL 8, which used a PRMLS
disturbance signal, it is noted that the PRBS generated the slightly better fits.
• Looking at the effect on model response accuracy when not using a dither disturbance signal (case NCL 9) and comparing this to the effect of not implementing
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setpoint changes (case NCL 7), it is found that the dither signal has a smaller contribution in persistently exciting system B. This is as the model identified from case
NCL 7 had a much worse response fit in comparison to that of the model generated
from NCL 9.
• Experiment NCL 5 is characterised as having varied controller parameters and
experiment NCL 6 is characterised as the same condition as NCL 5 but with constrained inputs. It is found from their insignificant difference in response percentage
fit values (specially for output y1 ) relative to each other and to that of case CL 3 (no
controller parameter variance and input constraints) that the variance of controller
parameters and implementation of input constraints had little effect on the extent
of informative data generation.
In addition to the observations made, it must be noted that there was no significant difference found between the validation results obtained from simulation and those
obtained from prediction.
6.4.3
Frequency Content Analysis
In the case of the of system A, the linear system, the frequency analysis presented in
section 6.3.4 was done such that the true model’s frequency response was assessed (bode
plots) and compared to the frequency responses of the identified models. In addition to
this the frequency content of the validation data was compared to the true frequency
response, this allowed for an indication of content of the validation data and the extent
at which the validation set represented the true system. In the case of the frequency
analysis of system B, presented in this section, an assessment is made into the frequency
content of the validation data and the simulated responses when validated against each
validation set.
As was established in the previous section, models generated from experiments where
larger disturbance signals were used had more accurate response fits than when validated
against the validation sets that were generated from similarly large signals, these being
sets NVaOLl 2 and NValCL 5. This condition was interpreted as an indication of the nonlinearity of the system being identified (system B). Figures 6.32(a)-(b) and 6.32(c)-(d)
respectively show the differences in the input signals used to generated validation data
for the open and closed-loop experiments. It clearly confirms the designed experiment
conditions in that the only difference between the signals for the respective open and
closed-loop experiments is the signal magnitude. Figure 6.33 on the other hand shows
the frequency content of the response of the true system, system B, to these input disturbances, these are thus the frequency representations of the outputs generated for the
validation data sets.
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CHAPTER 6. IDENTIFICATION OF SIMULATED SYSTEMS
(a) Frequency Content of u for NValOL 1 and 2
(c) Frequency Content of u for NValCL 4 and 5
1
0
149
1
0
10
10
NValOL 2
NValOL 1
NValCL 5
NValCL 4
-1
-1
10
Amplitude
Amplitude
10
-2
10
-3
-3
10
10
-4
10
-2
10
-4
-2
10
-1
0
10
10
1
10
10
-2
10
Frequency (rad/s)
0
2
0
10
NValOL 2
NValOL 1
NValCL 5
NValCL 4
-1
-1
10
Amplitude
10
Amplitude
10
(d) Frequency Content of u for NValCL 4 and 5
2
10
-2
10
-3
-2
10
-3
10
10
-4
10
1
10
Frequency (rad/s)
(b) Frequency Content of u for NValOL 1 and 2
0
-1
10
-4
-2
10
-1
0
10
10
1
10
10
-2
10
Frequency (rad/s)
-1
0
10
1
10
10
Frequency (rad/s)
Figure 6.32: Comparison of input signal frequency content between normal input range and
larger input range validation sets.
(a) Frequency Content of y for NValOL 1 and 2
(c) Frequency Content of y for NValCL 4 and 5
1
0
1
0
10
10
NValOL 2
NValOL 1
NValCL 5
NValCL 4
-1
-1
10
Amplitude
Amplitude
10
-2
10
-3
-3
10
10
-4
10
-2
10
-4
-2
10
-1
0
10
10
1
10
10
-2
10
Frequency (rad/s)
0
2
0
10
NValOL 2
NValOL 1
NValCL 5
NValCL 4
-1
-1
10
Amplitude
10
Amplitude
10
(d) Frequency Content of y for NValCL 4 and 5
2
10
-2
10
-3
-2
10
-3
10
10
-4
10
1
10
Frequency (rad/s)
(b) Frequency Content of y for NValOL 1 and 2
0
-1
10
-4
-2
10
-1
10
0
10
1
10
10
-2
10
Frequency (rad/s)
-1
10
0
10
1
10
Frequency (rad/s)
Figure 6.33: Comparison of output signal frequency content between normal input range and
larger input range validation sets.
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Analysing the differences between these responses, the first observation is that the
frequency content of responses for the open-loop experiments (figures 6.33(a)-(b)) are
less distinct than those of the closed-loop experiments (figures 6.33(c)-(d)). That is,
the effect of disturbing system B with a larger input signal, showed very little effect on
the frequency content of the response for the open-loop experiments. Looking at figures
6.33(c)-(d) it is observed that the frequency content of the responses obtained from the
closed-loop experiments are slightly more distinct from each other. This signifies that the
increase in disturbance signal magnitude under closed-loop conditions produced a more
distinct response than under open-loop conditions.
These findings imply that the validation sets obtained from closed-loop experiments
should do a better job at discriminating between models based on the range of data
that each model was identified from. Recalling the response percentage fit results for
the identified models when validated against the open-loop data sets and the closed-loop
data sets given by figures 6.27 and 6.28 respectively, this is exactly what is seen. This
is best noted when comparing the response fit results when validated against the openloop validation set NValOL 1 to those when validated against closed-loop set NValCL
4. As has been discussed before, the dips in the response fit profiles the fits of models
obtained from experiments that used larger signal disturbance signals in comparison to
the others. The corresponding experiments are NOL 4, NCL 4 and NCL 13. It is noted
that these dips are more pronounced when looking at the profile when validated against
the closed-loop set NValCL 4.
From this apparent improved ability to discriminate between models based on the
range of data that they were identified from, it may be stated that the closed-loop experiments did a better job at generating data that better discriminated between models
based the region of data that they were identified from and consequently how well they
approximate the non-linear dynamics presented in the validation set.
6.4.4
Residual Correlation Analysis
The residual correlations from the identified models approximating the non-linear system
(system B) were assessed in the same manner as was done for the linear system (system
A) in section 6.3.6. Correlation tests were done against all the validation sets besides
that generated from the open-loop step response, NValOL 3. In addition to this, higher
order correlation test were implemented on the identified models in efforts to obtain
higher levels of model discrimination based on how well they approximate the non-linear
dynamics of system B.
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Whiteness Test Results
The primary observation made while testing for whiteness was that all the models generated showed large extents of residual correlations regardless of the validation set used.
Neither of the validation sets obtained from open or closed-loop experiments could be attributed with showing for better model discrimination by producing different correlation
profiles for the different identified models.
The residual correlation profiles of each model when validated against the closedloop validation sets (NValCL 4 andOL 5) almost all resembled that shown in figure 6.34
revealing the results for the model identified from experiment NCL 8.
When validated against the open-loop validation set the residual correlation profiles
obtained for most of the models differed very little from that presented in figure 6.34.
Some however did seem to indicate slightly different residual correlation profiles. These
were those of the models obtained from experiments NOL 6, NCL 7 and NCL 16. With
experiment NOL 6 being characterised as the open-loop step disturbance experiment,
NCL 7 and NCL 16 being characterised as the closed-loop experiments where no setpoint
changes were imposed with NCL 16 having the larger signal-to-noise ratio.
Figure 6.35 illustrates the residual correlation plots for the model obtained from the
open-loop step disturbance experiment, NOL 6, validated against open-loop validation
sets NValOL 1 and NValOL 2. It is interesting to note that validation set NValOL 1 did
not reveal any difference in residual correlations for output y1 to that obtained from set
NValOL 2 (figure 6.35(a)). In contrast to this a slight difference between the residual
correlation profiles obtained between the two validation sets was revealed for output y2
(figure 6.35(b)). This suggests that the two validation sets detected the same extent of
unmodelled dynamics for output y1 but different extents of unmodelled dynamics for y2 .
This is also seen in figure 6.34. Given that the model being tested was obtained from an
open-loop step disturbance experiment, this attests to the step disturbance input signal’s
ability to show different dynamic responses for output y2 .
It must be said that while some slight differences in residual correlations where found,
these differences were very small. Almost all of the residual correlation profiles obtained
were very similar to each other, regardless of the validation set used. Thus the whiteness test shows that all the identified models had large extents of unmodelled dynamics.
Furthermore the lack of distinction in residual correlation profiles indicates that all the
validation sets did a bad job at discriminating between the models and consequently could
not verify whether some models better approximated the non-linear system (system B)
than others.
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Residual Auto-Correlation for ε against NValCL 4 and 5
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Figure 6.34: Residual correlation results for case NCL 8 using NValCL 4 and NValCL 5 as
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Residual Auto-Correlation for ε against NValOL 1 and 2
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Figure 6.35: Residual correlation results for case NOL 6 using NValOL 1 and NValOL 2 as
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Independence Test Results using Closed-Loop Validation Sets
In testing for the independence between inputs and residuals for the different models
generated by the different experiments and validated against the different validation sets,
it was noted that the results closely reflected the findings for the whiteness tests. All
the models generated showed high levels of residual-input correlations regardless of the
validation data set used. In addition to this it was found that most correlation profiles
changed very little between cases when validated against the different validation sets.
This suggests all the models showed large extents of incorrectly modelled dynamics and
that the validation sets did little to discriminate between most of the models based on
differences in the incorrectly modelled dynamics.
Figures 6.36(a), (b) and (c) (left column) show the input-residual correlations for the
models obtained from open-loop experiments NOP 1, 3, 5 and figures 6.36(d), (e) and (f)
(right column) show correlations of those obtained from closed-loop experiments NCL 1,
2, 11 between residual 1 and input u1 validated against the two closed-loop validation sets
(NValCL 4 and 5). While there are some small differences in the input-residual correlation
profiles, the figure clearly illustrates the point made regarding the little difference between
profiles when validated against the closed-loop data. To be more specific however, there
are differences in correlation profiles between models identified from open-loop cases (left
column) and those identified from closed-loop cases (right column) but within the models
generated under the same feedback condition there isn’t much to discriminate between.
This suggests that all the models obtained from the same feedback condition contain the
same incorrectly modelled dynamics.
While it has been established that there is little to discriminate between the correlation
results of most models when validated against the closed-loop data, figure 6.37 shows the
correlation profiles of the models that did distinctly show different correlation results
against the closed-loop validation sets. It is interesting to note that even though the
correlatin profiles presented here are different to the general trends they are very similar
to each other. These correlation profiles are those of the models obtained from the openloop step disturbance experiments (NOL 6 - figure 6.37(a)) and the two experiments
generated from closed-loop experiments where no setpoing changes were incurred (NCL
7 and 16 - figure 6.37(b) and (c)). Recall that the only difference between NCL 7 and
16 it that NCL 16 had a larger signal-to-noise ration. Note how the lack of distinction
in correlation profile suggest that the difference in signal-to-noise ratio did not affect the
extent of incorrectly modelled dynamics.
It is at this point appropriate to recall that input-residual correlations may not be
completely trusted since some correlations may be due to the feedback loop being closed
and not so much due to incorrectly modelled dynamics.
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Figure 6.36: Input-Residual correlation between u1 and 1 , validated against closed-loop data,
for some of the cases showing similar results
Figure 6.37: Input-Residual correlations between u1 and 1 , validated against closed-loop
data, for the cases showing results different correlations to the general trend
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Independence Test Results using Open-Loop Validation Sets
The residual-input correlation results obtained when using open-loop validation sets indicated the open-loop validation sets’ slightly better ability to discriminate between models
based on incorrectly modelled dynamics through input-residual correlation analysis.
Figures 6.38 and 6.39 are those very same cases assessed against the closed-loop
validation sets in figures 6.36 and 6.37 but validated against the open-loop validation sets.
Looking at figure 6.38, it is first observed that the magnitude of the correlations is smaller,
additionally it is evident that the models generated from the open-loop experiments are
better discriminated between when validated against the open-loop validation sets than
the closed-loop sets (shown in figure 6.36(a)-(c)). The correlations of models obtained
from closed-loop cases remain however indistinguishable.
Comparing the results shown in figure 6.39 to 6.37, the model obtained from open-loop
step responses, case NOL 6 figure (6.39(a)), shows completely different correlations when
validated against the open-loop sets to that obtained when validated against the closedloop validation sets. The same extent of incorrectly modelled dynamics with respect
to the two open-loop validation sets is revealed in contrast to two different correlation
profiles in figure 6.37(a).
Figure 6.38: Input-Residual correlation between u1 and 1 , validated against open-loop data
The results of cases NCL 7 and 16 (figures 6.39(b) and (c)), revealed correlation
profiles very similar to the general profiles obtained from models generated from closed-
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Figure 6.39: Input-Residual correlations between u1 and 1 , validated against open-loop data
loop experiments (figures 6.38(d), (e) and (f)). In contrast to the correlations obtained
when using the closed-loop validation sets, it is implied that the open-loop validation sets
did not detect different extents of incorrectly modelled dynamics from these to models
in comparison to the other models. However, in a similar fashion as the closed-loop
validation sets, the open-loop validation sets did not show an distinction in correlations
that suggests different dynamics identification due to different signal-to-noise ratios.
Independence Test Results - Other Observations and Comments
In investigating the differences in input-residual correlations between the different pairs
of residuals and inputs (u1 − 1 , u2 − 1 , u1 − 2 , u2 − 2 ). It was observed that all but one
of the models generated results with the correlation trends being near identical between
the different residuals and the same input (eg. u1 − 1 and u1 − 2 ) but dissimilar between
the different inputs and a residual (eg. u1 − 1 and u2 − 1 ).
Figure 6.40 shows all the input-residual correlations for the model identified from
experiment NOL 5 illustrating this point. In contrast, the only model to show correlation
result trends that were very similar for all input-residual pairs was that obtained from
experiment NOL 6, shown in figure 6.41, characterised by the open-loop step disturbance
experiment. The same was observed when using the open-loop validation set.
This observation is difficult to explain. Its implication is that the open-loop step
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Figure 6.40: Input-Residual correlations for case NOL 5 showing no common trend between
correlations
Figure 6.41: Input-Residual correlations for case NOL 6 showing common trends between
correlations with between residuals and the same inputs
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disturbance experiment, NOL 6, was the only experiment to generate a model with the
same incorrectly modelled dynamics for both outputs while all the other models had
incorrectly specified dynamics dependent on the output.
Higher Order Correlation Results
As explained in section 4.2.2, the higher order correlation tests are expected to provide
a higher order of model discrimination since each test discriminates the model according
to a different type of non-linear characteristic. Application of these tests to the identified
models and validated against the different validation sets gave some interesting results.
It is appropriate at this point to refer to table 4.1 which indicates the significance
of the correlation results. With the previous assessment into the correlations between
inputs, u, and the model residuals, , showing that all the cases have non-zero inputresidual correlations (Ru (τ ) 6= 0), it is evident that Ru2 2 will be non zero for all cases
(as per table 4.1), with only Ru2 to test for indications of whether the entire model is
inadequate or whether the model is only inadequate at representing the odd powered
non-linear dynamics.
Figure 6.42: Higher order correlation: Ru2 2 , of a select group of cases validated against closedloop data
Figure 6.42 shows the higher order cross-correlation, Ru2 2 , of representative models
validated against closed-loop validation sets. It is observed that the results are non-zero
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Figure 6.43: Higher order correlation: Ru2 , of a select group of cases validated against closedloop data
Figure 6.44: Higher order correlation: Ru2 , of a select group of cases validated against openloop data
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for all the cases and that the two closed-loop validation sets differently discriminated
between the models. Figure 6.43 shows the results of Ru2 similarly validated against the
closed-loop validation sets, it is noted that while there are significant correlations when
using validation set NValCL 5, the correlations are slightly less pronounced when using
NValCL 4. This suggests that validation set NValCL 5 finds the models generated by
these cases to be completely inadequate while NValCL 4 finds the models to be inadequate
more so in accurately modelling odd non-linearities. This is as per the rules given in table
4.1. From this one may deduce that the two different validation sets discriminate between
models on different terms.
When assessing the higher order correlations using open-loop data for the same cases,
it is found that the correlations are not as pronounced. Figure 6.44, shows the results for
the Ru2 assessments, the results obtained for Ru2 2 show very similar profiles. While the
results show smaller correlation magnitudes in comparison to the correlations obtained
when using closed-loop validation sets determining whether they are small enough to be
considered negligible is difficult
From these results it must be said that both higher order correlations tests, Ru2 and
Ru2 2 , revealed significant correlations with the results obtained from the open-loop validation sets being slightly debatable. When using the closed-loop validation sets these
correlations where larger in magnitude and more distinct correlation profiles were produced for each model than when using the open-loop validation sets. This implies that
the closed-loop validation sets better discriminated between models and indicated larger
extents of incorrectly modelled dynamics.
However, since the closed-loop validation sets are of questionable accuracy in inputresidual correlation tests due to the inherent correlations caused via the feedback condition, the results may not be entirely trusted. Looking at the results obtained when
using the open-loop validation sets, it must be said that all the models did exhibit correlations (although small in magnitude) implying that the models are inadequate in both
the noise model and the dynamics model. Furthermore, the open-loop validation did not
produce higher order correlation results for each model that were very distinct form each
other. This means that the higher order correlation tests did not improve the ability to
discriminate between models or give insight into how well each model approximated the
non-linear system.
6.4.5
Results Overview
The results and findings concerning the identification and validation efforts on system B,
the non-linear system, may be summarised as follows:
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Model Variance and Uncertainty
Variance due to Noise Sensitivity
• Models generated from closed-loop experiments where no setpoint changes were
incurred were found to be most sensitive to random realisations of noise.
• Increasing the signal-to-noise ratio proved to reduce this model parameter sensitivity to noise. This was however only noticed in the closed-loop experiments that did
not induce setpoint changes.
Model Uncertainty
• The model that revealed the largest measure of model uncertainty was that obtained
from the closed-loop experiments where no setpoint changes were incurred.
Simulation and Prediction
General Observations:
• Response percentage fit results showed that all the models were very sensitive to
the validation set used.
• Closed-loop validation sets showed indications of allowing for better discrimination
of models than the open-loop validation sets since they provided larger variances in
response fits amongst models and outputs (y1 and y2 )
Bias and Discrimination by PRBS Disturbance Validation Sets - Disturbance Signal Magnitude Sensitivities:
• The disturbance signal magnitude proved to be a very important variable.
• It was found that validation sets produced by disturbance signals of certain magnitudes were extremely biased towards models that were generated from disturbance
signals of similar magnitudes. This was thought to reflect the non-linear nature of
the dynamics of the system being identified
• The accuracy of response fits of output y2 was found to be the most sensitive of
the two outputs to disturbance signal magnitudes. This was thought to be due to
output y2 exhibiting larger extents of non-linear dynamics.
Bias and Discrimination by Step Disturbance Validation Sets - Disturbance Signal Magnitude Sensitivities:
• Validating the response fit results against the step disturbance validation set revealed the step response validation set to be extremely biased towards models identified from experiments that used disturbance signals with similar magnitudes.
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• The step disturbance validation set allowed for the confirmation that the increased
sensitivity of response fits of output y2 to disturbance signal magnitudes may be
attributed to larger extent on non-linear dynamics associated with output y2 .
• This validation set further revealed how using different disturbance signal magnitudes generates different response dynamics and how this can affect the validation
of a model.
• This further emphasised importance of validating models against validation sets
containing the dynamic behaviour most representative of the models purpose or intended use, especially when approximating a non-linear system with a linear model.
Other Experiment Condition Sensitivities:
• The closed-loop validations sets were found to better discriminate between models
based on differences in identification experiment conditions used to generate the
models.
• The changes in signal-to-noise ratio were only found effective in closed-loop experiments where no setpoint changes were incurred. This shows the importance of
setpoint changes in closed-loop experiments.
• From investigating the effect of removing disturbance signals and not inducing
setpoint changes, the greater value in setpoint changes was further established in
terms of its contribution to persistently exciting the system.
• PRBS disturbance signals were found to be most consistent for closed-loop identification efforts while no disturbance signal type given open-loop conditions could
clearly be stated as being best.
• No clear trend was observed in the effects of varying controller parameters and
implementing input constraints.
Frequency Content Analysis
• Analysing the frequency content of the validation data sets it was found that the
closed-loop validation sets were better suited for model discrimination based on the
magnitude of the disturbances signals used to generate the data from which the
models were identified than the open-loop validation sets.
• Since it was been established that disturbance signal magnitude greatly affects the
dynamics revealed by the non-linear system, the previous point implies that the
closed-loop validation sets are better at discriminating between models identified
from different dynamic responses from the non-linear system.
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• These finding, that the frequency characteristics of the closed-loop validation data
sets indicate them to better discriminate models, were found to reflect the simulated
response results.
Residual Correlation Analysis
Whiteness Test:
• Large residual correlations were found for all the models regardless of the validation
set used. This implies large extents of unmodelled dynamics.
• In addition to this the correlation profiles were mostly similar for all the models.
This indicates that the validation sets did little to discriminate between models
based on differences in unmodelled dynamics.
Independence Test Results Using Closed-Loop Validation Sets
• In a similar fashion as the whiteness test, all the models showed large input-residual
correlations indicating large extents of incorrectly modelled dynamics.
• It was found that the closed-loop validation sets only discriminated between models based on the feedback condition of the experiment from which the model was
identified. Otherwise little discrimination was evident.
Independence Test Results Using Open-Loop Validation Sets
• The input-residual correlation results obtained when validated against the openloop validation sets suggest the open-loop validation sets slightly better discriminated between models than the closed-loop validation sets.
• The input residual correlation tests could not detect any differences in modelled
dynamics that may be attributed to larger or smaller signal-to-noise values.
Higher Order Correlation Tests
• The higher order correlation tests did not produce a greater capacity to discriminate
between models based on how well the models approximated the non-linear system.
• When using closed-loop validation sets, higher order correlation results did indicate
a greater capacity to discriminate between models relative to previous validation
assessments, however, the results could not be trusted due to correlations created
via the feedback that are indistinguishable from correlations due to incorrectly
modelled dynamics.
• The results obtained when using the open-loop validation sets where inconclusive
in that the significance of correlation magnitudes were difficult to assess. The
correlations did however indicate that there open-loop validation sets did little to
discriminate between models.
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164
Discussion
This section aims to discuss the findings of the identification and validation efforts on
systems A and B relative to each other.
6.5.1
Identification
Determining which experiments generated better approximations of system A, the linear
system, was easier than for system B, the non-linear system. This was primarily due to
the added intricacies in validating the non-linear system approximations.
With respect to disturbance signal characteristics, it can generally be said that the
with the benefits or improvements in model accuracy from using signals other than PRBS
being relatively little to none for both open and closed-loop experiments, it must be stated
that the simple nature of PRBS disturbance signal is most applicable. This however is
only the case when the frequency characteristics of the signal are near to that of the
dominant time constant of the system
In terms of the effect of the feedback condition, both the linear and non-linear system
analyses showed how the accuracy of the models generated from open-loop conditions
were more sensitive to the properties of the disturbance signals used. While the accuracy
of the models generated from closed-loop data was not so sensitive to the disturbance
signal properties. This was thought to be accounted for by the fact that the feedback
loop dampens the effects of the disturbance signals consequently dampening the effects
of varying signal characteristics.
From this it must be said that this implies that the models identified from closedloop experiments are less sensitive to signals that are less persistently exciting or badly
designed in terms of frequency characteristics, however such models are also less capable
of producing more accurate models from better designed experiments. Thus closed-loop
experiments are found to allow for models that perform slightly worse but are more
robust, while open-loop experiments can produce models that perform better but are less
robust in terms of sensitivities to badly designed identification experiments.
6.5.2
Validation
Comparing the model cross-validation findings, specifically the effect of experimental
conditions on bias and discrimination characteristics of the validation sets, is not easily
done between the efforts on the linear system (system A) and non-linear system (system
B).
In general it may be stated that open-loop experiments produced validation sets that
better discriminated between models for the linear system while the closed-loop validation
set was found to best represent the system A. With respect to the non-linear system the
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validation sets obtained from closed-loop experiments were found to be most useful in
discriminating between models.
In terms of the different validation techniques, it was found for both the linear and
non-linear systems that response fit validations were the most sensitive to the differences
between validation data thus making the results extremely relative to the validation
set used. The application of residual correlation analysis on model approximating the
non-linear system was found to be very ineffective. While when applied to the models
approximating the linear system the results did contribute to better understanding on
the discrimination abilities of the validation sets based on different extents of unmodelled
dynamics.
It was quickly found from the linear system that the validation sets can seem to
be biased towards models obtained from similar experimental conditions as the models.
Validation sets obtained from closed-loop experiments seemed biased towards models obtained from closed-loop experiments. The same can be said about the other validation
sets. This was found to become much more complex when validating linear approximations of the non-linear system. Primarily since the disturbance signal magnitude and
dynamic region effects are introduced into the bias and prejudice of the validation sets.
From these issues it was make very evident that the question may not necessarily
be how representative is the validation set of the true system and is it discriminating
between models based on deviations from dynamic representation of the true system,
but how representative is the validation set of the intended use of the model. This is
especially the case if the model is of a different structure and order of the true system.
6.5.3
Other Issues
An important issue, while difficult to resolve but important to present, is that determining
whether these findings are characteristic results of identification and validation efforts of
linear systems and non-linear systems or characteristics of these specific systems being
identified. These results may not be attributed to being absolute findings, that is, while it
was evident that closed-loop experiments produced models that were less likely to produce
completely inaccurate responses than the open-loop experiments, this is not necessarily
a characteristic of closed-loop experiments but perhaps of the system being identified.
When assessing how sensitive each experiment and the respective identified models
were to noise in sections 6.3.1 and 6.4.1, for the linear and non-linear system respectively,
it was found that the experiment characterised by open-loop step disturbances was most
sensitive for system A while the closed-loop experiments where no setpoint changes were
incurred were most sensitive for system B. This means that the data generated from these
experiments was not informative enough. Consequently the estimation procedures that
used these data sets were more susceptible to modelling the noise, thus the identification
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sensitivity to random noise realisations.
This is very closely related to the fact that ARX model structures were used and how
these structures are prone to dynamic model corruption by noise due to an inability to
independently model noise and dynamics. Recalling that literature states that closed-loop
experiments are meant to be more susceptible to this problem. This makes it interesting
that the assessments on the linear system found the dynamics model obtained from an
open-loop experiment to most sensitive to noise. While assessments on the non-linear
system found the dynamics models obtained from closed-loop experiments to be most
sensitive. This might indicate that the greater sensitivity of closed-loop experiments to
noise might be more so when attempting to approximate a non-linear system.
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CHAPTER 7
Identification of a Distillation Column
In the previous chapter, known mathematical models and their simulated responses
to simulated disturbances allowed for an investigation into aspects of identification
experiment design variable sensitivities and cross-validation effectiveness. This
chapter aims to extent this investigation to a ’real’ system, a pilot scale distillation
column. A smaller scope of experimental designs, based on the findings of the
previous chapter, was used to generate data from the column for identification
and validation.
7.1
7.1.1
Investigative Approach and System Description
Investigative Approach
The investigative approach of the work presented in this chapter aims to study the same
concepts as those presented in the previous chapter but extended to a ’real’ system. That
is, the investigation into system identification and validation is now focused on a ’real’
system in search of pragmatic issues and difficulties that would not have been encountered
up to this point due to the controlled environment of simulated investigations. Thus the
investigative approach may be stated as follows:
• Investigate how accurately the ’real’ system may be identified using identification
experiments and the sensitivity of such accuracy to disturbance signal and feedback
conditions.
• Cross-validate the identified models using validation sets obtained from various experimental conditions. The focus is again made on the effects of varying disturbance
signals and feedback conditions.
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• Assess the results obtained from the different validation sets so as to obtain indications of how the different experimental conditions affect the identified model
accuracy and validation set discrimination.
These objectives are pursued through the use of a pilot scale distillation column with
unknown dynamics. The direct experimental objectives of this chapter may thus be
further refined as follows:
1. Study the pilot scale distillation column, and establish an operating range which is
to be identified. This range is to be established in consideration of normal operation
of the distillation column.
2. Design identification experiments to generate data sets around this established region of operation under different experimental conditions, with a primary focus on
varying feedback conditions.
3. Design experiments to generate data sets around and beyond the established region
of optimal operation to serve as validation data sets.
It is important to note that objective number 1 above distinguishes the identification efforts on the distillation column from those on the simulated systems. This point implies
that a specific output range is to be identified and that consequently both the open and
closed-loop experiments are output constrained. This is unlike the identification experiments on the simulated systems where open-loop experiments were input constrained and
closed-loop experiments were mostly only output constrained. This approach is taken in
view of the results obtained regarding identification of the non-linear system in the previous chapter, system B. That is, it was found that clearly establishing the range over
which a model is to approximate a non-linear system, is critical.
The following sections describe the distillation column used and further detail the
identification and validation experimental conditions.
7.1.2
Description of the Pilot Scale Distillation Column
The distillation column used, represented by the detailed diagram presented in figure 7.1,
is a ten plate glass distillation column which is not insulated, approximately 5 meters
tall and 15 centimetres in diameter. The column is run near atmospheric pressure and is
used to separate mixtures of water and ethanol. As figure 7.1 indicates, the column has
two feed points, one above plate 7 and the other above plate 3. Furthermore, the system
is a closed system in that the distillate and bottoms products are mixed and fed back
into the system.
The column boiler is fed with saturated steam from electric boilers with a supply
pressure between 600 and 700 kPa (gauge) and is throttled down to 40 - 100 kPa (gauge).
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To Drain
Condenser
T
F
T
Surge Tank
L
F
T
Column
Cooling Water
F
T
1
F
F
T
2
Feed 2
F
T
3
F
T
4
T
5
Section of Column
Internals
T
6
Feed 1
T
T
7
F
T
8
T
9
T
10
T
Cooling Water
Inlet
Mixing
Motor
F
Reboiler
To Drain
F
T
T
Mixing Tank
T
F
F
From Steam
Supply
To Drain
Bottoms
Cooler
F
P
F
Steam Trap
Feed Tank
Regulator
Feed Pump
Figure 7.1: Detailed illustration of the pilot scale distillation column
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
170
The throttling is done by a pneumatic control valve. The condenser, which is made
of glass, is fed with municipal water at ambient temperature. The reflux drum level,
measured by means of differential pressure of the liquid head, supplies the pressure head
for the reflux and distillate streams.
Equipment and Instrumentation Description
Figure 7.2 illustrates the piping and instrumentation of the column. The following may
be said regarding the instrumentation and measurements:
• Pneumatic control valves are placed on all the streams entering and leaving the
column.
• Temperature measurements are taken from Foundation Field Bus instruments on
each plate as well as on all feed and product streams.
• Gauge pressure measurements are taken for the steam in the boiler after throttling.
• Level measurements are taken for the boiler and reflux drum from pressure differential measurements of the liquid head.
• Mass flowrate measurements are taken for the boilup, bottoms product, reflux,
cooling water, feed and top products streams via Micro Motion Coriolis flowmeters.
• A gauge pressure measurement is made at the top of the column
It is important to note that the flowrate measurement devices cannot accurately measure any flowrate smaller than 3 kg/hr. In addition to this, on occasions when the steam
pressure is increased very quickly, the pressure in the column increases momentarily, this
causes the pressure in the reflux drum and boiler to increase causing the values of the
pressure differentials used to infer level to increase momentarily.
All the instrumentation and equipment are connected and operated via a DeltaV distributed control system. It is important to note that the sample rate of all the measuring
devices is once every second and that all data is stored in a DeltaV continuous historian.
Operation and Operating Ranges
A key issue in operating the distillation column is maintaining the top plate temperature
above the water-ethanol mixture azeotrope which is at 78o C. This introduces constraints
or limits on the amounts of energy addition and removal from the system via the boiler and
condenser. Given that the feed is sub-cooled as it is fed from a tank kept at atmospheric
conditions, the energy introduced via the boiler is closely related to the feed flowrate.
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
171
TI
16
PL-005-G-65 mm
FI
06
FT
06
TI
15
TT
15
TT
16
HX-03
CV-07
PL -006-SS-40mm
Cooling water
TIC
14
PT
02
Drain
LT
02
DM-01
PL-007-G-35mm
PL-002-G-35mm
Plate 3
CV-02
TT
12
TC
02
VL-01
FIC
01
TT
13
PL -001-G-35mm
Plate 6
CV-01
CIC
01
TT
02
TI
TT
03
TI
03
TT
04
TI
04
TT
05
TI
05
TT
06
TI
06
TT
07
TI
07
TT
08
TI
08
TT
09
TI
09
TT
14
PL-005-G-33mm
TT
01
FIC
02
CV-03
02
FT
03
LIC
02
FI
03
S-87
AV-03
PL-004-G-35mm
PL -007-G-35mm
SV-05
FI
04
FT
04
CV-04
fresh feed point
TT
10
PL-014-G-66mm
PL-014-G-35mm
TI
10
CV-06
LT
01
T
011-SS-25mm
011-SS-25 mm
PL-013-SS-72mm
AV-06
AV-05
AV-04
LIC
01
PL-004-G-35mm
PL-009-G-35 mm
FI
05
To atm
PT
01
M
HX-02
CV-05
PL-009-G-35 mm
PL-010-G-35 mm
HV-02
TT
11
PL-012-G-35 mm
FI
08
TT
17
DM-02
SV-03
FT
07
HX-01
AV-02
TI
18
To drain
PC
01
AV-01
TT
18
TI
17
HV-01
PL-001-G-35mm
PL-002-G-66mm
Steam supply
PL-008-G-35mm
Vent
FI
07
CV-08
Drain
Cooling water
DM-03
FI
PL-003-G-35mm
PL-003-G-35mm
SV-02
SV-01
PC-01
Figure 7.2: Diagram showing piping and Instrumentation of the pilot scale distillation column
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
172
C
LIC
02
L
D
FIC
03
T1
FIC
01
F
PC
01
T10
R
S
LIC
01
B
Figure 7.3: Simplified illustration of relevant control loops
Table 7.1 indicates the normal operating ranges of the different variables concerned
with the operation of the column. Note that all the pressure values indicated are gauge
pressure values.
The column is almost always operated with only one feed, typically the one feeding
onto plate 7. While it has been stated that the system is a closed system in that the
products are mixed and recycled back into the system via the feed tank, it has been
established that given the feed tank has a surplus volume of feed, the effects of varying
conditions or properties of the recycle streams (composition, temperature etc.) on the
feed stream are negligible. Thus for all purposes of this work the feed stream is considered
independent from the product streams.
Control and Dynamic Behaviour
The base layer control setup is primarily defined by the following control loops as illustrated in figure 7.2 and more clearly shown by figure 7.3:
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
Variable
Label
Units
Normal Range
Steam Pressure
Cooling Water Valve
Reflux Level
Boiler Level
Bottoms Flowrate
Distillate Flowrate
Reflux Flowrate
Feed Flowrate
Feed Temperature
Feed Composition
Top Plate Temperature
Bottom Plate Temperature
PS
XC
LR
LB
FB
FD
FR
FF
TF
CF
TP
TB
kPa
% Open
kPa
kPa
kg/hr
kg/hr
kg/hr
kg/hr
o
C
% mol
o
C
o
C
55-90
50-100
70-85
24-30
0-30
0-30
0-20
0-40
20-38
40-60
77-82
88-96
173
Table 7.1: Distillation column principle variables and normal operation ranges
LIC-01 : The boiler level is controlled by directly manipulating the bottoms product
stream valve.
LIC-02 : The relux drum level is controlled by directly manipulating the distillate product stream valve.
FIC-01 : The feed flow rate is controlled my manipulating the valve in this stream.
FIC-03 : This is a cascade control loop. The secondary loop is that which controls the
reflux flowrate by adjusting the valve in the reflux stream while the primary loop
controls the top plate temperature by adjusting the setpoint reflux flowrate value
for the secondary loop.
PC-01 : This is a cascade control loop with the secondary loop controlling the steam
pressure by manipulating the throttling valve in the steam feed line while the primary loop controls the bottom plate temperature by adjusting the setpoint steam
pressure value for the secondary loop.
These control configurations were chosen as they were found to perform better than
any other. While it is understood that typical distillation control theory dictates that
the condenser cooling water flowrate is a valuable manipulated variable as it allows for
reflux temperature control, early controllability studies found the effect of this variable to
saturate at a very low flowrate. This made it very ineffective as a manipulated variable.
All the control loops used PID controllers with most of the parameter tuning done by
trial and error and the occasional assistance by the DeltaV Tune Tool Kit. Control loop
FIC was the only loop of the five to use differential control action.
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
174
It must be said that improving the performance of the different controllers was a
difficult task. This problem was largely rooted in the feed pressure oscillations caused by
the electric boilers. The supply pressure before throttling was found to oscillate between
650 kPa and 700 kPa at a frequency of 0.18 Hz, the controller adjusting the throttling
valve in order to control the steam pressure in the boiler was unable to regulate these
disturbances. This consequently resulted in the steam pressure in the boiler oscillating by
5 kPa at a similar frequency. Figure 7.4 illustrates the performance obtained by the steam
pressure controllers. Figure 7.4(a) shows the achievable steam pressure control while
figure 7.4(b) shows the primary controllers efforts in manipulating the steam pressures
setpoint to control the bottom plate temperature.
(a) Control of steam pressure
85
Steam Pressure (kPa)
P
S
80
P setpoint
S
75
70
65
60
0
200
400
600
800
1000
Time (s)
1200
1400
1600
1800
(b) Manipulation of steam pressure setpoint for cascade control of T
B
Steam Pressure (kPa)
95
90
85
80
P
S
75
P setpoint
S
70
0
200
400
600
800
1000
Time (s)
1200
1400
1600
1800
Figure 7.4: Steam presssure control and manipulation
The effects of the pressure supply oscillation extended well beyond the boiler steam
pressure. Figure 7.5 shows the top and bottom plate temperature responses to steps in
reflux flowrates and steam pressure with the primary cascade loops open. Here the control
problem and multivariable nature of the column reveals itself in that the sensitivity of
the top plate temperature to the oscillations varies according to region of operation and
magnitude of system variables. Figure 7.5(a) shows a TP response to a 50 % step in reflux
flowrate magnitude, it is observed that the trend is smoother than when responding to a
50 % step in boiler steam pressure, 7.5(b).
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
(a) Response to 50% step in F
(b) Response to 50% step in P
R
S
81.5
Top Plate Temperature ( oC)
Top Plate Temperature ( oC)
79.5
79
78.5
78
77.5
77
76.5
76
0
100
200
300
Time (s)
400
81
80.5
80
79.5
79
500
0
(c) Response to 50% step in F
1500
S
Bottom Plate Temperature ( oC)
Bottom Plate Temperature ( oC)
1000
(d) Response to 50% step in P
R
94
93.5
93
92.5
100
500
Time (s)
94.5
92
0
175
200
300
Time (s)
400
95.2
95
94.8
94.6
94.4
94.2
94
500
0
500
1000
1500
Time (s)
Figure 7.5: Open-loop step response
(a) P Maniputlation for T control
S
(b) F Maniputlation for T control
B
R
90
80
70
60
0
1000
2000
3000
Time (s)
4000
10
8
6
4
2
0
0
5000
1000
(c) T Control
Bottom Plate Temperature (oC)
Top Plate Temperature (oC)
95
94
93.5
T
B
T setpoint
B
92
0
1000
2000
3000
Time (s)
4000
5000
4000
5000
P
94.5
92.5
2000
3000
Time (s)
(d)T Control
B
95.5
93
P
12
Reflux Flowrate (kg/hr)
Steam Pressure (kPa)
100
4000
5000
80.5
80
79.5
79
78.5
T
P
78
T setpoint
P
77.5
0
1000
2000
3000
Time (s)
Figure 7.6: Distillation column setpoint tracking
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
176
This phenomenon is seen again in the closed-loop responses in figure 7.6, where the top
plate temperature control (figure 7.6(d)) shows much larger oscillations when controlled
at 80 o C than at 78 o C. This dynamic behaviour is most likely due to fact that at larger
reflux flowrates the effects of the steam pressure oscillations are dampened. This type
of dynamic interaction between system variables and control loops made it difficult to
control for robustness and performance and was made more evident when operating the
column over larger temperature ranges.
It is clear that the column dynamics are such that the system variables display varying
dependencies to each other at different magnitudes. These observations reflect the nonlinear nature of the column. Higher order correlation tests for non-linear dynamics (as
done in the previous section, figure 6.5) could not be conducted as disturbing the column
under open loop conditions with white Gaussian disturbance signals resulted in extremely
unstable conditions.
Furthermore, it is clear that the column has several control problems. The use of controller gain scheduling would do very well in linearising the controller and system variable
interactions. An MPC controller would do even better and feed forward logic may have
been used to reduce the problems caused by the pressure supply fluctuations. However,
the problems that these solutions might resolve are characteristic of ’real’ systems and it
is with this perspective that the column is operated and identified.
7.2
7.2.1
Experimental Method : Framework and Design
System and Identification Problem Definition and Assumptions
As has been described in recent sections, the distillation column being used has 12 principal variables and 5 control loops all interacting and affecting each other. In defining the
identification problem presented for the distillation column, the key variables are defined
as follows:
Input 1 - u1 : Reflux flowrate - FR
Input 2 - u2 : Boiler steam pressure - PS
Output 1 - y1 : Top plate Temperature - TP
Output 2 - y2 : Bottom plate temperature - TB
This implies a 2×2 multivariable structure defines the distillation column to be identified
which is justified given the following assumptions:
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
Variable
Label
Units
Steady State Value
Steam Pressure
Cooling Water Valve
Reflux Level
Boiler Level
Bottoms Flowrate
Distillate Flowrate
Reflux Flowrate
Feed Flowrate
Feed Temperature
Feed Composition
Top Plate Temperature
Bottom Plate Temperature
PS
XC
LR
LB
FB
FD
FR
FF
TF
CF
TP
TB
kPa
% Open
kPa
kPa
kg/hr
kg/hr
kg/hr
kg/hr
o
C
% mol
o
C
o
C
75
63
79
27
8
15
7
20
36
50
79
94
177
Table 7.2: Steady state values used as initial states of experiments
• The other principal system variables not included in the model structure are the
reflux drum level, boiler level, feed properties and environmental conditions. These
variables will be maintained constant at relevant magnitudes (given in table 7.2)
by the appropriate controllers. The controllers for most of these variables perform
very well; it is thus assumed that there is no controller interaction through these
variables that adversely affect the variables in the model structure.
• Effects of recycle streams on the feed are dampened by a large feed surplus and are
thus assumed negligible.
• Both system inputs - FR and PS - are regulated by control loops manipulating the
reflux line valve and the steam supply line valve respectively to obtain the specified
setpoint values. It is understood that due to the steam pressure control difficulties
this input is to an extent continuously disturbed.
• It is assumed that the top and bottom plate temperatures can be used to infer composition values of the distillate and bottoms streams respectively. This is however
not expected to be so at temperature measurements bellow the azeotrope at 78 o C.
From this it is possible to define the condition and dynamic operating region of the
distillation column to be identified. This condition may be characterised as an optimal
region within the normal operating range. The region is defined by a 1 o C span above
and bellow the optimal steady state values of the top and bottom plate temperatures
with all else but the afore-mentioned inputs being constant and maintained at the steady
state values given in table 7.2.
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
7.2.2
178
Identification Framework and Validation Methods
The very same identification framework used in the simulated identification experiments
in chapter 6 was used on the distillation column. That is, the prediction error method
together with an unfiltered unweighted quadratic norm. These are detailed in section
2.5.2. When applied to closed-loop data the direct approach was used (see section 3.2.1).
With respect to the validation methods used to validate the models generated, once
again the cross-validation technique (as illustrated in figure 6.8) was the primary model
validation approach. As before, several approaches to cross-validation were used, these
may be categorised as follows :
• Simulation and Prediction percentage fits.
• Frequency Content Analysis
• Residual correlations
• Higher Order Residual Correlations
7.2.3
Model Structure and Order Selection
The same model structure and approach to selecting mode orders was taken in identifying
models of the distillation column as for the simulated systems, systems A and B . That
is, ARX structures defined by equations 6.3 through 6.5 were used with the order being
selected by trial and error considering the cost of introduced model complexity for the
improved response fit. The final order selected was
"
ni =
8 8
8 8
#
In terms of the response delays used in the models, step disturbances responses were
used to determine the response time delays. The delays used maybe given as follows (in
seconds):
"
#
5 32
d=
8 6
However, as will be mentioned in next section, all data for identification (and validation)
was sampled at intervals of 10 seconds, since the identified model structure is of discrete
from, this means the response delays need to be represented as factors of the sampling
interval. Thus, rounding off the time delays to the nearest product of the sampling
interval, the sample delays of the identified models are given as
"
d=
1 3
1 1
#
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
179
With respect to the scaling of the estimated models, all data generated was scaled
around the steady states given in table 7.2, thus allowing for deviation models.
7.2.4
Identification and Validation Experiment Conditions
As with the simulated experiments, open and closed-loop experiments were done to generate data for both identification and cross-validation. Recalling figures 2.3 and 3.1,
signals u(t) and r1 (t) were again used as disturbance signals for the open and closed-loop
cases respectively with the occasional setpoint changes through r2 (t) for the closed-loop
experiments. With the previous chapter establishing PRBS disturbance signals to be the
more consistent of the disturbance signals, this was the sole disturbance signal used in
efforts to identify models of the distillation column. Figure 7.7 illustrates the input signal
disturbances used to generate identification data for open-loop experiments.
It is at this point appropriate to reiterate that both the open-loop and closed-loop
experiments were output constrained so as to allow for the identification of a specific
region of dynamics. This means that while the closed-loop experiments where designed
in accordance to output ranges (setpoint ranges) the input ranges defining the open-loop
experiments were designed in consideration of constraints on the outputs or responses to
such inputs.
(a) PRBS disturbances of the F
R
Reflux Flowrate (kg/hr)
14
12
10
8
6
4
2
0
500
1000
1500
2000
Time (s)
2500
3000
3500
4000
(b) PRBS disturbances of P
S
85
P
Steam Pressure (kPa)
S
80
75
70
65
0
500
1000
1500
2000
Time (s)
2500
3000
3500
4000
Figure 7.7: Input disturbance for open-loop distillation column identification experiments
As discussed earlier, the distillation column does produce dynamic conditions that
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
180
were not seen in the simulated experiments in the previous chapter. This is primarily
the case in the continuous disturbance of one of the inputs, the boiler steam pressure
(u2 ). This introduces an interesting challenge in identifying models of the column in that
the column is expected to exhibit non-linear dynamics, making it important to identify
models over certain dynamic regions. With one of the inputs continuously disturbing the
column at a certain frequency and over a span of magnitudes, specifying a region over
which to generate data for identification and validation becomes more difficult.
The conditions defining the different experiments used to generate data for identification and validation are defined in tables 7.3 and 7.4 which detail the open and closed-loop
identification experiments respectively, and tables 7.5 and 7.6 which detail the open and
closed-loop experiments used to generate validation data. It must be said that the same
approach used in designing the identification and validation experiments for the simulated systems was taken. That is, the experimental design common practises presented in
chapter 5 are used to determine the average constant signal intervals (B values), experiment duration and disturbance signal magnitudes. The remainder of this section details
the experimental conditions and further elaborates on the information presented in tables
7.3, 7.4, 7.5 and 7.6.
Case
Tag
Input Signal (u(t))
Signal Type Range [u1 ],[u2 ]
[kg/hr],[kPa]
DOL 1 PBRS
[4, 14],[70,77]
B
[s]
200
Table 7.3: Open-Loop cases used to generated data for identification of the distillation column
Case
Tag
Dither Signal (r1 (t))
Signal Type
Range [r11 ],[r12 ]
[kg/hr],[kPa]
B
[s]
Setpoint Signal (r2 (t))
SP Values [SP1 ],[SP2 ]
[o C],[o C]
DCL 1 PRBS
[-0.6, 0.6],[-1.2, 1.2]
200
[78,80,79],[93,95,94]
DCL 2 PRBS No Dith
[-0.6, 0.6],[-1.2, 1.2]
200
[78,80,79],[93,95,94]
DCL 3 PRBS no ∆ SP
[-0.6, 0.6],[-1.2, 1.2]
200
[79,79,79],[94,94,94]
Table 7.4: Closed-Loop cases used to generated data for identification of the distillation column
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
Case
Tag
Input Signal (u(t))
Signal Type Range [u1 ],[u2 ]
[kg/hr],[kPa]
181
B
[s]
DVOL 1 PBRS
[4, 14],[70,77]
180
DVOL 2 PBRS L
[5, 18],[62,77]
180
Table 7.5: Open-Loop cases used to generated data for validation of the distillation column
Case
Dither Signal (r1 (t))
Signal Type
Range [r11 ],[r12 ]
[kg/hr],[kPa]
B
[s]
Setpoint Signal (r2 (t))
SP Values [SP1 ],[SP2 ]
[o C],[o C]
DVCL 1 PRBS
[-0.6, 0.6],[-1.2, 1.2]
180
[78,80,79],[93,95,94]
DVCL 2 PRBS
[-0.6, 0.6],[-1.2, 1.2]
180
[80,78,79],[95,93,94]
DVCL 3 PRBS L
[-0.6, 0.6],[-1.2, 1.2]
180
[77,76,79],[94,92,93]
Table 7.6: Closed-Loop cases used to generated data for validation of the distillation column
The following further elaborates on the information presented in tables 7.3, 7.4, 7.5
and 7.6:
Range : Specifies the range of the input and dither signals used in the open and closedloop experiments respectively. In the case of the open-loop experiments the ranges
of the input signals themselves are given, while in the case of the closed-loop experiments the ranges of the dither signals disturbing each input are given.
SP Values : Specifies the changes in setpoint values made for each output. [SP1 ] are
the setpoint changes made to output y1 , [SP2 ] are the setpoint changes made to
output y2 .
no ∆SP : is the condition where no setpoint changes were incurred.
No Dither Signal : is the condition where no dither signal was used, only setpoint
changes.
L : The indicator L, was assigned to show where the experiment was operated over
a larger input range (open-loop experiments) or larger output range (closed-loop
experiments) than the typical identification experiments.
The following can be said regarding the operation of the distillation column in conducting identification experiments and experiments to generate validation data:
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
182
• The column was always brought to the same steady state before any experiment
was conducted. All experimental cases were generated around this steady state
with the primary region of interest being 1o C around the steady state temperatures
of the top and bottom plates. Table 7.2 shows the steady state values.
• All data was sampled from the continuous historian every 10 seconds. This sample
rate was chosen as it was found to be most optimal in terms of information content
and the consequent computation intensity during model estimation procedures.
• Data pre-treatment for use as validation and identification data involved de-trending
by subtracting the steady state values from the measured values, thus creating
deviation from steady-state data.
• As was established in previous chapter, identification theory dictates that the average constant signal time interval (B) of the disturbance signal be taken as 30%
of the largest time constant. Step tests found the largest time constant to be 650
seconds, thus the PRBS disturbance signals used in identification experiments had
B values of 200 seconds. The B value for the disturbance signals used generate data
for validation were taken as 180 seconds.
• The duration of each of the experiments was taken as several times the largest time
constant, 5000 seconds.
• Both input variables were disturbed simultaneously but not at the exact same
moment in time. Figure 7.7 illustrates this.
• All experiments besides those designed to be carried out over larger ranges (experiments indicated with ”L”) where maintained
• The input signal magnitudes for the open-loop experiments (not including the ”L”
experiments - larger range) were designed to disturb the system such that the output
responses maintained themselves within 1o C of the steady state values. Figure 7.8
illustrates this for input u1 and output y2 .
• The magnitude of the dither signals used during closed-loop experiments was taken
as 10 % of the input range that maintains the temperature values within 1o C around
the steady state values.
• The setpoint changes for output y1 and y2 ([SP1 ],[SP2 ]), were made at 1400, 3200
and 4700 seconds and 200, 2200 and 4000 seconds respectively.
• Four data sets were generated within a tight range of operation considered to be
optimal in terms of maximising the temperature differences between the top and
bottom plate. One of these sets was generated under open-loop conditions (DOL
© University of Pretoria
CHAPTER 7. IDENTIFICATION OF A DISTILLATION COLUMN
183
1) while the other three were under closed-loop conditions (DOL 1-3). These data
sets were used for identification.
• Five data sets were generated as validation sets, two open-loop (DVOL 1-2) and
three closed-loop (DVCL 1-3). All but one closed-loop and open-loop experiment
were done over the normal operating range (1o C around the steady state temperatures). The experiments that were not done over the normal operating ranges were
operated over a range about 2o C around the steady state values. Note that with the
azeotrope at 78o C and the steady state temperature at 79o C the the azeotrope tempaerature was crossed for these experiments implying that equilibrium conditions
were not always maintained.
• The only difference between validation experiments DVCL 1 and DVCL 2 is that
the setpoint changes made for DVCL 1 are the same as those made for identification
experiment DCL 1 while the setpoint changes for case DVCL 2 are not.
(a) Closed-loop input range - u
(c) Open-loop input range - u
2
2
10
15
∆ Steam Pressure (kPa)
∆ Steam Pressure (kPa)
20
10
5
0
-5
-10
-15
0
1000
2000
3000 4000
Time (s)
5000
-5
-10
0
1
0.5
0
-0.5
-1
-1.5
1000
2000
3000 4000
Time (s)
5000
1000
2000
3000 4000
Time (s)
5000
(d) Open-loop output range - y
2
1.5
-2
0
0
o
∆ Bottom Plate Temperature ( C)
o
∆ Bottom Plate Temperature ( C)
(b) Closed-loop output range - y
6000
5
6000
6000
2
2
1
0
-1
-2
0
1000
2000
3000 4000
Time (s)
5000
6000
Figure 7.8: (a)-(b) u2 and y2 values generated from closed-loop identification experiment DCL
2. (c)-(d) u2 and y2 values generated from open-loop identification experiment
DOL 1
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General Execution
As mentioned before, the column was operated via the DeltaV operating system with
the data being accessed via the continuous historian. While the setpoint changes and
disturbance signal scheduling were done using the sequential flow functions in DeltaV, all
other aspects of the generating signals, models and validation were done through Matlab.
The same points made in section 6.2.5 regarding the execution and code used in the
identification of models for the simulated systems and the validation of said models extend
to the identification and validation efforts for the distillation column.
7.3
Identification and Validation Results
This section presents and discusses the results of the identification of linear models approximating the pilot scale distillation column and the validation of such models. Note
that all the parameters of each identified model are presented in appendix A.2.
7.3.1
Simulation and Prediction
The results presented here regarding the model response simulation and prediction analyses were generated in a very similar manner to the simulation and prediction results
presented for the simulated systems, systems A and B, in sections 6.3.3 and 6.4.2 respectively. That is, each identified model was cross-validated against each validation set.
This was done by disturbing each identified model by the input contained in each validation set, from this the model response to this input is compared against the response
contained in the validation set (i.e. the response of the distillation column). Figure 6.8
illustrates this cross-validation approach. The accuracy of the model response relative to
the distillation column response (validation set response) is expressed as a percentage fit
value (see equation 6.7).
The percentage fit values of the simulated responses of each model when validated
against open-loop validation sets and closed-loop validation sets are given in figures 7.9
and 7.10 respectively.
General Observations
In comparing the differences between figures 7.9 and 7.10, it is clear that all the models
generated extremely inaccurate responses when validated against the open-loop validation
sets (figure 7.9). In fact most of the response percentage fit values are negative. In
contrast, the response fit values obtained by each model when validated against the
closed-loop validation sets showed much better percentage fits in that the values were
generally much larger. Furthermore it is noted that both open and closed-loop validation
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(a) Response Percentage Fit Against Open-Loop Validation Data for y
1
20
0
Percentage Fit
-20
-40
-60
-80
-100
Validated against DVOL 1
Validated against DVOL 2
(b) Response Percentage Fit Against Open-Loop Validation Data for y
2
Percentage Fit
20
0
-20
-40
-60
Validated against DVOL 1
DOL 1
Validated against DVOL 2
DCL 1
DCL 2
DCL 3
Figure 7.9: Percentage fit values when validating against open-loop data sets
(a) Response Percentage Fit Against Closed-Loop Validation Data for y
1
Percentage Fit
50
0
-50
-100
-150
Validated against DVCL 1
Validated against DVCL 2
Validated against DVCL 3
(b) Response Percentage Fit Against Closed-Loop Validation Data for y
2
80
Percentage Fit
60
40
20
0
-20
-40
Validated against DVCL 1
Validated against DVCL 2
DOL 1
DCL 1
DCL 2
Validated against DVCL 3
DCL 3
Figure 7.10: Percentage fit values when validating against closed-loop data sets
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sets reveal that all the models generated more accurate response fits for output y2 in
comparison to those for output y1 . As will be discussed later, this may be attributed to
the continuous disturbance of input u2
Validation Set Model Discrimination and Bias - Closed-loop Validation Sets
The closed-loop validation sets proved to be more informative in terms of providing insight
into the identification and validation problem. This is due to the larger contrast in results
in comparison to those obtained when using the open-loop results which simply indicated
that all the models produced terribly inaccurate responses.
In interpreting the significance of the general observations it is important to consider
the different discrimination capacities by the different validation sets given certain experimental conditions. As was established in section 5.5, closed-loop experiments are
expected to produce more informative data in comparison to open-loop experiments only
under the implementation of output constraints. This is exactly the case here, figure 7.8
shows this effect of implementing output constraints clearly. The closed-loop conditions
allowed for a larger input signal magnitude range while maintaining the output with in
bounds.
Looking at the response percentage fit values for the different identified models when
validated against the closed-loop validation sets (figure 7.10), it is very clear that the
models obtained from closed-loop experiments (DCL 1-3) almost always generated more
accurate responses to that obtained from the open-loop experiment, DOL 1. This discrimination against the model obtained from open-loop experiments is certainly due to
the differences in information content between open-loop and closed-loop experiments
based on the afore mentioned condition caused by output constraints.
In observing the differences in response fits when validating the models against closedloop validation set DVCL 1 and DVCL 2 some interesting observations were made. With
the only difference between the two validation sets being that set DVCL 1 was generated
from experiments where the setpoint changes were precisely the same as those used to
generate the identification data for cases DCL 1-2, while set DVCL 2 used a different
sequence of setpoint changes (see tables 7.4 and 7.6). It is observed that the models all
produced slightly less accurate responses when validated against DVCL 2. This implies
that the closed-loop validation sets are slightly biased towards models obtained from
similar setpoint direction changes.
Validation Set Model Discrimination and Bias - Signal Magnitude
The differences in response fit results when using validation sets obtained over the normal
operating region to those obtained when using validation sets generated by larger input
signals and output ranges are of particular interest. This is as the larger the effect
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of signal magnitude on the results the more the column dynamics may be attributed as
being non-linear. Additionally, response fit sensitivity to validation set signal magnitudes
also imply the validation sets are discriminating based on differences in dynamic region
exposure caused by different signal magnitudes.
Relative to validation sets obtained from closed-loop experiments, set DVCL 3 differs
from DVCL 1 only in that the setpoint magnitudes are larger. Relative to the validation
sets obtained from open-loop experiments, set DVOL 2 differs from DVOL 1 only in that
larger input signals magnitudes were used.
Looking at the closed-loop validation results in figure 7.10 it is clear that all the models
performed worse when validated against the validation set generated over a larger dynamic
region (DVCL 3). That is, the accuracy of the model responses greatly diminished when
validated against DVCL 3 in comparison to DVCL 1. The difference between figures
7.11 and 7.12 further shows this difference in response accuracy as figure 7.11 displays
each model response when validated against DVCL 1 and figure 7.12 shows the responses
validated against DVCL 3.
The results obtained from the open-loop validation sets regarding the effects of larger
disturbance signals were very interesting. It may generally be stated that the response
accuracy of the models obtained from closed-loop experiments (DCL 1-3) improved when
validated against the open-loop validation set generated from larger disturbance signals
(DVOL 2) in comparison to the respons accuracy obtained when using set DVOL 1. This
while the model obtained from the open-loop experiment (DOL 1) generally revealed
worse response fits. This may be justified by the point made earlier regarding the closedloop experiments ability to use a larger range of the input signal while maintaining its
output constraint (figure 7.8). This means that the closed-loop experiments identified
a larger range of dynamics in comparison to the open-loop experiments given the same
output constraints. Thus its improved fit when validated against the open-loop validation
set obtained from larger input signal magnitudes.
Experimental Condition Sensitivities
In determining which experimental condition generated the model that produced the
most inaccurate responses it was found that the results depended on the output. It was
found that the model obtained from the open-loop experiments, DOL 1, produced the
least accurate (smallest percentage fits) responses regardless of the validation set used
for output y1 . While the same may be said for the model obtained from closed-loop
experiments where no setpoint changes were made, DCL 3, for output y2 .
In actuality, as mentioned earlier, it may be said that all the models produced more
accurate responses for output y2 in comparison to y1 . These improved response fits for
output y2 may be explained by the fact that input u2 in consistently being disturbed by
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Response fit for y using DVCL 1
1
o
∆ Top Plate Temperature ( C)
2
1
0
-1
-2
-3
0
DVCL 1 (100%)
DOL 1: -178 %
DCL 1: 40
DCL 2: 53
500
1000
1500
2000
Time (s)
2500
3000
3500
4000
3000
3500
4000
Response fit for y using DVCL 1
o
∆ Bottom Plate Temperature ( C)
2
2
1
0
-1
-2
0
DVCL 1 (100%)
DOL 1: 55.2 %
DCL 1: 72
DCL 2: 79
500
1000
1500
2000
Time (s)
2500
Figure 7.11: Validation simulations against set DVCL 1
Response fit for y using DVCL 3
1
o
∆ Top Plate Temperature ( C)
3
2
1
DVCL 3 (100%)
DOL 1: -106 %
DCL 1: -6.7 %
DCL 2: -4.5 %
0
-1
-2
-3
0
500
1000
1500
2000
Time (s)
2500
3000
3500
4000
Response fit for y using DVCL 3
o
∆ Bottom Plate Temperature ( C)
2
2
DVCL 3 (100%)
DOL 1: -32.1 %
DCL 1: -8.6 %
DCL 2: -4.3 %
1
0
-1
-2
0
500
1000
1500
Time (s)
2000
2500
Figure 7.12: Validation simulations against set DVCL 3
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an upstream condition. This, and the fact that input u2 has a larger relative gain with
output y2 than y1 , means that output y2 is being persistently excited to a greater extent.
This condition is further discussed in the frequency analyses section that follows.
Experiments DCL 1 and DCL 2 are both closed-loop experiments with the only difference between the two being that DCL 2 did not use dither signal disturbances and
DCL 1 did. Observing the differences in response fits of the models obtained from these
experiments it is noted that there is no clear distinction trend between the two. In some
cases, such as output y2 when validated against DVCL 2 (figure 7.10(b)), the results suggest that the use of dither signal disturbances allowed for improved system excitation and
information generation and the identification of a model with a more accurate response.
However, most cases the difference in response fits were not so pronounced, in fact in
some cases, such as output y1 when validated against DVCL 1 (figure 7.10(a)), DCL 2
showed larger response fit values than DCL 1 suggesting that not using a dither signal
allowed for more informative data and a more accurate model.
(a) y control with with dither disturbance - DCL 1
(c) y control with without dither disturbance - DCL 2
1
0.5
0
-0.5
y
-1
1
y setpoint
1
-1.5
0
1000
1
o
∆ Top Plate Temperature ( C)
o
∆ Top Plate Temperature ( C)
1
1.5
2000
Time (s)
3000
(b) y control with with dither disturbance - DCL 1
0.5
0
-0.5
-1
y
2
y setpoint
-1.5
0
2
1000
2000
Time (s)
3000
0.5
0
-0.5
y
-1
1
y setpoint
1
-1.5
0
1000
2000
Time (s)
3000
2
o
∆ Bottom Plate Temperature ( C)
o
∆ Bottom Plate Temperature ( C)
1
1
(d) y control with without dither disturbance - DCL 2
2
1.5
1.5
1.5
1
0.5
0
-0.5
-1
y
2
y setpoint
-1.5
0
2
1000
2000
Time (s)
3000
Figure 7.13: Experimentation with and without dither signals
This variance in effectiveness of the dither signal may be explained by the fact that
the dither signal did little to additionally disturb the system on top of the disturbances
caused by the continuous disturbance of input u2 by an upstream condition. Figure
7.13 shows the output response differences between experimental case DCL 1 which used
dither signals and case DCL 2 which did not. It is noted that the dither signals did
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little to further excite or disturb the system. The persistent disturbance of input u2 by
the upstream condition was such that the contrast of the effects of the dither signals
were diminished. This explains the small difference in model accuracy since both cases
produced similarly informative data.
Prediction
Analysing the predictive accuracies of each model against each validation set, it was found
that no significant difference in percentage fit could be found using prediction horizons
larger than 20 steps. These 20 steps translate to 200 seconds given the 10 second data
sampling rate and the consequent step size defining the discretised models identified.
It is at horizons smaller than 200 seconds that the model fits begin to improve and
the differences between the results generated by the different validation sets begin to
diminish. Figure 7.14 shows the prediction percentage fits using a 100 second horizon
when validated against the closed-loop validation sets.
Prediction Percentage Fit Against Closed-Loop Validation Data for y
1
100
DOL 1
DCL 1
DCL 2
DCL 3
Percentage Fit
80
60
40
20
0
-20
Validated against DVCL 1
Validated against DVCL 2
Validated against DVCL 3
Prediction Percentage Fit Against Closed-Loop Validation Data for y
2
100
DOL 1
DCL 1
DCL 2
DCL 3
Percentage Fit
80
60
40
20
0
-20
Validated against DVCL 1
Validated against DVCL 2
Validated against DVCL 3
Figure 7.14: Validation predictions against the closed-loop validation sets
In comparing the prediction results given in figure 7.14 to the simulation results given
in figure 7.10 it must be said that the models are more accurate in predictions than
in simulations given small enough prediction horizons are used. Furthermore the effect
of increasing the signal magnitudes and output range has a much smaller effect on the
prediction accuracy than the simulation accuracy. This is seen by the much improved
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results for the prediction when validated against validation set DVCL 3 in comparison
to the simulation result. Furthermore the model obtained from experiment DCL 3, characterised as a closed-loop experiment where no setpoint changes were incurred, clearly
produced the least accurate predictions of all the models.
7.3.2
Frequency Content Analyses
Analyses of the identified models approximating the distillation column in the frequency
domain were done from the perspective of cross-validation. That is, the models themselves
were not assessed but their responses and how they fit the validation data responses. In
other words, each model’s simulated response was assessed in the frequency domain and
compared to that of the validation set response. Thus the frequency content of the
response fit is presented in this section.
It was found that most of the frequency content analyses of the response fits did
generally reflect the simulated response fit results given in the previous section. Some
interesting observations were however made.
(a) Response Fit Frequency Analysis for DOL 1 for y
2
1
10
Amplitude
DVCL 1 (100 %)
DOL 1: -178 %
0
10
-2
10
-2
10
-1
0
10
1
10
10
Frequency (rad/s)
(b) Response Fit Frequency Analysis for DOL 1 for y
2
2
10
Amplitude
DVCL 1 (100 %)
DOL 1: 55.2 %
0
10
-2
10
-2
10
-1
0
10
10
1
10
Frequency (rad/s)
Figure 7.15: Frequency analysis of simulation validation for DOL 1 against DVCL 1
Recalling that the identified models showed slightly improved response fits for output
y2 in comparison to output y1 and that this improved fit was thought to be due to the
continuous disturbance of input u2 . This contrast in response fit for the different outputs
was most evident for the model obtained from the open-loop identification experiment,
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DOL 1 (see figure 7.10), thus it was most fitting to use this model and its responses
to investigate these conditions in the frequency domain. In assessing this case and its
response fits in the frequency domain it was found that output y2 has improved fits at
certain frequencies in comparison to output y1 . This is shown in figure 7.15. Figure
7.15(a) shows the frequency fit between the response generated by DOL 1 for output y1
and the validation set response given by DVCL 1. Figure 7.15(b) shows the same but for
output y2 . It is noted that the frequency content fit shows output y2 to have a slightly
better fit at both the lower and higher end of the frequency band.
It is interesting to note that the always present disturbance of input u2 occurred at
an approximate frequency of 0.18 Hz (1.13 rad/s), one might postulate from this that the
improved fit for output y2 could be due to the generation of more informative data due
to these disturbances and the fact the relative gain is larger between this input-output
pair.
7.3.3
Residual Correlation Analysis
As was done in sections 6.3.5 and 6.4.4 for the models approximating systems A and
B respectively, the residual correlation analyses presented here was done so with the
purpose of obtaining indications of unmodelled and incorrectly modelled dynamics in each
identified model. Additionally, the differences between correlation profiles obtained when
using different validation sets were studied to provide insight into how the validation sets
differently discriminate. Furthermore higher order residual correlation tests were used to
gain insight into how well the linear models approximated the non-linear dynamics.
Whiteness Test Results
In assessing the residual independence of each model when validated against each validation set it was found that the results did not clearly reflect the simulated response fit
findings. Figure 7.16 shows the residual auto-correlation plots obtained for the model
generated by closed-loop experiment DOL 1 validated against validation sets DVCL 1,
DVCL 3 and DVOL 1. In a similar fashion figure 7.17 shows the correlation plots for the
model generated by the open-loop experiment DCL 1.
It is at this point appropriate to recall the response fit results obtained by these two
models (DOL 1 and DCL 1) in figures 7.9 and 7.10. As has been established, one of the
key observations made regarding the response fits was that all the models produced more
accurate response fits for output y2 , this was especially the case for the model obtained
from the open-loop experiments (DOL 1). In addition to this, all the models produced
extremely inaccurate response fits for both outputs when validated against the open-loop
validation set (DVOL 1) in comparison to the response fits found when validated against
the closed-loop validation set (DVCL 1). The last critical observation made was that all
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Figure 7.16: Residual auto-correlation for a case DOL 1 against validation sets DVCL 1,
DVCL 3 and DVOL 1
Figure 7.17: Residual auto-correlation for a case DCL 1 against validation sets DVCL 1, DVCL
3 and DVOL 1
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the models produced very inaccurate responses when validated against the closed-loop
validation set obtained by larger disturbance signals (DVCL 3).
Looking at the residual correlation profiles of cases DOL 1 and DCL 1 in figures 7.16
and 7.17 respectively, the first observation made is the greater distinction in correlations
between output y1 and y2 for case DOL 1 in comparison to DCL 1. This reflects the
findings that the model obtained from open-loop experiments did show the greatest differences in simulated response fits between output y1 and y2 of all the identified models.
Further observing these figures it is noted that while the response fits obtained when
validated against the open-loop validation set DVOL 1 (figure 7.9) show extremely inaccurate fits that suggest large extents of unmodelled dynamics. The correlation profiles
obtained when using DVOL 1 do not suggest significantly different correlation profiles
compared to the others. Figure 7.18 shows this point clearly in that figure 7.18(a) shows
the correlations profiles obtained when validating the models obtained from experiments
DCL 1, DCL 3 and DOL 1 against DVOL 1, while figure 7.18(b) shows the same models
validated against DVCL 1.
Figure 7.18: Residual auto-correlation for a cases DCL 1, DCL 2 and DOL 1 against validation
set DVOL 1
It is additionally observed, looking at figure 7.18, that the correlation profiles obtained
for the different models are more distinct when validated against the closed-loop validation
set, figure 7.18(b), in comparison to the correlations obtained by the open-loop validation
set, figure 7.18(b). This suggest that the closed-loop validation set did a better job at
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discriminating between models based on differences in unmodelled dynamics.
Independence Test Results
The input-residual independence test results were found to be more agreeable with the
simulated response fit results. Specifically in showing little differences in the correlation
profiles between the models generated from experiments DCL 1 and DCL 2 further suggesting the redundancy of the dither disturbance signal due to the continuous disturbance
of u2 . It must however be said that all the models showed relatively large amounts of
input-residual correlations regardless of the validation set, suggesting incorrectly modelled dynamics.
Figures 7.19(a)-(c) show the input-residual correlation profiles for the model generated
by closed-loop experiment DCL 1 validated against validation sets DVCL 1, DVCL 2
and DVOL 1 respectively. The correlations strongly indicate large extents of incorrectly
modelled dynamics in that several peaks where found regardless of the validation set.
The same extent of input-residual correlations were found for the other input-residual
pairs suggesting that no input-output dynamic relationship was better modelled than the
other.
Figure 7.19: Input-residual correaltions for DCL 1
Figures 7.20(a)-(c) show the input-residual correlation profiles in the same fashion as
figure 7.19(a)-(c) but for the model generated by the open-loop experiment DOL 1. It
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is observed that profiles obtained when validated against the open-loop validation set
(DVOL 1 -figure 7.20) are very similar to those obtained when using the closed-loop
validation set (DVCL 1 -figure 7.20) except when validated against DVCL 2 (figures
7.19(b)-7.20(b)). This implies that closed-loop validation set DVCL 2 better discriminated between the models based on differences in incorrectly modelled dynamics.
Figure 7.20: Input-residual correlations for DOL 1
It is additionally observed that the correlation profiles using validation data obtained
from closed-loop experiments indicate larger correlation magnitudes in comparison to the
correlations obtained when using the open-loop validation sets. As has been mentioned
before, the closed-loop validation sets are prone to produce input-residual correlations
that are rooted in input-output correlations caused by the feedback and not unmodelled
dynamics. Determining whether these difference correlation magnitudes are due to differences in models discrimination or due to the correlations caused by feedback is difficult
to determine.
Higher Order Correlation Results
The higher order correlation results were relatively uninformative. All the results for all
the models against all the validations sets revealed significant higher order correlations
between most the residuals - inputs pairs for both Ru2 and Ru2 2 . This implies, by
table 4.1, that both the system and noise models are inadequate and completely failed
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to estimate both even and odd non-linear dynamics.
An interesting result was however found in comparing the correlation profiles for Ru2 to that of Ru2 2 for the model obtained from the open-loop experiment DOL 1 when
validated against the closed-loop validation set DVCL 1 and the open-loop validation set
DVOL 1. Figures 7.21 and 7.22 show the correlations profiles obtained for Ru2 for the
model obtained from DOL 1 validated against DVCL 1 and DVOL 1 respectively. It
is noted that the correlation profiles concerned with input u2 are less pronounced when
using the open-loop validation set DVOL 1 (figure 7.22(b) and (d)) than when using
the closed-loop validation set DVCL 1 (figure 7.21(b) and (d)). This is not seen when
analysing Ru2 2 which, according to table 4.1, implies that the open-loop validation set
DVOL 1 indicates the unmodelled dynamics concerned with output u2 for the model
identified from case DOL 1 to be attributed to an inaccurate noise model more so than
an inaccurate dynamics model. This observation suggests that the dynamics concerned
Figure 7.21: Higher order correlation profile, Ru2 , for DOL 1 against DVCL 1
with input u2 has been more accurately modelled. If this is so then it implies that
the open-loop validation set is more informative as it has better discriminated between
models so as to detect this condition. Additionally it is very likely that the closed-loop
validation set did not detect this conditions in the higher order input-residual correlation
due to correlation corruption through the correlations caused by the feedback loop.
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Figure 7.22: Higher order correlation profile, Ru2 , for DOL 1 against DVOL 1
7.3.4
Results Overview
Simulation and Prediction
General Observations:
• The responses of the identified models were found to be very inaccurate when
validated against open-loop validation sets while when validated against the closedloop validation sets the responses were much more accurate.
• The responses of the identified models were additionally found to always be more
accurate for output y2 .
Validation Set Model Discrimination and Bias - Closed-loop Validation Sets:
• The closed-loop validation sets were found to be bias towards models obtained
from the same feedback condition. This was expected as the output constraints
implemented on experiments resulted in a significant difference between open and
closed-loop experiments in that the closed-loop experiments were able to use larger
input signal magnitudes. This suggested the closed-loop experiments generated
more informative data.
• The closed-loop validation sets additionally indicated bias towards models obtained
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from experiments that used a similar sequence of setpoint changes to used to generate the validation set.
Validation Set Model Discrimination and Bias - Signal Magnitude:
• When validating the model responses against the closed-loop validation set obtained
over a larger output range than that used to generate the normal closed-loop validation set, the accuracy of all the model responses worsened.
• When validating the model responses against the open-loop validation set obtained
using larger input signal magnitudes than the normal open-loop validation set, the
response accuracy of the model obtained from the open-loop experiment worsened
while the response accuracy of the models obtained from closed-loop experiments
improved.
• This reduced sensitivity to the signal magnitudes used to generate the open-loop
validation set by the models obtained from closed-loop experiments was found to
be due to the fact that the closed-loop validation sets used a larger input signal
range than the open-loop experiments. This implies that the closed-loop validation
experiments identified a larger range of dynamics in comparison to the open-loop
validation experiments given the same output constraints.
Experimental Condition Sensitivities:
• As mentioned earlier, there were significant differences in response accuracy between
the two outputs. That is, all the models produced responses that were more accurate
in y2 than y1 . The increased excitation of input u2 by an upstream condition and
the consequent generation of more informative data for the output with which this
input shared a larger relative gain with (y2 ) explains the difference in response fits.
• Further studying this effect however, lead to the finding that the dither signal was
to some extent made redundant by this continuous disturbance. Thus experiments
that did disturb the system via the dither signals did no not produce models with
response accuracies that distinguished themselves from those experiments that did
not use dither signal disturbances.
Prediction:
• Prediction results indicated that the identified models produced more accurate responses through prediction than simulation given the prediction horizon was smaller
than 200 seconds.
• Prediction response accuracy was addition found to be less sensitive to the experiment conditions used to produce the different validation sets.
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Frequency Content Analysis
• Analysing the model response fits validated against each validation set in the frequency domain reflected what the simulated response fit results indicated. The
analysis was how ever useful in gaining insight into some differences in response
accuracy.
• It was found that the better response fit for output y2 generated by all the models
may be attributed to a better fit of a high frequency component in comparison
to the fit for output y1 . This higher frequency corresponds with the frequency of
the continuous disturbance of input u2 . Thus affirming the contribution of the
continuous disturbance of the system by an upstream condition.
Residual Correlation Analysis
Whiteness Test:
• All the identified models produced responses that generated significant residual
auto-correlations regardless of the validation set used. This implies that all the
models had large extents of unmodelled dynamics.
• Correlation profile differences were found that further indicate distinctions in response accuracy between output y1 and y2 and that suggest that the dynamics
concerned with output y2 were better modelled.
• Differences between correlation profiles when validated against the different validation sets suggested that the closed-loop validation sets better discriminated between
models than the open-loop validation sets.
Independence Test:
• As with the whiteness test, the independence tests revealed that all the models identified had residuals with significant input dependencies regardless of the validation
set used. This implies that all the models have incorrectly modelled dynamics.
• Differences between correlation profiles obtained by the different validation sets
again suggested that the closed-loop validation sets better discriminated between
models. Confidence in this finding however is uncertain due to the corruption of
the correlation profiles by correlations caused by the feedback loop.
Higher Order Correlation Tests
• The higher order correlation results were found to be less informative than expected.
• All the models generated responses with significant higher order correlations. This
implies that all the identified models had inadequate noise and dynamic models.
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• The open-loop validation set did however produce correlation profiles that suggest
that output y2 was more accurately modelled.
7.4
7.4.1
Discussion
A ’Real’ Identification Problem
Considering the results and findings of the response accuracy, frequency and residual
correlation assessments it must be said that the distillation column did serve its purpose
in providing an understanding of a few ’real’ problems that are not always evident in
simulated identification experiments. These ’real’ problems predominantly manifested
themselves as the effects of the continuous disturbance of input u2 , the steam pressure,
by an upstream condition. This disturbance had large effects on the information being
generated by the different identification and validation experiments.
It is important to note that while validation results and interpretations suggested
these continuous disturbances may have allowed for the identification of models with more
accurate responses, it cannot be stated with absolute confidence that these disturbances
contributed positively or negatively to the successful identification and validation of the
distillation column. This is mostly due to the fact that the validation data sets used
to validate these models may have been bias towards the conditions caused by these
disturbances and thus not representative of the dominant dynamics of the column.
This is perhaps the greater lesson in studying a ’real’ system. Since the system
dynamics are unknown, there is no way of comparing the identified models directly with
the dynamics of the distillation column, only the validation sets. It is perhaps in this
notion that the concept of a useful model being a good model is most applicable. Perhaps
it is not necessary to know how well the model represents the true dynamics of the
distillation column. Perhaps the understanding that certain models did well to represent
certain validation sets is adequate, regardless of how bias the validation set is. That is,
given that the column is continuously characterised by these continuous disturbances,
then the models found to accurately represent the validation sets will be useful.
7.4.2
Experimental Conditions
In terms of the effects of the different experimental conditions there is a point that
has been made relatively clear. That is, the result of implementing constraints on the
outputs so as to allow for the identification of a dynamic region most representative of
normal operation gave closed-loop experiments the advantage. The models identified from
closed-loop experiments were found to produce much more accurate responses while the
validation sets generated from closed-loop experiments were found to better discriminate
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between models.
7.4.3
Validation Techniques
It must be said that the residual correlation assessments were not very informative. While
this was expected of the whiteness and independence tests due to the non-linear nature
of the system being identified, it was not expected of the higher order correlation tests.
A greater ability of discrimination between models was expected form the higher order
tests, a discrimination that would have allowed for insight into how well each model
approximated the non-linear dynamics.
The response fit analysis was relatively informative as it clearly indicated how well
models represented the different validation sets. As has be stated, it is very clear that
the response fit results were very sensitive to the validation sets and the experimental
conditions used to generate these validation sets.
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CHAPTER 8
Discussion and Conclusions
This chapter presents key discussions relevant to the work presented and the results obtained in preceding chapters. Additionally the general conclusions gathered
from the simulated and pilot scale distillation column identification and validation
efforts are presented.
8.1
8.1.1
Discussion
Sensitivity to Noise and Model Uncertainty
The topic of model sensitivity to noise and model uncertainty while very briefly addressed
in this work is acknowledged as being relatively important and worthy of further discussion. The manner in which both noise sensitivity and model uncertainty were assessed
was through the simulated experiments on systems A and B.
In the case of noise sensitivity, identification experiments were repeated several times
generating different identification data sets with different noise realisation from which
several models were identified. The parameters of such models were assessed for variance,
the experiments that produced models with the largest measures of parameter variance
were labelled as most sensitive to noise. This has a serious implication. If an experiment
was found to be most sensitive to noise then it implies that such an experiment produced
identification data that was less informative of system dynamics and consequently more
susceptible to corruption by noise. The experiments found to be most sensitive to noise
were the open-loop step disturbance experiment and the closed-loop experiment where
no setpoint changes were incurred. Both of which were characterised with conditions that
were expected to be less informative than other experiments.
In terms of the model uncertainty, this measure was obtained from the parameter
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estimation procedure, to be more specific, it was derived from a property of the asymptotic parameter convergence variance, the asymptotic covariance matrix. This parameter
was used to infer parameter uncertainty in the identified models. It is important to note
that this measure of uncertainty is independent of the actual accuracy of the identified
model and solely dependent on the asymptotic variance of parameter convergence. This
essentially makes this property a measure of how conducive the data generated from a
specific experiment is to efficient parameter convergence. It is important to note however
that in the case of identifying models approximating the non-linear system, system B,
these uncertainty measures did coincide with models that were expected to be uninformative and did produce inaccurate responses relative to certain validation sets. These
models being those obtained from closed-loop experiments where not setpoint changes
were incurred.
8.1.2
Closed-loop Vs. Open-Loop
With one of the primary focal points of this work being the difference between open-loop
identification and closed-loop identification it is at this point appropriate to discuss the
insight obtained during this work with regard to such differences.
The simulated experiments on the linear system (system A) were quick to show that
the closed-loop experiments produced informative data such that the models identified
from them were more robust than those obtained from open-loop experiments.
To further elaborate on this it is recalled that in identifying the simulated linear
system, experimental design variables were varied substantially so as to allow for an idea of
their effects on the accuracy of the identified models and discrimination of the validation
sets. The primary variables being the type of disturbance signal and the frequency
characteristics of the disturbance signal. These variables were varied between values
expected to produce informative data, to those expected to produce very uninformative
data.
It was found that the models generated by the open-loop experiments were much
more sensitive to these design variables while models generated from closed-loop experiment were less sensitive. This meant that optimising the design variables so as to allow
for the most informative data had a larger effect on the open-loop cases, they thus produced better models and more discriminative validation sets. However, this also meant
that using design variables expected to produce uninformative experiments affected the
models obtained from open-loop experiments more than those obtained from closed-loop
experiments.
Thus, given the same variance of quality in experimental design variables, open-loop
experiments were found to generate the most accurate models and the least accurate
models while the closed-loop experiments maintained relatively consistent. It must be
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noted however that this is assuming that setpoint changes are made during closed-loop
experiments.
The key point from this finding being that given uncertainty in experimental design
variables and how optimal they are, closed-loop experiments are more likely to produce
more accurate models.
8.1.3
Linear Approximation of Non-linear Systems
In efforts to identify the non-linear system, system B, it was established that model
accuracy and validation set discrimination was not only more sensitive to the experimental
conditions used to generate them, but also more relative. That is, designing identification
and validation experiments was less about generating data that was informative about the
true dynamics of the system and more about generating data that reflected the relative
purpose of the model. This was primarily due to the fact that the non-linear nature of the
system meant it was impossible to truly represent it with a linear model. Additionally, the
increased sensitivity to experimental design variables meant that the condition mentioned
in the previous section, where models obtained from open-loop experiments were more
sensitive to experimental design variances, was compounded.
These conditions characterising non-linear dynamics identification were thus found
to create an environment that was best suited for closed-loop experiments. A dominant
concept through out the assessment of non-linear system identification was how varying
disturbance signal magnitudes and output ranges affected the dynamics exposed during
the experiments. Thus, given a linear approximation is to be obtained, there is the clear
requirement to generate informative data over a constrained region so as to consistently
expose the model estimation procedures to only the relevant dynamics. This meant
that closed-loop experiments had the advantage. Closed-loop experiments could more
effectively use the input signal to disturb the system while maintaining the output within
its constraints.
This was exactly what was found when identifying approximations of the distillation
column. Models identified from closed-loop experiments produces responses that more
accurately approximated the distillation column responses. Furthermore, the validation
sets obtained from closed-loop validation sets were found to better discriminate between
models based on the dynamic regions approximated.
8.1.4
Identification and Control
An issue that has had limited attention in this work is that of the direct effects of the
controllers themselves during closed-loop experiments. Some interesting considerations
did however arise from the work done.
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It was established in section 3.3 that literature suggests a non-linear controller would
allow for more persistent disturbances of the system to be identified. In efforts to investigate this, non-linear controller action was implemented in specific simulated experiments on system A and system B through varying the PID controller parameters over
the duration of the identification experiments. Validation techniques failed to detect
any indications or account for any improved accuracy in the models identified from such
experiments.
It is important to note that a repeated observation in this work has been that the
closed-loop experiments have reduced sensitivity to variances in experimental variables
and that this is most likely due to the controller doing its job in dampening the effects of
system disturbances. It is interesting to consider the role played by controller performance
with respect to this effect. This is of specific relevance to the experiments done on the
distillation column. Recalling that the column’s steam pressure was being continuously
disturbed by an upstream condition, much effort went into reducing the effects of this
disturbance in tuning the controllers to perform better. In doing so the controllers were
made slightly less robust. This lack of robustness meant a slight variance in controller
performance over the 2 o C range over which the respective outputs were being identified.
This variable controller performance effectively means variable extents of output disturbance which implies variances in the information generated over the identification
range. This means, especially in the case where non-linear dynamics is concerned, that
the model approximated from this data will be biased towards the regions where the
controllers performed worst as they were most disturbed.
Taking this a step further, implementing such a model in a controller would thus
result in very poor robustness since the model is most accurate over an even smaller
region than initially designed. From this it can be understood that controller robustness
and consistent performance are important in producing unbiased models and validation
data.
8.1.5
Simulation Vs. Experimentation and Beyond
The contrast between investigating system identification and validation through simulated experiments to experiments on a real pilot scale system was found to be very large.
While the simulated experiments allowed for consistent and unfaltering specification of
all the variables, the pilot scale system presented many more variables that were beyond
control of the experiment.
This is exemplified in the continuous disturbance of the steam pressure supply to the
distillation column which nullified the effects of designed dither signals. Furthermore
in order to define the identification problem regarding the distillation column it was
necessary to assume any dynamic interactions with variables external to those included
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in the model structure being identified were negligible.
This points to the key difference found between simulation and real experimentation,
that is, in simulation the system being identified was isolated and closed, while the pilot
scale system had external disturbances, some known and possibly some unknown. Determining whether an identified model response inaccuracy is due to unmodelled dynamics
between accounted for variables or between unaccounted for variables it difficult. From
this it is clear in the efforts presented here to an industrial scale system, which is part of
a large process, isolating the dynamics and variables and defining the system becomes a
key component in the identification problem.
8.2
Conclusions
The following were successfully achieved as a means to satisfy the objectives of this work:
• Different system identification approaches, specifically those concerned with closedloop system identification, were surveyed together with different model validation
techniques and identification experiment design.
• Identification and validation experiments were simulated on known linear and nonlinear systems so as to obtain insight into the sensitivities between identified model
accuracy and experimental variables. A specific focus was made on feedback conditions and disturbance signal characteristics.
• Different validation techniques were assessed relative to each other through these
simulated experiments, a specific focus was made on cross-validation techniques
using data sets generated under open and closed-loop conditions.
• A pilot scale distillation column was used to obtain understanding regarding pragmatic issues of implementing identification and validation techniques on a real system.
The following sections detail some of the key conclusions made considering literature
survey findings and the simulated and pilot scale identification and validation experiments.
8.2.1
Model Accuracy Sensitivity to Identification Experiment
Conditions
Feedback Condition
• Given no output constraints, models identified from open-loop experiments produced more informative data and consequently more accurate models.
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• Open-loop experiment were however found to be more sensitive to experimental
conditions. Thus the improved model accuracy is subject to assuring the correct
specification of identification experiment variables.
• Given output constraints, closed-loop experiments produced more informative data
and identified a larger range of dynamics.
• Closed-loop experiments were found to produce models with improved accuracy at
lower frequencies.
Disturbance Signal Type and Characteristics
• Step disturbance signals under open-loop conditions were found to produce the
models most sensitive to noise when identifying a linear system
• No specific signal type or characteristic was found to allow for significantly larger
measures of model uncertainty when identifying a linear system.
• Experiments where setpoint changes were not incurred were found to produce models with the largest measure of model uncertainty and sensitivity to noise when
idenitfying a non-linear system.
• PRMLS disturbance signals were found to be most informative (closely followed by
PRBS) for open-loop experiments when identifying a linear system.
• PRBS disturbance signals were found to be most informative for closed-loop experiments when identifying a linear system.
• PRBS disturbance signals were found to be the most consistent and informative for
closed and open-loop experiments when identifying a non-linear system.
• Step disturbance signals were found to be the least informative and produced models
with the worst response accuracy when validated against validation sets generated
from any experiments but open-loop step disturbance experiments.
• Step disturbance signals were however useful in extracting information regarding
the extent of non-linear responses and their sensitivity to signal magnitudes.
• Dither signal disturbances were found to be slightly more informative than setpoint
changes for closed-loop experiments when identifying a linear system.
• Effects from increased dither signal magnitudes (signal-to-noise ratio) were mostly
only revealed in closed-loop experiments were no setpoint changes were incurred
when identifying the non-linear system. Increasing the signal magnitude in these
cases increased the model response accuracy.
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• Setpoint changes were found to be vastly more informative than dither signals for
closed-loop experiments when identifying a non-linear system.
• The accuracy of models identified from closed-loop experiments was found to be
dependent on the sequence and direction of setpoint changes made.
• The better lower frequency accuracy of models identified from closed-loop experiments was attributed to information generated by setpoint changes.
• Designing the frequency characteristic of the disturbance signal in accordance with
dominant time constant (i.e. 30 % the dominant time constant) was found to allow
for the identification of models accurate over a larger frequency bandwidth.
8.2.2
Validation Set Bias and Discrimination Sensitivity to Validation Experiment Conditions
Feedback Condition
• Open-loop experiments were found to be most discriminative of models approximating the linear system.
• Closed-loop experiments were found to be most discriminative of models approximating a system revealing non-linear dynamics.
• Validation sets produced from both open and closed-loop experiments indicated
bias towards models obtained from similar feedback conditions.
Disturbance Signal Type and Characteristics
• Step disturbance signals for open-loop experiments were found to produce the most
biased validation sets.
• Designing the frequency characteristic of the disturbance signal in accordance with
dominant time constant (i.e. 30 % the dominant time constant) was found to allow
for the generation of validation data more representative of the true system.
• The bias and prejudice of the validation sets obtained from closed-loop experiments
was found to be sensitive to the sequence and direction of setpoint changes made.
8.2.3
Validation Techniques
The dominant approach used in this work to validate the models was that of crossvalidation. While other methods were used in the case of the linear system, they were
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mostly in efforts to validate the validation results to obtain indications of how bias or
prejudice they were.
Measuring the cross-validation results as response percentage fits was useful in that
quantitative measures were generated representing model accuracy. In some cases however these measures did not allow for sufficient discrimination between models as the
response fits were very similar.
The use of discrete Fourier transforms did prove very valuable in translating the simulated response fit assessments into the frequency domain. From these assessments model
accuracy at specific frequencies and validation set discrimination at specific frequencies
were made evident.
Model validation through residual correlation was found to be useful in discriminating between models based on unmodelled and incorrectly modelled dynamics. However,
this was only the case for approximations of the linear system and when using openloop validation sets. Closed-loop validation sets did produce residual correlations that
suggested a greater capacity to discriminate between models compared to the open-loop
validation sets but this could not always be trusted due to correlation profile corruption
by correlations rooted in the feedback loop.
Of all the validation approaches the higher order correlation tests were expected to
be most informative and most discriminative based on how each linear model approximated the non-linear system. Unfortunately this was not the case to the extent that this
approach was most uninformative.
8.2.4
Linear Approximation of Non-linear Dynamics
As the points made in the previous sections indicate, the closed-loop experiments were
quick to establish themselves as the better alternative over open-loop experiments when
attempting to identify an approximation of a non-linear system. This was found to be
due to the fact that disturbance signal magnitudes and output ranges had large effects on
the dynamics revealed from non-linear systems. This is such that in order to successfully
approximate non-linear dynamics with a linear model, the output signal magnitudes and
ranges and the consequent dynamic regions to be approximated must be clearly stated
and the data for identification and validation definitively generated within this range.
The closed-loop experiment’s capacity to generate more informative data over open-loop
experiments given output constraints make it the best candidate for non-linear system
approximation.
8.2.5
Identification and Validation of a real System
The primary conclusions obtained from identification and validation efforts on the pilot
scale distillation column are such that there are many pragmatic and real problems in
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211
identifying a real system. First and foremost system isolation and defining the identification problem and all the respective variables concerned is not so easily done. Real
systems will always by disturbed by some upstream condition of have some unforeseen
interaction with an unaccounted for variable.
8.2.6
General Conclusions
For the sake of generalising it may be said that the improved persistence of system
excitation of open-loop experiments was found to produce the most accurate models
given well defined and designed disturbance signals. This was especially the case if the
system was well represented by a linear approximation and not restricted by system
output constraints.
In examining the identification of non-linear systems, it was found that if the system
has a high degree of non-linear dynamics and thus can not be well represented by a linear
model, closed-loop experiments were most effective in that they allowed for accurate
implementation of bounds on outputs allowing for data generation over strict regions
where linear approximations accurately represented the non-linear system.
In terms of cross-validation efforts, it is concluded that when identifying a system
that may accurately be represented by a linear model, there is merit in assuring the
validation set is representative of the true system. However, if approximating a nonlinear system with a linear model it is more valuable to determine whether the validation
set is representative of the model purpose.
© University of Pretoria
CHAPTER 9
Recommendations and Further Investigation
In this chapter recommendations are made with respect to system identification
and validation approaches based on results and conclusions presented in this work.
Additionally suggestions into further studies are made in terms of advancing the
successful identification of the pilot scale distillation column and other topics of
relevance that would be worthy of further investigation.
9.1
Recommendations
Given the conclusions stated in the previous chapter, an understanding towards an approach to system identification and validation is evident.
It is recommended that in cases where the linearity of the system is well known and
the purpose of the model is well defined, open-loop experimentation be used to generate
data for validation and identification of models. This must however be coupled with
an extensive design of the characteristics of the disturbance signals with respect to the
model’s intended use.
In cases where the system is expected to be non-linear, or the extent of linearity is unknown, closed-loop experiments are suggested given an accurate definition of the dynamic
range of interest. It is further recommended that when using closed-loop experimental
conditions that setpoint changes be made where possible.
In terms of the validation of identified models, given that the identification and validation of any ’real’ system will more often that not imply that the true dynamics of
the system are unknown, cross-validation is recommended. In which case the same recommendations to identification experiments are made. If the system may be accurately
represented by linear dynamics then use open-loop experiments. If the system being
approximated is non-linear or of an unknown extent of non-linearity, then it becomes
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213
important to establish the purpose of the model and designed closed-loop validation experiments that reflects such a purpose.
9.2
9.2.1
Further Investigation
Identification of Linear Approximations of Non-linear Systems
While this work attempted to uncover some issues of identifying linear models of nonlinear systems and the validation of such systems, there is much to investigate further.
This work concentrated on experimental design variables, specifically feedback conditions and disturbance signals and how these variables affected the approximation of
non-linear systems. There are however many other variables and conditions to further
investigate. Of specific interest would be variables defining the structure of the linear
model used. Investigating the effect of independence between the noise model and the
dynamic model on the accuracy of the linear approximation would probably produce
some interesting results.
In addition to looking at other model structures and orders assessing the model accuracy sensitivities to different model parameter estimation routines would be useful.
Investigating the optimisation and sensitivities of different prediction error minimisation norms and frequency weighting functions could produce greater understandings of
the ideal identification and estimation conditions that would allow for the convergence
towards the best linear approximations.
Furthermore, while section 4.2.1 presented the LTI second order equivalent as the
best linear approximation of a non-linear system on a theoretical basis, investigating the
incorporation of this concept of the best possible linear approximation of a non-linear
system into validation techniques would be extremely valuable.
9.2.2
Identification of the Pilot Scale Distillation Column
With respect to the efforts in modelling the pilot scale distillation column, it must be made
evident that the less intensive, easily adaptable parametric black box model structures
can be most accommodating provided a purpose for the model is kept in mind. Deriving
an accurate model of the column based on first principles will be difficult and would in
any case most likely require the incorporation of adjustable parameters to account for
the many inefficiencies and unique dynamic characteristics.
It is noted however that a black box parametric model does have its limitations in
identifying the column. That is, parametric estimates would be limited to a certain range
of initial conditions. Attempting to develop a parametric model that accurately describes
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214
the dynamics of the column given a mixture besides water and ethanol is being separated
would be very difficult, if at all possible.
In terms of using linear models to represent the column’s dynamics, establishing the
different ranges that are accurately represented by different linear approximations would
be very valuable. However, a non-linear model would do well in identifying the column.
9.2.3
MPCI and Other Recommendations
From the perspective of control, the ultimate model validation would be how well the
model predictive controller performs with an identified model. Extending the understanding of experimental design for models to be used in MPC is extremely relevant to
closed-loop system identification and validation. A more in depth study would also investigate adaptive model predictive control, or MPCI, where system identification and
validation techniques are implemented online in efforts to continuously update the model
being used.
From this further investigation into MPCI it is almost certain that model validation
will need to be emphasised. Validating the different models being generated from a
continuous stream of data while varying the identification variables is an interesting
problem, specifically if using a linear approximation of a non-linear system.
Last but not least, studying the effects of closed-loop identification experiments on
controller performance and further establishing clear boundaries beyond which closedloop identification would no longer be successful would allow for an understanding of the
limitations of implementing this approach.
© University of Pretoria
APPENDIX A
Software and Identified Models
A.1
Description of Software Used and Compiled
All the programs written and used in this dissertation are provided in the attached
DVD in the ’SOFTWARE’ folder. Within the folder there are two subsequent folders,
’FUNDAMENTAL’ and ’EVERYTHING’. The folder entitled ’EVERYTHING’ contains
all the matlab m-file and data files used and generated while the ’FUNDAMENTAL’ folder
contains all the matlab m-files and data files that form the fundamental components of
the work.
The ’EVERYTHING’ folder is further divided into three folders containing files used
and generated when working on linear, non-linear and distillation column respectively.
To illustrate the use of the different functions and software, the fundamental components contained in the ’FUNDAMENTAL’ folder are illustrated by figure A.1 and further
described in following sections.
A.1.1
Signal Generation
The file Inputgen.m was the function used to generate the disturbance signals. The
function calls on the Matlab function idinput to generate the various disturbance signals
given specified signal characteristics i.e. type, magnitude, frequency, etc.
A.1.2
Identification and Validation Data Generation
Once the disturbance signals were identified they were used to generate data for identification and validation. While no programming was required to do this when using the
pilot scale distillation column, the simulated experiments on systems A and B did.
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APPENDIX A. SOFTWARE AND IDENTIFIED MODELS
Function
Signal Generation
Data Generation
File
InputGen
ArxModel/ArxControl/
NarxModel/NarxControl
Description
Used to Generate
Disturbance
Signals:
•
•
•
White Gaussian
PRBS
PRML
Used to Simulate
Open and Closedloop responses to
disturbances for
the linear and
non-linear
systems, systems
A and B
respectively.
216
Model Generation
Model Validation
SimuFit/Frequency/
Correl/HighCorrel
ModelGen
Used to estimate
the ARX models
approximating the
system being
identified from the
identification data
Used to Conduct
the various
validation
techniques:
•
•
•
•
Response Simulation
and Prediction % fit
Frequency Response
Analysis
Response fit frequency
content analysis
Residual correlations
Figure A.1: Illustration of the fundamental software components and the associated matlab
m-files.
With system A being an ARX model and system B being a NARX model, the files
used to generate responses for the two models under open and closed-loop conditions are
defined as follows:
ARXmodel.m : Generates open-loop responses for system A given specifications of
disturbances signals (inputs) and noise.
NARXcontrol.m : Generates closed-loop responses for system A given specifications
of disturbances signals (inputs), noise, setpoint changes, time of setpoint changes,
controller parameters.
ARXmodel.m : Generates open-loop responses for system B given specifications of
disturbances signals (inputs) and noise.
NARXcontrol.m : Generates closed-loop responses for system B given specifications
of disturbances signals (inputs), noise, setpoint changes, time of setpoint changes,
controller parameters..
A.1.3
Model Generation
The m-file ModelGen.m illustrates the use of the arx function used to generate the parametric model given input and output data, model structure specifications and output
delays.
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APPENDIX A. SOFTWARE AND IDENTIFIED MODELS
A.1.4
217
Model Validation
Simulation and Prediction Fit
The m-file SimuFit.m illustrates the use of the compare function used to generate model
responses and response percentage fit values for simulated responses and prediction responses. The last parameter in the function, n, specifies the prediction horizon in number
of simples, if this is omitted the prediction horizon is made very large and thus the response is a simulation.
Frequency Analysis
Recalling that two types of frequency analysis were used. One being the analysis of the
frequency response of the identified model in the same fashion as that generated by bode
diagrams. The other being the analysis of the simulated response in the frequency domain
via the use of discrete Fourier transforms.
The m-file FreqAnal.m illustrates both frequency analysis types.
Residual Correlations
Recalling that the both normal residual correlations and higher order correlations were
done, the m-file Correl.m and HighCorrel.m illustrate respectively. Both correlation mfiles call the correlation generation function crossco.
A.2
Model Parameters of Identified Models
See attached DVD, model parameter tables are given in ModelParameterTables.pdf found
in the ’DOCUMENTATION’ folder.
© University of Pretoria
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