Model Predictive Control Toolbox User`s Guide

Model Predictive Control Toolbox User`s Guide
Model Predictive Control Toolbox™
User's Guide
Alberto Bemporad
Manfred Morari
N. Lawrence Ricker
R2015a
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Model Predictive Control Toolbox™ User's Guide
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Revision History
October 2004
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March 2015
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New for Version 2.1 (Release 14SP1)
Revised for Version 2.2 (Release 14SP2)
Revised for Version 2.2.1 (Release 14SP3)
Revised for Version 2.2.2 (Release 2006a)
Revised for Version 2.2.3 (Release 2006b)
Revised for Version 2.2.4 (Release 2007a)
Revised for Version 2.3 (Release 2007b)
Revised for Version 2.3.1 (Release 2008a)
Revised for Version 3.0 (Release 2008b)
Revised for Version 3.1 (Release 2009a)
Revised for Version 3.1.1 (Release 2009b)
Revised for Version 3.2 (Release 2010a)
Revised for Version 3.2.1 (Release 2010b)
Revised for Version 3.3 (Release 2011a)
Revised for Version 4.0 (Release 2011b)
Revised for Version 4.1 (Release 2012a)
Revised for Version 4.1.1 (Release 2012b)
Revised for Version 4.1.2 (Release R2013a)
Revised for Version 4.1.3 (Release R2013b)
Revised for Version 4.2 (Release R2014a)
Revised for Version 5.0 (Release R2014b)
Revised for Version 5.0.1 (Release 2015a)
Contents
1
Introduction
Specifying Scale Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Defining Scale Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-2
1-2
1-2
Choosing Sample Time and Horizons . . . . . . . . . . . . . . . . . . .
Sample Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prediction Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Control Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-5
1-5
1-6
1-7
Specifying Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Input and Output Constraints . . . . . . . . . . . . . . . . . . . . . . .
Constraint Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-8
1-8
1-9
Tuning Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Initial Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Testing and Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . .
Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1-13
1-13
1-15
1-16
Model Predictive Control Problem Setup
Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Standard Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alternative Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
QP Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Unconstrained Model Predictive Control . . . . . . . . . . . . . . .
2-2
2-2
2-2
2-6
2-7
2-8
2-13
v
3
vi
Contents
Adjusting Disturbance and Noise Models . . . . . . . . . . . . . . .
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output Disturbance Model . . . . . . . . . . . . . . . . . . . . . . . . .
Measurement Noise Model . . . . . . . . . . . . . . . . . . . . . . . . .
Input Disturbance Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Disturbance Rejection Tuning . . . . . . . . . . . . . . . . . . . . . . .
2-14
2-14
2-14
2-15
2-16
2-17
2-17
Custom State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-19
Time-Varying Weights and Constraints . . . . . . . . . . . . . . . .
Time-Varying Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time-Varying Constraints . . . . . . . . . . . . . . . . . . . . . . . . . .
2-20
2-20
2-20
Terminal Weights and Constraints . . . . . . . . . . . . . . . . . . . .
2-22
Constraints on Linear Combinations of Inputs and
Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-25
Manipulated Variable Blocking . . . . . . . . . . . . . . . . . . . . . . .
2-26
QP Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-27
Controller State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . .
Controller State Variables . . . . . . . . . . . . . . . . . . . . . . . . . .
State Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Built-in Steady-State Kalman Gains Calculation . . . . . . . . .
Output Variable Prediction . . . . . . . . . . . . . . . . . . . . . . . . .
2-29
2-29
2-30
2-31
2-33
2-34
Model Predictive Control Simulink Library
MPC Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-2
MPC Controller Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MPC Controller Block Mask . . . . . . . . . . . . . . . . . . . . . . . . .
MPC Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . . .
Connect Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optional Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
3-3
3-4
3-5
3-6
4
Input Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Output Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Look Ahead and Signals from the Workspace . . . . . . . . . . .
Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-9
3-10
3-11
3-12
Generate Code and Deploy Controller to Real-Time
Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-13
Multiple MPC Controllers Block . . . . . . . . . . . . . . . . . . . . . .
Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-14
3-14
3-14
Relationship of Multiple MPC Controllers to MPC Controller
Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Listing the controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Designing the controllers . . . . . . . . . . . . . . . . . . . . . . . . . . .
Defining controller switching . . . . . . . . . . . . . . . . . . . . . . .
Improving prediction accuracy . . . . . . . . . . . . . . . . . . . . . . .
3-15
3-15
3-15
3-15
3-16
Case-Study Examples
Servomechanism Controller . . . . . . . . . . . . . . . . . . . . . . . . . . .
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Control Objectives and Constraints . . . . . . . . . . . . . . . . . . . .
Defining the Plant Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
Controller Design Using MPCTOOL . . . . . . . . . . . . . . . . . . .
Using Model Predictive Control Toolbox Commands . . . . . .
Using MPC Tools in Simulink . . . . . . . . . . . . . . . . . . . . . . .
4-2
4-2
4-3
4-4
4-5
4-19
4-22
Paper Machine Process Control . . . . . . . . . . . . . . . . . . . . . .
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linearizing the Nonlinear Model . . . . . . . . . . . . . . . . . . . . .
MPC Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Controlling the Nonlinear Plant in Simulink . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-27
4-27
4-28
4-30
4-36
4-39
Bumpless Transfer Between Manual and Automatic
Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Open Simulink Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-40
4-40
vii
Define Plant and MPC Controller . . . . . . . . . . . . . . . . . . . .
Configure MPC Block Settings . . . . . . . . . . . . . . . . . . . . . .
Examine Switching Between Manual and Automatic
Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Turn off Manipulated Variable Feedback . . . . . . . . . . . . . .
viii
Contents
4-41
4-42
4-43
4-45
Switching Controller Online and Offline with Bumpless
Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-48
Coordinate Multiple Controllers at Different Operating
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-54
Using Custom Constraints in Blending Process . . . . . . . . .
About the Blending Process . . . . . . . . . . . . . . . . . . . . . . . . .
MPC Controller with Custom Input/Output Constraints . . .
4-61
4-61
4-62
Providing LQR Performance Using Terminal Penalty . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-68
4-73
Real-Time Control with OPC Toolbox . . . . . . . . . . . . . . . . . .
4-74
Simulation and Code Generation Using Simulink Coder . .
4-79
Simulation and Structured Text Generation Using PLC
Coder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-86
Setting Targets for Manipulated Variables . . . . . . . . . . . . .
4-90
Specifying Alternative Cost Function with Off-Diagonal
Weight Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-94
Review Model Predictive Controller for Stability and
Robustness Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-98
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-117
5
Adaptive MPC Design
Adaptive MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
When to Use Adaptive MPC . . . . . . . . . . . . . . . . . . . . . . . . .
Plant Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nominal Operating Point . . . . . . . . . . . . . . . . . . . . . . . . . . .
State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-2
5-2
5-2
5-4
5-4
Model Updating Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Other Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-6
5-6
5-6
Adaptive MPC Control of Nonlinear Chemical Reactor Using
Successive Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-8
Adaptive MPC Control of Nonlinear Chemical Reactor Using
Online Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . .
5-21
6
Explicit MPC Design
Explicit MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-2
Design Workflow for Explicit MPC . . . . . . . . . . . . . . . . . . . . .
Traditional (Implicit) MPC Design . . . . . . . . . . . . . . . . . . . .
Explicit MPC Generation . . . . . . . . . . . . . . . . . . . . . . . . . . .
Explicit MPC Simplification . . . . . . . . . . . . . . . . . . . . . . . . .
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-4
6-4
6-5
6-6
6-6
6-7
Explicit MPC Control of a Single-Input-Single-Output
Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-9
Explicit MPC Control of an Aircraft with Unstable Poles . .
6-21
Explicit MPC Control of DC Servomotor with Constraint on
Unmeasured Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-30
ix
7
Gain Scheduling MPC Design
Gain-Scheduled MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-2
Design Workflow for Gain Scheduling . . . . . . . . . . . . . . . . . .
General Design Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-3
7-3
7-3
Gain Scheduled MPC Control of Nonlinear Chemical
Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-5
Gain Scheduled MPC Control of Mass-Spring System . . . .
8
x
Contents
7-26
Reference for the Design Tool GUI
Working with the Design Tool . . . . . . . . . . . . . . . . . . . . . . . . .
Opening the MPC Design Tool . . . . . . . . . . . . . . . . . . . . . . .
Creating a New MPC Design Task . . . . . . . . . . . . . . . . . . . .
Menu Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Toolbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tree View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Importing a Plant Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Importing a Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exporting a Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Signal Definition View . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plant Models View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Controllers View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation Scenarios List . . . . . . . . . . . . . . . . . . . . . . . . . .
Controller Specifications View . . . . . . . . . . . . . . . . . . . . . . .
Simulation Scenario View . . . . . . . . . . . . . . . . . . . . . . . . . .
8-2
8-2
8-3
8-4
8-6
8-6
8-8
8-12
8-15
8-16
8-20
8-23
8-26
8-29
8-49
Weight Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
Defining the Performance Metric . . . . . . . . . . . . . . . . . . . .
Baseline Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sensitivities and Tuning Advice . . . . . . . . . . . . . . . . . . . . .
Refine Controller Tuning Weights . . . . . . . . . . . . . . . . . . . .
Updating the Controller . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-57
8-59
8-60
8-61
8-63
8-67
Restoring Baseline Tuning . . . . . . . . . . . . . . . . . . . . . . . . .
Modal Dialog Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scenarios for Performance Measurement . . . . . . . . . . . . . . .
8-67
8-68
8-68
Customize Response Plots . . . . . . . . . . . . . . . . . . . . . . . . . . .
Data Markers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Displaying Multiple Scenarios . . . . . . . . . . . . . . . . . . . . . . .
Viewing Selected Variables . . . . . . . . . . . . . . . . . . . . . . . . .
Grouping Variables in a Single Plot . . . . . . . . . . . . . . . . . .
Normalizing Response Amplitudes . . . . . . . . . . . . . . . . . . .
8-69
8-69
8-71
8-72
8-72
8-73
xi
1
Introduction
• “Specifying Scale Factors” on page 1-2
• “Choosing Sample Time and Horizons” on page 1-5
• “Specifying Constraints” on page 1-8
• “Tuning Weights” on page 1-13
1
Introduction
Specifying Scale Factors
In this section...
“Overview” on page 1-2
“Defining Scale Factors” on page 1-2
Overview
Recommended practice includes specification of scale factors for each plant input and
output variable. This is especially important when certain variables will have much
larger or smaller magnitudes than others.
The scale factor should equal (or approximate) the variable’s span. Span is the difference
between its maximum and minimum value in engineering units, i.e., the unit of measure
specified in the plant model. Internally, MPC divides each plant input and output signal
by its scale factor to generate dimensionless signals.
The potential benefits of scaling are as follows:
• Default MPC tuning weights work best when all signals are of order unity.
Appropriate scale factors make the default weights a good starting point for controller
tuning and refinement.
• When choosing cost function weights you can focus on the relative priority of each
term rather than a combination of priority and signal scale.
• Improved numerical conditioning. When values are scaled, round-off errors have less
impact on calculations.
Once you have tuned the controller, changing a scale factor is likely to affect performance
and might necessitate retuning. Best practice is to establish scale factors at the
beginning of controller design and hold them constant thereafter.
Defining Scale Factors
To identify scale factors, estimate the span of each plant input and output variable in
engineering units.
• If the signal has known bounds, use the difference between the upper and lower limit.
1-2
Specifying Scale Factors
• If you do not know the signal bounds, consider running open-loop plant model
simulations in which you vary the inputs over their likely ranges and record output
signal spans.
• If you have no idea, use the default scale factor (=1).
After you create the MPC controller object using the mpc command), set the scale factor
property for each plant input and output variable.
For example, the following commands define a random plant, specifies the signal types,
and creates and specifies a scale factor for each signal.
% Random plant for illustrative purposes: 5 inputs, 3 outputs
Plant = drss(4,3,5);
Plant.InputName = {'MV1','UD1','MV2','UD2','MD'};
Plant.OutputName = {'UO','MO1','MO2'};
% Example signal spans
Uspan = [2, 20, 0.1, 5, 2000];
Yspan = [0.01, 400, 75];
% Example signal type specifications
iMV = [1 3];
iMD = 5;
iUD = [2 4]; iDV = [iMD, iUD];
Plant = setmpcsignals(Plant, 'MV', iMV, 'MD', iMD, 'UD', iUD, ...
'MO', [2 3], 'UO', 1);
Plant.d(:,iMV) = 0;
% MPC requires zero direct MV feed-through
% Controller object creation.
MPCobj = mpc(Plant, 0.3);
Ts = 0.3 for illustration.
% Override default scale factors using specified spans
for i = 1:2
MPCobj.MV(i).ScaleFactor = Uspan(iMV(i));
end
% NOTE: DV sequence is MD followed by UD
for i = 1:3
MPCobj.DV(i).ScaleFactor = Uspan(iDV(i));
end
for i = 1:3
MPCobj.OV(i).ScaleFactor = Yspan(i);
end
Once you have set the scale factors and have begun to tune controller performance, hold
the scale factors constant.
1-3
1
Introduction
See Also
mpc
Related Examples
•
Using Scale Factor to Facilitate Weight Tuning
More About
•
1-4
“Choosing Sample Time and Horizons”
Choosing Sample Time and Horizons
Choosing Sample Time and Horizons
In this section...
“Sample Time” on page 1-5
“Prediction Horizon” on page 1-6
“Control Horizon” on page 1-7
Sample Time
Duration
Recommended practice is to choose the control interval duration (controller property Ts)
initially, and then hold it constant as you tune other controller parameters. If it becomes
obvious that the original choice was poor, you can revise Ts. If you do so, you might then
need to re-tune other settings.
Qualitatively, as Ts decreases, rejection of unknown disturbance usually improves
and then plateaus. The Ts value at which performance plateaus depends on the plant
dynamic characteristics.
On the other hand, as Ts becomes small, the computational effort increases dramatically.
Thus, the optimal choice is a balance of performance and computational effort.
In Model Predictive Control, the prediction horizon, p is also an important consideration.
If one chooses to hold the prediction horizon duration (the product p*Ts) constant, p
must vary inversely with Ts. Many array sizes are proportional to p. Thus, as p increases
memory requirements increase, as does the time needed to solve each QP.
Consider the following when choosing Ts:
• Rough guideline: establish the minimum desired closed-loop response time and set Ts
between 0.1 and 0.25 of this.
• Run at least one simulation test to see whether unmeasured disturbance rejection
improves significantly when Ts is halved. If so, consider revising Ts.
• For process control, Ts >> 1 s is common, especially when MPC supervises lower-level
single-loop controllers. Other applications (e.g., automotive, aerospace) can require
Ts < 1 s. If the time needed to solve the QP in real time exceed the desired control
interval, consider the Explicit MPC option.
1-5
1
Introduction
• Plants with delay: the number of state variables needed to model delays is inversely
proportional to Ts.
• Open-loop unstable plants: if p*Ts is too large, such that the plant step responses
become infinite during this amount of time, key parameters needed for MPC
calculations become undefined, generating an error message.
Units
The controller inherits its time unit from the plant model. Specifically, the controller uses
the TimeUnit property of the plant model LTI object. This property defaults to seconds.
Prediction Horizon
Suppose the current control interval is k. The prediction horizon, p, is the number of
future control intervals the MPC controller must evaluate by prediction when optimizing
its MVs at control interval k.
Tips
• Recommended practice is to choose p early in the controller design and then hold it
constant while tuning other controller settings, such as the cost function weights. In
other words, you should not use p adjustments for controller tuning. Rather, the value
of p should be such that the controller is internally stable and anticipates constraint
violations early enough to allow corrective action.
• If the desired closed-loop response time is T and the control interval is Ts, try p such
that T ≈ pTs.
• Plant delays impose a lower bound on the possible closed-loop response times. Choose
p accordingly. Use the review command to check for a violation of this condition.
• Recommended practice is to increase p until further increases have a minor impact
on performance. If the plant is open-loop unstable, the maximum possible p is the
number of control intervals required for the plant’s open-loop step response to become
infinite. p > 50 is rarely necessary unless Ts is too small.
• Unfavorable plant characteristics combined with a small p can generate an internally
unstable controller. Use review to check for this condition, and increase p if possible.
If p is already large, consider the following.
• Increase Ts.
• Increase the cost function weights on MV increments.
1-6
Choosing Sample Time and Horizons
• Modify the control horizon and/or use MV blocking (see “Manipulated Variable
Blocking”).
• Use a small p in combination with terminal weighting to approximate LQR
behavior. (See “Terminal Weights and Constraints”.)
Control Horizon
The control horizon, m, is the number of MV moves to be optimized at control interval k.
The control horizon falls between 1 and the prediction horizon p. The default is m = 2.
Regardless of your choice for m, when the controller operates, the optimized MV move at
the beginning of the horizon is used and any others are discarded.
Tips
Reasons to keep m << p are as follows:
• Small m means fewer variables to compute in the QP solved at each control interval,
which promotes faster computations.
• If the plant includes delays, m < p is essential. Otherwise, some MV moves might not
affect any of the plant outputs prior to the end of the prediction horizon, leading to a
singular QP Hessian matrix. Use review to check for a violation of this condition.
• Small m promotes (but does not guarantee) an internally-stable controller.
More About
•
“Specifying Constraints”
1-7
1
Introduction
Specifying Constraints
In this section...
“Input and Output Constraints” on page 1-8
“Constraint Softening” on page 1-9
Input and Output Constraints
By default, when you create a controller object using the mpc command, no constraints
exist. To include a constraint, set the appropriate controller property. The following table
summarizes the controller properties used to define most MPC Toolbox constraints. (MV
= plant manipulated variable; OV = plant output variable; MV increment = u(k) – u(k –
1).
To include this constraint
Set this controller property
Soften constraint by setting
Lower bound on ith MV
MV(i).Min > -Inf
MV(i).MinECR > 0
Upper bound on ith MV
MV(i).Max < Inf
MV(i).MaxECR > 0
Lower bound on ith OV
OV(i).Min > -Inf
OV(i).MinECR > 0
Upper bound on ith OV
OV(i).Max < Inf
OV(i).MaxECR > 0
Lower bound on ith MV
increment
MV(i).RateMin > -Inf
MV(i).RateMinECR > 0
Upper bound on ith MV
increment
MV(i).RateMax < Inf
MV(i).RateMaxECR > 0
See “Constraints” for the equations describing the corresponding constraints.
Tips
For MV bounds:
• Include known physical limits on the plant MVs as hard MV bounds.
• Include MV increment bounds when there is a known physical limit on the rate of
change, or your application requires you to prevent large increments for some other
reason.
• Do not include both hard MV bounds and hard MV increment bounds on the same
MV. They might conflict. If both types of bounds are important, soften one.
1-8
Specifying Constraints
For OV bounds:
• Do not include OV bounds unless they are essential to your application. As an
alternative to setting an OV bound, you can define an OV reference and set its cost
function weight to keep the OV close to its reference value.
• All OV constraints should be softened.
• Consider leaving the OV unconstrained for some prediction horizon steps. See “TimeVarying Weights and Constraints”.
• Consider a time-varying OV constraint that is easy to satisfy early in the horizon,
gradually tapering to a more strict constraint. See “Time-Varying Weights and
Constraints”.
• Do not include OV constraints that are impossible to satisfy. Even if soft, such
constraints can cause unexpected controller behavior. For example, consider a
SISO plant with five sampling periods of delay. An OV constraint prior to the sixth
prediction horizon step is, in general, impossible to satisfy. You can use the review
command to check for such impossible constraints, and use a time-varying OV bound
instead. See “Time-Varying Weights and Constraints”.
Constraint Softening
Hard constraints are constraints that must be satisfied by the quadratic programming
(QP) solution. If it is mathematically impossible to satisfy a hard constraint at a given
control interval, k, the QP is infeasible. In this case, the controller returns an error
status, and sets the manipulated variables (MVs) to u(k) = u(k–1), i.e., no change. If the
condition leading to infeasibility is not resolved, infeasibility can continue indefinitely,
leading to a loss of control.
Disturbances and prediction errors are inevitable in practice. Thus, a constraint violation
could occur in the plant even though the controller predicts otherwise. A feasible QP
solution does not guarantee that all hard constraints will be satisfied when the optimal
MV is used in the plant.
If the only constraints in your application are bounds on MVs, the MV bounds can be
hard constraints, as they are by default. MV bounds alone cannot cause infeasibility. The
same is true when the only constraints are on MV increments.
On the other hand, a hard MV bound in combination with a hard MV increment
constraint can lead to infeasibility. For example, an upset or operation under manual
control could cause the actual MV used in the plant to exceed the specified bound during
1-9
1
Introduction
interval k–1. If the controller is in automatic during interval k, it must return the MV
to a value within the hard bound. If the MV exceeds the bound by too much, the hard
increment constraint might make correcting the bound violation in the next interval
impossible.
When there are hard constraints on plant outputs or hard custom constraints (on linear
combinations of plant inputs and outputs) and the plant is subject to disturbances, QP
infeasibility is a distinct possibility.
All MPC toolbox constraints (except slack variable nonnegativity) can be soft. When
a constraint is soft, the controller may deem an MV optimal even though it predicts a
violation of that constraint. If all plant output, MV increment, and custom constraints
are soft (as they are by default), QP infeasibility does not occur. However, controller
performance might be sub-standard.
To soften a constraint, set the corresponding ECR value to a positive value (zero implies
a hard constraint). The larger the ECR value, the more likely the controller will deem
it optimal to violate the constraint in order to satisfy your other performance goals. The
Model Predictive Control Toolbox™ software provides default ECR values but, as for
the cost function weights, you might need to tune the ECR values in order to achieve
acceptable performance.
To better understand how constraint softening works, suppose your cost function uses
wui, j = wiD, uj = 0 , giving both the MV and MV increments zero weight in the cost function.
Only the output reference tracking and constraint violation terms are nonzero. In this
case, the cost function is:
2
Ï wy
¸
Ô i, j È
Ô
2
J ( zk ) =
Ì y Î rj ( k + i|k ) - y j ( k + i|k ) ˘˚ ˝ + r k .
s
Ô˛
j =1i =1 Ô
Ó j
ny p
ÂÂ
Suppose you have also specified hard MV bounds with V uj ,min ( i ) = 0 and V uj ,max ( i) = 0 .
Then these constraints simplify to:
u j,min ( i )
suj
1-10
£
u j ( k + i - 1|k )
suj
£
u j,max ( i )
suj
, i = 1 : p,
j = 1 : nu .
Specifying Constraints
Thus, the slack variable, ∊k, no longer appears in the above equations. You have also
specified soft constraints on plant outputs with V jy,min ( i ) > 0 and V jy,max ( i) > 0 .
y j ,min ( i )
s yj
-
y
k V j ,min
(i) £
y j ( k + i|k )
s yj
£
y j ,max ( i )
s yj
+
y
kV j, max
( i ) , i = 1 : p,
j = 1 : ny .
Now suppose a disturbance has pushed a plant output above its specified upper bound,
but the QP with hard output constraints would be feasible, i.e., all constraint violations
could be avoided in the QP solution. The QP involves a trade-off between output
reference tracking and constraint violation. The slack variable, ∊k, must be nonnegative.
Its appearance in the cost function discourages, but does not prevent, an optimal ∊k > 0.
A larger ρ weight, however, increases the likelihood that the optimal ∊k will be small or
∊
zero.
If the optimal ∊k > 0, at least one of the bound inequalities must be active (at equality). A
relatively large V jy,max ( i) makes it easier to satisfy the constraint with a small
case,
. In that
∊k
y j ( k + i|k )
s yj
can be larger, without exceeding
y j ,max ( i )
s yj
+
y
k V j ,max (i).
Notice that V jy,max ( i) does not set an upper limit on the constraint violation. Rather, it is
a tuning factor determining whether a soft constraint is easy or difficult to satisfy.
Tips
• Use of dimensionless variables simplifies constraint tuning. Define appropriate scale
factors for each plant input and output variable. See “Specifying Scale Factors” on
page 1-2.
1-11
1
Introduction
• Use the ECR parameter associated with each constraint (see above table) to indicate
the relative magnitude of a tolerable violation. Rough guidelines are as follows:
• 0: no violation allowed (hard constraint)
• 0.05: very small violation allowed (nearly hard)
• 0.2: small violation allowed (quite hard)
• 1: average softness
• 5: greater-than-average violation allowed (quite soft)
• 20: large violation allowed (very soft)
• Use the controller’s overall constraint softening parameter (controller object property:
Weights.ECR) to penalize a tolerable soft constraint violation relative to the other
cost function terms. Set the Weights.ECR property such that the corresponding
penalty is 1 to 2 orders of magnitude greater than the typical sum of the other three
cost function terms. If constraint violations seem too large during simulation tests, try
increasing Weights.ECR by a factor of 2 to 5.
Be aware, however, that an excessively large Weights.ECR distorts MV optimization,
leading to inappropriate MV adjustments when constraint violations occur. To check
for this, display the cost function value during simulations. If its magnitude increases
by more than 2 orders of magnitude when a constraint violation occurs, consider
decreasing Weights.ECR.
• Disturbances and prediction errors will lead to unexpected constraint violations in a
real system. Attempting to prevent this by making constraints harder often degrades
controller performance.
See Also
review
More About
1-12
•
“Time-Varying Weights and Constraints”
•
“Terminal Weights and Constraints”
•
“Optimization Problem”
Tuning Weights
Tuning Weights
In this section...
“Initial Tuning” on page 1-13
“Testing and Refinement” on page 1-15
“Robustness” on page 1-16
A Model Predictive Controller design usually requires some tuning of the cost function
weights. This topic provides tuning tips. See “Optimization Problem” for details on the
cost function equations.
Initial Tuning
• Prior to tuning the cost function weights, specify scale factors for each plant input
and output variable “Specifying Scale Factors” on page 1-2. Hold these scale factors
constant as you tune the controller.
• At any point during tuning, use the sensitivity and review commands to obtain
diagnostic feedback. The sensitivity command is specifically intended to help with
cost function weight selection.
• Change a weight by setting the appropriate controller property, as follows:
To change this weight
Set this controller property
Array size
OV reference tracking (wy) Weights.OV
p-by-ny
MV reference tracking (wu) Weights.MV
p-by-nu
MV increment suppression Weights.MVRate
(wΔu)
p-by-nu
Here, MV is a plant manipulated variable, and nu is the number of MVs. OV is a plant
output variable, and ny is the number of OVs. Finally,p is the number of steps in the
prediction horizon.
If a weight array contains n < p rows, the controller duplicates the last row to obtain
a full array of p rows. The default (n = 1) minimizes the number of parameters to be
tuned, and is therefore recommended. See “Time-Varying Weights and Constraints” for
an alternative.
1-13
1
Introduction
Tips for Setting OV Weights
• Considering the ny OVs, suppose that nyc must be held at or near a reference value
(setpoint). If the ith OV is not in this group, set Weights.OV(:,i) = 0.
• If nu ≥ nyc, it is usually possible to achieve zero OV tracking error at steady state,
provided that at least nyc MVs are not at a bound. The default Weights.OV =
ones(1,ny) is a good starting point in this case.
If nu > nyc, however, you have excess degrees of freedom. Unless you take preventive
measures, therefore, the MVs may drift even when the OVs are near their reference
values.
• The most common preventive measure is to define reference values (targets) for
the number of excess MVs you have, nu – nyc MVs. Such targets can represent
economically or technically desirable steady-state values.
• An alternative measure is to set w∆u > 0 for at least nu – nyc MVs to discourage the
controller from changing them.
• If nu < nyc, you do not have enough degrees of freedom to keep all required OVs
at a setpoint. In this case, consider prioritizing reference tracking. To do so, set
Weights.OV(:,i) > 0 to specify the priority for the ith OV. Rough guidelines for
this are as follows:
• 0.05 Low priority: large tracking error acceptable
• 0.2 Below-average priority
• 1 Average priority – the default. Use this if nyc = 1.
• 5 Above average priority
• 20 High priority: small tracking error desired
Tips for Setting MV Weights
By default, Weights.MV = zeros(1,nu). If some MVs have targets, the corresponding
MV reference tracking weights must be nonzero. Otherwise, the targets are ignored. If
the number of MV targets is less than (nu – nyc), try using the same weight for each. A
suggested value is 0.2, the same as below-average OV tracking. This value allows the
MVs to move away from their targets temporarily to improve OV tracking.
Otherwise, the MV and OV reference tracking goals are likely to conflict. Prioritize
by setting the Weights.MV(:,i) values in a manner similar to that suggested for
1-14
Tuning Weights
Weights.OV (see above). Typical practice sets the average MV tracking priority lower
than the average OV tracking priority (e.g., 0.2 < 1).
If the ith MV does not have a target, set Weights.MV(:,i) = 0 (the default).
Tips for Setting MVRate Weights
• By default, Weights.MVRate = 0.1*ones(1,nu). The reasons for this default
include:
• If the plant is open-loop stable, large increments are unnecessary and probably
undesirable. For example, when model predictions are imperfect, as is always the
case in practice, more conservative (smaller) increments usually provide more
robust controller performance, although also but poorer reference tracking.
• These values force the QP Hessian matrix to be positive-definite, such that the QP
has a unique solution if no constraints are active.
To encourage the controller to use even smaller increments for the ith MV, increase
the Weights.MVRate(:,i) value.
• If the plant is open-loop unstable, you might need to decrease the average
Weight.MVRate value to allow sufficiently rapid response to upsets.
Tips for Setting ECR Weights
See “Constraint Softening” on page 1-9 for tips regarding the Weights.ECR property.
Testing and Refinement
To focus on tuning individual cost function weights, perform closed-loop simulation tests
under the following conditions:
• No constraints.
• No prediction error. The controller’s prediction model should be identical to the plant
model.
Both the design tool and the sim function provide the option to simulate under these
conditions.
Use changes in the reference and measured disturbance signals (if any) to force a
dynamic response. Based on the results of each test, consider changing the magnitudes of
selected weights.
1-15
1
Introduction
One suggested approach is to use constant Weights.OV(:,i) = 1 to signify “average
OV tracking priority,” and adjust all other weights to be relative to this value. Use the
sensitivity command for guidance. Use the review command to check for typical
tuning issues, such as lack of closed-loop stability.
See “Adjusting Disturbance and Noise Models” for tests focusing on the controller’s
ability to reject arbitrary disturbances.
Robustness
Once you have weights that work well under the above conditions, check for sensitivity to
prediction error. There are several ways to do so:
• If you have a nonlinear plant model of your system, such as a Simulink® model,
simulate the closed-loop performance at operating points other than that for which
the LTI prediction model applies.
• Alternatively, run closed-loop simulations in which the LTI model representing the
plant differs (such as in structure or parameter values) from that used at the MPC
prediction model. Both the design tool and the sim function provide the option to
simulate under these conditions.
If controller performance seems to degrade significantly in comparison to tests with
no prediction error, for an open-loop stable plant, consider making the controller less
aggressive. In the design tool, you can do so using the performance/robustness trade-off
slider adjustment. At the command line, you can make the following changes to increase
controller aggressiveness:
• Increase all Weight.MVRate values by a multiplicative factor of order 2.
• Decrease all Weight.OV and Weight.MV values by dividing by the same factor.
After making these adjustments, reevaluate performance with and without prediction
error.
• If both are now acceptable, stop tuning the weights
• If there is improvement but still too much degradation with model error, repeat the
above weight adjustments.
• If the change does noticeably improve performance, restore the original weights and
focus on state estimator tuning (see “Adjusting Disturbance and Noise Models”.
Finally, if tuning changes do not provide adequate robustness, consider one of the
following options:
1-16
Tuning Weights
• Adaptive MPC control
• Gain-scheduled MPC control
Related Examples
•
Tuning Controller Weights
•
“Setting Targets for Manipulated Variables”
More About
•
“Optimization Problem”
•
“Specifying Constraints” on page 1-8
•
“Adjusting Disturbance and Noise Models”
1-17
2
Model Predictive Control Problem
Setup
• “Optimization Problem” on page 2-2
• “Adjusting Disturbance and Noise Models” on page 2-14
• “Custom State Estimation” on page 2-19
• “Time-Varying Weights and Constraints” on page 2-20
• “Terminal Weights and Constraints” on page 2-22
• “Constraints on Linear Combinations of Inputs and Outputs” on page 2-25
• “Manipulated Variable Blocking” on page 2-26
• “QP Solver” on page 2-27
• “Controller State Estimation” on page 2-29
2
Model Predictive Control Problem Setup
Optimization Problem
In this section...
“Overview” on page 2-2
“Standard Cost Function” on page 2-2
“Alternative Cost Function” on page 2-6
“Constraints” on page 2-7
“QP Matrices” on page 2-8
“Unconstrained Model Predictive Control” on page 2-13
Overview
Model Predictive Control solves an optimization problem – specifically, a quadratic
program (QP) – at each control interval. The solution determines the manipulated
variables (MVs) to be used in the plant until the next control interval.
This QP problem includes the following features:
• The objective, or “cost”, function — A scalar, nonnegative measure of controller
performance to be minimized.
• Constraints — Conditions the solution must satisfy, such as physical bounds on MVs
and plant output variables.
• Decision — The MV adjustments that minimizes the cost function while satisfying the
constraints.
The following sections describe these features in more detail.
Standard Cost Function
The standard cost function is the sum of four terms, each focusing on a particular aspect
of controller performance, as follows:
J ( zk ) = J y ( zk ) + Ju ( zk ) + JD u ( zk ) + J
( zk ) .
Here, zk is the QP decision. As described below, each term includes weights that help
you balance competing objectives. MPC controller provides default weights but you will
usually need to adjust them to tune the controller for your application.
2-2
Optimization Problem
Output Reference Tracking
In most applications, the controller must keep selected plant outputs at or near specified
reference values. MPC controller uses the following scalar performance measure:
2
Ï wy
¸
Ô i, j È
Ô
J y ( zk ) =
Ì y Î rj ( k + i|k ) - y j ( k + i| k )˘˚ ˝ .
s
Ô˛
j =1 i =1 Ô
Ó j
ny
p
ÂÂ
Here,
• k — Current control interval.
• p — Prediction horizon (number of intervals).
• ny — Number of plant output variables.
• zk — QP decision, given by:
T
È
zT
k = Îu( k| k)
u( k + 1 |k) T
L u( k + p - 1| k) T
˘
k ˚.
• yj(k+i|k) — Predicted value of jth plant output at ith prediction horizon step, in
engineering units.
• rj(k+i|k) — Reference value for jth plant output at ith prediction horizon step, in
engineering units.
•
•
s yj — Scale factor for jth plant output, in engineering units.
wiy, j — Tuning weight for jth plant output at ith prediction horizon step
(dimensionless).
The values ny, p, s yj , and wiy, j are controller specifications, and are constant. The
controller receives rj(k+i|k) values for the entire prediction horizon. The controller
uses the state observer to predict the plant outputs. At interval k, the controller state
estimates and MD values are available. Thus, Jy is a function of zk only.
Manipulated Variable Tracking
In some applications, i.e. when there are more manipulated variables than plant outputs,
the controller must keep selected manipulated variables (MVs) at or near specified target
values. MPC controller uses the following scalar performance measure:
2-3
2
Model Predictive Control Problem Setup
Ju ( zk ) =
nu p-1 Ï wu
Ô i, j
Ì u
j =1 i =0 Ô
Ó sj
ÂÂ
2
¸
È u j ( k + i|k ) - u j,target ( k + i|k ) ˘ Ô˝ .
Î
˚
Ô˛
Here,
• k — Current control interval.
• p — Prediction horizon (number of intervals).
• nu — Number of manipulated variables.
• zk — QP decision, given by:
T
È
zT
k = Îu( k| k)
u( k + 1 |k) T
L u( k + p - 1| k) T
˘
k ˚.
• uj,target(k+i|k) — Target value for jth MV at ith prediction horizon step, in engineering
units.
•
•
suj — Scale factor for jth MV, in engineering units.
wui, j — Tuning weight for jth MV at ith prediction horizon step (dimensionless).
The values nu, p, suj , and wui, j are controller specifications, and are constant. The
controller receives uj,target(k+i|k) values for the entire horizon. The controller uses the
state observer to predict the plant outputs. Thus, Ju is a function of zk only.
Manipulated Variable Move Suppression
Most applications prefer small MV adjustments (moves). MPC uses the following scalar
performance measure:
JD u ( zk ) =
Here,
Here,
2-4
nu p-1 Ï w Du
Ô i, j
Ì u
j =1 i=0 Ô
Ó sj
ÂÂ
2
¸
Èu j ( k + i|k ) - u j ( k + i - 1|k ) ˘ Ô˝ .
Î
˚
Ô˛
Optimization Problem
• k — Current control interval.
• p — Prediction horizon (number of intervals).
• nu — Number of manipulated variables.
• zk — QP decision, given by:
T
È
zT
k = Îu( k| k)
•
•
u( k + 1 |k) T
L u( k + p - 1| k) T
˘
k ˚.
suj — Scale factor for jth MV, in engineering units.
wDi, uj — Tuning weight for jth MV movement at ith prediction horizon step
(dimensionless).
The values nu, p, suj , and wDi, uj are controller specifications, and are constant. u(k–1|k) =
u(k–1), which are the known MVs from the previous control interval. JΔu is a function of
zk only.
In addition, a control horizon m < p (or MV blocking) constrains certain MV moves to be
zero.
Constraint Violation
In practice, constraint violations might be unavoidable. Soft constraints allow a feasible
QP solution under such conditions. MPC controller employs a dimensionless, nonnegative
slack variable, εk, which quantifies the worst-case constraint violation. (See “Constraints”
on page 2-7) The corresponding performance measure is:
J
( zk ) = r
2
k.
Here,
• zk — QP decision, given by:
T
È
zT
k = Îu( k| k)
u( k + 1 |k) T
L u( k + p - 1| k) T
˘
k ˚.
• εk — Slack variable at control interval k (dimensionless).
2-5
2
Model Predictive Control Problem Setup
• ρ — Constraint violation penalty weight (dimensionless).
∊
Alternative Cost Function
You can elect to use the following alternative to the standard cost function:
J ( zk ) =
p-1
 {ÈÎ eTy ( k + i) Qe y ( k + i)˘˚ + ÈÎeTu ( k + i) Rueu ( k + i)˘˚ + ÈÎ DuT (k + i ) RDuDu (k + i)˘˚} + r
i =0
Here, Q (ny-by-ny), Ru, and RΔu (nu-by-nu) are positive-semi-definite weight matrices, and:
e y ( i + k ) = S-y 1 ÈÎ r ( k + i + 1|k ) - y(k + i + 1| k) ˘˚
eu ( i + k ) = Su-1 ÈÎ utarget ( k + i|k ) - u( k + i | k) ˘˚
D u ( k + i ) = Su-1 ÈÎ u ( k + i|k ) - u( k + i - 1 | k) ˘˚ .
Also,
• Sy — Diagonal matrix of plant output variable scale factors, in engineering units.
• Su — Diagonal matrix of MV scale factors in engineering units.
• r(k+1|k) — ny plant output reference values at the ith prediction horizon step, in
engineering units.
• y(k+1|k) — ny plant outputs at the ith prediction horizon step, in engineering units.
• zk — QP decision, given by:
T
È
zT
k = Îu( k| k)
u( k + 1 |k) T
L u( k + p - 1| k) T
˘
k ˚.
• utarget(k+i|k) — nu MV target values corresponding to u(k+i|k), in engineering units.
Output predictions use the state observer, as in the standard cost function.
The alternative cost function allows off-diagonal weighting, but requires the weights to
be identical at each prediction horizon step.
The alternative and standard cost functions are identical if the following conditions hold:
2-6
2
k.
Optimization Problem
•
The standard cost functions employs weights wiy, j , wui, j , and wDi, uj that are constant
with respect to the index, i = 1:p.
• The matrices Q, Ru, and RΔu are diagonal with the squares of those weights as the
diagonal elements.
Constraints
Certain constraints are implicit. For example, a control horizon m < p (or MV blocking)
forces some MV increments to be zero, and the state observer used for plant output
prediction is a set of implicit equality constraints. Explicit constraints that you can
configure are described below.
Bounds on Plant Outputs, MVs, and MV Increments
The most common MPC constraints are bounds, as follows.
y j ,min ( i )
y
sj
u j ,min ( i )
suj
D u j ,min ( i )
suj
- k V jy,min ( i ) £
- kV uj,min ( i) £
u
- kV Dj,min
( i) £
y j ( k + i|k )
y
sj
£
y j, max ( i )
u j ( k + i - 1|k )
suj
y
sj
£
Du j ( k + i - 1|k )
suj
+ k V jy,m ax (i),
u j ,max ( i )
suj
£
j = 1 : ny
+ k V ju,max (i), i = 1 : p,
Du j ,max ( i )
suj
i = 1 : p,
j = 1 : nu
+ k V jD,umax (i), i = 1 : p,
j = 1 : nu .
Here, the V parameters (ECR values) are dimensionless controller constants analogous to
the cost function weights but used for constraint softening (see “Constraint Softening”).
Also,
• ∊k — Scalar QP slack variable (dimensionless) used for constraint softening.
•
•
s yj — Scale factor for jth plant output, in engineering units.
suj — Scale factor for jth MV, in engineering units.
• yj,min(i), yj,max(i) — lower and upper bounds for jth plant output at ith prediction
horizon step, in engineering units.
2-7
2
Model Predictive Control Problem Setup
• uj,min(i), uj,max(i) — lower and upper bounds for jth MV at ith prediction horizon step,
in engineering units.
• Δuj,min(i), Δuj,max(i) — lower and upper bounds for jth MV increment at ith prediction
horizon step, in engineering units.
Except for the slack variable non-negativity condition, all of the above constraints
are optional and are inactive by default (i.e., initialized with infinite limiting values).
To include a bound constraint, you must specify a finite limit when you design the
controller.
QP Matrices
This section describes the matrices associated with the model predictive control
optimization problem described in “Optimization Problem” on page 2-2.
Prediction
Assume that the disturbance models described in “Input Disturbance Model” is unit gain,
for example, d(k)=nd(k) is a white Gaussian noise). You can denote this problem as
ÈA
È x˘
x ¨ Í ˙,A ¨ Í
x
Î d˚
Î0
È Bd D˘
È Bu ˘
È Bv˘
Bd C˘
˙ , Bu ¨ Í ˙ , Bv ¨ Í ˙ , Bd ¨ Í
˙ C ¨ ÈÎC
A ˚
Î0 ˚
Î0 ˚
Î B ˚
Dd C ˘˚
Then, the prediction model is:
x(k+1) = Ax(k) +Buu(k) +Bvv(k)+Bdnd(k)
y(k) = Cx(k) +Dvv(k) +Ddnd(k)
Next, consider the problem of predicting the future trajectories of the model performed at
time k=0. Set nd(i)=0 for all prediction instants i, and obtain
i-1
h
È
Ê Ê
ˆ˘
ˆ
y( i| 0) = C Í Ai x(0) +
Ai -1 Á Bu Á u(- 1) +
Du( j) ˜ + Bvv( h) ˜ ˙ + Dv v( i)
˜
Í
Á Á
˜˙
h =0
j =0
¯
Ë Ë
¯˚
Î
Â
This equation gives the solution
2-8
Â
Optimization Problem
È y(1) ˘
È Du(0) ˘
È v(0) ˘
Í L ˙ = S x(0) + S u(-1) + S Í L
˙+H Í L ˙
x
u1
uÍ
vÍ
Í
˙
˙
˙
ÍÎ y( p) ˙˚
ÍÎ Du( p - 1) ˙˚
ÍÎ v( p) ˙˚
where
CBu
È
˘
È CA ˘
ÍCB + CAB ˙
Í 2˙
u˙
Í u
Í CA ˙
pn y¥ nx
Í
˙ Œ ¬ pny ¥ nu
L
Sx = Í
Œ
¬
,
S
=
u1
Í p-1
˙
L ˙
Í
˙
Í
˙
h
p
ÍCA ˙
CA Bu ˙
Í
Î
˚
ÍÎ h =0
˙˚
CBu
0
L
0 ˘
È
Í
˙
CB
+
CAB
CB
L
0 ˙
u
u
Í u
L
L
L L ˙ Œ ¬ pny¥ pnu
Su = Í
Í
˙
p-2
Í p-1
˙
h
h
CA Bu
CA Bu L CBu ˙
Í
ÍÎ h =0
˙˚
h =0
Dv
0
L 0 ˘
È CBv
Í CAB
CBv
Dv
L 0 ˙˙
v
pn ¥( p+1) nv
Í
Hv =
Œ¬ y
.
Í
L
L
L
L L˙
Í
˙
ÍÎ CA p -1 Bv CA p-2 Bv CA p-3 Bv L Dv ˙˚
Â
Â
Â
Optimization Variables
Let m be the number of free control moves, and let z= [z0; ...; zm–1]. Then,
È Du(0) ˘
Í L
˙ =J
M
Í
˙
ÍÎDu( p - 1) ˙˚
È z0 ˘
Í L ˙
Í
˙
ÍÎ zm-1 ˙˚
where JM depends on the choice of blocking moves. Together with the slack variable ɛ,
vectors z0, ..., zm–1 constitute the free optimization variables of the optimization problem.
In the case of systems with a single manipulated variables, z0, ..., zm–1 are scalars.
2-9
2
Model Predictive Control Problem Setup
Consider the blocking moves depicted in the following graph.
Blocking Moves: Inputs and Input Increments for moves = [2 3 2]
This graph corresponds to the choice moves=[2 3 2], or, equivalently,
u(0)=u(1), u(2)=u(3)=u(4), u(5)=u(6), Δ u(0)=z0, Δ u(2)=z1, Δ u(5)=z2, Δ u(1)=Δ u(3)=Δ
u(4)=Δ u(6)=0.
Then, the corresponding matrix JM is
JM
2-10
ÈI
Í0
Í
Í0
Í
= Í0
Í0
Í
Í0
Í0
Î
0˘
0 ˙˙
0˙
˙
0˙
0 0˙
˙
0 I˙
0 0 ˙˚
0
0
I
0
Optimization Problem
Cost Function
• “Standard Form” on page 2-11
• “Alternative Cost Function” on page 2-12
Standard Form
The function to be optimized is
T
Ê È u( 0) ˘ È utarget ( 0) ˘ ˆ
Ê È u(0) ˘ È ut arget (0) ˘ ˆ È D u(0) ˘ T
È Du( 0) ˘
ÁÍ
Í
˙˜
˙˜ Í
˙
2ÁÍ
˙ -Í
˙ W2 Í L
˙
J ( z, e ) = Á Í L ˙ - Í
L
W
L
L
+
L
˙˜
u ÁÍ
˙˜ Í
Du Í
˙ Í
˙
˙
Á Íu( p - 1) ˙ Íu
˜
Á
˜
˙
Íu( p - 1) ˙˚ Íutarget ( p - 1) ˙ ÍÎ D u( p - 1) ˙˚
ÍÎDu( p - 1) ˙˚
˚ Î target ( p - 1) ˚ ¯
Î
˚¯
ËÎ
ËÎ
T
Ê È y(1) ˘ È r(1) ˘ ˆ
Ê È y(1) ˘ È r(1) ˘ ˆ
Á
˜
Á
˜
+ Á ÍÍ L ˙˙ - ÍÍ L ˙˙ ˜ Wy2 Á ÍÍ L ˙˙ - ÍÍ L ˙˙ ˜ + re e 2
Á Í y( p) ˙ Ír( p) ˙ ˜
Á Í y( p) ˙ Í r( p) ˙ ˜
ËÎ
˚ Î
˚¯
ËÎ
˚ Î
˚¯
where
(
)
= diag ( w0D,u1, w0D,u2 ,..., w0D,un ,..., wDpu-1,1 , wDpu-1,,2 ,..., wDpu-1,n )
u
u
u
u
Wu = diag w0u,1, wu
0,2 ,..., w0,nu ,..., w p-1,1, w p-1,2,..., w p-1, nu
WD u
u
(
Wy = diag w1y,1, w1y,2 ,..., w1y,n ,..., w yp,1, w py,2,..., w yp,n
y
y
u
)
Finally, after substituting u(k), Δu(k), y(k), J(z) can be rewritten as
T
T
Ê
T
ˆ
È utarget ( 0) ˘
È v(0) ˘
Á È r(1) ˘
˜
Í
˙
Í
˙
Í
˙
T
T
J ( z, e ) = re e + z K Du z + 2 Á Í L ˙ K r + Í L ˙ K v + u( -1) K u + Í
L
Kut + x( 0) K x ˜ z
˙
Á
˜
Íu
˙
Á ÍÎr( p) ˙˚
ÍÎv( p) ˙˚
˜
Î target ( p - 1) ˚
Ë
¯
+ constant
2
T
Note You may want the QP problem to remain strictly convex. If the condition number
of the Hessian matrix KΔU is larger than 1012, add the quantity 10*sqrt(eps)
on each diagonal term. You can use this solution only when all input rates are
unpenalized (WΔu=0) (see “Weights” in the Model Predictive Control Toolbox reference
documentation).
2-11
2
Model Predictive Control Problem Setup
Alternative Cost Function
If you are using the alternative cost function shown in “Alternative Cost Function” on
page 2-6, Equation 2-3, then Equation 2-2 is replaced by the following:
Wu = blkdiag ( Ru ,..., Ru )
WD u = blkdiag ( RD u ,..., RDu )
Wy = blkdiag ( Q,...,Q )
In this case, the block-diagonal matrices repeat p times, for example, once for each step
in the prediction horizon.
You also have the option to use a combination of the standard and alternative forms. See
“Weights” in the Model Predictive Control Toolbox reference documentation for more
details.
Constraints
Next, consider the limits on inputs, input increments, and outputs along with the
constraint ɛ≥ 0.
y
y
È
˘
È
˘
ymin (1) - e Vmin
(1)
ymax (1) + e Vmax
(1)
Í
˙ È y(1) ˘ Í
˙
L
L
Í
˙ Í L
˙ Í
˙
˙ Í
Í
˙ Í
˙
y
y
ymin ( p) - e Vmin ( p)
ymax ( p) + e Vmax ( p)
Í
˙ Í y( p) ˙ Í
˙
Í
˙ Í
˙ Í
˙
u
u
u
(
0
)
V
(
0
)
u
(
0
)
u
(
0
)
+
V
(
0
)
e
e
min
min
˙ Í
max
max
Í
˙ Í
˙
˙£Í
Í
˙£Í L
˙
L
L
Í
˙ Í
˙ Í
˙
u
u
Í umin ( p - 1) - e Vmin
( p - 1) ˙ Í u( p - 1) ˙ Í umax ( p - 1) + e Vmax
( p - 1) ˙
Í
˙ Í Du(0) ˙ Í
˙
Du
Du
Í
Dumin (0) - e Vmin
(0)
˙ Í
˙ Í
Dumax (0) + e Vmax
( 0)
˙
˙ Í
Í
˙ Í L
˙
L
L
Í
˙ Í Du( p - 1) ˙ Í
˙
˚ Í
ÍDu
Du
˙ Î
Du
˙
(
p
1
)
V
(
p
1
)
D
u
(
p
1
)
+
V
(
p
1
)
e
e
max
Î min
min
˚
Î max
˚
Note To reduce computational effort, the controller automatically eliminates extraneous
constraints, such as infinite bounds. Thus, the constraint set used in real time may be
much smaller than that suggested in this section.
2-12
Optimization Problem
Similar to what you did for the cost function, you can substitute u(k), Δu(k), y(k), and
obtain
M z z + M e e £ M lim
È v(0) ˘
+ M v ÍÍ L ˙˙ + M uu( -1) + M x x(0)
ÍÎv( p) ˙˚
In this case, matrices Mz,M ,Mlim,Mv,Mu,Mx are obtained from the upper and lower
ɛ
bounds and ECR values.
Unconstrained Model Predictive Control
The optimal solution is computed analytically
T
T
Ê
T
ˆ
È utarget ( 0) ˘
È v(0) ˘
Á È r(1) ˘
˜
Í
˙
T
˜
z* = - KD-1u Á Í L ˙ K r + Í L ˙ Kv + u(-1) T K u + Í
L
K
+
x
(
0
)
K
˙
ut
x
˙
Í
˙
ÁÍ
˜
Íu
˙
Á ÍÎ r( p) ˙˚
ÍÎv( p) ˙˚
˜
Î target ( p - 1) ˚
Ë
¯
and the model predictive controller sets Δu(k)=z*0, u(k)=u(k–1)+Δu(k).
More About
•
“Adjusting Disturbance and Noise Models” on page 2-14
•
“Time-Varying Weights and Constraints” on page 2-20
•
“Terminal Weights and Constraints” on page 2-22
2-13
2
Model Predictive Control Problem Setup
Adjusting Disturbance and Noise Models
Model Predictive Control requires the following in order to reject unknown disturbances
effectively:
• Disturbance modeling tailored to the application
• Feedback from the measurements to update controller state estimates
This section topic guidance for these issues, focusing on design-tool options.
In this section...
“Overview” on page 2-14
“Output Disturbance Model” on page 2-14
“Measurement Noise Model” on page 2-15
“Input Disturbance Model” on page 2-16
“Restrictions” on page 2-17
“Disturbance Rejection Tuning” on page 2-17
Overview
MPC attempts to predict how known and unknown events will affect the plant output
variables (OVs). Known events are changes in the measured plant input variables (MV
and MD inputs). The controller’s plant model predicts their impact (see “MPC Modeling”),
and such predictions can be quite accurate.
The impacts of unknown events appear as errors in the predictions of known events.
These errors are, by definition, impossible to predict accurately. An ability to anticipate
trends can improve disturbance rejection, however.
For example, suppose the control system has been operating at a near-steady condition
with all measured OVs near their predicted values. There are no known events, but one
or more of these OVs suddenly deviates from its prediction. How should the controller
react? Its disturbance and measurement models allow you to provide guidance.
Output Disturbance Model
For the moment, suppose your plant model includes no unmeasured disturbance inputs.
The MPC controller then models unknown events using an output disturbance model.
2-14
Adjusting Disturbance and Noise Models
As shown in “MPC Modeling”, the output disturbance model is independent of the plant
model, and its output adds directly to that of the plant model.
The design tool requires one of the following assumptions regarding the disturbances
affecting a given plant OV (signal-by-signal option):
• Any prediction error is a realization of white noise with zero mean. This assumption
implies that the of impact of noise is short-lived, calling for a modest, short-term
controller response.
• Any prediction error is due to a step-like disturbance (the default), which lasts
indefinitely, maintaining a roughly constant magnitude. This assumption calls for a
more aggressive, sustained controller response.
• Any prediction error is due to a ramp-like disturbance, which lasts indefinitely
and tending to grow with time. This assumption calls for an even more aggressive
controller response.
Each assumption can be represented by a model in which white noise with unit variance,
zero mean enters a SISO dynamic system consisting of one of the following:
• A static gain, for white noise disturbance
• An integrator in series with a static gain, for step-like disturbance
• Two integrators in series with a static gain, for ramp-like disturbance
The design tool also allows you to specify the white noise input magnitude, overriding
the assumption of unit variance. As you increase this, the controller will respond more
aggressively to a given prediction error. (Your magnitude specification determines the
gain value associated with your chosen model form: white, step, or ramp.)
Measurement Noise Model
MPC also attempts to distinguish disturbances, which require a controller response,
from measurement noise, which the controller should ignore. To guide it, you specify the
expected measurement noise magnitude and character. The design tool options parallel
the output disturbance model case, but the character must be either white (random)
or step (sustained). In nearly all applications, the white noise option should provide
adequate performance and is the default.
When you include a measurement noise model, the controller considers each prediction
error to be a combination of disturbance and noise effects. Qualitatively, as you increase
2-15
2
Model Predictive Control Problem Setup
the specified noise magnitude, the controller attributes a larger fraction of each
prediction error to noise, and it responds less aggressively. Ultimately, the controller
stops responding to prediction errors and only changes its MVs when you change the OV
or MV reference signals.
Input Disturbance Model
When your plant model includes unmeasured disturbance (UD) inputs, the controller
can employ an input disturbance model in addition to the standard output disturbance
model. The former provides more flexibility and is generated automatically by default.
(If the chosen input disturbance model does not appear to allow complete elimination of
sustained disturbances, the Toolbox adds an output disturbance model by default.)
As shown in “MPC Modeling”, the input disturbance model consists of one or more white
noise signals (unit variance, zero mean) entering a dynamic system, whose outputs are
the plant model’s UD inputs.
As with the output disturbance model, the design tool allows you to specify that a UD
signal generated by the input disturbance model has a white (random), step (sustained
– the default), or ramp (growing) character. In contrast to the output disturbance model,
these UD disturbances then affect the outputs in a more complex way as they pass
through the plant model dynamics.
A popular approach is to model unknown events as disturbances adding to the plant
MVs. (These are termed load disturbances in many texts, and are realistic in that some
unknown events are failures to set the MVs to the values requested by the controller.)
You can create a load disturbance model as follows:
1
Begin with an LTI plant model in which all inputs are known (MVs and MDs).
Suppose this model is called Plant.
2
Obtain the state-space matrices of Plant. For example:
[A,B,C,D] = ssdata(Plant);
3
Suppose there are nu MVs. Set Bu = columns of B corresponding to the MVs. Also set
Du = columns of D corresponding to the MVs.
4
Redefine the plant model to include nu additional inputs. For example:
set(Plant,‘b’,[B, Bu],‘d’,[D, Du])
5
2-16
Use setmpcsignals, or set the Plant.InputGroup property, to indicate that the
new inputs are unmeasured disturbances.
Adjusting Disturbance and Noise Models
This procedure adds load disturbance inputs without increasing the number of states in
the plant model.
By default, given a plant model containing load disturbances, the Model Predictive
Control Toolbox software creates an input disturbance model that generates nym steplike load disturbances. If nym > nu, it will also create an output disturbance model with
integrated white noise adding to (nym – nu) measured outputs. If nym < nu, the last (nu –
nym) load disturbances will be zero by default. You can modify these defaults using the
design tool.
Restrictions
As discussed in “Controller State Estimation” on page 2-29, the plant, disturbance,
and noise models combine to form a state observer, which must be detectable using the
measured plant outputs. If not, the software displays a command-window error message
when you attempt to use the controller.
This restricts the form of the disturbance and noise models. If these models are other
than a static gain, their model states must be detectable.
For example, an integrated white noise disturbance adding to an unmeasured OV would
be undetectable. The design tool prevents you from choosing such a model. Similarly, the
number of measured disturbances, nym, limits the number of step-like UD inputs from an
input disturbance model.
By default, the Model Predictive Control Toolbox software creates detectable models. If
you modify the default assumptions (or change nym) and encounter a detectability error,
you can revert to the default case.
Disturbance Rejection Tuning
The following suggestions derive from the above qualitative description of the
disturbance and measurement noise models.
• Prior to any controller tuning, define scale factors for each plant input and output
variable (see “Specifying Scale Factors”). In the context of disturbance and noise
modeling, this makes the default assumption of unit-variance white noise inputs more
likely to yield good performance.
• Allow the toolbox to create default disturbance and measurement models.
2-17
2
Model Predictive Control Problem Setup
• After tuning the cost function weights (see “Tuning Weights”), test your controller’s
response to an unmeasured disturbance input other than a step disturbance at
the plant output. Specifically, if your plant model includes UD inputs, simulate
a disturbance using one or more of these. Otherwise, simulate one or more load
disturbances, i.e., a step disturbance added to a designated MV. Both the design tool
and the sim command support such simulations.
• If the response in the simulations seems too sluggish, try one or more of the following:
• Increase all disturbance model gains by a multiplicative factor (e.g., 5). In the
design tool, do this by increasing the specified magnitude of each disturbance. If
this helps but is insufficient, repeat.
• Decrease the measurement noise gains by a multiplicative factor. In the design
tool, decrease the specified measurement noise magnitude. If this helps but is
insufficient, repeat.
• Change the character of one or more disturbances (from white to step, or from step
to ramp). Note, however, that if a disturbance is white by default, changing it to
step might violate the state observer detectability restriction.
• If the response is too aggressive, and in particular, if the controller is not robust when
its prediction of known events is inaccurate, try reversing the above steps.
Related Examples
•
“Design Controller Using the Design Tool”
More About
2-18
•
“MPC Modeling”
•
“Controller State Estimation” on page 2-29
Custom State Estimation
Custom State Estimation
The Model Predictive Control Toolbox software allows the following alternatives to the
default state estimation approach:
• You can override the default Kalman gains, L and M. Obtain the default values using
getEstimator. Then, use setEstimator to override those values. These commands
assume that the columns of L and M are in the engineering units for the measured
plant outputs. Internally, the software converts them to dimensionless form.
• You can use the custom estimation option. This skips all Kalman gain calculations.
When the controller operates, at each control interval you must use an external
procedure to estimate the controller states, xc(k|k), providing this to the controller.
Related Examples
•
Using Custom State Estimation
More About
•
“Controller State Estimation” on page 2-29
2-19
2
Model Predictive Control Problem Setup
Time-Varying Weights and Constraints
In this section...
“Time-Varying Weights” on page 2-20
“Time-Varying Constraints” on page 2-20
Time-Varying Weights
As explained in “Optimization Problem” on page 2-2, the wy, wu and w∆u weights can
change from one step in the prediction horizon to the next. Such a time-varying weight is
an array containing p rows, where p is the prediction horizon, and either ny or nu columns
(number of OVs or MVs).
Using time-varying weights provides additional tuning possibilities. However, it
complicates tuning. Recommended practice is to use constant weights unless your
application includes unusual characteristics. For example, an application requiring
terminal weights must employ time-varying weights. See “Terminal Weights and
Constraints” on page 2-22.
Time-Varying Constraints
When bounding an MV, OV, or MV increment, you have the option to use a different
bound value at each prediction-horizon step. To do so, specify the bound as a vector of up
to p values, where p is the prediction horizon length (number of control intervals). If you
specify n < p values, the nth value applies for the remaining p – n steps.
You can remove constraints at selected steps by specifying Inf (or -Inf).
If plant delays prevent the MVs from affecting an OV during the first d steps of the
prediction horizon and you must include bounds on that OV, leave the OV unconstrained
for the first d steps.
Related Examples
•
Varying Input and Output Constraints
More About
•
2-20
“Optimization Problem” on page 2-2
Time-Varying Weights and Constraints
•
“Terminal Weights and Constraints” on page 2-22
2-21
2
Model Predictive Control Problem Setup
Terminal Weights and Constraints
Terminal weights are the quadratic weights Wy on y(t+p) and Wu on u(t + p – 1). The
variable p is the prediction horizon. You apply the quadratic weights at time k +p only,
such as the prediction horizon’s final step. Using terminal weights, you can achieve
infinite horizon control that guarantees closed-loop stability. However, before using
terminal weights, you must distinguish between problems with and without constraints.
Terminal constraints are the constraints on y(t + p) and u(t + p – 1), where p is the
prediction horizon. You can use terminal constraints as an alternative way to achieve
closed-loop stability by defining a terminal region.
Note: You can use terminal weights and constraints only at the command-line. See
setterminal.
For the relatively simple unconstrained case, a terminal weight can make the finitehorizon Model Predictive Controller behave as if its prediction horizon were infinite. For
example, the MPC controller behavior is identical to a linear-quadratic regulator (LQR).
The standard LQR derives from the cost function:
•
J (u) =
 x(k + i)T Qx(k + i) + u(k + i - 1)T Ru(k + i - 1)
i =1
where x is the vector of plant states in the standard state-space form:
x(k + 1) = Ax + Bu(k)
The LQR provides nominal stability provided matrices Q and R meet certain conditions.
You can convert the LQR to a finite-horizon form as follows:
p -1
J (u) =
 [ x(k + i)T Qx(k + i) + u(k + i - 1)T Ru(k + i - 1)] + x(k + p)T Qp x(k + p)
i =1
where Qp , the terminal penalty matrix, is the solution of the Riccati equation:
Q p = AT Q p A - AT Q p B( BT Q p B + R)-1 BT Q p A + Q
2-22
Terminal Weights and Constraints
You can obtain this solution using the lqr command in Control System Toolbox™
software.
In general, Qp is a full (symmetric) matrix. You cannot use the “Standard Cost Function”
to implement the LQR cost function. The only exception is for the first p – 1 steps if Q
and R are diagonal matrices. Also, you cannot use the alternative cost function because
it employs identical weights at each step in the horizon. Thus, by definition, the terminal
weight differs from those in steps 1 to p – 1. Instead, use the following steps:
1
Augment the model (Equation 2-7) to include the weighted terminal states as
auxiliary outputs:
yaug(k) = Qcx(k)
where Qc is the Cholesky factorization of Qp such that Qp = QcTQc.
2
Define the auxiliary outputs yaug as unmeasured, and specify zero weight to them.
3
Specify unity weight on yaug at the last step in the prediction horizon using
setterminal.
To make the Model Predictive Controller entirely equivalent to the LQR, use a control
horizon equal to the prediction horizon. In an unconstrained application, you can use
a short horizon and still achieve nominal stability. Thus, the horizon is no longer a
parameter to be tuned.
When the application includes constraints, the horizon selection becomes important. The
constraints, which are usually softened, represent factors not considered in the LQR cost
function. If a constraint becomes active, the control action deviates from the LQR (state
feedback) behavior. If this behavior is not handled correctly in the controller design, the
controller may destabilize the plant.
For an in-depth discussion of design issues for constrained systems see [2]. Depending
on the situation, you might need to include terminal constraints to force the plant states
into a defined region at the end of the horizon, after which the LQR can drive the plant
signals to their targets. Use setterminal to add such constraints to the controller
definition.
The standard (finite-horizon) Model Predictive Controller provides comparable
performance, if the prediction horizon is long. You must tune the other controller
parameters (weights, constraint softening, and control horizon) to achieve this
performance.
2-23
2
Model Predictive Control Problem Setup
Tip Robustness to inaccurate model predictions is usually a more important factor than
nominal performance in applications.
Related Examples
• Implementing Infinite-Horizon LQR by Setting Terminal Weights in a Finite-Horizon
MPC Formulation
• “Providing LQR Performance Using Terminal Penalty”
2-24
Constraints on Linear Combinations of Inputs and Outputs
Constraints on Linear Combinations of Inputs and Outputs
You can also constrain linear combinations of plant input and output variables. For
example, you might want a particular manipulated variable (MV) to be greater than
a linear combination of two other MVs. The general form of such constraints is the
following:
Eu ( k + i|k ) + Fy ( k + i|k ) + Sv( k + i | k) £ G +
kV .
Here,
• ∊k — QP slack variable used for constraint softening (See “Constraint Softening”).
• u(k+i|k) — nu MV values, in engineering units
• y(k+i|k) — ny predicted plant outputs, in engineering units
• v(k+i|k) — nv measured plant disturbance inputs, in engineering units
• E, F, S, G, and V are constants
As with the QP cost function, output prediction using the state observer makes these
constraints a function of the QP decision.
Custom constraints are dimensional by default. It is good practice to define the constant
coefficients (E, F, S, G, V) such that each term is dimensionless and of order unity. This
requires application-specific analysis.
See Also
getconstraint | setconstraint
Related Examples
•
Using Custom Input and Output Constraints
•
Nonlinear Blending Process with Custom Constraints
More About
•
“Optimization Problem” on page 2-2
2-25
2
Model Predictive Control Problem Setup
Manipulated Variable Blocking
Blocking is an alternative to the simpler control horizon concept (see “Choosing Sample
Time and Horizons”). It has many of the same benefits. It also provides more tuning
flexibility and potential to smooth MV adjustments. A recommended approach to
blocking is as follows:
• Divide the prediction horizon into 3-5 blocks.
• Try the following alternatives
• Equal block sizes (one-fifth to one-third of the prediction horizon, p)
• Block sizes increasing. Example, with p = 20, three blocks of duration 3, 7 and 10
intervals.
Test the resulting controller in the same way as you test cost function weights. See
“Tuning Weights”.
Related Examples
•
2-26
“Design Controller for Plant with Delays”
QP Solver
QP Solver
The model predictive controller QP solver converts an MPC optimization problem to the
general QP form
Min( f
x
T
x+
1 T
x Hz )
2
such that
Ax £ b
where xT=[zT ε] are the decisions, H is the Hessian matrix, A is a matrix of linear
constraint coefficients, and b and f are vectors. The H and A matrices are constants.
The controller computes these during initialization and retrieves them from computer
memory when needed. It computes the time-varying b and f vectors at the beginning of
each control instant.
The toolbox uses the KWIK algorithm [1] to solve the QP problem, which requires the
Hessian to be positive definite. In the very first control step, KWIK uses a cold start, in
which the initial guess is the unconstrained solution described in “Unconstrained Model
Predictive Control” on page 2-13. If this x satisfies the constraints, it is the optimal QP
solution, x*, and the algorithm terminates. Otherwise this means that at least one of the
linear inequality constraints must be satisfied as an equality. In this case, KWIK uses an
efficient, numerically robust strategy to determine the active constraint set satisfying the
standard optimality conditions. In the following control steps, KWIK uses a warm start.
In this case, the active constraint set determined at the previous control step becomes the
initial guess for the next.
Although KWIK is robust, you should consider the following:
• One or more linear constraints might be violated slightly due to numerical round-off
errors. The toolbox employs a nonadjustable relative tolerance. This tolerance allows
a constraint to be violated by 10-6 times the magnitude of each term. Such violations
are considered normal and do not generate warning messages.
• The toolbox also uses a nonadjustable tolerance when it tests a solution for optimality.
• The search for the active constraint set is an iterative process. If the iterations reach a
problem-dependent maximum, the algorithm terminates.
2-27
2
Model Predictive Control Problem Setup
• If your problem includes hard constraints, these constraints might be infeasible
(impossible to satisfy). If the algorithm detects infeasibility, it terminates
immediately.
In the last two situations, with an abnormal outcome to the search, the controller
will retain the last successful control output. For more information, see, the mpcmove
command. You can detect an abnormal outcome and override the default behavior as you
see fit.
References
[1] Schmid, C. and L.T. Biegler, “Quadratic programming methods for reduced Hessian
SQP,” Computers & Chemical Engineering, Vol. 18, Number 9, 1994, pp. 817–832.
More About
•
2-28
“Optimization Problem” on page 2-2
Controller State Estimation
Controller State Estimation
In this section...
“Controller State Variables” on page 2-29
“State Observer” on page 2-30
“State Estimation” on page 2-31
“Built-in Steady-State Kalman Gains Calculation” on page 2-33
“Output Variable Prediction” on page 2-34
Controller State Variables
As the controller operates, it uses its current state, xc, as the basis for predictions. By
definition, the state vector is the following:
È T
xT
c ( k ) = Î x p (k)
T
xid
(k)
T
˘
xT
od ( k) x n (k) ˚ .
Here,
• xc is the controller state, comprising nxp + nxid + nxod + nxn state variables.
• xp is the plant model state vector, of length nxp.
• xid is the input disturbance model state vector, of length nxid.
• xod is the output disturbance model state vector, of length nxod.
• xn is the measurement noise model state vector, of length nxn.
Thus, the variables comprising xc represent the models appearing in the following
diagram of the MPC system.
2-29
2
Model Predictive Control Problem Setup
Some of the state vectors may be empty. If not, they appear in the sequence defined
within each model.
By default, the controller updates its state automatically using the latest plant
measurements. See “State Estimation” on page 2-31 for details. Alternatively, the
custom state estimation feature allows you to update the controller state using an
external procedure, and then supply these values to the controller. See “Custom State
Estimation” on page 2-19 for details.
State Observer
Combination of the models shown in the diagram yields the state observer:
xc ( k + 1 ) = Axc ( k ) + Buo ( k )
y ( k ) = Cxc ( k ) + Duo ( k ) .
MPC controller uses the state observer in the following ways:
• To estimate values of unmeasured states needed as the basis for predictions (see
“State Estimation” on page 2-31).
2-30
Controller State Estimation
• To predict how the controller’s proposed manipulated variable (MV) adjustments will
affect future plant output values (see “Output Variable Prediction” on page 2-34).
The observer’s input signals are the dimensionless plant manipulated and measured
disturbance inputs, and the white noise inputs to the disturbance and noise models:
T
T
T
T
È T
˘
uT
o ( k ) = Îu ( k ) v ( k ) wid ( k ) wod ( k ) wn ( k ) ˚ .
The observer’s outputs are the ny dimensionless plant outputs.
In terms of the parameters defining the four models shown in the diagram, the observer’s
parameters are:
È Ap
Í
0
A=Í
Í 0
Í
ÍÎ 0
È
C = ÍC p
Î
Bpd Cid
0
Aid
0
0
0
Aod
0
D pd Cid
Cod
0 ˘
˙
0 ˙
,
0 ˙
˙
An ˙˚
È Bpu
Í
0
B= Í
Í 0
Í
ÍÎ 0
È
È Cn ˘ ˘
Í 0 ˙ ˙ , D = Í0
Î ˚˚
Î
Bpv
Bpd Did
0
0
0
0
Bid
0
0
0
Bod
0
D pv
D pd Did
Dod
0 ˘
˙
0 ˙
0 ˙
˙,
Bn ˙˚
È Dn ˘ ˘
Í 0 ˙˙ .
Î ˚˚
Here, the plant and output disturbance models are resequenced so that the measured
outputs precede the unmeasured outputs.
State Estimation
In general, the controller states are unmeasured and must be estimated. By default, the
controller uses a steady state Kalman filter that derives from the state observer. (See
“State Observer” on page 2-30.)
At the beginning of the kth control interval, the controller state is estimated with the
following steps:
1
Obtain the following data:
• xc(k|k–1) — Controller state estimate from previous control interval, k–1
• uact(k–1) — Manipulated variable (MV) actually used in the plant from k–1 to k
(assumed constant)
2-31
2
Model Predictive Control Problem Setup
• uopt(k–1) — Optimal MV recommended by MPC and assumed to be used in the
plant from k–1 to k
• v(k) — Current measured disturbances
• ym(k) — Current measured plant outputs
• Bu, Bv — Columns of observer parameter B corresponding to u(k) and v(k) inputs
• Cm — Rows of observer parameter C corresponding to measured plant outputs
• Dmv — Rows and columns of observer parameter D corresponding to measured
plant outputs and measured disturbance inputs
• L, M — Constant Kalman gain matrices
Plant input and output signals are scaled to be dimensionless prior to use in
calculations.
2
Revise xc(k|k–1) when uact(k–1) and uopt(k–1) are different:
È act ( k - 1 ) - uopt ( k - 1 ) ˘ .
xrev
c ( k|k - 1 ) = xc ( k|k - 1 ) + Bu Î u
˚
3
Compute the innovation:
e ( k ) = ym ( k ) - È Cm xcrev ( k| k - 1) + Dmv v ( k ) ˘ .
Î
˚
4
Update the controller state estimate to account for the latest measurements.
xc ( k|k ) = xrev
c ( k|k - 1 ) + Me ( k ) .
Then, the software uses the current state estimate xc(k|k) to solve the quadratic
program at interval k. The solution is uopt(k), the MPC-recommended manipulatedvariable value to be used between control intervals k and k+1.
Finally, the software prepares for the next control interval assuming that the
unknown inputs, wid(k), wod(k), and wn(k) assume their mean value (zero) between
times k and k+1. The software predicts the impact of the known inputs and the
innovation as follows:
2-32
xc ( k + 1| k ) = Axc ( k| k ) + Bu uopt ( k ) + Bv v ( k ) + Le ( k ) .
Controller State Estimation
Built-in Steady-State Kalman Gains Calculation
Model Predictive Control Toolbox software uses the kalman command to calculate
Kalman estimator gains L and M. The following assumptions apply:
• State observer parameters A, B, C, D are time-invariant.
• Controller states, xc, are detectable. (If not, or if the observer is numerically close to
undetectability, the Kalman gain calculation fails, generating an error message.)
• Stochastic inputs wid(k), wod(k), and wn(k) are independent white noise, each with zero
mean and identity covariance.
• Additional white noise wu(k) and wv(k) with the same characteristics adds to the
dimensionless u(k) and v(k) inputs respectively. This improves estimator performance
in certain cases, such as when the plant model is open-loop unstable.
Without loss of generality, set the u(k) and v(k) inputs to zero. The effect of the stochastic
inputs on the controller states and measured plant outputs is:
xc ( k + 1 ) = Axc ( k ) + Bw ( k )
ym ( k ) = Cm xc ( k ) + Dm w ( k ) .
Here,
T
T
˘
wT ( k ) = È wuT ( k ) wvT ( k ) wT
id ( k ) wod ( k ) wn ( k ) ˚ .
Î
Inputs to the kalman command are the state observer parameters A, Cm, and the
following covariance matrices:
{
}
T
R = E { Dm wwT Dm
} = Dm DmT
T
N = E{ BwwT Dm
} = BDmT.
Q = E BwwT BT = BBT
Here, E{...} denotes the expectation.
2-33
2
Model Predictive Control Problem Setup
Output Variable Prediction
Model Predictive Control requires prediction of noise-free future plant outputs used in
optimization. This is a key application of the state observer (see “State Observer” on page
2-30).
In control interval k, the required data are as follows:
• p — Prediction horizon (number of control intervals, which is greater than or equal to
1)
• xc(k|k) — Controller state estimates (see “State Estimation” on page 2-31)
• v(k) — Current measured disturbance inputs (MDs)
• v(k+i|k) — Projected future MDs, where i=1:p–1. If you are not using MD previewing,
then v(k+i|k) = v(k).
• A, Bu, Bv, C, Dv — State observer constants, where Bu, Bv, and Dv denote columns of
the B and D matrices corresponding to inputs u and v. Du is a zero matrix because of
no direct feedthrough
Predictions assume that unknown white noise inputs are zero (their expectation).
Also, the predicted plant outputs are to be noise-free. Thus, all terms involving
the measurement noise states disappear from the state observer equations. This is
equivalent to zeroing the last nxn elements of xc(k|k).
Given the above data and simplifications, for the first step the state observer predicts:
xc ( k + 1| k ) = Axc ( k|k ) + Bu u ( k|k ) + Bv v ( k ) .
Continuing for successive steps, i = 2:p, the state observer predicts:
xc ( k + i|k ) = Axc ( k + i - 1|k ) + Buu ( k + i - 1|k ) + Bv v ( k + i - 1| k ) .
At any step, i = 1:p, the predicted noise-free plant outputs are:
y ( k + i | k ) = Cxc ( k + i | k ) + Dv v ( k + i| k ) .
All of these equations employ dimensionless plant input and output variables. See
“Specifying Scale Factors”. The equations also assume zero offsets. Inclusion of nonzero
offsets is straightforward.
2-34
Controller State Estimation
For faster computations, the MPC controller uses an alternative form of the above
equations in which constant terms are computed and stored during controller
initialization. See “QP Matrices” on page 2-8.
More About
•
“MPC Modeling”
•
“Optimization Problem” on page 2-2
2-35
3
Model Predictive Control Simulink
Library
• “MPC Library” on page 3-2
• “MPC Controller Block” on page 3-3
• “Generate Code and Deploy Controller to Real-Time Targets” on page 3-13
• “Multiple MPC Controllers Block” on page 3-14
• “Relationship of Multiple MPC Controllers to MPC Controller Block” on page 3-15
3
Model Predictive Control Simulink Library
MPC Library
The MPC Simulink Library provides two blocks you can use to implement MPC control in
Simulink, MPC Controller, and Multiple MPC Controllers.
Access the library using the Simulink Library Browser or by typing mpclib at the
command prompt. The following figure shows the library contents.
MPC Simulink Library
Once you have access to the library, you can add one of its blocks to your Simulink model
by clicking-and-dragging or copying-and-pasting.
3-2
MPC Controller Block
MPC Controller Block
In this section...
“MPC Controller Block Mask” on page 3-3
“MPC Controller Parameters” on page 3-4
“Connect Signals” on page 3-5
“Optional Ports” on page 3-6
“Input Signals” on page 3-9
“Output Signals” on page 3-10
“Look Ahead and Signals from the Workspace” on page 3-11
“Initialization” on page 3-12
The MPC Controller block represents a single MPC controller. You can adjust plant
inputs to control plant outputs, accounting for constraints, including actuator limits.
MPC Controller Block Mask
Insert an MPC Controller block in your Simulink model and then specify its properties.
Double-click on the block to open its mask. The following figure shows the mask's default
settings. This section shows you how to configure the controller by:
• Specifying required parameters.
• Enabling optional inports and outputs.
• Connecting appropriate signals.
3-3
3
Model Predictive Control Simulink Library
MPC Controller Block Mask
MPC Controller Parameters
Specify the MPC Controller
Specify a valid MPC object in the MPC Controller field in one of two ways:
• Type the name of an MPC object saved in your workspace. To review its settings
before running a simulation, click Design to load the named object into the MPC
design tool.
• If your installation includes Simulink Control Design™ software, connect the MPC
Controller block to the plant it controls. Leaving the MPC controller field empty, click
Design. The block constructs a default MPC object by linearizing the plant at the
3-4
MPC Controller Block
default operating point and exports this defaults controller to the base workspace. If
the default operating point is not the one you want, see “Importing a Plant Model” for
help with importing a new plant model.
Note: You can run closed-loop simulations with a linear plant in the design tool, allowing
you to tune the controller parameters. To test the controller in Simulink, export it from
the design tool to the workspace and run the simulation in Simulink.
Indicate Initial Controller State
If you do not specify an initial controller state, the controller uses a default initial
condition in simulations. To change the initial state, specify an mpcstate object. See
“MPC Simulation Options Object” in the Model Predictive Control Toolbox Reference.
Connect Signals
The MPC Controller requires at least two inputs and generates at least one output.
There are also optional inputs and outputs. In most cases, the signals are vectors
comprised of plant variables. Verify that the signal dimensions and the sequence of
elements within each vector signal are consistent with your controller definition.
Required inputs
Connect the following to the indicated controller inports:
• Measured output variables (mo). Connect the measured plant output variables to
the MPC Controller’s mo inport as a vector signal, length n ym ≥ 1 . This provides the
controller’s feedback.
• References (ref). Your controller’s prediction model contains ny ≥ nym output variables.
Each must have a reference target or setpoint value. Connect this 1-by-ny vector
signal to the controller’s ref inport. This inport defines the reference values at the first
step in the controller’s prediction horizon. By default, the controller assumes these
values hold for the entire horizon.
You can also preview reference signals at run-time. To activate reference previewing,
supply the reference input as an Nxny signal, where 1 < N £ p , and p is the prediction
horizon length. The rows 1 to N define the reference values for time instants tk+1 to tk+N
3-5
3
Model Predictive Control Simulink Library
in the prediction horizon. If N< p, the last row is used for steps tk+n+1 to tk+p. See the
mpcpreview example.
Required outport
During operation, the controller updates the manipulated variable (mv) outport at each
control interval. The mv outport defines adjustments to plant input variables. Connect
this vector signal to the plant.
Sample Time
Every Dt( > 0) time units, the MPC Controller samples its input signals and updates its
output signals. This control interval is defined in the Ts property of the MPC object.
Optional Ports
Feedforward compensation for measured disturbances
If your prediction model includes measured disturbances, ensure that Measured
disturbance is selected, and click Apply. Selecting this parameter adds an inport
labeled md. Connect an N x nmd signal, where:
•
n md ≥ 1 is the number of measured disturbances coming from the plant
•
N ≥ 1 is the number of sampling instants for which you are supplying measured
disturbance values.
If N=1, the previewing is disabled and the signal must contain the vector of measured
disturbances at the current controller sampling instant, tk. N>1 activates previewing.
In this case, rows 2 to N must contain estimates of the measured disturbances at future
sampling instants tk+1, ..., tk+N-1. These estimates allows the controller to
preview, or, anticipate future changes in these disturbances and compensate for them at
time tk. See mpcpreview for an illustration of the improved performance this approach
can provide.
Externally supplied MV signals
Ideally, the manipulated variable values used in the plant are always identical to those
specified by the controller’s mv outport. The controller makes this assumption by default.
In practice, however, unexpected constraints, disturbances, and plant nonlinearities
can prevent these values from being exactly alike. If the actual values of manipulated
3-6
MPC Controller Block
variables are measured and fed back, the controller’s predictions improve. This feature
can also smooth the transition between manual and automatic operation. For more
information, see “Bumpless Transfer Between Manual and Automatic Operations”.
Select the Externally supplied MV signals check box, and click Apply to add an
inport labeled ext.mv. Connect the feedback signal, length nmv.
The controller’s default behavior is the feedback of its mv outport to the ext.mv inport.
Input and output limits
By default, the controller employs the constraints specified in your controller design as
constants. You have the option to update the upper and lower bounds on manipulated
and output variables at each controller sampling instant.
Select Input and output limits, and click Apply to add the following four inports:
• umin defines nmv lower bounds on the manipulated variables (the controller’s mv
outport).
• umax is a corresponding vector of nmv upper bounds.
• ymin defines ny lower bounds on the output variables (the same variables to which
the ref inport applies).
• ymax is a corresponding vector of ny upper bounds.
After adding the inports, connect the appropriate signals. See mpcvarbounds for more
information.
Disabling optimization
Select this option to control whether or not the block performs its optimization
calculations at each sampling instant. For example, if the controller’s mv outport is being
ignored because the plant is manually controlled, then turning off optimization reduces
the computational load. When optimization is off, the mv output is zero.
To activate this option, select Optimization enabling switch, and click Apply.
Selecting this option adds an inport labeled QP switch. Connect a scalar signal. When
the signal is zero, optimization occurs. Setting it to nonzero disables optimization.
If you select this option, the Externally supplied MV signals option also activates
automatically. To prevent "bumps" when optimization is deactivated temporarily and
then reactivates, you must feed the actual MV signal back to the ext.mv inport.
3-7
3
Model Predictive Control Simulink Library
For more information and examples, see “Bumpless Transfer Between Manual and
Automatic Operations”.
Monitoring the optimal cost
This option allows you to monitor the objective function value (optimal cost) obtained by
solving the controller’s quadratic program (QP). See “Optimization Problem” for more
information.
Select the Optimal cost and click Apply to add a new scalar outport labeled cost.
If the optimization problem is infeasible, that is, one or more hard constraints cannot be
satisfied or the solver experiences numerical difficulties. The returned cost is –1.
Monitoring the optimal control sequence
To monitor the controller’s planned sequence of future adjustments, select Optimal
control sequence, and click Apply. This option adds an outport labeled mv.seq. At each
control instant, the outport contains a p X nmv matrix signal. The rows correspond to the
p steps of the prediction horizon, and the columns to the nmv manipulated variables. The
first row represents current time, k=0. The last row represents time k+p-1.
Monitoring the QP status
Selecting this option allows you to take application-specific actions if the controller’s
optimization calculation (QP) terminates abnormally. For example, you can set an alarm.
Select Optimization status, and click Apply to add an outport labeled qp.status.
This action generates a scalar signal.
If the QP terminates normally, qp.status (> 0) is the number of QP iterations
required. This value is proportional to calculational effort. A large value indicates a
difficult QP, and might prevent the controller from making its adjustments in timely
fashion.
Possible abnormal terminations are:
• = 0: The QP could not be solved in the maximum number of iterations specified in the
controller definition.
• = -1: The QP solver detected an infeasible QP, that is, it was impossible to satisfy all
the hard constraints imposed in the controller definition.
3-8
MPC Controller Block
• = -2: The QP solver encountered severe numerical difficulties, such as a poorly
conditioned QP.
If the QP terminates abnormally, the controller’s mv outport contains the most recent
successful solution.
Online tuning
Three related options allow you to adjust controller performance during operation. To
activate one, select the appropriate check box, and click Apply. This adds a new inport
with the functionality:
Weights on plant outputs
The added inport is labeled y.wt. Connect a vector signal, length ny, which defines
the nonnegative real weight on reference tracking for each of the ny output variables.
This signal overrides the controller’s MPCobj.Weights.OV property. A larger weight
increases the importance of accurate tracking for the corresponding output variable.
Weights on manipulated variables rate
The added inport is labeled du.wt. Connect a vector signal, length nmv,
which defines the nonnegative real weight on the adjustment (increment) for
each of the nmv manipulated variables. This signal overrides the controller’s
MPCobj.Weights.MVRate property. A larger weight discourages the controller from
making large-magnitude adjustments in the corresponding plant variable.
Weight on overall constraint softening
The added inport is labeled ECR.wt. Connect a scalar nonnegative real signal
specifying the weight on the slack variable used to soften constraints. This signal
overrides the controller’s MPCobj.Weights.ECR property. A larger weight makes all
soft constraints harder.
Input Signals
You must connect appropriate Simulink signals to the MPC Controller block's inports.
The measured output (mo) and reference (ref) inports are required. You can add optional
inports by selecting check boxes at the bottom of the mask.
As shown in the figure, MPC Controller Block Mask , Enable measured disturbances
is a default selection and the corresponding inport (md) appears in figure MPC Simulink
Library. This provides feedforward compensation for measured disturbances.
3-9
3
Model Predictive Control Simulink Library
Enable externally supplied MV signals allows you to keep the controller informed
of the actual manipulated variable values. Ideally, the actual manipulated variables are
those specified by the controller block output mv. In practice, unexpected constraints,
disturbances, or plant nonlinearities can modify the values actually implemented in
the plant. If the actual values are known and fed back to the controller, its predictions
improve. This feature can also improve the transition between manual and automatic
operation. See “Bumpless Transfer Between Manual and Automatic Operations” on page
4-40.
Enable input and output limits allows you to specify constraints that vary during
a simulation. Otherwise, the block uses the constant constraint values stored within
its MPC Controller object. The example mpcvarbounds shows how this option works.
It enables inports for lower and upper bounds on the manipulated variables (inports
umin and umax) and lower and upper bounds on the controlled outputs (inports ymin
and ymax). An unconnected constraint inport causes the corresponding variable to be
unconstrained.
Enable optimization switch allows you to control whether or not the block performs
its optimization calculations at each sampling instant during a simulation. If the
controller is output is being ignored during the simulation, e.g., due to a switch to
manual control, turning off the optimization reduces the computational load. When
optimization is off, the controller output is zero. To turn the optimization off, set the
switch input signal to a nonzero value. When the switch input is zero or disconnected, the
optimization occurs and the controller output varies in the normal way.
If you select the switching option, the Enable externally supplied MV signals
option must also be activated. See “Bumpless Transfer Between Manual and Automatic
Operations” on page 4-40 for an example application.
Output Signals
The block updates its output(s) at regular intervals. The MPC object named in the block’s
MPC controller field contains the control interval (its Ts property). The object also
specifies the number of manipulated variables, nu, to be calculated and sent to the plant
at each control instant. The default outport (labeled mv) provides these as a vector signal
of dimension nu.
There are two optional outputs:
• The Enable optimal cost outport option adds an outport labeled cost, which
contains the value of the optimal objective function obtained when calculating the
3-10
MPC Controller Block
manipulated variables. See “Optimization Problem” on page 2-2 for the various forms
this can take. If the optimization problem is infeasible, i.e., some constraints can’t be
satisfied or the solver experiences numerical difficulties, the returned cost is –1 .
• The Enable control sequence outport option creates a new outport labeled
mv.seq. At each control instant, this outport contains the calculated optimal
manipulated variable sequence for the prediction horizon specified in the MPC object.
For more information, see the MPC Controller block reference page.
Look Ahead and Signals from the Workspace
The mask's Input signals section allows you to define the reference and/or measured
disturbance signals as variables in the workspace. In this case, the block ignores the
signals connected to its corresponding inports.
You must create such a signal as a MATLAB® structure with two fields: time and
signals. The Simulink From Workspace and To Workspace blocks use the same
format.
For example, to specify a sinusoidal reference signal sin(t) over a time horizon of 10
seconds, use the following MATLAB commands:
time=(0:Ts:10);
ref.time=time;
ref.signals.values=sin(time);
where Ts is the controller sampling period. After the variable is created, select the Use
custom reference signal check box and enter the variable name in the edit box.
An alternative would be to run a Simulink simulation in which you connect an
appropriate block (Sine, in the above example) to a To Workspace block.
The Look ahead check box enables an anticipative action on the corresponding signal.
This option becomes available when you define reference and measured disturbance
signals in the workspace. For example, if you define the reference signal as described
above, the Look ahead option becomes available. Selecting it causes the controller to
compensate for the known future reference variations, which usually improves setpoint
tracking. When Look ahead is deselected, the controller assumes that the current
reference (or measured disturbance) value applies throughout its prediction horizon.
See the mpcpreview example for an illustrative example of enabling preview and
reading signals from the workspace.
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3
Model Predictive Control Simulink Library
Initialization
If Initial controller state is unspecified, as in MPC Controller Block Mask, the
controller uses a default initial condition in simulations. You can change the initial
condition by specifying an mpcstate object. See “MPC Simulation Options Object” in the
Model Predictive Control Toolbox Reference.
3-12
Generate Code and Deploy Controller to Real-Time Targets
Generate Code and Deploy Controller to Real-Time Targets
After designing a controller in Simulink software using the MPC Controller block, you
can generate code and deploy it for real-time control. You can deploy the controller to all
targets supported by the following products:
• Simulink Coder™
• Embedded Coder®
• Simulink PLC Coder™
• Simulink Real-Time™
The sampling rate that a controller can achieve in real-time environment is system
dependent. For example, for a typical small MIMO control application running on
Simulink Real-Time, the sampling rate may go as low as 1–10ms. To determine the
sampling rate, first test a less aggressive controller whose sampling rate produces
acceptable performance on the target. Next, increase the sampling rate and monitor the
execution time used by the controller. You can further decrease the sampling rate as long
as the optimization safely completes within each sampling period under the normal plant
operations.
Note: The MPC Controller block is implemented using the MATLAB Function block. To
see the structure, right-click the block and select Mask > Look Under Mask. Open the
MPC subsystem underneath.
See Also
MPC Controller | Multiple MPC Controllers | review
Related Examples
•
“Simulation and Code Generation Using Simulink Coder”
•
“Simulation and Structured Text Generation Using PLC Coder”
3-13
3
Model Predictive Control Simulink Library
Multiple MPC Controllers Block
In this section...
“Limitations” on page 3-14
“Examples” on page 3-14
The Multiple MPC Controllers block allows you to achieve better control of a nonlinear
plant over a range of operating conditions.
A controller that works well initially can degrade if the plant is nonlinear and its
operating point changes. In conventional feedback control, you might compensate for this
degradation by gain scheduling.
In a similar manner, the Multiple MPC Controllers block allows you to transition
between multiple MPC controllers in real time in a preordained manner. You design
each controller to work well in a particular region of the operating space. When the plant
moves away from this region, you instruct another MPC controller to take over.
Limitations
The Multiple MPC Controllers block does not provide all the optional features found in
the MPC Controller block. The following ports are currently not available:
• Optional outports such as optimal cost, optimal control sequence, and optimization
status
• Optional inports for online tuning
Examples
See the mpcswitching and mpccstr examples for applications of the Multiple MPC
Controllers block.
3-14
Relationship of Multiple MPC Controllers to MPC Controller Block
Relationship of Multiple MPC Controllers to MPC Controller Block
The key difference between the Multiple MPC Controllers and the MPC Controller blocks
is the way in which you designate the controllers to be used.
Listing the controllers
You must provide an ordered list containing N names, where N is the number of
controllers and each name designates a valid MPC object in your base workspace. Each
named controller must use the identical set of plant signals (for example, the same
measured outputs and manipulated variables). See the Multiple MPC Controllers
reference for more information on creating lists.
Designing the controllers
Use your knowledge of the process to identify distinct operating regions and design
a controller for each. One convenient approach is to use the Simulink Control Design
product to calculate each nominal operating point (typically a steady-state condition).
Then, obtain a linear prediction model at this condition. To learn more, see the Simulink
Control Design documentation. You must have Simulink Control Design product license
to use this approach.
After the prediction models have been defined for each operating region, design each
corresponding MPC Controller and give it a unique name in your base workspace.
Defining controller switching
Next, define the switching mechanism that will select among the controllers in real time.
Add this mechanism to your Simulink model. For example, you could use one or more
selected plant measurements to determine when each controller becomes active.
Your mechanism must define a scalar switching signal in the range 1 to N, where N is
the number of controllers in your list. Connect this signal to the block’s switch inport.
Set it to 1 when you want the first controller in your list to become active, to 2 when the
second is to become active, and so on.
Note: The Multiple MPC Controllers block automatically rounds the switching signal
to the nearest integer. If the signal is outside the range 1 to N, none of the controllers
activate and the block output is zero.
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3
Model Predictive Control Simulink Library
Improving prediction accuracy
During operation, all inactive controllers receive the current manipulated variable
and measured output signals so they can update their state estimates. These updates
minimize bumps during controller transitions. See “Bumpless Transfer Between Manual
and Automatic Operations” for more information. It is good practice to enable the
Externally supplied MV signal option and feedback the actual manipulated variables
measured in the plant to the ext.mv inport. This signal is provided to all the controllers
in the block’s list.
3-16
4
Case-Study Examples
• “Servomechanism Controller” on page 4-2
• “Paper Machine Process Control” on page 4-27
• “Bumpless Transfer Between Manual and Automatic Operations” on page 4-40
• “Switching Controller Online and Offline with Bumpless Transfer” on page 4-48
• “Coordinate Multiple Controllers at Different Operating Points” on page 4-54
• “Using Custom Constraints in Blending Process” on page 4-61
• “Providing LQR Performance Using Terminal Penalty” on page 4-68
• “Real-Time Control with OPC Toolbox” on page 4-74
• “Simulation and Code Generation Using Simulink Coder” on page 4-79
• “Simulation and Structured Text Generation Using PLC Coder” on page 4-86
• “Setting Targets for Manipulated Variables” on page 4-90
• “Specifying Alternative Cost Function with Off-Diagonal Weight Matrices” on page
4-94
• “Review Model Predictive Controller for Stability and Robustness Issues” on page
4-98
• “Bibliography” on page 4-117
4
Case-Study Examples
Servomechanism Controller
In this section...
“System Model” on page 4-2
“Control Objectives and Constraints” on page 4-3
“Defining the Plant Model” on page 4-4
“Controller Design Using MPCTOOL” on page 4-5
“Using Model Predictive Control Toolbox Commands” on page 4-19
“Using MPC Tools in Simulink” on page 4-22
System Model
A position servomechanism consists of a DC motor, gearbox, elastic shaft, and a load.
Position Servomechanism Schematic
The differential equations representing this system are
k Ê
q ˆ b
w& L = - q Áq L - M ˜ - L w L
JL Ë
r ¯ JL
k Ê V - kT w M ˆ b M w M
k Ê
q ˆ
+ q ÁqL - M ˜
w& M = T Á
˜JM Ë
R
J
r
J
r ¯
¯
M
MË
where V is the applied voltage, T is the torque acting on the load, wL = q&L is the load's
angular velocity, w = q& is the motor shaft's angular velocity, and the other symbols
M
4-2
M
Servomechanism Controller
represent constant parameters (see Parameters Used in the Servomechanism Model for
more information on these).
If you define the state variables as x p = [q L
wL q M
w < ] , then you can convert the
T
above model to an LTI state-space form:
x& p
qL
1
0
È 0
Í k
kq
bL
Í q
Í JL
JL
rJ L
Í
0
0
=Í 0
Í
Í
kq
Í kq
0 Í rJ
2
r JM
Î M
= [1 0 0 0] x p
È
T = Í kq
Î
0
kq
r
˘
˙
È 0 ˘
˙
0
Í 0 ˙
˙
Í
˙
˙
1
˙ xp + Í 0 ˙ V
Í
˙
˙
k2
Í kT ˙
bM + T ˙
Í RJ ˙
R ˙
Î
M˚
JM ˙
˚
0
˘
0˙ x p
˚
Parameters Used in the Servomechanism Model
Symbol
Value (SI Units)
Definition
kθ
1280.2
Torsional rigidity
kT
10
Motor constant
JM
0.5
Motor inertia
JL
50JM
Load inertia
ρ
20
Gear ratio
βM
0.1
Motor viscous friction coefficient
βL
25
Load viscous friction coefficient
R
20
Armature resistance
Control Objectives and Constraints
The controller must set the load's angular position, θL, at a desired value by adjusting the
applied voltage, V. The only measurement available for feedback is θL.
4-3
4
Case-Study Examples
The elastic shaft has a finite shear strength, so the torque, T, must stay within specified
limits
|T| ≤ 78.5Nm
Also, the applied voltage must stay within the range
|V| ≤ 220V
From an input/output viewpoint, the plant has a single input, V, which is manipulated
by the controller. It has two outputs, one measured and fed back to the controller, θL, and
one unmeasured, T.
The specifications require a fast servo response despite constraints on a plant input and
a plant output.
Defining the Plant Model
The first step in a design is to define the plant model.
% DC-motor with elastic shaft
%
%Parameters (MKS)
%----------------------------------------------------------Lshaft=1.0;
%Shaft length
dshaft=0.02;
%Shaft diameter
shaftrho=7850;
%Shaft specific weight (Carbon steel)
G=81500*1e6;
%Modulus of rigidity
tauam=50*1e6;
%Shear strength
Mmotor=100;
%Rotor mass
Rmotor=.1;
%Rotor radius
Jmotor=.5*Mmotor*Rmotor^2;
%Rotor axial moment of inertia
Bmotor=0.1;
%Rotor viscous friction coefficient (A CASO)
R=20;
%Resistance of armature
Kt=10;
%Motor constant
gear=20;
%Gear ratio
Jload=50*Jmotor;
%Load inertia
Bload=25;
%Load viscous friction coefficient
Ip=pi/32*dshaft^4;
%Polar momentum of shaft (circular) section
Kth=G*Ip/Lshaft;
%Torsional rigidity (Torque/angle)
Vshaft=pi*(dshaft^2)/4*Lshaft;
%Shaft volume
Mshaft=shaftrho*Vshaft;
%Shaft mass
Jshaft=Mshaft*.5*(dshaft^2/4);
%Shaft moment of inertia
JM=Jmotor;
JL=Jload+Jshaft;
Vmax=tauam*pi*dshaft^3/16;
%Maximum admissible torque
Vmin=-Vmax;
%Input/State/Output continuous time form
%----------------------------------------------------------
4-4
Servomechanism Controller
AA=[0
-Kth/JL
0
Kth/(JM*gear)
1
-Bload/JL
0
0
0
Kth/(gear*JL)
0
-Kth/(JM*gear^2)
0;
0;
1;
-(Bmotor+Kt^2/R)/JM];
BB=[0;0;0;Kt/(R*JM)];
Hyd=[1 0 0 0];
Hvd=[Kth 0 -Kth/gear 0];
Dyd=0;
Dvd=0;
% Define the LTI state-space model
sys=ss(AA,BB,[Hyd;Hvd],[Dyd;Dvd]);
Controller Design Using MPCTOOL
The servomechanism model is linear, so you can use the Model Predictive Control
Toolbox design tool (mpctool) to configure a controller and test it.
Note To follow this example on your own system, first create the servomechanism model
as explained in “Servomechanism Controller” on page 4-2. This defines the variable
sys in your MATLAB workspace.
Opening MPCTOOL and Importing a Model
To begin, open the design tool by typing the following at the MATLAB prompt:
mpctool
Once the design tool has appeared, click the Import Plant button. The Plant Model
Importer dialog box appears (see the following figure).
By default, the Import from option buttons are set to import from the MATLAB
workspace, and the box at the upper right lists all LTI models defined there. In the
following figure, sys is the only available model, and it is selected. The Properties area
lists the selected model's key attributes.
4-5
4
Case-Study Examples
Import Dialog Box with the Servomechanism Model Selected
Make sure your servomechanism model, sys, is selected. Then click the Import button.
You won't be importing more models, so close the import dialog box.
Meanwhile, the model has loaded, and tables now appear in the design tool's main
window (see the figure below). Note the previous diagram enumerates the model's input
and output signals.
4-6
Servomechanism Controller
Design Tool After Importing the Plant Model
Specifying Signal Properties
It's essential to specify signal types before going on. By default, the design tool assumes
all plant inputs are manipulated, which is correct in this case. But it also assumes all
outputs are measured, which is not. Specify that the second output is unmeasured by
clicking on the appropriate table cell and selecting the Unmeasured option.
You also have the option to change the default signal names (In1, Out1, Out2) to
something more meaningful (e.g., V, ThetaL, T), enter descriptive information in the
4-7
4
Case-Study Examples
blank Description and Units columns, and specify a nominal initial value for each
signal (the default is zero).
After you've entered all your changes, the design tool resembles the following figure.
Design Tool After Specifying Signal Properties
Navigation Using the Tree View
Now consider the design tool's left-hand frame. This tree is an ordered arrangement of
nodes. Selecting (clicking) a node causes the corresponding view to appear in the righthand frame. When the design tool starts, it creates a root node named MPC Design
Task and selects it, as in Design Tool After Importing the Plant Model.
The Plant models node is next in the hierarchy. Click on it to list the plant models
being used in your design. (Each model name is editable.) The middle section displays the
selected model's properties. There is also a space to enter notes describing the model's
special features. Buttons allow you to import a new model or delete one you no longer
need.
The next node is Controllers. You might see a + sign to its left, indicating that it
contains subnodes. If so, click on the + sign to expand the tree (as shown in Design Tool
After Importing the Plant Model). All the controllers in your design will appear here.
By default, you have one: MPC1. In general, you might opt to design and test several
alternatives.
Select Controllers to see a list of all controllers, similar to the Plant models view.
The table columns show important controller settings: the plant model being used, the
controller sampling period, and the prediction and control horizons. All are editable. For
now, leave them at their default values.
4-8
Servomechanism Controller
The buttons on the Controllers view allow you to:
• Import a controller designed previously and stored either in your workspace or in a
MAT-file.
• Export the selected controller to your workspace.
• Create a New controller, which will be initialized to the Model Predictive Control
Toolbox defaults.
• Copy the selected controller to create a duplicate that you can modify.
• Delete the selected controller.
Specifying Controller Properties
Select the MPC1 subnode. The main pane should change to the controller design.
If the selected Prediction model is continuous-time, as in this example, the Control
interval (sampling period) defaults to 1. You need to change this to an applicationappropriate value. Set it to 0.1 seconds (as shown in Controller Design View, Models and
Horizons Pane). Leave the other values at their defaults for now.
Controller Design View, Models and Horizons Pane
4-9
4
Case-Study Examples
Specifying Constraints
Next, click the Constraints tab. The view shown in Controller Design View, Constraints
Pane appears. Enter the appropriate constraint values. Leaving a field blank implies
that there is no constraint.
Controller Design View, Constraints Pane
In general, it's good practice to include all known manipulated variable constraints, but
it's unwise to enter constraints on outputs unless they are an essential aspect of your
application. The limit on applied torque is such a constraint, as are the limits on applied
voltage. The angular position has physical limits but the controller shouldn't attempt to
enforce them, so you should leave the corresponding fields blank (see Controller Design
View, Constraints Pane).
The Max down rate should be nonpositive (or blank). It limits the amount a
manipulated variable can decrease in a single control interval. Similarly, the Max up
rate should be nonnegative. It limits the increasing rate. Leave both unconstrained (i.e.,
blank).
The shaded columns can't be edited. If you want to change this descriptive information,
select the root node view and edit its tables. Such changes apply to all controllers in the
design.
Weight Tuning
Next, click the Weight Tuning tab.
4-10
Servomechanism Controller
The weights specify trade-offs in the controller design. First consider the Output
weights. The controller will try to minimize the deviation of each output from its
setpoint or reference value. For each sampling instant in the prediction horizon, the
controller multiplies predicted deviations for each output by the output's weight,
squares the result, and sums over all sampling instants and all outputs. One of the
controller's objectives is to minimize this sum, i.e., to provide good setpoint tracking. (See
“Optimization Problem” on page 2-2 for more details.)
Here, the angular position should track its setpoint, but the applied torque can vary,
provided that it stays within the specified constraints. Therefore, set the torque's weight
to zero, which tells the controller that setpoint tracking is unnecessary for this output.
Similarly, it's acceptable for the applied voltage to deviate from nominal (it must in order
to change the angular position). Its weight should be zero (the default for manipulated
variables). On the other hand, it's probably undesirable for the controller to make drastic
changes in the applied voltage. The Rate weight penalizes such changes. Use the
default, 0.1.
When setting the rates, the relative magnitudes are more important than the absolute
values, and you must account for differences in the measurement scales of each variable.
For example, if a deviation of 0.1 units in variable A is just as important as a deviation
of 100 units in variable B, variable A's weight must be 1000 times larger than that for
variable B.
Controller Design View, Weight Tuning Pane
4-11
4
Case-Study Examples
The tables allow you to weight individual variables. The slider at the top adjusts an
overall trade-off between controller aggressiveness and setpoint tracking. Moving
the slider to the left places a larger overall penalty on manipulated variable changes,
making them smaller. This usually increases controller robustness, but setpoint tracking
becomes more sluggish.
The Estimation (Advanced) tab allows you to adjust the controller's response to
unmeasured disturbances (not used in this example).
Defining a Simulation Scenario
If you haven't already done so, expand the Scenarios node to show the Scenario1
subnode (see Design Tool After Importing the Plant Model). Select Scenario1.
A scenario is a set of simulation conditions. As shown in Simulation Settings View for
“Scenario1”, you choose the controller to be used (from among controllers in your design),
the model to act as the plant, and the simulation duration. You must also specify all
setpoints and disturbance inputs.
Duplicate the settings shown in Simulation Settings View for “Scenario1”, which will test
the controller's servo response to a unit-step change in the angular position setpoint. All
other inputs are being held constant at their nominal values.
4-12
Servomechanism Controller
Simulation Settings View for “Scenario1”
Note The ThetaL and V unmeasured disturbances allow you to simulate additive
disturbances to these variables. By default, these disturbances are turned off, i.e., zero.
The Look ahead option designates that all future setpoint variations are known. In that
case, the controller can adjust the manipulated variable(s) in advance to improve setpoint
tracking. This would be unusual in practice, and is not being used here.
Running a Simulation
Once you're ready to run the scenario, click the Simulate button or the green arrow on
the toolbar.
4-13
4
Case-Study Examples
Note The green arrow tool is available from any view once you've defined at least one
scenario. It runs the active scenario, i.e., the one most recently selected or modified.
We obtain the results shown in Response to Unit Step in the Angular Position Setpoint.
The blue curves are the output signals, and the gray curves are the corresponding
setpoints. The response is very sluggish, and hasn't settled within the 30-second
simulation period.
Response to Unit Step in the Angular Position Setpoint
4-14
Servomechanism Controller
Note The window shown in Response to Unit Step in the Angular Position Setpoint
provides many of the customization features available in Control System Toolbox
linearSystemAnalyzer andcontrolSystemDesigner displays. Try clicking a curve
to obtain the numerical characteristics of the selected point, or right-clicking in the plot
area to open a customization menu.
The corresponding applied voltage adjustments appear in a separate window and are also
very sluggish.
On the positive side, the applied torque stays well within bounds, as does the applied
voltage.
Retuning to Achieve a Faster Servo Response
To obtain a more rapid servo response, navigate to the MPC1 Weight Tuning pane
(select the MPC1 node to get the controller design view, then click the Weight Tuning
tab) and move the slider all the way to the right. Then click the green arrow in the
toolbar. Your results should now resemble Faster Servo Response and Manipulated
Variable Adjustments.
4-15
4
Case-Study Examples
The angular position now settles within 10 seconds following the step. The torque
approaches its lower limit, but doesn't exceed it (see Faster Servo Response) and the
applied voltage stays within its limits (see Manipulated Variable Adjustments).
Faster Servo Response
4-16
Servomechanism Controller
Manipulated Variable Adjustments
Modifying the Scenario
Finally, increase the step size to π radians (select the Scenario1 node and edit the
tabular value).
As shown in Servo Response for Step Increase of π Radians and Voltage Adjustments,
the servo response is essentially as good as before, and we avoid exceeding the torque
constraint at –78.5 Nm, even though the applied voltage is saturated for about 2.5
seconds (see Voltage Adjustments).
4-17
4
Case-Study Examples
Servo Response for Step Increase of π Radians
Voltage Adjustments
4-18
Servomechanism Controller
Saving Your Work
Once you're satisfied with a controller's performance, you can export it to the workspace,
for use in a Simulink block diagram or for analysis (or you can save it in a MAT-file).
To export a controller, right-click its node and select Export from the resulting menu
(or select the Controllers node, select the controller in the list, and click the Export
button). A dialog box like that shown in Exporting a Controller to the Workspace will
appear.
The Controller source is the design from which you want to extract a controller.
There's only one in this example, but in general you might be working on several
simultaneously. The Controller to export choice defaults to the controller most
recently selected. Again, there's no choice in this case, but there could be in general. The
Name to assign edit box allows you to rename the exported controller. (This will not
change its name in the design tool.)
Exporting a Controller to the Workspace
Note When you exit the design tool, you will be prompted to save the entire design in a
MAT file. This allows you to reload it later using the File/Load menu option or the Load
icon on the toolbar.
Using Model Predictive Control Toolbox Commands
Once you've become familiar with the toolbox, you may find it more convenient to build a
controller and run a simulation using commands.
For example, suppose that you've defined the model as discussed in “Defining the Plant
Model” on page 4-4. Consider the following command sequence:
4-19
4
Case-Study Examples
ManipulatedVariables = struct('Min', -220, 'Max', 220, 'Units', 'V');
OutputVariables(1) = struct('Min', -Inf, 'Max', Inf, 'Units', 'rad');
OutputVariables(2) = struct('Min', -78.5, 'Max', 78.5, 'Units', 'Nm');
Weights = struct('Input', 0, 'InputRate', 0.05, 'Output', [10 0]);
Model.Plant = sys;
Model.Plant.OutputGroup = {[1], 'Measured' ; [2], 'Unmeasured'};
Ts = 0.1;
PredictionHorizon = 10;
ControlHorizon = 2;
This creates several structure variables. For example, ManipulatedVariables defines
the display units and constraints for the applied voltage (the manipulated plant input).
Weights defines the tuning weights shown in Controller Design View, Weight Tuning
Pane (but the numerical values used here provide better performance). Model designates
the plant model (stored in sys, which we defined earlier). The code also sets the
Model.Plant.OutputGroup property to designate the second output as unmeasured.
Constructing an MPC Object
Use the mpc command to construct an MPC object called ServoMPC:
ServoMPC = mpc(Model, Ts, PredictionHorizon, ControlHorizon);
Like the LTI objects used to define linear, time-invariant dynamic models, an MPC object
contains a complete definition of a controller.
Setting, Getting, and Displaying Object Properties
Once you've constructed an MPC object, you can change its properties as you would for
other objects. For example, to change the prediction horizon, you could use one of the
following commands:
ServoMPC.PredictionHorizon = 12;
set(ServoMPC, 'PredictionHorizon', 12);
For a listing of all the object's properties, you could type:
get(ServoMPC)
To access a particular property (e.g., the control horizon), you could type either:
M = get(ServoMPC, 'ControlHorizon');
M = ServoMPC.ControlHorizon;
You can also set multiple properties simultaneously.
4-20
Servomechanism Controller
Set the following properties before continuing with this example:
set(ServoMPC, 'Weights', Weights, ...
'ManipulatedVariables', ManipulatedVariables, ...
'OutputVariables', OutputVariables);
Typing the name of an object without a terminating semicolon generates a formatted
display of the object's properties. You can achieve the same effect using the display
command:
display(ServoMPC)
Running a Simulation
The sim command performs a linear simulation. For example, the following code
sequence defines constant setpoints for the two outputs, then runs a simulation:
TimeSteps = round(10/Ts);
r = [pi 0];
[y, t, u] = sim(ServoMPC, TimeSteps, r);
By default, the model used to design the controller (stored in ServoMPC) also represents
the plant.
The sim command saves the output and manipulated variable sequences in variables y
and u. For example,
subplot(311)
plot(t, y(:,1), [0 t(end)], pi*[1 1])
title('Angular Position (radians)');
subplot(312)
plot(t, y(:,2), [0 t(end)], [-78.5 -78.5])
title('Torque (nM)')
subplot(313)
stairs(t, u)
title('Applied Voltage (volts)')
xlabel('Elapsed Time (seconds)')
produces the custom plot shown in Plotting the Output of the sim Command. The plot
includes the angular position's setpoint. The servo response settles within 5 seconds with
no overshoot. It also displays the torque's lower bound, which becomes active after about
0.9 seconds but isn't exceeded. The applied voltage saturates between about 0.5 and 2.8
seconds, but the controller performs well despite this.
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Case-Study Examples
Plotting the Output of the sim Command
Using MPC Tools in Simulink
Block Diagram for the Servomechanism Example is a Simulink block diagram for the
servomechanism example. Most of the blocks are from the standard Simulink library.
There are two exceptions:
• Servomechanism Model is an LTI System block from the Control System Toolbox
library. The LTI model sys (which must exist in the workspace) defines its dynamic
behavior. To review how to create this model, see “Defining the Plant Model” on page
4-4.
• MPC Controller is from the MPC Blocks library. Model Predictive Control Toolbox
Simulink Block Dialog Box shows the dialog box obtained by double-clicking this
block. You need to supply an MPC object, and ServoMPC is being used here. It must
be in the workspace before you run a simulation. The Design button opens the design
tool, which allows you to create or modify the object. To review how to use commands
to create ServoMPC, see “Constructing an MPC Object” on page 4-20.
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Servomechanism Controller
Block Diagram for the Servomechanism Example
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Case-Study Examples
Model Predictive Control Toolbox Simulink Block Dialog Box
The key features of the diagram are as follows:
• The MPC Controller output is the plant input. The Voltage Scope block plots it (yellow
curve). Minimum and maximum voltage values are shown as magenta and cyan
curves.
• The plant output is a vector signal. The first element is the measured angular
position. The second is the unmeasured torque. A Demux block separates them. The
angular position feeds back to the controller and plots on the Angle scope (yellow
curve). The torque plots on the Torque scope (with its lower and upper bounds).
• The position setpoint varies sinusoidally with amplitude π radians and frequency 0.4
rad/s. It also appears on the Angle scope (magenta curve).
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Servomechanism Controller
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Case-Study Examples
The angular position tracks the sinusoidal setpoint variations well despite saturation of
the applied voltage. The setpoint variations are more gradual than the step changes used
previously, so the torque stays well within its bounds.
4-26
Paper Machine Process Control
Paper Machine Process Control
In this section...
“System Model” on page 4-27
“Linearizing the Nonlinear Model” on page 4-28
“MPC Design” on page 4-30
“Controlling the Nonlinear Plant in Simulink” on page 4-36
“References” on page 4-39
System Model
Ying et al. [1] studied the control of consistency (percentage pulp fibers in aqueous
suspension) and liquid level in a paper machine headbox, a schematic of which is shown
in Schematic of Paper Machine Headbox Elements.
Schematic of Paper Machine Headbox Elements
The process is nonlinear, and has three outputs, two manipulated inputs, and two
disturbance inputs, one of which is measured for feedforward control.
The process model is a set of ordinary differential equations (ODEs) in bilinear form. The
states are
x = [ H1
H2
N1
T
N2 ]
where H1 is the liquid level in the feed tank, H2 is the headbox liquid level, N1 is the feed
tank consistency, and N2 is the headbox consistency. The measured outputs are:
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Case-Study Examples
y = [ H2
T
N2 ]
N1
The primary control objectives are to hold H2 and N2 at setpoints. There are two
manipulated variables
u = ÈÎG p
T
Gw ˘˚
where Gp is the flow rate of stock entering the feed tank, and Gw is the recycled white
water flow rate. The consistency of stock entering the feed tank, Np, is a measured
disturbance.
v = Np
The white water consistency is an unmeasured disturbance.
d = Nw
Variables are normalized. All are zero at the nominal steady state and have comparable
numerical ranges. Time units are minutes. The process is open-loop stable.
The mpcdemos folder contains the file mpc_pmmodel.m, which implements the nonlinear
model equations as a Simulink S-function. The input sequence is Gp, Gw, Np, Nw, and the
output sequence is 2, N1, N2.
Linearizing the Nonlinear Model
The paper machine headbox model is easy to linearize analytically, yielding the following
state space matrices:
A = [-1.9300
0
0
0
0.3940
-0.4260
0
0
0
0
-0.6300
0
0.8200
-0.7840
0.4130
-0.4260];
B = [1.2740
1.2740
0
0
0
0
0
0
1.3400
-0.6500
0.2030
0.4060
0
0
0
0];
C = [0
1.0000
0
0
0
0
1.0000
0
4-28
Paper Machine Process Control
0
D = zeros(3,4);
0
0
1.0000];
Use these to create a continuous-time LTI state-space model, as follows:
PaperMach = ss(A, B, C, D);
PaperMach.InputName = {'G_p', 'G_w', 'N_p', 'N_w'};
PaperMach.OutputName = {'H_2', 'N_1', 'N_2'};
(The last two commands are optional; they improve plot labeling.)
As a quick check of model validity, plot its step responses as follows:
step(PaperMach);
The results appear in the following figure. Note the following:
• The two manipulated variables affect all three outputs.
• They have nearly identical effects on H2.
• The Gw→N2 pairing exhibits an inverse response.
These features make it difficult to achieve accurate, independent control of H2 and N2.
Linearized Paper Machine Model's Step Responses
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Case-Study Examples
MPC Design
Type
mpctool
to open the MPC design tool. Import your LTI Paper Mach model as described in
“Opening MPCTOOL and Importing a Model” on page 4-5.
Next, define signal properties, being sure to designate Np and Nw as measured and
unmeasured disturbances, respectively. Your specifications should resemble Signal
Properties for the Paper Machine Application.
Signal Properties for the Paper Machine Application
Initial Controller Design
If necessary, review “Specifying Controller Properties” on page 4-9. Then click the MPC1
node and specify the following controller parameters, leaving others at their default
values:
• Models and Horizons. Control interval = 2 minutes
• Constraints. For both Gp and Gw, Minimum = –10, Maximum = 10, Max down rate =
–2, Max up rate = 2.
• Weight Tuning. For both Gp and Gw, Weight = 0, Rate weight = 0.4.
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Paper Machine Process Control
For N1, Weight = 0. (Other outputs have Weight = 1.)
Servo Response
Finally, select the Scenario1 node and define a servo-response test:
• Duration = 30
• H2 setpoint = 1 (constant)
Simulate the scenario. You should obtain results like those shown in Servo Response for
Unit Step in Headbox Level Setpoint and Manipulated Variable Moves.
Servo Response for Unit Step in Headbox Level Setpoint
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Manipulated Variable Moves
Weight Tuning
The response time is about 8 minutes. We could reduce this by decreasing the control
interval, reducing the manipulated variable rate weights, and/or eliminating the up/
down rate constraints. The present design uses a conservative control effort, which would
usually improve robustness, so we will continue with the current settings.
Note the steady-state error in N1 (it's about –0.25 units in Servo Response for Unit Step
in Headbox Level Setpoint). There are only two manipulated variables, so it's impossible
to hold three outputs at setpoints. We don't have a setpoint for N1 so we have set its
weight to zero (see controller settings in “Initial Controller Design” on page 4-30).
Otherwise, all three outputs would have exhibited steady-state error (try it).
Consistency control is more important than level control. Try decreasing the H2 weight
from 1 to 0.2. You should find that the peak error in N2 decreases by almost an order of
magnitude, but the H2 response time increases from 8 to about 18 minutes (not shown).
Use these modified output weights in subsequent tests.
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Paper Machine Process Control
Feedforward Control
To configure a test of the controller's feedforward response, define a new scenario by
clicking the Scenarios node, clicking the New button, and renaming the new scenario
Feedforward (by editing its name in the tree or the summary list).
In the Feedforward scenario, define a step change in the measured disturbance, Np,
with Initial value = 0, Size = 1, Time = 10. All output setpoints should be zero. Set the
Duration to 30 time units.
If response plots from the above servo response tests are still open, close them. Simulate
the Feedforward scenario. You should find that the H2 and N2 outputs deviate very
little from their setpoints (not shown).
Experiment with the “look ahead” feature. First, observe that in the simulation just
completed the manipulated variables didn't begin to move until the disturbance occurred
at t = 10 minutes. Return to the Feedforward scenario, select the Look ahead option
for the measured disturbance, and repeat the simulation.
Notice that the manipulated variables begin changing in advance of the disturbance.
This happens because the look ahead option uses known future values of the disturbance
when computing its control action. For example, at time t = 0 the controller is using
a prediction horizon of 10 control intervals (20 time units), so it “sees” the impending
disturbance at t = 10 and begins to prepare for it. The output setpoint tracking improves
slightly, but it was already so good that the improvement is insignificant. Also, it's
unlikely that there would be advanced knowledge of a consistency disturbance, so clear
the Look ahead check box for subsequent simulations.
Unmeasured Input Disturbance
To test the response to unmeasured disturbances, define another new scenario called
Feedback. Configure it as for Feedforward, but set the measured disturbance, Np, to
zero (constant), and the unmeasured disturbance, Nw, to 1.0 (constant). This simulates a
sudden, sustained, unmeasured disturbance occurring at time zero.
Running the simulation should yield results like those in Feedback Scenario:
Unmeasured Disturbance Rejection. The two controlled outputs (H2 and N2 ) exhibit
relatively small deviations from their setpoints (which are zero). The settling time is
longer than for the servo response (compare to Servo Response for Unit Step in Headbox
Level Setpoint) which is typical.
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Case-Study Examples
Feedback Scenario: Unmeasured Disturbance Rejection
One factor limiting performance is the chosen control interval of 2 time units. The
controller can't respond to the disturbance until it first appears in the outputs, i.e., at t =
2. If you wish, experiment with larger and smaller intervals (modify the specification on
the controller's Model and Horizons tab).
Effect of Estimator Assumptions
Another factor influencing the response to unmeasured disturbances (and model
prediction error) is the estimator configuration. The results shown in Feedback Scenario:
Unmeasured Disturbance Rejection are for the default configuration.
To view the default assumptions, select the controller node (MPC1), and click
its Estimation tab. The resulting view should be as shown in Default Estimator
Assumptions: Output Disturbances. The status message (bottom of figure) indicates that
Model Predictive Control Toolbox default assumptions are being used.
4-34
Paper Machine Process Control
Default Estimator Assumptions: Output Disturbances
Now consider the upper part of the figure. The Output Disturbances tab is active, and
its Signal-by-signal option is selected. According to the tabular data, the controller is
assuming independent, step-like disturbances (i.e., integrated white noise) in the first
two outputs.
Click the Input Disturbances tab. Verify that the controller is also assuming
independent step-like disturbances in the unmeasured disturbance input.
Thus, there are a total of three independent, sustained (step-like) disturbances. This
allows the controller to eliminate offset in all three measured outputs.
The disturbance magnitudes are unity by default. Making one larger than the rest would
signify a more important disturbance at that location.
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Case-Study Examples
Click the Measurement Noise tab. Verify that white noise (unit magnitude) is
being added to each output. The noise magnitude governs how much influence each
measurement has on the controller's decisions. For example, if a particular measurement
is relatively noisy, the controller will give it less weight, relying instead upon the model
predictions of that output. This provides a noise filtering capability.
In the paper machine application, the default disturbance assumptions are reasonable. It
is difficult to improve disturbance rejection significantly by modifying them.
Controlling the Nonlinear Plant in Simulink
It's good practice to run initial tests using the linear plant model as described in “Servo
Response” on page 4-31 and “Unmeasured Input Disturbance” on page 4-33. Such
tests don't introduce prediction error, and are a useful benchmark for more demanding
tests with a nonlinear plant model. The controller's prediction model is linear, so such
tests introduce prediction error.
Open the paper machine headbox control Simulink model by typing:
mpc_papermachine
Paper Machine Headbox Control Using MPC Tools in Simulink
Paper Machine Headbox Control Using MPC Tools in Simulink is a Simulink model
in which the Model Predictive Control Toolbox controller is being used to regulate the
nonlinear paper machine headbox model. The block labeled S-Function embodies the
nonlinear model, which is coded in a file called mpc_pmmodel.m.
As shown in the following dialog box, the MPC block references a controller design called
MPC1, which was exported to the MATLAB workspace from the design tool. Note also
4-36
Paper Machine Process Control
that the measured disturbance inport is enabled, allowing the measured disturbance to
be connected as shown in Paper Machine Headbox Control Using MPC Tools in Simulink.
Test, Output Variables shows the scope display from the “Outputs” block for the setup
of Paper Machine Headbox Control Using MPC Tools in Simulink, i.e., an unmeasured
disturbance. The yellow curve is H2, the magenta is N1, and the cyan is N2. Comparing to
Feedback Scenario: Unmeasured Disturbance Rejection, the results are almost identical,
indicating that the effects of nonlinearity and prediction error were insignificant in
this case. Simulink Test, Manipulated Variables shows the corresponding manipulated
variable moves (from the “MVs” scope in Paper Machine Headbox Control Using MPC
Tools in Simulink) which are smooth yet reasonably fast.
As disturbance size increases, nonlinear effects begin to appear. For a disturbance size
of 4, the results are still essentially the same as shown in Test, Output Variables and
Simulink Test, Manipulated Variables (scaled by a factor of 4), but for a disturbance size
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Case-Study Examples
of 6, the setpoint deviations are relatively larger, and the curve shapes differ (not shown).
There are marked qualitative and quantitative differences when the disturbance size is
8. When it is 9, deviations become very large, and the MVs saturate. If such disturbances
were likely, the controller would have to be retuned to accommodate them.
Test, Output Variables
4-38
Paper Machine Process Control
Simulink Test, Manipulated Variables
References
[1] Ying, Y., M. Rao, and Y. Sun “Bilinear control strategy for paper making process,”
Chemical Engineering Communications (1992), Vol. 111, pp. 13–28.
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4
Case-Study Examples
Bumpless Transfer Between Manual and Automatic Operations
In this section...
“Open Simulink Model” on page 4-40
“Define Plant and MPC Controller” on page 4-41
“Configure MPC Block Settings” on page 4-42
“Examine Switching Between Manual and Automatic Operation” on page 4-43
“Turn off Manipulated Variable Feedback” on page 4-45
This example shows how to bumplessly transfer between manual and automatic
operations of a plant.
During startup of a manufacturing process, operators adjust key actuators manually
until the plant is near the desired operating point before switching to automatic control.
If not done correctly, the transfer can cause a bump, that is, large actuator movement.
In this example, you simulate a Simulink model that contains a single-input singleoutput LTI plant and an MPC Controller block.
A model predictive controller monitors all known plant signals, even when it is not
in control of the actuators. This monitoring improves its state estimates and allows a
bumpless transfer to automatic operation.
Open Simulink Model
Open the Simulink model.
open_system('mpc_bumpless');
4-40
Bumpless Transfer Between Manual and Automatic Operations
To simulate switching between manual and automatic operation, the Switching block
sends either 1 or 0 to control a switch. When it sends 0, the system is in automatic mode,
and the output from the MPC Controller block goes to the plant. Otherwise, the system is
in manual mode, and the signal from the Operator Commands block goes to the plant.
In both cases, the actual plant input feeds back to the controller ext.mv inport, unless
the plant input saturates at –1 or 1. The controller constantly monitors the plant output
and updates its estimate of the plant state, even when in manual operation.
This model also shows the optimization switching option. When the system switches to
manual operation, a nonzero signal enters the switch inport of the controller block. The
signal turns off the optimization calculations, which reduces computational effort.
Define Plant and MPC Controller
Create the plant model.
num=[1 1];
den=[1 3 2 0.5];
sys=tf(num,den);
The plant is a stable single-input single-output system as seen in its step response.
step(sys)
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4
Case-Study Examples
Create an MPC controller.
Ts=0.5;
% sampling time (seconds)
p=15;
% prediction horizon
m=2;
% control horizon
MPC1=mpc(sys,Ts,p,m);
MPC1.Weights.Output=0.01;
MPC1.MV=struct('Min',-1,'Max',1);
Tstop=250;
Configure MPC Block Settings
Open the Function Block Parameters: MPC Controller dialog box.
• Specify MPC1 in the MPC Controller box.
• Verify that the External Manipulated Variable (ext.mv) option in the General
tab is selected. This option adds the ext.mv inport to the block to enable the use of
external manipulated variables.
• Verify that the Use external signal to enable or disable optimization (switch)
option in the Others tab is selected. This option adds the switch inport to the
controller block to enable switching off the optimization calculations.
4-42
Bumpless Transfer Between Manual and Automatic Operations
Click OK.
Examine Switching Between Manual and Automatic Operation
Click Run in the Simulink model window to simulate the model.
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4
Case-Study Examples
For the first 90 time units, the Switching Signal is 0, which makes the system
operate in automatic mode. During this time, the controller smoothly drives the
controlled plant output from its initial value, 0, to the desired reference value, –0.5.
4-44
Bumpless Transfer Between Manual and Automatic Operations
The controller state estimator has zero initial conditions as a default, which is
appropriate when this simulation begins. Thus, there is no bump at startup. In general,
start the system running in manual mode long enough for the controller to acquire an
accurate state estimate before switching to automatic mode.
At time 90, the Switching Signal changes to 1. This change switches the system to
manual operation and sends the operator commands to the plant. Simultaneously, the
nonzero signal entering the switch inport of the controller turns off the optimization
calculations. While the optimization is turned off, the MPC Controller block passes the
current ext.mv signal to the Controller Output.
Once in manual mode, the operator commands set the manipulated variable to –0.5 for
10 time units, and then to 0. The Plant Output plot shows the open-loop response
between times 90 and 180 when the controller is deactivated.
At time 180, the system switches back to automatic mode. As a result, the plant output
returns to the reference value smoothly, and a similar smooth adjustment occurs in the
controller output.
Turn off Manipulated Variable Feedback
Delete the signals entering the ext.mv and switch inports of the controller block.
Delete the Unit Delay block and the signal line entering its inport.
Open the Function Block Parameters: MPC Controller dialog box.
Deselect the External Manipulated Variable (ext.mv) option in the General tab to
remove the ext.mv inport from the controller block.
Deselect the Use external signal to enable or disable optimization (switch) option
in the Others tab to remove the switch inport from the controller block.
Click OK. The Simulink model now resembles the following figure.
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Case-Study Examples
Click Run to simulate the model.
4-46
Bumpless Transfer Between Manual and Automatic Operations
The behavior is identical to the original case for the first 90 time units.
When the system switches to manual mode at time 90, the plant behavior is the same
as before. However, the controller tries to hold the plant at the setpoint. So, its output
increases and eventually saturates, as seen in Controller Output. Since the controller
assumes that this output is going to the plant, its state estimates become inaccurate.
Therefore, when the system switches back to automatic mode at time 180, there is a large
bump in the Plant Output.
Such a bump creates large actuator movements within the plant. By smoothly
transferring from manual to automatic operation, a model predictive controller
eliminates such undesired movements.
Related Examples
•
“Switching Controller Online and Offline with Bumpless Transfer” on page 4-48
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Case-Study Examples
Switching Controller Online and Offline with Bumpless Transfer
This example shows how to obtain bumpless transfer when switching model predictive
controller from manual to automatic operation or vice versa.
In particular, it shows how the EXT.MV input signal to the MPC block can be used to
keep the internal MPC state up to date when the operator or another controller is in
control.
Define Plant Model
The linear open-loop dynamic plant model is as follows:
num = [1 1];
den = [1 3 2 0.5];
sys = tf(num,den);
Design MPC Controller
Construct MPC controller
Create an MPC controller with plant model, sample time and horizons.
Ts = 0.5;
% Sampling time
p = 15;
% Prediction horizon
m = 2;
% Control horizon
mpcobj = mpc(sys,Ts,p,m);
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
MV Constraints
Define constraints on the manipulated variable.
mpcobj.MV=struct('Min',-1,'Max',1);
Weights
Change the output weight.
mpcobj.Weights.Output=0.01;
Simulate Using Simulink®
To run this example, Simulink® is required.
4-48
Switching Controller Online and Offline with Bumpless Transfer
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
Simulate closed-loop control of the linear plant model in Simulink. Controller "mpcobj" is
specified in the block dialog.
mdl = 'mpc_bumpless';
open_system(mdl)
sim(mdl)
-->Converting the "Model.Plant" property of "mpc" object to state-space.
-->Converting model to discrete time.
-->Integrated white noise added on measured output channel #1.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
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Case-Study Examples
4-50
Switching Controller Online and Offline with Bumpless Transfer
Simulate without Using External MV Signal
Without using the external MV signal, MPC controller is no longer able to provide
bumpless transfer because the internal controller states are not estimated correctly.
delete_line(mdl,'Switch/1','Unit Delay/1');
delete_line(mdl,'Unit Delay/1','MPC Controller/3');
delete_block([mdl '/Unit Delay']);
delete_line(mdl,'Switching/1','MPC Controller/4');
set_param([mdl '/MPC Controller'],'mv_inport','off');
set_param([mdl '/MPC Controller'],'switch_inport','off');
set_param([mdl '/Yplots'],'Ymin','-1~-0.1')
set_param([mdl '/Yplots'],'Ymax','3~1.1')
set_param([mdl '/MVplots'],'Ymin','-1.1~-5')
set_param([mdl '/MVplots'],'Ymax','1.1~10')
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Case-Study Examples
sim(mdl);
4-52
Switching Controller Online and Offline with Bumpless Transfer
Now the transition from manual to automatic control is much less smooth. Note the large
"bump" between time = 180 and 200.
bdclose(mdl)
Related Examples
•
“Bumpless Transfer Between Manual and Automatic Operations” on page 4-40
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4
Case-Study Examples
Coordinate Multiple Controllers at Different Operating Points
Chemical reactors can exhibit strongly nonlinear behavior due to the exponential effect
of temperature on reaction rate. If the primary reaction is exothermic, an increase in
reaction rate causes an increase in reactor temperature. This positive feedback can lead
to open-loop unstable behavior.
Reactors operate in either a continuous or a batch mode. In batch mode, operating
conditions can change dramatically during a batch as the reactants disappear. Although
continuous reactors typically operate at steady state, they must often move to a new
steady state. In other words, both batch and continuous reactors need to operate safely
and efficiently over a range of conditions.
If the reactor behaves nonlinearly, a single linear controller might not be able to manage
such transitions. One approach is to develop linear models that cover the anticipated
operating range, design a controller based on each model, and then define a criterion by
which the control system switches from one such controller to another. Gain scheduling is
an established technique. The challenge is to move the reactor operating conditions from
an initial steady-state point to a much different condition. The transition passes through
a region in which the plant is open-loop unstable. This example illustrates an alternative
— coordination of multiple MPC controllers. The solution uses the Simulink Multiple
MPC Controller block to coordinate the use of three controllers, each of which has been
designed for a particular operating region.
The subject process is a constant-volume continuous stirred-tank reactor (CSTR). The
model consists of two nonlinear ordinary differential equations (see [1]). The model
states are the reactor temperature and the rate-limiting reactant concentration. For the
purposes of this example, both are assumed to be measured plant outputs.
There are three inputs:
• Concentration of the limiting reactant in the reactor feed stream, kmol/m3
• The reactor feed temperature, K
• The coolant temperature, K
The control system can adjust the coolant temperature in order to regulate the reactor
state and the rate of the exothermic main reaction. The other two inputs are independent
unmeasured disturbances.
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Coordinate Multiple Controllers at Different Operating Points
The Simulink diagram for this example appears below. The CSTR model is a masked
subsystem. The feed temperature and composition are constants. As discussed above, the
control system adjusts the coolant temperature (the Tc input on the CSTR block).
The two CSTR outputs are the reactor temperature and composition respectively. These
are being sent to a scope display and to the control system as feedback.
The reference signal (i.e. setpoint) is coming from variable CSTR_Setpoints, which is in
the base workspace. As there is only one manipulated variable (the coolant temperature)
the control objective is to force the reactor concentration to track a specified trajectory.
The concentration setpoint also goes to the Plant State scope for plotting. The control
system receives a setpoint for the reactor temperature too but the controller design
ignores it.
In that case why supply the temperature measurement to the controller? The main
reason is to improve state estimation. If this were not done, the control system would
have to infer the temperature value from the concentration measurement, which would
introduce an estimation error and degrade the model's predictive accuracy.
The rationale for the Switch 1 and Switch 2 blocks appears below.
The figure below shows the Multi MPC Controller mask. The block is coordinating
three controllers (MPC1, MPC2 and MPC3 in that sequence). It is also receiving the setpoint
signal from the workspace, and the Look ahead option is active. This allows the
controller to anticipate future setpoint values and usually improves setpoint tracking.
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Case-Study Examples
In order to designate which one of the three controllers is active at each time instant, we
send the Multi MPC Controllers block a switching signal (connected to its switch
input port). If it is 1, MPC1 is active. If it is 2, MPC2 is active, and so on.
In the diagram, Switch 1 and Switch 2 perform the controller selection function as
follows:
• If the reactor concentration is 8 kmol/m3 or greater, Switch 1 sends the constant 1 to
its output. Otherwise it sends the constant 2.
• If the reactor concentration is 3 kmol/m3 or greater, Switch 2 passes through the
signal coming from Switch 1 (either 1 or 2). Otherwise is sends the constant 3.
4-56
Coordinate Multiple Controllers at Different Operating Points
Thus, each controller handles a particular composition range. The simulation begins
with the reactor at an initial steady state of 311K and 8.57 kmol/m3. The feed
concentration is 10 kmol/m3 so this is a conversion of about 15%, which is low.
The control objective is to transition smoothly to 80% conversion with the reactor
concentration at 2 kmol/m3. The simulation will start with MPC1 active, transition to
MPC2, and end with MPC3.
We decide to design the controllers around linear models derived at the following three
reactor compositions (and the corresponding steady-state temperature): 8.5, 5.5, and 2
kmol/m3.
In practice, you would probably obtain the three models from data. This example
linearizes the nonlinear model at the above three conditions (for details see “Using
Simulink to Develop LTI Models” in the Getting Started Guide).
Note As shown later, we need to retain at the unmeasured plant inputs in the
model. This prevents us from using the Model Predictive Control Toolbox automatic
linearization feature. In the current toolbox, the automatic linearization feature can
linearize with respect to manipulated variable and measured disturbance inputs only.
The following code obtains the linear models and designs the three controllers
[sys, xp] = CSTR_INOUT([],[],[],'sizes');
up = [10 298.15 298.15]';
yp = xp;
Ts = 1;
Nc = 3;
Controllers = cell(1,3);
Concentrations = [8.5 5.5 2];
Y = yp;
for i = 1:Nc
clear Model
Y(2) = Concentrations(i);
[X,U,Y,DX]=trim('CSTR_INOUT',xp(:),up(:),Y(:),[],[1,2]',2)
[a,b,c,d]=linmod('CSTR_INOUT', X, U );
Plant = ss(a,b,c,d);
Plant.InputGroup.MV = 3;
Plant.InputGroup.UD = [1,2];
Model.Plant = Plant;
Model.Nominal.U = [0; 0; up(3)];
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Case-Study Examples
Model.Nominal.X = xp;
Model.Nominal.Y = yp;
MPCobj = mpc(Model, Ts);
MPCobj.Weight.OV = [0 1];
D = ss(getindist(MPCobj));
D.b = D.b*10;
set(D,'InputName',[],'OutputName',[],'InputGroup',[], ...
'OutputGroup',[]);
setindist(MPCobj, 'model', D);
Controllers{i} = MPCobj;
end
MPC1 = Controllers{1};
MPC2 = Controllers{2};
MPC3 = Controllers{3}
The key points regarding the designs are as follows:
• All three controllers use the same nominal condition, the values of the plant inputs
and outputs at the initial steady-state. Exception: all unmeasured disturbance inputs
must have zero nominal values.
• Each controller employs a different prediction model. The model structure is the same
in each case (input and outputs are identical in number and type) but each model
represents a particular steady-state reactor composition.
• It turns out that the MPC2 plant model obtained at 5 kmol/m3 is open-loop unstable.
We must use a model structure that promotes a stable Kalman state estimator. If
we include the unmeasured disturbance inputs in the prediction model, the default
estimator assumes integrated white noise at each such input, which produces a stable
estimator in this case.
• The default estimator signal-to-noise settings are inappropriate, however. If you
use them and monitor the state estimates (not shown), the internally estimated
temperature and composition can be far from the measured values. To overcome
this, we increase the signal-to-noise ratio in each disturbance channel. See the use of
getindist and setindist above. The default signal to noise is being increased by a
factor of 10.
• We are using a zero weight on the measured temperature. See the above discussion of
control objectives for the rationale.
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Coordinate Multiple Controllers at Different Operating Points
The above plots show the simulation results. The Multi MPC Controller block uses the
three controllers sequentially as expected (see the switching signal). Tracking of the
concentration setpoint is excellent and the reactor temperature is also controlled well.
To achieve this, the control system starts by increasing the coolant temperature,
causing the reaction rate to increase. Once the reaction has achieved a high rate, it
generates substantial heat and the coolant temperature must decrease to keep the
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Case-Study Examples
reactor temperature under control. As the reactor concentration depletes, the reaction
rate slows and the control system must again raise the coolant temperature, finally
settling at 305 K, about 7 K above the initial condition.
For comparison the plots below show the results for the same scenario if we force MPC3 to
be active for the entire simulation. The CSTR eventually stabilizes at the desired steadystate but both the reactor temperature and composition exhibit large excursions away
from the desired conditions.
4-60
Using Custom Constraints in Blending Process
Using Custom Constraints in Blending Process
In this section...
“About the Blending Process” on page 4-61
“MPC Controller with Custom Input/Output Constraints” on page 4-62
About the Blending Process
A continuous blending process combines three feeds in a well-mixed container to produce
a blend having desired properties. The dimensionless governing equations are:
3
dv
= Â fi - f
dt i =1
V
dg j
dt
3
(
)
= Â g ij - g j fi
i =1
where V is the mixture inventory (in the container), ϕi is the flow rate of the ith feed, ϕ
is the demand, i.e., the rate at which the blend is being removed from inventory, γij is the
concentration of constituent j in feed i, γj is the concentration of j in the blend, and τ is
time. In this example, there are two important constituents, j = 1 and 2.
The control objectives are targets for the blend’s two constituent concentrations and the
mixture inventory. The challenge is that the demand ϕ and feed compositions γij vary.
The inventory, blend compositions, and demand are measured, but the feed compositions
are unmeasured.
At the nominal operating condition:
• Feed 1 ϕ1 (mostly constituent 1) is 80% of the total inflow ϕ.
• Feed 2 ϕ2 (mostly constituent 2) is 20%.
• Feed 3 ϕ3 (pure constituent 1) is not used.
The process design allows manipulation of the total feed entering the mixing chamber
and the individual rates of feeds 2 and 3. In other words, the rate of feed 1 is:
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4
Case-Study Examples
ϕ1 = ϕT – ϕ2 – ϕ3
Each feed has limited availability:
0 ≤ ϕi ≤ ϕi,max
The equations are normalized such that—at the nominal steady state—the mean
residence time in the mixing container is τ = 1. The target inventory is V = 1, and the
target blend composition is γ1 = γ2 = 1.
The constraints ϕi,max = 0.8 is imposed by an upstream process and ϕ2,max = ϕ3,max = 0.6 is
imposed by physical limits.
MPC Controller with Custom Input/Output Constraints
1
Open the Simulink model mpc_blendingprocess:
In the model, an MPC controller controls the blending process. The block labeled
Blending incorporates the previously described model equations and includes
unmeasured (step) disturbances in the feed compositions.
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Using Custom Constraints in Blending Process
Signal u represents controller adjustments, i.e.,
u = [ϕT ϕ2 ϕ3].
Signal v represents the demand, ϕ which is a measured disturbance. The operator
can vary the demand, and the resulting signal goes to the process and controller.
Consider the following scenario:
• At τ = 0, the process is operating at its nominal steady state.
• At τ = 0.5, the demand decreases from ϕ = 1 to ϕ = 0.9.
• At τ = 1, there is a large increase in the concentration of constituent 1 in feed 1
γ11 from 1.17 to 2.17.
The plant is mildly nonlinear. You can derive a linear model at the nominal steady
state. This approach is quite accurate unless the (unmeasured) feed compositions
change. If the change is sufficiently large, the steady-state gains of the nonlinear
process change sign and the closed-loop system can become unstable.
2
Define a linear model.
% Create a linear approximation -- a state-space model based on the nominal
% operating point:
ni = 3;
% number of feed streams
nc = 2;
% number of components
Fin_nom = [1.6, 0.4, 0];
% Nominal flow rate for the ith feed stream
F_nom = sum(Fin_nom);
% Nominal flow rate for the exit stream (demand)
cin_nom = [0.7 0.2 0.8
% Nominal composition for jth constituent in the ith feed flow
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Case-Study Examples
0.3 0.8
0];
cout_nom = cin_nom*Fin_nom'/F_nom;
% Nominal product composition
% Normalize the linear model such that the target demand is 1 and the
% product composition is 1:
fin_nom = Fin_nom/F_nom;
gij = [cin_nom(1,:)/cout_nom(1)
cin_nom(2,:)/cout_nom(2)];
% Create the state-space model with feed flows |[F1, F2, F3]| as MVs:
A = [ zeros(1,nc+1)
zeros(nc,1) -eye(nc)];
Bu = [ones(1,ni)
gij-1];
% Change MV definition to [FT, F2, F3] where F1 = FT - F2 - F3
Bu = [Bu(:,1), Bu(:,2)-Bu(:,1), Bu(:,3)-Bu(:,1)];
% Add the blend demand as the 4th model input, a measured disturbance
Bv = [-1
zeros(nc,1)];
B = [Bu Bv];
% All the states (inventory and compositions) are measurable
C = eye(nc+1);
% No direct feed-through term
D = zeros(nc+1,ni+1);
% Construct the plant model
Model = ss(A, B, C, D);
Model.InputName = {'F_T','F_2','F_3','F'};
Model.InputGroup.MV = 1:3;
Model.InputGroup.MD = 4;
Model.OutputName = {'V','c_1','c_2'};
3
Specify an MPC controller.
% Create the controller object with sampling period, prediction and control
% horizons:
Ts = 0.1;
p=10;
m=3;
MPCobj = mpc(Model, Ts, p, m);
% The outputs are the inventory |y(1)| and the constituent concentrations
% |y(2)| and |y(3)|. Specify nominal values of unity after normalization:
MPCobj.Model.Nominal.Y = [1 1 1];
% The manipulated variables are |u1 = FT|, |u2 = F2|, |u3 = F3|.
% nominal values after normalization:
MPCobj.Model.Nominal.U = [1 fin_nom(2) fin_nom(3) 1];
Specify
% Specify output tuning weights. Larger weights are assigned to the first
% two outputs because we pay more attention to controlling the inventory
% and composition of the first blending material:
MPCobj.Weights.OV = [1 1 0.5];
% Specify the hard bounds (physical limits) on the manipulated variables:
umin = [0 0 0];
umax = [2 0.6 0.6];
for i = 1:3
MPCobj.MV(i).Min = umin(i);
MPCobj.MV(i).Max = umax(i);
MPCobj.MV(i).RateMin = -0.1;
MPCobj.MV(i).RateMax = 0.1;
end
4-64
Using Custom Constraints in Blending Process
The total feed rate and the rates of feed 2 and feed 3 have upper bounds. Feed 1 also
has an upper bound, determined by the upstream unit supplying it. Under normal
conditions, the plant operates far from these bounds but for the scenario outlined
previously, the controller must reduce the rate of feed 1 drastically, as it is bringing
in excess constituent 1. To do this, the controller must increase the rates of feeds 2
and 3 (keeping the total feed rate close to the demand rate to maintain the target
inventory.)
4
Specify constraints.
Given the specified bounds on the feed 2 and 3 rates (= 0.6), it is possible that their
sum could be as much as 1.2. Because the total feed rate is of order 0.9 to 1.0, the
controller can request a physically impossible condition in which the sum of feeds 2
and 3 exceeds the total feed rate. This implies a negative feed 1 rate.
The constraint
0 ≤ ϕ1 = ϕT –
ϕ2
– ϕ3 ≤ 0.8
prevents the controller from requesting an unrealistic ϕ1 value.
To specify this constraint, enter:
E = [-1 1 1; 1 -1 -1];
% No outputs are specified in the mixed constraints, so set their
% coefficients to zero:
F = zeros(2,3);
% Specify vector g in E*u + F*y <= g:
g = [0; 0.8];
% Specify that both constraints are hard (ECR = 0):
v = zeros(2,1);
% Specify zero coefficients for the measured disturbance:
h = zeros(2,1);
% Include the mixed constraints in the controller object:
setconstraint(MPCobj, E, F, g, v, h);
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4
Case-Study Examples
v = zeros(2,1) defines hard constraints, which are reasonable here because the
constraints involve manipulated variables only. If the constraints involved a mixture
of input and output variables, use soft constraints.
5
Simulate the model and plot the input and output signals.
sim('mpc_blendingprocess')
figure
plot(MVs.time,[MVs.signals(1).values(:,2), ...
(MVs.signals(2).values + MVs.signals(3).values), ...
(MVs.signals(1).values(:,2)-MVs.signals(2).values-MVs.signals(3).values)])
grid
legend('FT','F2+F3','F1')
The plot shows the evolution of the total feed rate (blue curve) and the sum of feeds 2
and 3 (green curve). They coincide between τ = 1.7 and τ = 2.2.
If the custom input constraints had not been included, the controller would have
requested a negative feed 1 rate during this period, as shown in by the red curve.
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Using Custom Constraints in Blending Process
The controller maintains the inventory very close to its setpoint, but the severe
disturbance in the feed composition causes a prediction error and a large disturbance
in the blend composition (especially for constituent 1). Despite this, the controller
recovers and drives the blend composition back to its setpoint, as shown in the
following output of the CVs scope.
Related Examples
• MPC Control with Constraints on a Combination of Input and Output Signals
• MPC Control of a Nonlinear Blending Process
More About
“Constraints on Linear Combinations of Inputs and Outputs” on page 2-25
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Case-Study Examples
Providing LQR Performance Using Terminal Penalty
This example, from Scokaert and Rawlings [1], shows how to make a finite-horizon Model
Predictive Controller equivalent to an infinite-horizon linear quadratic regulator (LQR).
The standard MPC cost function is similar to that used in an LQR controller with output
weighting, as shown in the following equation:
•
J (u) =
 y(k + i)T Qy(k + i) + u(k + i - 1)T Ru(k + i - 1)
i =1
The LQR and MPC cost functions differ in the following ways:
• The LQR cost function forces y and u towards zero whereas the MPC cost function
forces y and u toward nonzero setpoints.
You can shift the MPC prediction model’s origin to eliminate this difference and
achieve zero setpoints at nominal condition.
• The LQR cost function uses an infinite prediction horizon in which the manipulated
variable changes at each sampling instant. In the standard MPC cost function, the
horizon length is p, and the manipulated variable changes m times, where m is the
control horizon.
The two cost functions are equivalent if the MPC cost function is:
p -1
J (u) =
 y(k + i)T Qy(k + i) + u(k + i - 1)T Ru(k + i - 1) + x(k + p)T Qp x(k + p)
i =1
where Qp is a penalty applied at the last (i.e., terminal) prediction horizon step, and
the prediction and control horizons are equal, i.e., p = m. The required Qp is the
Ricatti matrix that you can calculate using the Control System Toolbox lqr and lqry
commands. The value is a positive definite symmetric matrix.
The following procedure shows how to design an unconstrained MPC controller that
provides performance equivalent to a LQR controller:
1
Define a plant with one input and two outputs.
The plant is a double-integrator, represented as a state-space model in discrete-time
with sampling interval 0.1 seconds.
4-68
Providing LQR Performance Using Terminal Penalty
A = [1 0;0.1 1];
B = [0.1;0.005];
C = eye(2);
D = zeros(2,1);
Ts = 0.1;
Plant = ss(A,B,C,D,Ts);
Plant.InputName = {'u'};
Plant.OutputName = {'x_1','x_2'};
2
Design an LQR controller with output feedback for the plant.
Q = eye(2);
R = 1;
[K,Qp] = lqry(Plant,Q,R);
Q and R are output and input weight matrices, respectively. Qp is the Ricatti matrix.
3
Design an MPC controller equivalent to the LQR controller.
To implement Equation 4-2, compute L, the Cholesky decomposition of Qp, such that
LTL= Qp. Then, define auxiliary unmeasured output variables ya(k) = Lx(k) such
that yaTya = xTQpx. For the first p-1 prediction horizon steps, the standard Q and
R weights apply to the original u and y, and ya has a zero penalty. On step p, the
original u and y have zero penalties, and ya has a unity penalty.
a
Augment the plant model, and specify the augmented outputs as unmeasured.
NewPlant = Plant;
cholP = chol(Qp);
set(NewPlant,'C',[C;cholP],'D',[D;zeros(2,1)],...
'OutputName',{'x_1','x_2','Cx_1','Cx_2'});
NewPlant.InputGroup.MV = 1;
NewPlant.OutputGroup.MO = [1 2];
NewPlant.OutputGroup.UO = [3 4];
b
Create an MPC controller with equal prediction and control horizons.
P = 3;
M = 3;
MPCobj = mpc(NewPlant,Ts,P,M);
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assumi
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. As
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming de
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4
Case-Study Examples
for output(s) y1 and zero weight for output(s) y2 y3 y4
When there are no constraints, you can use a rather short horizon (in this case, p
≥ 1 gives identical results).
c
Specify weights for manipulated variables (MV) and output variables (OV).
ywt = sqrt(diag(Q))';
uwt = sqrt(diag(R))';
MPCobj.Weights.OV = [ywt 0 0];
MPCobj.Weights.MV = uwt;
MPCobj.Weights.MVrate = 1e-6;
The two augmented outputs have zero weights during the prediction horizon.
d
Specify terminal weights.
To obtain the desired effect, define unity weights for these at the final point in
the horizon.
U = struct('Weight', uwt);
Y = struct('Weight', [0 0 1 1]);
setterminal(MPCobj, Y, U);
The first two states receive zero weight at the terminal point, and the input
weight is unchanged.
e
Remove default state estimator.
The model states are measured directly, so the default MPC state estimator is
unnecessary.
setoutdist(MPCobj,'model',tf(zeros(4,1)));
setEstimator(MPCobj,[],C);
The setoutdist command removes the output disturbances from the output
channels, and the setEstimator command sets the controller state estimates
equal to the measured output values.
4
Compare the control performance of LQR, MPC with terminal weights, and a
standard MPC.
a
Compute closed-loop response with LQR controller.
clsys = feedback(Plant,K);
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Providing LQR Performance Using Terminal Penalty
Tstop = 6;
x0 = [0.2; 0.2];
[yLQR tLQR] = initial(clsys,x0,Tstop);
b
Compute closed-loop response with MPC with terminal weights.
SimOptions = mpcsimopt(MPCobj);
SimOptions.PlantInitialState = x0;
r = zeros(1,4);
[y, t, u] = sim(MPCobj,ceil(Tstop/Ts),r,SimOptions);
Cost = sum(sum(y(:,1:2)*diag(ywt).*y(:,1:2))) + sum(u*diag(uwt).*u);
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white nois
c
Compute closed-loop response with standard MPC controller.
MPCobjSTD = mpc(Plant,Ts); % Default P = 10, M = 2;
MPCobjSTD.Weights.MV = uwt;
MPCobjSTD.Weights.MVrate = 1e-6;
MPCobjSTD.Weights.OV = ywt;
SimOptions = mpcsimopt(MPCobjSTD);
SimOptions.PlantInitialState = x0;
r = zeros(1,2);
[ySTD,tSTD,uSTD] = sim(MPCobjSTD,ceil(Tstop/Ts),r,SimOptions);
CostSTD = sum(sum(ySTD*diag(ywt).*ySTD)) + sum(uSTD*uwt.*uSTD);
-->The "PredictionHorizon" property of "mpc" object is empty. Trying Prediction
-->The "ControlHorizon" property of the "mpc" object is empty. Assuming 2.
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assumi
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. As
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming de
for output(s) y1 and zero weight for output(s) y2
-->Integrated white noise added on measured output channel #1.
Assuming unmeasured input disturbance #2 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white nois
d
Compare the responses.
figure;
h1 = line(tSTD,ySTD,'color','r');
Annotation = get(h1,'Annotation');
set(get(Annotation{2},'LegendInformation'),'IconDisplayStyle','off');
h2 = line(t,y(:,1:2),'color','b');
Annotation = get(h2,'Annotation');
set(get(Annotation{2},'LegendInformation'),'IconDisplayStyle','off');
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Case-Study Examples
h3 = line(tLQR,yLQR,'color','m','marker','o','linestyle','none');
Annotation = get(h3,'Annotation');
set(get(Annotation{2},'LegendInformation'),'IconDisplayStyle','off');
xlabel('Time');
ylabel('Plant Outputs');
legend('Standard MPC','MPC with Terminal Weights','LQR','Location','NorthEast')
The plot shows that the MPC controller with the terminal weights provides
faster settling to the origin than the standard MPC. The LQR controller and
MPC with terminal weights provide identical control performance.
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Providing LQR Performance Using Terminal Penalty
As reported by Scokaert and Rawlings [1], the computed Cost value is 2.23,
identical to that provided by the LQR controller. The computed CostSTD value
for the standard MPC is 4.82, more than double compared to Cost.
You can improve the standard MPC by retuning. For example, use the same
state estimation strategy. If the prediction and control horizons are then
increased, it provides essentially the same performance.
This example shows that using a terminal penalty can eliminate the need to tune the
MPC prediction and control horizons for the unconstrained case. If your application
includes constraints, using a terminal weight is insufficient to guarantee nominal
stability. You must also choose appropriate horizons and possibly add terminal
constraints. For an in-depth discussion, see Rawlings and Mayne [2].
Although you can design and implement such a controller in Model Predictive Control
Toolbox software, you might find designing the standard MPC controller more
convenient.
References
[1] Scokaert, P. O. M. and J. B. Rawlings “Constrained linear quadratic regulation” IEEE
Transactions on Automatic Control (1998), Vol. 43, No. 8, pp. 1163-1169.
Related Examples
•
“Designing Model Predictive Controller Equivalent to Infinite-Horizon LQR”
More About
•
“Terminal Weights and Constraints”
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4
Case-Study Examples
Real-Time Control with OPC Toolbox
This example shows how to implement an online model predictive controller application
using the OPC client supplied with the OPC Toolbox™.
The example uses the Matrikon™ Simulation OPC server to simulate the behavior of an
industrial process on Windows® operating system.
Download the Matrikon™ OPC Simulation Server from "www.matrikon.com"
Download and install the server and set it running either as a service or as an
application.
This example needs OPC Toolbox™.
if ~mpcchecktoolboxinstalled('opc')
disp('The example needs OPC Toolbox(TM).')
end
The example needs OPC Toolbox(TM).
Establish a Connection to the OPC Server
Use OPC Toolbox commands to connect to the Matrikon OPC Simulation Server.
if mpcchecktoolboxinstalled('opc')
% Clear any existing opc connections.
opcreset
% Flush the callback persistent variables.
clear mpcopcPlantStep;
clear mpcopcMPCStep;
try
h = opcda('localhost','Matrikon.OPC.Simulation.1');
connect(h);
catch ME
disp('The Matrikon(TM) OPC Simulation Server must be running on the local machi
return
end
end
Set up the Plant OPC I/O
In practice the plant would be a physical process, and the OPC tags which define its
I/O would already have been created on the OPC server. However, since in this case
4-74
Real-Time Control with OPC Toolbox
a simulation OPC server is being used, the plant behavior must be simulated. This is
achieved by defining tags for the plant manipulated and measured variables and creating
a callback (mpcopcPlantStep) to simulate plant response to changes in the manipulated
variables. Two OPC groups are required, one to represent the two manipulated variables
to be read by the plant simulator and another to write back the two measured plant
outputs storing the results of the plant simulation.
if mpcchecktoolboxinstalled('opc')
% Build an opc group for 2 plant inputs and initialize them to zero.
plant_read = addgroup(h,'plant_read');
imv1 = additem(plant_read,'Bucket Brigade.Real8', 'double');
writeasync(imv1,0);
imv2 = additem(plant_read,'Bucket Brigade.Real4', 'double');
writeasync(imv2,0);
% Build an opc group for plant outputs.
plant_write = addgroup(h,'plant_write');
opv1 = additem(plant_write,'Bucket Brigade.Time', 'double');
opv2 = additem(plant_write,'Bucket Brigade.Money', 'double');
plant_write.WriteAsyncFcn = []; % Suppress command line display.
end
Specify the MPC Controller Which Will Control the Simulated Plant
Create plant model.
plant_model = ss([-.2 -.1; 0 -.05],eye(2,2),eye(2,2),zeros(2,2));
disc_plant_model = c2d(plant_model,1);
% We assume no model mismatch, a control horizon 6 samples and
% prediction horizon 20 samples.
mpcobj = mpc(disc_plant_model,1,20,6);
mpcobj.weights.ManipulatedVariablesRate = [1 1];
% Build an internal MPC object structure so that the MPC object
% is not rebuilt each callback execution.
state = mpcstate(mpcobj);
y1 = mpcmove(mpcobj,state,[1;1]',[1 1]');
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
-->Integrated white noise added on measured output channel #1.
-->Integrated white noise added on measured output channel #2.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
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Build the OPC I/O for the MPC Controller
Build two OPC groups, one to read the two measured plant outputs and the other to
write back the two manipulated variables.
if mpcchecktoolboxinstalled('opc')
% Build an opc group for MPC inputs.
mpc_read = addgroup(h,'mpc_read');
impcpv1 = additem(mpc_read,'Bucket Brigade.Time', 'double');
writeasync(impcpv1,0);
impcpv2 = additem(mpc_read,'Bucket Brigade.Money', 'double');
writeasync(impcpv2,0);
impcref1 = additem(mpc_read,'Bucket Brigade.Int2', 'double');
writeasync(impcref1,1);
impcref2 = additem(mpc_read,'Bucket Brigade.Int4', 'double');
writeasync(impcref2,1);
% Build an opc group for mpc outputs.
mpc_write = addgroup(h,'mpc_write');
additem(mpc_write,'Bucket Brigade.Real8', 'double');
additem(mpc_write,'Bucket Brigade.Real4', 'double');
% Suppress command line display.
mpc_write.WriteAsyncFcn = [];
end
Build OPC Groups to Trigger Execution of the Plant Simulator & Controller
Build two opc groups based on the same external opc timer to trigger execution of both
plant simulation and MPC execution when the contents of the OPC time tag changes.
if mpcchecktoolboxinstalled('opc')
gtime = addgroup(h,'time');
time_tag = additem(gtime,'Triangle Waves.Real8');
gtime.UpdateRate = 1;
gtime.DataChangeFcn = {@mpcopcPlantStep plant_read plant_write disc_plant_model};
gmpctime = addgroup(h,'mpctime');
additem(gmpctime,'Triangle Waves.Real8');
gmpctime.UpdateRate = 1;
gmpctime.DataChangeFcn = {@mpcopcMPCStep mpc_read mpc_write mpcobj};
end
Log Data from the Plant Measured Outputs
Log the plant measured outputs from tags 'Bucket Brigade.Money' and 'Bucket
Brigade.Money'.
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if mpcchecktoolboxinstalled('opc')
mpc_read.RecordsToAcquire = 40;
start(mpc_read);
while mpc_read.RecordsAcquired < mpc_read.RecordsToAcquire
pause(3)
fprintf('Logging data: Record %d / %d',mpc_read.RecordsAcquired,mpc_read.Records
end
stop(mpc_read);
end
Extract and Plot the Logged Data
if mpcchecktoolboxinstalled('opc')
[itemID, value, quality, timeStamp, eventTime] = getdata(mpc_read,'double');
plot((timeStamp(:,1)-timeStamp(1,1))*24*60*60,value)
title('Measured Outputs Logged from Tags Bucket Brigade.Time,Bucket Brigade.Money')
xlabel('Time (secs)');
end
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Case-Study Examples
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Simulation and Code Generation Using Simulink Coder
Simulation and Code Generation Using Simulink Coder
This example shows how to simulate and generate real-time code for an MPC Controller
block with Simulink Coder. Code can be generated in both single and double precisions.
Required Products
To run this example, Simulink® and Simulink® Coder™ are required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
if ~mpcchecktoolboxinstalled('simulinkcoder')
disp('Simulink(R) Coder(TM) is required to run this example.');
return
end
Simulink(R) Coder(TM) is required to run this example.
Setup Environment
You must have write-permission to generate the relevant files and the executable.
So, before starting simulation and code generation, change the current directory to a
temporary directory.
cwd = pwd;
tmpdir = tempname;
mkdir(tmpdir);
cd(tmpdir);
Define Plant Model and MPC Controller
Define a SISO plant.
plant = ss(tf([3 1],[1 0.6 1]));
Define the MPC controller for the plant.
Ts = 0.1;
%Sampling time
p = 10;
%Prediction horizon
m = 2;
%Control horizon
Weights = struct('MV',0,'MVRate',0.01,'OV',1); % Weights
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MV = struct('Min',-Inf,'Max',Inf,'RateMin',-100,'RateMax',100); % Input constraints
OV = struct('Min',-2,'Max',2); % Output constraints
mpcobj = mpc(plant,Ts,p,m,Weights,MV,OV);
Simulate and Generate Code in Double-Precision
By default, MPC Controller blocks use double-precision in simulation and code
generation.
Simulate the model in Simulink.
mdl1 = 'mpc_rtwdemo';
open_system(mdl1);
sim(mdl1);
The controller effort and the plant output are saved into base workspace as variables u
and y, respectively.
Build the model with the rtwbuild command.
disp('Generating C code... Please wait until it finishes.');
set_param(mdl1,'RTWVerbose','off');
rtwbuild(mdl1);
On a Windows system, an executable file named "mpc_rtwdemo.exe" appears in the
temporary directory after the build process finishes.
Run the executable.
if ispc
disp('Running executable...');
status = system(mdl1);
else
disp('The example only runs the executable on Windows system.');
end
After the executable completes successfully (status=0), a data file named
"mpc_rtwdemo.mat" appears in the temporary directory.
Compare the responses from the generated code (rt_u and rt_y) with the responses from
the previous simulation in Simulink (u and y).
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They are numerically equal.
Simulate and Generate Code in Single-Precision
You can also configure the MPC block to use single-precision in simulation and code
generation.
mdl2 = 'mpc_rtwdemo_single';
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open_system(mdl2);
To do that, open the MPC block dialog and select "single" as the "output data type" at the
bottom of the dialog.
open_system([mdl2 '/MPC Controller']);
Simulate the model in Simulink.
close_system([mdl2 '/MPC Controller']);
sim(mdl2);
The controller effort and the plant output are saved into base workspace as variables u1
and y1, respectively.
Build the model with the rtwbuild command.
disp('Generating C code... Please wait until it finishes.');
set_param(mdl2,'RTWVerbose','off');
rtwbuild(mdl2);
On a Windows system, an executable file named "mpc_rtwdemo_single.exe" appears in
the temporary directory after the build process finishes.
Run the executable.
if ispc
disp('Running executable...');
status = system(mdl2);
else
disp('The example only runs the executable on Windows system.');
end
After the executable completes successfully (status=0), a data file named
"mpc_rtwdemo_single.mat" appears in the temporary directory.
Compare the responses from the generated code (rt_u1 and rt_y1) with the responses
from the previous simulation in Simulink (u1 and y1).
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Simulation and Code Generation Using Simulink Coder
They are numerically equal.
Close the Simulink model.
bdclose(mdl1);
bdclose(mdl2);
cd(cwd)
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Simulation and Structured Text Generation Using PLC Coder
This example shows how to simulate and generate Structured Text for an MPC
Controller block using PLC Coder software. The generated code uses single-precision.
Required Products
To run this example, Simulink® and Simulink® PLC Coder™ are required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
if ~mpcchecktoolboxinstalled('plccoder')
disp('Simulink(R) PLC Coder(TM) is required to run this example.');
return
end
Simulink(R) PLC Coder(TM) is required to run this example.
Setup Environment
You must have write-permission to generate the relevant files and the executable.
So, before starting simulation and code generation, change the current directory to a
temporary directory.
cwd = pwd;
tmpdir = tempname;
mkdir(tmpdir);
cd(tmpdir);
Define Plant Model and MPC Controller
Define a SISO plant.
plant = ss(tf([3 1],[1 0.6 1]));
Define the MPC controller for the plant.
Ts = 0.1;
%Sampling time
p = 10;
%Prediction horizon
m = 2;
%Control horizon
Weights = struct('MV',0,'MVRate',0.01,'OV',1); % Weights
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MV = struct('Min',-Inf,'Max',Inf,'RateMin',-100,'RateMax',100); % Input constraints
OV = struct('Min',-2,'Max',2); % Output constraints
mpcobj = mpc(plant,Ts,p,m,Weights,MV,OV);
Simulate and Generate Structured Text
Open the Simulink model.
mdl = 'mpc_plcdemo';
open_system(mdl);
To generate structured text for the MPC Controller block, complete the following two
steps:
• Configure the MPC block to use single precision. Select "single" in the "Output data
type" combo box in the MPC block dialog.
open_system([mdl '/Control System/MPC Controller']);
• Put MPC block inside a subsystem block and treat the subsystem block as an atomic
unit. Select the "Treat as atomic unit" checkbox in the subsystem block dialog.
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Simulate the model in Simulink.
close_system([mdl '/Control System/MPC Controller']);
open_system([mdl '/Outputs//References']);
open_system([mdl '/Inputs']);
sim(mdl);
To generate code with the PLC Coder, use the plcgeneratecode command.
disp('Generating PLC structure text... Please wait until it finishes.');
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Simulation and Structured Text Generation Using PLC Coder
plcgeneratecode([mdl '/Control System']);
The Message Viewer dialog box shows that PLC code generation was successful.
Close the Simulink model.
bdclose(mdl);
cd(cwd)
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Case-Study Examples
Setting Targets for Manipulated Variables
This example shows how to design a model predictive controller for a plant with two
inputs and one output with target set-point for a manipulated variable.
Define Plant Model
The linear plant model has two inputs and two outputs.
N1 = [3 1];
D1 = [1 2*.3 1];
N2 = [2 1];
D2 = [1 2*.5 1];
plant = ss(tf({N1,N2},{D1,D2}));
A = plant.a;
B = plant.b;
C = plant.c;
D = plant.d;
x0 = [0 0 0 0]';
Design MPC Controller
Create MPC controller.
Ts = 0.4;
mpcobj = mpc(plant,Ts,20,5);
% Sampling time
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
Specify weights.
mpcobj.weights.manipulated = [0.3 0]; % weight difference MV#1 - Target#1
mpcobj.weights.manipulatedrate = [0 0];
mpcobj.weights.output = 1;
Define input specifications.
mpcobj.MV = struct('RateMin',{-0.5;-0.5},'RateMax',{0.5;0.5});
Specify target set-point u=2 for the first manipulated variable.
mpcobj.MV(1).Target=2;
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Simulation Using Simulink®
To run this example, Simulink® is required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
Simulate.
mdl = 'mpc_utarget';
open_system(mdl)
sim(mdl);
% Open Simulink(R) Model
% Start Simulation
-->Converting model to discrete time.
-->Integrated white noise added on measured output channel #1.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
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Setting Targets for Manipulated Variables
bdclose(mdl)
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Specifying Alternative Cost Function with Off-Diagonal Weight
Matrices
This example shows how to use non-diagonal weight matrices in a model predictive
controller.
Define Plant Model and MPC Controller
The linear plant model has two inputs and two outputs.
plant = ss(tf({1,1;1,2},{[1 .5 1],[.7 .5 1];[1 .4 2],[1 2]}));
[A,B,C,D] = ssdata(plant);
Ts = 0.1;
% sampling time
plant = c2d(plant,Ts); % convert to discrete time
Create MPC controller.
p=20;
% prediction horizon
m=2;
% control horizon
mpcobj = mpc(plant,Ts,p,m);
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
Define constraints on the manipulated variable.
mpcobj.MV = struct('Min',{-3;-2},'Max',{3;2},'RateMin',{-100;-100},'RateMax',{100;100})
Define non-diagonal output weight. Note that it is specified inside a cell array.
OW = [1 -1]'*[1 -1];
% Non-diagonal output weight, corresponding to ((y1-r1)-(y2-r2))^2
mpcobj.Weights.OutputVariables = {OW};
% Non-diagonal input weight, corresponding to (u1-u2)^2
mpcobj.Weights.ManipulatedVariables = {0.5*OW};
Simulate Using SIM Command
Specify simulation options.
Tstop = 30;
Tf = round(Tstop/Ts);
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% simulation time
% number of simulation steps
Specifying Alternative Cost Function with Off-Diagonal Weight Matrices
r = ones(Tf,1)*[1 2];
% reference trajectory
Run the closed-loop simulation and plot results.
[y,t,u] = sim(mpcobj,Tf,r);
subplot(211)
plot(t,y(:,1)-r(1,1)-y(:,2)+r(1,2));grid
title('(y_1-r_1)-(y_2-r_2)');
subplot(212)
plot(t,u);grid
title('u');
-->Integrated white noise added on measured output channel #1.
-->Integrated white noise added on measured output channel #2.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
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Simulate Using Simulink®
To run this example, Simulink® is required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this part of the example.')
return
end
Now simulate closed-loop MPC in Simulink®.
mdl = 'mpc_weightsdemo';
open_system(mdl);
sim(mdl)
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Specifying Alternative Cost Function with Off-Diagonal Weight Matrices
bdclose(mdl);
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Review Model Predictive Controller for Stability and Robustness
Issues
This example shows how to use the review command to detect potential issues with a
model predictive controller design.
The Fuel Gas Blending Process
The example application is a fuel gas blending process. The objective is to blend six gases
to obtain a fuel gas, which is then burned to provide process heating. The fuel gas must
satisfy three quality standards in order for it to burn reliably and with the expected
heat output. The fuel gas header pressure must also be controlled. Thus, there are four
controlled output variables. The manipulated variables are the six feed gas flow rates.
Inputs:
1.
2.
3.
4.
5.
6.
Natural Gas (NG)
Reformed Gas (RG)
Hydrogen (H2)
Nitrogen (N2)
Tail Gas 1 (T1)
Tail Gas 2 (T2)
Outputs:
1.
2.
3.
4.
High Heating Value (HHV)
Wobbe Index (WI)
Flame Speed Index (FSI)
Header Pressure (P)
The fuel gas blending process was studied by Muller et al.: "Modeling, validation, and
control of an industrial fuel gas blending system", C.J. Muller, I.K. Craig, N.L. Ricker, J.
of Process Control, in press, 2011.
Linear Plant Model
Use the following linear plant model as the prediction model for the controller. This
state-space model, applicable at a typical steady-state operating point, uses the time unit
of hours.
a = diag([-28.6120, -28.6822, -28.5134,
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Review Model Predictive Controller for Stability and Robustness Issues
-22.9377,
-23.0890,
-27.0793,
b = zeros(24,6);
b( 1: 4,1) = [4, 4,
b( 5: 8,2) = [2, 2,
b( 9:12,3) = [2, 2,
b(13:16,4) = [4, 4,
b(17:20,5) = [2, 2,
b(21:24,6) = [1, 2,
c = [diag([ 6.1510,
diag([-2.2158,
diag([-2.5223,
diag([-3.3187,
diag([-1.6583,
diag([-1.6807,
d = zeros(4,6);
Plant = ss(a, b, c,
- 0.0101, -26.4877, -26.7950, -27.2210, -0.0083, ...
-23.0062, -22.9349, -0.0115, -25.8581, -25.6939, ...
-0.0117, -22.8975, -22.8233, -21.1142, -0.0065]);
8, 32]';
4, 32]';
4, 32]';
8, 32]';
4, 32]';
1, 32]';
7.6785, -5.9312, 34.2689]), ...
-3.1204, 2.6220, 35.3561]), ...
1.1480, 7.8136, 35.0376]), ...
-7.6067, -6.2755, 34.8720]), ...
-2.0249, 2.5584, 34.7881]), ...
-1.2217, 1.0492, 35.0297])];
d);
By default, all the plant inputs are manipulated variables.
Plant.InputName = {'NG', 'RG', 'H2', 'N2', 'T1', 'T2'};
By default, all the plant outputs are measured outputs.
Plant.OutputName = {'HHV', 'WI', 'FSI', 'P'};
Transport delay is added to plant outputs to reflect the delay in the sensors.
Plant.OutputDelay = [0.00556
0.0167
0.00556
0];
Initial Controller Design
Construct an initial model predictive controller based on design requirements.
Specify sampling time, horizons and steady-state values.
The sampling time is that of the sensors (20 seconds). The prediction horizon is
approximately equal to the plant settling time (39 intervals). The control horizon uses
four blocked moves that have lengths of 2, 6, 12 and 19 intervals respectively. The
nominal operating conditions are non-zero. The output measurement noise is white noise
with magnitude of 0.001.
MPC_verbosity = mpcverbosity('off'); % Disable MPC message displaying at command line
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Ts = 20/3600;
% Time units are hours.
Obj = mpc(Plant, Ts, 39, [2, 6, 12, 19]);
Obj.Model.Noise = ss(0.001*eye(4));
Obj.Model.Nominal.Y = [16.5, 25, 43.8, 2100];
Obj.Model.Nominal.U = [1.4170, 0, 2, 0, 0, 26.5829];
Specify lower and upper bounds on manipulated variables.
Since all the manipulated variables are flow rates of gas streams, their lower bounds are
zero. All the MV constraints are hard (MinECR and MaxECR = 0) by default.
MVmin = zeros(1,6);
MVmax = [15, 20, 5, 5, 30, 30];
for i = 1:6
Obj.MV(i).Min = MVmin(i);
Obj.MV(i).Max = MVmax(i);
end
Specify lower and upper bounds on manipulated variable increments.
The bounds are set large enough to allow full range of movement in one interval. All the
MV rate constraints are hard (RateMinECR and RateMaxECR = 0) by default.
for i = 1:6
Obj.MV(i).RateMin = -MVmax(i);
Obj.MV(i).RateMax = MVmax(i);
end
Specify lower and upper bounds on plant outputs.
All the OV constraints are soft (MinECR and MaxECR = 0) by default.
OVmin = [16.5, 25, 39, 2000];
OVmax = [18.0, 27, 46, 2200];
for i = 1:4
Obj.OV(i).Min = OVmin(i);
Obj.OV(i).Max = OVmax(i);
end
Specify weights on manipulated variables.
MV weights are specified based on the known costs of each feed stream. This tells MPC
controller how to move the six manipulated variables in order to minimize the cost of the
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Review Model Predictive Controller for Stability and Robustness Issues
blended fuel gas. The weights are normalized so the maximum weight is approximately
1.0.
Obj.Weights.MV = [54.9, 20.5, 0, 5.73, 0, 0]/55;
Specify weights on manipulated variable increments.
They are small relative to the maximum MV weight so the MVs are free to vary.
Obj.Weights.MVrate = 0.1*ones(1,6);
Specify weights on plant outputs.
The OV weights penalize deviations from specified setpoints and would normally be
"large" relative to the other weights. Let us first consider the default values, which equal
the maximum MV weight specified above.
Obj.Weights.OV = [1, 1, 1, 1];
Using the review Command to Improve the Initial Design
Review the initial controller design.
review(Obj)
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The summary table shown above lists three warnings and one error. Let's consider these
in turn. Click QP Hessian Matrix Validity and scroll down to display the warning. It
indicates that the plant signal magnitudes differ significantly. Specifically, the pressure
response is much larger than the others.
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Examination of the specified OV bounds shows that the spans are quite different, and
the pressure span is two orders of magnitude larger than the others. It is good practice
to specify MPC scale factors to account for the expected differences in signal magnitudes.
We are already weighting MVs based on relative cost, so we focus on the OVs only.
Calculate OV spans
OVspan = OVmax - OVmin;
%
% Use these as the specified scale factors
for i = 1:4
Obj.OV(i).ScaleFactor = OVspan(i);
end
% Use review to verify that the scale factor warning has disappeared.
review(Obj);
%
% <<reviewDemo03.png>>
The next warning indicates that the controller does not drive the OVs to their targets at
steady state. Click Closed-Loop Steady-State Gains to see a list of the non-zero gains.
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Review Model Predictive Controller for Stability and Robustness Issues
The first entry in the list shows that adding a sustained disturbance of unit magnitude
to the HHV output would cause the HHV to deviate 0.0860 units from its steadystate target, assuming no constraints are active. The second entry shows that a unit
disturbance in WI would cause a steady-state deviation ("offset") of -0.0345 in HHV, etc.
Since there are six MVs and only four OVs, excess degrees of freedom are available and
you might be surprised to see non-zero steady-state offsets. The non-zero MV weights we
have specified in order to drive the plant toward the most economical operating condition
are causing this.
Non-zero steady-state offsets are often undesirable but are acceptable in this application
because: # The primary objective is to minimize the blend cost. The gas quality (HHV,
etc.) can vary freely within the specified OV limits. # The small offset gain magnitudes
indicate that the impact of disturbances would be small. # The OV limits are soft
constraints. Small, short-term violations are acceptable.
View the second warning by clicking Hard MV Constraints, which indicates a potential
hard-constraint conflict.
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Review Model Predictive Controller for Stability and Robustness Issues
If an external event causes the NG to go far below its specified minimum, the constraint
on its rate of increase might make it impossible to return the NG within bounds in one
interval. In other words, when you specify both MV.Min and MV.RateMax, the controller
would not be able to find an optimal solution if the most recent MV value is less than
(MV.Min - MV.RateMax). Similarly, there is a potential conflict when you specify both
MV.Max and MV.RateMin.
An MV constraint conflict would be extremely unlikely in the gas blending application,
but it's good practice to eliminate the possibility by softening one of the two constraints.
Since the MV minimum and maximum values are physical limits and the increment
bounds are not, we soften the increment bounds as follows:
for i = 1:6
Obj.MV(i).RateMinECR = 0.1;
Obj.MV(i).RateMaxECR = 0.1;
end
Review the new controller design.
review(Obj)
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The MV constraint conflict warning has disappeared. Now click Soft Constraints to
view the error message.
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We see that the delay in the WI output makes it impossible to satisfy bounds on that
variable until at least three control intervals have elapsed. The WI bounds are soft but it
is poor practice to include unattainable constraints in a design. We therefore modify the
WI bound specifications such that it is unconstained until the 4th prediction horizon step.
Obj.OV(2).Min = [-Inf(1,3), OVmin(2)];
Obj.OV(2).Max = [ Inf(1,3), OVmax(2)];
% Ee-issuing the review command to verifies that this eliminates the
% error message (see the next step).
Diagnosing the Impact of Zero Output Weights
Given that the design requirements allow the OVs to vary freely within their limits,
consider zeroing their penalty weights:
Obj.Weights.OV = zeros(1,4);
Review the impact of this design change.
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review(Obj)
A new warning regarding QP Hessian Matrix Validity has appeared. Click QP Hessian
Matrix Validity warning to see the details.
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The review has flagged the zero weights on all four output variables. Since the zero
weights are consistent with the design requirement and the other Hessian tests indicate
that the quadratic programming problem has a unique solution, this warning can be
ignored.
Click Closed-Loop Steady-State Gains to see the second warning. It shows another
consequence of setting the four OV weights to zero. When an OV is not penalized by
a weight, any output disturbance added to it will be ignored, passing through with no
attenuation.
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Since it is a design requirement, non-zero steady-state offsets are acceptable provided
that MPC is able to hold all the OVs within their specified bounds. It is therefore a good
idea to examine how easily the soft OV constraints can be violated when disturbances are
present.
Reviewing Soft Constraints
Click Soft Constraints to see a list of soft constraints -- in this case an upper and lower
bound on each OV.
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The Impact Factor column shows that using the default MinECR and MaxECR values
give the pressure (P) a much higher priority than the other OVs. If we want the priorities
to be more comparable, we should increase the pressure constraint ECR values and
adjust the others too. For example, we consider
Obj.OV(1).MinECR
Obj.OV(1).MaxECR
Obj.OV(3).MinECR
Obj.OV(3).MaxECR
Obj.OV(4).MinECR
Obj.OV(4).MaxECR
=
=
=
=
=
=
0.5;
0.5;
3;
3;
80;
80;
Review the impact of this design change.
review(Obj)
Notice from the Sensitivity Ratio column that all the sensitivity ratios are now less than
unity. This means that the soft constraints will receive less attention than other terms in
the MPC objective function, such as deviations of the MVs from their target values. Thus,
it is likely that an output constraint violation would occur.
4-114
Review Model Predictive Controller for Stability and Robustness Issues
In order to give the output constraints higher priority than other MPC objectives,
increase the Weights.ECR parameter from default 1e5 to a higher value to harden all the
soft OV constraints.
Obj.Weights.ECR = 1e8;
Review the impact of this design change.
review(Obj)
The controller is now a factor of 100 more sensitive to output constraint violations than
to errors in target tracking.
Reviewing Data Memory Size
Click Memory Size for MPC Data to see the estimated memory size needed to store the
MPC data matrices used on the hardware.
4-115
4
Case-Study Examples
In this example, if the controller is running using single precision, it requires 250 KB
of memory to store its matrices. If the controller memory size exceeds the memory
available on the target system, you must redesign the controller to reduce its memory
requirements. Alternatively, increase the memory available on the target system.
mpcverbosity(MPC_verbosity);
[~, hWebBrowser] = web;
close(hWebBrowser);
4-116
Bibliography
Bibliography
[1] Seborg, D. E., T. F. Edgar, and D. A. Mellichamp Process Dynamics and Control, 2nd
Edition (2004), Wiley, pp. 34–36.
[2] Rawlings, J. B., and David Q. Mayne “Model Predictive Control: Theory and Design”
Nob Hill Publishing, 2010.
4-117
5
Adaptive MPC Design
• “Adaptive MPC” on page 5-2
• “Model Updating Strategy” on page 5-6
• “Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive
Linearization” on page 5-8
• “Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model
Estimation” on page 5-21
5
Adaptive MPC Design
Adaptive MPC
In this section...
“When to Use Adaptive MPC” on page 5-2
“Plant Model” on page 5-2
“Nominal Operating Point” on page 5-4
“State Estimation” on page 5-4
When to Use Adaptive MPC
MPC control predicts future behavior using a linear-time-invariant (LTI) dynamic model.
In practice, such predictions are never exact, and a key tuning objective is to make MPC
insensitive to prediction errors. In many applications, this approach is sufficient for
robust controller performance.
If the plant is strongly nonlinear or its characteristics vary dramatically with time,
LTI prediction accuracy might degrade so much that MPC performance becomes
unacceptable. Adaptive MPC can address this degradation by adapting the prediction
model for changing operating conditions. As implemented in the Model Predictive Control
Toolbox software, adaptive MPC uses a fixed model structure, but allows the model’s
parameters to evolve with time. Ideally, whenever the controller requires a prediction
(at the beginning of each control interval) it uses a model appropriate for the current
conditions.
After you design an MPC controller for the average or most likely operating conditions
of your control system, you can implement an adaptive MPC controller based on that
design. For information about designing that initial controller, see “Controller Creation”.
An alternative option for controlling a nonlinear or time-varying plant is to use gainscheduled MPC control. See “Gain-Scheduled MPC”.)
Plant Model
The plant model used as the basis for Adaptive MPC must be an LTI discrete-time,
state-space model. See “Basic Models” in the Control System Toolbox documentation or
“Linearization Basics” in the Simulink Control Design documentation for information
about creating and modifying such systems. The plant model structure is as follows:
5-2
Adaptive MPC
x p ( k + 1 ) = A p x p ( k ) + Bpu u ( k ) + Bpv v ( k ) + B pd d ( k )
y ( k ) = Cp x p ( k ) + D pv v ( k ) + D pd d ( k ) .
Here, the matrices Ap, Bpu, Bpv, Bpd, Cp, Dpv and Dpd are the parameters that can vary
with time. The other variables in the expression are:
• k — Time index (current control interval).
• xp — nxp plant model states.
• u — nu manipulated inputs (MVs). These are the one or more inputs that are adjusted
by the MPC controller.
• v — nv measured disturbance inputs.
• d — nd unmeasured disturbance inputs.
• y — ny plant outputs, including nym measured and nyu unmeasured outputs. The
total number of outputs, ny = nym + nyu. Also, nym ≥ 1 (there is at least one measured
output).
Additional requirements for the plant model in adaptive MPC control are:
• Sample time (Ts) is a constant and identical to the MPC control interval.
• Time delay (if any) is absorbed as discrete states (see, e.g., the Control System
Toolbox absorbDelay command).
• nxp, nu, ny, nd, nym, and nyu are all constants.
• Adaptive MPC prohibits direct feed-through from any manipulated variable to any
plant output. Thus, Du = 0 in the above model.
When you create the plant model, you specify it as an LTI state-space model with
parameters Ap, Bp, Cp, Dp, and Ts. The sampling time is Ts > 0. The InputGroup and
OutputGroup properties of the model designate input and output signal types (such as
manipulated or measured). Each column in Bp and Dp represents a particular plant input
variable. Grouping these columns according to input signal type yields the matrices Bpu,
Bpv, Bpd, Dpv and Dpd. Similarly, each row in Cp, and Dp represent a particular output
variable. Once you create these column and row assignments, you cannot change them as
the system evolves in time.
For more details about creation of plant models for MPC control, see “Plant
Specification”.
5-3
5
Adaptive MPC Design
Nominal Operating Point
A traditional MPC controller includes a nominal operating point at which the plant
model applies, such as the condition at which you linearize a nonlinear model to obtain
the LTI approximation. The Model.Nominal property of the controller contains this
information.
In adaptive MPC, as time evolves you should update the nominal operating point to be
consistent with the updated plant model.
You can write the plant model in terms of deviations from the nominal conditions:
(
)
x p ( k + 1 ) = x p + A p x p ( k ) - x p + Bp ( ut ( k ) - ut ) + Dx p
(
)
y ( k ) = y + C p x p ( k ) - xp + D p ( ut ( k ) - ut ) .
Here, the matrices Ap, Bp, Cp, and Dp are the parameter matrices to be updated. ut is the
combined plant input variable, comprising the u, v, and d variables defined above. The
nominal conditions to be updated are:
•
x p — nxp nominal states
•
Dx p — nxp nominal state increments
•
ut — nut nominal inputs
•
y — ny nominal outputs
State Estimation
By default, MPC uses a static Kalman filter (KF) to update its controller states, which
include the nxp plant model states, nd (≥ 0) disturbance model states, and nn (≥ 0)
measurement noise model states. This KF requires two gain matrices, L and M. By
default, the MPC controller calculates them during initialization. They depend upon
the plant, disturbance, and noise model parameters, and assumptions regarding the
stochastic noise signals driving the disturbance and noise models. For more details about
state estimation in traditional MPC, see “Controller State Estimation”.
Adaptive MPC uses a Kalman filter by default, and adjusts the gains, L and M, at each
control interval to maintain consistency with the updated plant model. The result is a
linear-time-varying Kalman filter (LTVKF):
5-4
Adaptive MPC
(
)(
T
Lk = Ak Pk|k-1CT
m,k + N Cm ,k Pk|k-1Cm ,k + R
(
T
T
M k = Pk|k-1Cm
,k Cm ,k Pk|k-1Cm ,k + R
-1
)
-1
)
(
)
T
T
Pk+1|k = Ak Pk|k-1 AkT - Ak Pk|k-1Cm
,k + N Lk + Q.
Here, Q, R, and N are constant covariance matrices defined as in MPC state estimation.
Ak and Cm,k are state-space parameter matrices for the entire controller state, defined
as for traditional MPC but with the portions affected by the plant model updated to
time k. The value Pk|k–1 is the state estimate error covariance matrix at time k based
on information available at time k–1. Finally, Lk and Mk are the updated KF gain
matrices. For details on the KF formulation used in traditional MPC, see “Controller
State Estimation”. By default, the initial condition, P0|–1, is the static KF solution prior to
any model updates.
The KF gain and the state error covariance matrix depend upon the model parameters
and the assumptions leading to the constant Q, R, and N matrices. If the plant model
is constant, the expressions for Lk and Mk converge to the equivalent static KF solution
used in traditional MPC.
The equations for the controller state evolution at time k are identical to the KF
formulation of traditional MPC described in “Controller State Estimation”, but with the
estimator gains and state space matrices updated to time k.
You have the option to update the controller state using a procedure external to the MPC
controller, and then supply the updated state to MPC at each control instant, k. In this
case, the MPC controller skips all KF and LTVKF calculations.
Related Examples
•
“Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive
Linearization”
•
“Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model
Estimation”
More About
•
“Model Updating Strategy” on page 5-6
•
“Controller State Estimation”
5-5
5
Adaptive MPC Design
Model Updating Strategy
In this section...
“Overview” on page 5-6
“Other Considerations” on page 5-6
Overview
Typically, to implement adaptive MPC control, you employ one of the following modelupdating strategies:
Successive linearization. Given a mechanistic plant model, e.g., a set of nonlinear
ordinary differential and algebraic equations, derive its LTI approximation at the
current operating condition. For example, Simulink Control Design software provides
linearization tools for this purpose.
Using a Linear Parameter Varying (LPV) model. Control System Toolbox software
provides a LPV System Simulink block that allows you to specify an array of LTI models
with scheduling parameters. You can perform batch linearization offline to obtain an
array of plant models at the desired operating points and then use them in the LPV
System block to provide model updating to the Adaptive MPC Controller Simulink block.
Online parameter estimation. Given an empirical model structure and initial
estimates of its parameters, use the available real-time plant measurements to estimate
the current model parameters. For example, the System Identification Toolbox™
software provides real-time parameter estimation tools.
Other Considerations
There are several factors to keep in mind when designing and implementing an adaptive
MPC controller.
• Before attempting adaptive MPC, define and tune an MPC controller for the most
typical (nominal) operating condition. Make sure the system can tolerate some
prediction error. Test this tolerance via simulations in which the MPC prediction
model differs from the plant. See “MPC Design”.
• An adaptive MPC controller requires more real-time computations than traditional
MPC. In addition to the state estimation calculation, you must also implement and
test a model-updating strategy, which might be computationally intensive.
5-6
Model Updating Strategy
• You must determine MPC tuning constants that provide robust performance over the
expected range of model parameters. See “Tuning Weights”.
• Model updating via online parameter estimation is most effective when parameter
variations occur gradually.
• When implementing adaptive MPC control, adapt only parameters defining the
Model.Plant property of the controller. The disturbance and noise models, if any,
remain constant.
See Also
Adaptive MPC Controller
Related Examples
•
“Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive
Linearization”
•
“Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model
Estimation”
More About
•
“Adaptive MPC” on page 5-2
5-7
5
Adaptive MPC Design
Adaptive MPC Control of Nonlinear Chemical Reactor Using
Successive Linearization
This example shows how to use an Adaptive MPC controller to control a nonlinear
continuous stirred tank reactor (CSTR) as it transitions from low conversion rate to high
conversion rate.
A first principle nonlinear plant model is available and being linearized at each control
interval. The adaptive MPC controller then updates its internal predictive model with
the linearized plant model and achieves nonlinear control successfully.
About the Continuous Stirred Tank Reactor
A Continuously Stirred Tank Reactor (CSTR) is a common chemical system in the
process industry. A schematic of the CSTR system is:
5-8
Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive Linearization
This is a jacketed non-adiabatic tank reactor described extensively in Seborg's book,
"Process Dynamics and Control", published by Wiley, 2004. The vessel is assumed to be
perfectly mixed, and a single first-order exothermic and irreversible reaction, A --> B,
takes place. The inlet stream of reagent A is fed to the tank at a constant volumetric rate.
The product stream exits continuously at the same volumetric rate and liquid density is
constant. Thus the volume of reacting liquid is constant.
The inputs of the CSTR model are:
and the outputs (y(t)), which are also the states of the model (x(t)), are:
The control objective is to maintain the concentration of reagent A,
at its desired
setpoint, which changes over time when reactor transitions from low conversion rate to
high conversion rate. The coolant temperature
is the manipulated variable used by
the MPC controller to track the reference as well as reject the measured disturbance
arising from the inlet feed stream temperature . The inlet feed stream concentration,
, is assumed to be constant. The Simulink model mpc_cstr_plant implements the
nonlinear CSTR plant.
We also assume that direct measurements of concentrations are unavailable or
infrequent, which is the usual case in practice. Instead, we use a "soft sensor" to estimate
CA based on temperature measurements and the plant model.
About Adaptive Model Predictive Control
It is well known that the CSTR dynamics are strongly nonlinear with respect to reactor
temperature variations and can be open-loop unstable during the transition from
one operating condition to another. A single MPC controller designed at a particular
operating condition cannot give satisfactory control performance over a wide operating
range.
To control the nonlinear CSTR plant with linear MPC control technique, you have a few
options:
5-9
5
Adaptive MPC Design
• If a linear plant model cannot be obtained at run time, first you need to obtain several
linear plant models offline at different operating conditions that cover the typical
operating range. Next you can choose one of the two approaches to implement MPC
control strategy:
(1) Design several MPC controllers offline, one for each plant model. At run time, use
Multiple MPC Controller block that switches MPC controllers from one to another
based on a desired scheduling strategy. See "Gain Scheduled MPC Control of Nonlinear
Chemical Reactor" for more details. Use this approach when the plant models have
different orders or time delays.
(2) Design one MPC controller offline at the initial operating point. At run time, use
Adaptive MPC Controller block (updating predictive model at each control interval)
together with Linear Parameter Varying (LPV) System block (supplying linear plant
model with a scheduling strategy). See "Adaptive MPC Control of Nonlinear Chemical
Reactor Using Linear Parameter Varying System" for more details. Use this approach
when all the plant models have the same order and time delay.
• If a linear plant model can be obtained at run time, you should use Adaptive MPC
Controller block to achieve nonlinear control. There are two typical ways to obtain a
linear plant model online:
(1) Use successive linearization as shown in this example. Use this approach when a
nonlinear plant model is available and can be linearized at run time.
(2) Use online estimation to identify a linear model when loop is closed. See "Adaptive
MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation" for more
details. Use this approach when linear plant model cannot be obtained from either an
LPV system or successive linearization.
Obtain Linear Plant Model at Initial Operating Condition
To linearize the plant, Simulink® and Simulink Control Design® are required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
if ~mpcchecktoolboxinstalled('slcontrol')
disp('Simulink Control Design(R) is required to run this example.')
return
end
5-10
Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive Linearization
To implement an adaptive MPC controller, first you need to design a MPC controller at
the initial operating point where CAi is 10 kgmol/m^3, Ti and Tc are 298.15 K.
Create operating point specification.
plant_mdl = 'mpc_cstr_plant';
op = operspec(plant_mdl);
Feed concentration is known at the initial condition.
op.Inputs(1).u = 10;
op.Inputs(1).Known = true;
Feed temperature is known at the initial condition.
op.Inputs(2).u = 298.15;
op.Inputs(2).Known = true;
Coolant temperature is known at the initial condition.
op.Inputs(3).u = 298.15;
op.Inputs(3).Known = true;
Compute initial condition.
[op_point, op_report] = findop(plant_mdl,op);
Operating Point Search Report:
--------------------------------Operating Report for the Model mpc_cstr_plant.
(Time-Varying Components Evaluated at time t=0)
Operating point specifications were successfully met.
States:
---------(1.) mpc_cstr_plant/CSTR/Integrator
x:
311
dx:
8.12e-11 (0)
(2.) mpc_cstr_plant/CSTR/Integrator1
x:
8.57
dx:
-6.87e-12 (0)
Inputs:
---------(1.) mpc_cstr_plant/CAi
5-11
5
Adaptive MPC Design
u:
10
(2.) mpc_cstr_plant/Ti
u:
298
(3.) mpc_cstr_plant/Tc
u:
298
Outputs:
---------(1.) mpc_cstr_plant/T
y:
311
(2.) mpc_cstr_plant/CA
y:
8.57
[-Inf Inf]
[-Inf Inf]
Obtain nominal values of x, y and u.
x0 = [op_report.States(1).x;op_report.States(2).x];
y0 = [op_report.Outputs(1).y;op_report.Outputs(2).y];
u0 = [op_report.Inputs(1).u;op_report.Inputs(2).u;op_report.Inputs(3).u];
Obtain linear plant model at the initial condition.
sys = linearize(plant_mdl, op_point);
Drop the first plant input CAi because it is not used by MPC.
sys = sys(:,2:3);
Discretize the plant model because Adaptive MPC controller only accepts a discrete-time
plant model.
Ts = 0.5;
plant = c2d(sys,Ts);
Design MPC Controller
You design an MPC at the initial operating condition. When running in the adaptive
mode, the plant model is updated at run time.
Specify signal types used in MPC.
plant.InputGroup.MeasuredDisturbances = 1;
plant.InputGroup.ManipulatedVariables = 2;
plant.OutputGroup.Measured = 1;
5-12
Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive Linearization
plant.OutputGroup.Unmeasured = 2;
plant.InputName = {'Ti','Tc'};
plant.OutputName = {'T','CA'};
Create MPC controller with default prediction and control horizons
mpcobj = mpc(plant);
-->The
-->The
-->The
-->The
-->The
for
"PredictionHorizon" property of "mpc" object is empty. Trying PredictionHorizon
"ControlHorizon" property of the "mpc" object is empty. Assuming 2.
"Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
"Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
"Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
output(s) y1 and zero weight for output(s) y2
Set nominal values in the controller
mpcobj.Model.Nominal = struct('X', x0, 'U', u0(2:3), 'Y', y0, 'DX', [0 0]);
Set scale factors because plant input and output signals have different orders of
magnitude
Uscale = [30 50];
Yscale = [50 10];
mpcobj.DV(1).ScaleFactor
mpcobj.MV(1).ScaleFactor
mpcobj.OV(1).ScaleFactor
mpcobj.OV(2).ScaleFactor
=
=
=
=
Uscale(1);
Uscale(2);
Yscale(1);
Yscale(2);
Let reactor temperature T float (i.e. with no setpoint tracking error penalty), because
the objective is to control reactor concentration CA and only one manipulated variable
(coolant temperature Tc) is available.
mpcobj.Weights.OV = [0 1];
Due to the physical constraint of coolant jacket, Tc rate of change is bounded by degrees
per minute.
mpcobj.MV.RateMin = -2;
mpcobj.MV.RateMax = 2;
Implement Adaptive MPC Control of CSTR Plant in Simulink (R)
Open the Simulink model.
5-13
5
Adaptive MPC Design
mdl = 'ampc_cstr_linearization';
open_system(mdl);
The model includes three parts:
1
The "CSTR" block implements the nonlinear plant model.
2
The "Adaptive MPC Controller" block runs the designed MPC controller in the
adaptive mode.
3
The "Successive Linearizer" block in a MATLAB Function block that linearizes a
first principle nonlinear CSTR plant and provides the linear plant model to the
"Adaptive MPC Controller" block at each control interval. Double click the block to
see the MATLAB code. You can use the block as a template to develop appropriate
linearizer for your own applications.
Note that the new linear plant model must be a discrete time state space system with the
same order and sample time as the original plant model has. If the plant has time delay,
it must also be same as the original time delay and absorbed into the state space model.
Validate Adaptive MPC Control Performance
Controller performance is validated against both setpoint tracking and disturbance
rejection.
5-14
Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive Linearization
• Tracking: reactor concentration CA setpoint transitions from original 8.57 (low
conversion rate) to 2 (high conversion rate) kgmol/m^3. During the transition, the
plant first becomes unstable then stable again (see the poles plot).
• Regulating: feed temperature Ti has slow fluctuation represented by a sine wave with
amplitude of 5 degrees, which is a measured disturbance fed to the MPC controller.
Simulate the closed-loop performance.
open_system([mdl '/Concentration'])
open_system([mdl '/Temperature'])
open_system([mdl '/Pole'])
sim(mdl);
-->Integrated white noise added on measured output channel #1.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
5-15
5
Adaptive MPC Design
5-16
Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive Linearization
The tracking and regulating performance is very satisfactory. In an application to a real
reactor, however, model inaccuracies and unmeasured disturbances could cause poorer
tracking than shown here. Additional simulations could be used to study these effects.
Compare with Non-Adaptive MPC Control
Adaptive MPC provides superior control performance than a non-adaptive MPC. To
illustrate this point, the control performance of the same MPC controller running in the
non-adaptive mode is shown below. The controller is implemented with a MPC Controller
block.
mdl1 = 'ampc_cstr_no_linearization';
open_system(mdl1);
open_system([mdl1 '/Concentration'])
open_system([mdl1 '/Temperature'])
sim(mdl1);
5-17
5
Adaptive MPC Design
5-18
Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive Linearization
5-19
5
Adaptive MPC Design
As expected, the tracking and regulating performance is unacceptable.
bdclose(mdl)
bdclose(mdl1)
See Also
Adaptive MPC Controller
Related Examples
•
“Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model
Estimation”
More About
•
5-20
“Adaptive MPC” on page 5-2
Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation
Adaptive MPC Control of Nonlinear Chemical Reactor Using
Online Model Estimation
This example shows how to use an Adaptive MPC controller to control a nonlinear
continuous stirred tank reactor (CSTR) as it transitions from low conversion rate to high
conversion rate.
A discrete time ARX model is being identified online by the Recursive Polynomial Model
Estimator block at each control interval. The adaptive MPC controller uses it to update
internal plant model and achieves nonlinear control successfully.
About the Continuous Stirred Tank Reactor
A Continuously Stirred Tank Reactor (CSTR) is a common chemical system in the
process industry. A schematic of the CSTR system is:
5-21
5
Adaptive MPC Design
This is a jacketed non-adiabatic tank reactor described extensively in Seborg's book,
"Process Dynamics and Control", published by Wiley, 2004. The vessel is assumed to be
perfectly mixed, and a single first-order exothermic and irreversible reaction, A --> B,
takes place. The inlet stream of reagent A is fed to the tank at a constant volumetric rate.
The product stream exits continuously at the same volumetric rate and liquid density is
constant. Thus the volume of reacting liquid is constant.
The inputs of the CSTR model are:
and the outputs (y(t)), which are also the states of the model (x(t)), are:
The control objective is to maintain the reactor temperature at its desired setpoint,
which changes over time when reactor transitions from low conversion rate to high
conversion rate. The coolant temperature
is the manipulated variable used by the
MPC controller to track the reference as well as reject the measured disturbance arising
from the inlet feed stream temperature . The inlet feed stream concentration,
,
is assumed to be constant. The Simulink model mpc_cstr_plant implements the
nonlinear CSTR plant.
About Adaptive Model Predictive Control
It is well known that the CSTR dynamics are strongly nonlinear with respect to reactor
temperature variations and can be open-loop unstable during the transition from
one operating condition to another. A single MPC controller designed at a particular
operating condition cannot give satisfactory control performance over a wide operating
range.
To control the nonlinear CSTR plant with linear MPC control technique, you have a few
options:
• If a linear plant model cannot be obtained at run time, first you need to obtain several
linear plant models offline at different operating conditions that cover the typical
5-22
Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation
operating range. Next you can choose one of the two approaches to implement MPC
control strategy:
(1) Design several MPC controllers offline, one for each plant model. At run time, use
Multiple MPC Controller block that switches MPC controllers from one to another
based on a desired scheduling strategy. See "Gain Scheduled MPC Control of Nonlinear
Chemical Reactor" for more details. Use this approach when the plant models have
different orders or time delays.
(2) Design one MPC controller offline at the initial operating point. At run time, use
Adaptive MPC Controller block (updating predictive model at each control interval)
together with Linear Parameter Varying (LPV) System block (supplying linear plant
model with a scheduling strategy). See "Adaptive MPC Control of Nonlinear Chemical
Reactor Using Linear Parameter Varying System" for more details. Use this approach
when all the plant models have the same order and time delay.
• If a linear plant model can be obtained at run time, you should use Adaptive MPC
Controller block to achieve nonlinear control. There are two typical ways to obtain a
linear plant model online:
(1) Use successive linearization. See "Adaptive MPC Control of Nonlinear Chemical
Reactor Using Successive Linearization" for more details. Use this approach when a
nonlinear plant model is available and can be linearized at run time.
(2) Use online estimation to identify a linear model when loop is closed, as shown in this
example. Use this approach when linear plant model cannot be obtained from either an
LPV system or successive linearization.
Obtain Linear Plant Model at Initial Operating Condition
To linearize the plant, Simulink® and Simulink Control Design® are required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
if ~mpcchecktoolboxinstalled('slcontrol')
disp('Simulink Control Design(R) is required to run this example.')
return
end
To implement an adaptive MPC controller, first you need to design a MPC controller at
the initial operating point where CAi is 10 kgmol/m^3, Ti and Tc are 298.15 K.
5-23
5
Adaptive MPC Design
Create operating point specification.
plant_mdl = 'mpc_cstr_plant';
op = operspec(plant_mdl);
Feed concentration is known at the initial condition.
op.Inputs(1).u = 10;
op.Inputs(1).Known = true;
Feed temperature is known at the initial condition.
op.Inputs(2).u = 298.15;
op.Inputs(2).Known = true;
Coolant temperature is known at the initial condition.
op.Inputs(3).u = 298.15;
op.Inputs(3).Known = true;
Compute initial condition.
[op_point, op_report] = findop(plant_mdl,op);
Operating Point Search Report:
--------------------------------Operating Report for the Model mpc_cstr_plant.
(Time-Varying Components Evaluated at time t=0)
Operating point specifications were successfully met.
States:
---------(1.) mpc_cstr_plant/CSTR/Integrator
x:
311
dx:
8.12e-11 (0)
(2.) mpc_cstr_plant/CSTR/Integrator1
x:
8.57
dx:
-6.87e-12 (0)
Inputs:
---------(1.) mpc_cstr_plant/CAi
u:
10
(2.) mpc_cstr_plant/Ti
5-24
Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation
u:
298
(3.) mpc_cstr_plant/Tc
u:
298
Outputs:
---------(1.) mpc_cstr_plant/T
y:
311
(2.) mpc_cstr_plant/CA
y:
8.57
[-Inf Inf]
[-Inf Inf]
Obtain nominal values of x, y and u.
x0 = [op_report.States(1).x;op_report.States(2).x];
y0 = [op_report.Outputs(1).y;op_report.Outputs(2).y];
u0 = [op_report.Inputs(1).u;op_report.Inputs(2).u;op_report.Inputs(3).u];
Obtain linear plant model at the initial condition.
sys = linearize(plant_mdl, op_point);
Drop the first plant input CAi and second output CA because they are not used by MPC.
sys = sys(1,2:3);
Discretize the plant model because Adaptive MPC controller only accepts a discrete-time
plant model.
Ts = 0.5;
plant = c2d(sys,Ts);
Design MPC Controller
You design an MPC at the initial operating condition. When running in the adaptive
mode, the plant model is updated at run time.
Specify signal types used in MPC.
plant.InputGroup.MeasuredDisturbances = 1;
plant.InputGroup.ManipulatedVariables = 2;
plant.OutputGroup.Measured = 1;
plant.InputName = {'Ti','Tc'};
plant.OutputName = {'T'};
5-25
5
Adaptive MPC Design
Create MPC controller with default prediction and control horizons
mpcobj = mpc(plant);
-->The
-->The
-->The
-->The
-->The
"PredictionHorizon" property of "mpc" object is empty. Trying PredictionHorizon
"ControlHorizon" property of the "mpc" object is empty. Assuming 2.
"Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
"Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
"Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
Set nominal values in the controller
mpcobj.Model.Nominal = struct('X', x0, 'U', u0(2:3), 'Y', y0(1), 'DX', [0 0]);
Set scale factors because plant input and output signals have different orders of
magnitude
Uscale = [30 50];
Yscale = 50;
mpcobj.DV.ScaleFactor = Uscale(1);
mpcobj.MV.ScaleFactor = Uscale(2);
mpcobj.OV.ScaleFactor = Yscale;
Due to the physical constraint of coolant jacket, Tc rate of change is bounded by 2 degrees
per minute.
mpcobj.MV.RateMin = -2;
mpcobj.MV.RateMax = 2;
Reactor concentration is not directly controlled in this example. If reactor temperature
can be successfully controlled, the concentration will achieve desired performance
requirement due to the strongly coupling between the two variables.
Implement Adaptive MPC Control of CSTR Plant in Simulink (R)
To run this example with online estimation, System Identification® is required.
if ~mpcchecktoolboxinstalled('ident')
disp('System Identification(R) is required to run this example.')
return
end
Open the Simulink model.
mdl = 'ampc_cstr_estimation';
5-26
Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation
open_system(mdl);
The model includes three parts:
1
The "CSTR" block implements the nonlinear plant model.
2
The "Adaptive MPC Controller" block runs the designed MPC controller in the
adaptive mode.
3
The "Recursive Polynomial Model Estimator" block estimates a two-input (Ti and Tc)
and one-output (T) discrete time ARX model based on the measured temperatures.
The estimated model is then converted into state space form by the "Model Type
Converter" block and fed to the "Adaptive MPC Controller" block at each control
interval.
In this example, the initial plant model is used to initialize the online estimator with
parameter covariance matrix set to 1. The online estimation method is "Kalman Filter"
with noise covariance matrix set to 0.01. The online estimation result is sensitive to these
parameters and you can further adjust them to achieve better estimation result.
Both "Recursive Polynomial Model Estimator" and "Model Type Converter" are provided
by System Identification Toolbox. You can use the two blocks as a template to develop
appropriate online model estimation for your own applications.
The initial value of A(q) and B(q) variables are populated with the numerator and
denominator of the initial plant model.
5-27
5
Adaptive MPC Design
[num, den] = tfdata(plant);
Aq = den{1};
Bq = num;
Note that the new linear plant model must be a discrete time state space system with the
same order and sample time as the original plant model has. If the plant has time delay,
it must also be same as the original time delay and absorbed into the state space model.
Validate Adaptive MPC Control Performance
Controller performance is validated against both setpoint tracking and disturbance
rejection.
• Tracking: reactor temperature T setpoint transitions from original 311 K (low
conversion rate) to 377 K (high conversion rate) kgmol/m^3. During the transition,
the plant first becomes unstable then stable again (see the poles plot).
• Regulating: feed temperature Ti has slow fluctuation represented by a sine wave with
amplitude of 5 degrees, which is a measured disturbance fed to MPC controller.
Simulate the closed-loop performance.
open_system([mdl '/Concentration'])
open_system([mdl '/Temperature'])
sim(mdl);
-->Integrated white noise added on measured output channel #1.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
5-28
Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation
5-29
5
Adaptive MPC Design
The tracking and regulating performance is very satisfactory.
Compare with Non-Adaptive MPC Control
Adaptive MPC provides superior control performance than a non-adaptive MPC. To
illustrate this point, the control performance of the same MPC controller running in the
non-adaptive mode is shown below. The controller is implemented with a MPC Controller
block.
mdl1 = 'ampc_cstr_no_estimation';
open_system(mdl1);
open_system([mdl1 '/Concentration'])
open_system([mdl1 '/Temperature'])
sim(mdl1);
5-30
Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation
5-31
5
Adaptive MPC Design
5-32
Adaptive MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation
As expected, the tracking and regulating performance is unacceptable.
bdclose(mdl)
bdclose(mdl1)
See Also
Adaptive MPC Controller
Related Examples
•
“Adaptive MPC Control of Nonlinear Chemical Reactor Using Successive
Linearization”
More About
•
“Adaptive MPC” on page 5-2
5-33
6
Explicit MPC Design
• “Explicit MPC” on page 6-2
• “Design Workflow for Explicit MPC” on page 6-4
• “Explicit MPC Control of a Single-Input-Single-Output Plant” on page 6-9
• “Explicit MPC Control of an Aircraft with Unstable Poles” on page 6-21
• “Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output” on
page 6-30
6
Explicit MPC Design
Explicit MPC
A traditional model predictive controller solves a quadratic program (QP) at each
control interval to determine the optimal manipulated variable (MV) adjustments.
Mathematically, these adjustments are an implicit nonlinear function of the current
controller state and other variables, such as the current output reference values. Suppose
that all the independent variables affecting the QP solution form a vector, x. The
optimal MV adjustments are then u = f(x), where f is the implicit function to be solved
by quadratic programming. The Model Predictive Control Toolbox software imposes
restrictions that force the QP to have a unique solution.
Evaluating the solution of u = f(x) via QP can be time consuming, however, and the
required time can vary significantly from one control interval to the next. In applications
that require the solution to be obtained consistently within a certain elapsed time, which
could be of order milliseconds or microseconds, the implicit MPC approach might be
unsuitable.
As shown in “Optimization Problem”, if x is such that none of the QP’s inequality
constraints are active at the solution, the MPC control law reduces to a linear function of
x:
u = Fx + G.
Here, F and G are constants. Similarly, if x stays within a region for which a fixed subset
of the inequality constraints are active, the QP solution is again a linear function of x but
with different F and G constants.
Explicit MPC uses offline computations to determine all polyhedral regions in which
the optimal MV adjustments are a linear function of x and the corresponding controllaw constants. When the controller operates in real time, the explicit MPC controller
performs the following steps at each control instant, k:
6-2
1
Uses the available measurements to estimate the controller state, as in traditional
MPC.
2
Uses this estimated state and current values of the other independent variables to
form x(k).
3
Identifies the region in which x(k) resides.
4
Looks up the predetermined F and G constants for this region.
5
Evaluates the linear function u(k) = Fx(k) + G.
Explicit MPC
You can establish a tight upper bound for the time required in each step. The total
computational time can be quite small of the number of regions is not too large. In
fact, the time required in step 3 dominates, and the memory needed to store all the
linear control laws and polyhedral regions becomes excessive. The number of regions
characterizing u = f(x) depends primarily on the combinations of QP inequality constraint
that could be active at the solution. Thus, an explicit MPC involving many constraints
might require too much computational effort or memory. In that case, a traditional
(implicit) implementation may be preferable.
Related Examples
•
“Explicit MPC Control of a Single-Input-Single-Output Plant”
•
“Explicit MPC Control of an Aircraft with Unstable Poles”
•
“Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output”
More About
•
“Design Workflow for Explicit MPC” on page 6-4
6-3
6
Explicit MPC Design
Design Workflow for Explicit MPC
In this section...
“Traditional (Implicit) MPC Design” on page 6-4
“Explicit MPC Generation” on page 6-5
“Explicit MPC Simplification” on page 6-6
“Implementation” on page 6-6
“Simulation” on page 6-7
To create an explicit MPC controller, you must first design a traditional (implicit)
MPC controller. You then generate an explicit MPC controller based on the traditional
controller design.
Traditional (Implicit) MPC Design
First design a traditional (implicit) MPC for your application and test it in simulations.
Key considerations are as follows:
• The Model Predictive Control Toolbox software currently supports the following as
independent variables for explicit MPC:
• nxc controller state variables (plant, disturbance, and measurement noise model
states).
• ny (≥ 1) output reference values, where ny is the number of plant output variables.
• nv (≥ 0) measured plant disturbance signals.
Thus, you must fix most MPC design parameters prior to determining an explicit
MPC. Fixed parameters include prediction models (plant, disturbance and
measurement noise), scale factors, horizons, penalty weights, manipulated variable
targets, and constraint bounds.
For information about designing a traditional MPC controller, see “Controller
Creation”.
For information about tuning traditional MPC controllers, see “Refinement”.
• Reference and measured disturbance previewing are not supported. At each control
interval, the current ny reference and nv measured disturbance signals apply for the
entire prediction horizon.
6-4
Design Workflow for Explicit MPC
• To limit the number of regions needed by explicit MPC, include only essential
constraints.
• When including a constraint on a manipulated variable (MV) use a short control
horizon or MV blocking. See “Choosing Sample Time and Horizons”.
• Avoid constraints on plant outputs. If such a constraint is essential, consider
imposing it for selected prediction horizon steps rather than the entire prediction
horizon.
• Establish upper and lower bounds for each of the nx = nxc + ny + nv independent
variables. You might know some of these bounds a priori. However, you must run
simulations that record at least the nxc controller states as the system operates over
the range of expected conditions. It is very important that you not understimate
this range, because the explicit MPC control function is not defined for independent
variables outside the range.
For information about specifying bounds, see generateExplicitRange.
For information about simulating a traditional MPC controller, see “Simulation”.
Explicit MPC Generation
Given the constant MPC design parameters and the nx upper and lower bounds on the
control law’s independent variables, i.e.,
xl £ x(k) £ xu ,
the generateExplicitMPC command determines nr regions. Each of these regions is
defined by an inequality constraint and the corresponding control law constants:
Hi x ( k ) £ Ki , i = 1, nr
u ( k ) = Fi x ( k ) + Gi , i = 1, nr .
The Explicit MPC Controller object contains the constants Hi, Ki, Fi, and Gi for each
region. The Explicit MPC Controller object also holds the original (implicit) design and
independent variable bounds. Provided that x(k) stays within the specified bounds and
you retain all nr regions, the explicit MPC object should provide the same optimal MV
adjustments, u(k), as the equivalent implicit MPC object.
6-5
6
Explicit MPC Design
For details about explicit MPC, see [1]. For details about how the explicit MPC controller
is generated, see [2].
Explicit MPC Simplification
Even a relatively simple explicit MPC controller might need nr >> 100 to characterize the
QP solution completely. If the number of regions is large, consider the following:
• Visualize the solution using the plotSection command.
• Use the simplify command to reduce the number of regions. In some cases, this can
be done with no (or negligible) impact on control law optimality. For example, pairs
of adjacent regions might employ essentially the same Fi and Ki constants. If so, and
if the union of the two regions forms a convex set, they can be merged into a single
region.
Alternatively, you can eliminate relatively small regions or retain selected regions
only. If during operation the current x(k) is not contained in any of the retained
regions, the explicit MPC will return a suboptimal u(k), as follows:
u ( k ) = Fj x ( k ) + G j .
Here, j is the index of the region whose bounding constraint, Hjx(k) ≤ Kj, is least
violated.
Implementation
During operation, for a given x(k), the explicit MPC controller performs the following
steps:
1
Verifies that x(k) satisfies the specified bounds, xl ≤ x(k) ≤ xu. If not, the controller
returns an error status and sets u(k) = u(k–1).
2
Beginning with region i = 1, tests the regions one by one to determine whether x(k)
belongs. If Hix(k) ≤ Ki, then x(k) belongs to region i. If x(k) belongs to region i, then
the controller:
• Obtains Fi and Gi from memory, and computes u(k) = Fix(k) + Gi.
• Signals successful completion, by returning a status code and the index i.
• Returns without testing the remaining regions.
6-6
Design Workflow for Explicit MPC
If x(k) does not belong to region i, the controller:
• Computes the violation term vi, which is the largest (positive) component of the
vector (Hix(k) – Ki).
• If vi is the minimum violation for this x(k), the controller sets j = i, and sets vmin =
vi.
• The controller then increments i and tests the next region.
3
If all regions have been tested and x(k) does not belong to any region (for example,
due to a numerical precision issue), the controller:
• Obtains Fj and Gj from memory, and computes u(k) = Fjx(k) + Gj.
• Sets status to indicate a suboptimal solution and returns.
Thus, the maximum computational time per control interval is that needed to test each
region, computing the violation term in each case, and then calculating the suboptimal
control adjustment.
Simulation
You can perform command-line simulations using the sim or mpcmoveExplicit
commands.
You can use the Explicit MPC Controller block to connect an explicit MPC to a plant
modeled in Simulink.
References
[1] A. Bemporad, M. Morari, V. Dua, and E.N. Pistikopoulos, “The explicit linear
quadratic regulator for constrained systems,” Automatica, vol. 38, no. 1, pp. 3–20,
2002.
[2] A. Bemporad, “A multi-parametric quadratic programming algorithm with polyhedral
computations based on nonnegative least squares,” 2014, Submitted for
publication.
See Also
Explicit MPC Controller | generateExplicitMPC | mpcmoveExplicit
6-7
6
Explicit MPC Design
Related Examples
•
“Explicit MPC Control of a Single-Input-Single-Output Plant”
•
“Explicit MPC Control of an Aircraft with Unstable Poles”
•
“Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output”
More About
•
6-8
“Explicit MPC” on page 6-2
Explicit MPC Control of a Single-Input-Single-Output Plant
Explicit MPC Control of a Single-Input-Single-Output Plant
This example shows how to control a double integrator plant under input saturation in
Simulink® using explicit MPC.
See also MPCDOUBLEINT.
Define Plant Model
The linear open-loop dynamic model is a double integrator:
plant = tf(1,[1 0 0]);
Design MPC Controller
Create the controller object with sampling period, prediction and control horizons:
Ts = 0.1;
p = 10;
m = 3;
mpcobj = mpc(plant, Ts, p, m);
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
Specify actuator saturation limits as MV constraints.
mpcobj.MV = struct('Min',-1,'Max',1);
Generate Explicit MPC Controller
Explicit MPC executes the equivalent explicit piecewise affine version of the MPC control
law defined by the traditional MPC. To generate an Explicit MPC from a traditional
MPC, you must specify range for each controller state, reference signal, manipulated
variable and measured disturbance so that the multi-parametric quadratic programming
problem is solved in the parameter space defined by these ranges.
Obtain a range structure for initialization
Use generateExplicitRange command to obtain a range structure where you can
specify range for each parameter afterwards.
6-9
6
Explicit MPC Design
range = generateExplicitRange(mpcobj);
-->Converting the "Model.Plant" property of "mpc" object to state-space.
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
Specify ranges for controller states
MPC controller states include states from plant model, disturbance model and noise
model in that order. Setting the range of a state variable is sometimes difficult when the
state does not correspond to a physical parameter. In that case, multiple runs of openloop plant simulation with typical reference and disturbance signals are recommended in
order to collect data that reflect the ranges of states.
range.State.Min(:) = [-10;-10];
range.State.Max(:) = [10;10];
Specify ranges for reference signals
Usually you know the practical range of the reference signals being used at the nominal
operating point in the plant. The ranges used to generate Explicit MPC must be at least
as large as the practical range.
range.Reference.Min = -2;
range.Reference.Max = 2;
Specify ranges for manipulated variables
If manipulated variables are constrained, the ranges used to generate Explicit MPC must
be at least as large as these limits.
range.ManipulatedVariable.Min = -1.1;
range.ManipulatedVariable.Max = 1.1;
Construct the Explicit MPC controller
Use generateExplicitMPC command to obtain the Explicit MPC controller with the
parameter ranges previously specified.
mpcobjExplicit = generateExplicitMPC(mpcobj, range);
display(mpcobjExplicit);
6-10
Explicit MPC Control of a Single-Input-Single-Output Plant
Regions found / unexplored:
19/
0
Explicit MPC Controller
--------------------------------------------Controller sample time:
0.1 (seconds)
Polyhedral regions:
19
Number of parameters:
4
Is solution simplified:
No
State Estimation:
Default Kalman gain
--------------------------------------------Type 'mpcobjExplicit.MPC' for the original implicit MPC design.
Type 'mpcobjExplicit.Range' for the valid range of parameters.
Type 'mpcobjExplicit.OptimizationOptions' for the options used in multi-parametirc QP c
Type 'mpcobjExplicit.PiecewiseAffineSolution' for regions and gain in each solution.
Use simplify command with the "exact" method to join pairs of regions whose
corresponding gains are the same and whose union is a convex set. This practice
can reduce memory footprint of the Explicit MPC controller without sacrifice any
performance.
mpcobjExplicitSimplified = simplify(mpcobjExplicit, 'exact');
display(mpcobjExplicitSimplified);
Regions to analyze:
15/
15
Explicit MPC Controller
--------------------------------------------Controller sample time:
0.1 (seconds)
Polyhedral regions:
15
Number of parameters:
4
Is solution simplified:
Yes
State Estimation:
Default Kalman gain
--------------------------------------------Type 'mpcobjExplicitSimplified.MPC' for the original implicit MPC design.
Type 'mpcobjExplicitSimplified.Range' for the valid range of parameters.
Type 'mpcobjExplicitSimplified.OptimizationOptions' for the options used in multi-param
Type 'mpcobjExplicitSimplified.PiecewiseAffineSolution' for regions and gain in each so
The number of piecewise affine region has been reduced.
6-11
6
Explicit MPC Design
Plot Piecewise Affine Partition
You can review any 2-D section of the piecewise affine partition defined by the Explicit
MPC control law.
Obtain a plot parameter structure for initialization
Use generatePlotParameters command to obtain a parameter structure where you
can specify which 2-D section to plot afterwards.
params = generatePlotParameters(mpcobjExplicitSimplified);
Specify parameters for a 2-D plot
In this example, you plot the 1th state variable vs. the 2nd state variable. All the other
parameters must be fixed at a value within its range.
params.State.Index = [];
params.State.Value = [];
Fix other reference signals
params.Reference.Index = 1;
params.Reference.Value = 0;
Fix manipulated variables
params.ManipulatedVariable.Index = 1;
params.ManipulatedVariable.Value = 0;
Plot the 2-D section
Use plotSection command to plot the 2-D section defined previously.
plotSection(mpcobjExplicitSimplified, params);
axis([-4 4 -4 4]);
grid
xlabel('State #1');
ylabel('State #2');
6-12
Explicit MPC Control of a Single-Input-Single-Output Plant
Simulate Using MPCMOVE Command
Compare closed-loop simulation between tradition MPC (as referred as Implicit MPC)
and Explicit MPC using mpcmove and mpcmoveExplicit commands respectively.
Prepare to store the closed-loop MPC responses.
Tf = round(5/Ts);
YY = zeros(Tf,1);
YYExplicit = zeros(Tf,1);
UU = zeros(Tf,1);
UUExplicit = zeros(Tf,1);
Prepare the real plant used in simulation
6-13
6
Explicit MPC Design
sys = c2d(ss(plant),Ts);
xsys = [0;0];
xsysExplicit = xsys;
Use MPCSTATE object to specify the initial states for both controllers
xmpc = mpcstate(mpcobj);
xmpcExplicit = mpcstate(mpcobjExplicitSimplified);
Simulate closed-loop response in each iteration.
for t = 0:Tf
% update plant measurement
ysys = sys.C*xsys;
ysysExplicit = sys.C*xsysExplicit;
% compute traditional MPC action
u = mpcmove(mpcobj,xmpc,ysys,1);
% compute Explicit MPC action
uExplicit = mpcmoveExplicit(mpcobjExplicit,xmpcExplicit,ysysExplicit,1);
% store signals
YY(t+1)=ysys;
YYExplicit(t+1)=ysysExplicit;
UU(t+1)=u;
UUExplicit(t+1)=uExplicit;
% update plant state
xsys = sys.A*xsys + sys.B*u;
xsysExplicit = sys.A*xsysExplicit + sys.B*uExplicit;
end
fprintf('\nDifference between traditional and Explicit MPC responses using MPCMOVE comm
Difference between traditional and Explicit MPC responses using MPCMOVE command is 1.80
Simulate Using SIM Command
Compare closed-loop simulation between tradition MPC and Explicit MPC using sim
commands respectively.
Tf = 5/Ts;
% simulation iterations
[y1,t1,u1] = sim(mpcobj,Tf,1); % simulation with tradition MPC
[y2,t2,u2] = sim(mpcobjExplicitSimplified,Tf,1);
% simulation with Explicit MPC
-->Converting the "Model.Plant" property of "mpc" object to state-space.
-->Converting model to discrete time.
6-14
Explicit MPC Control of a Single-Input-Single-Output Plant
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty.
-->Converting the "Model.Plant" property of "mpc" object to
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty.
-->Converting the "Model.Plant" property of "mpc" object to
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty.
Assuming white noise on eac
state-space.
Assuming white noise on eac
state-space.
Assuming white noise on eac
The simulation results are identical.
fprintf('\nDifference between traditional and Explicit MPC responses using SIM command
Difference between traditional and Explicit MPC responses using SIM command is 1.80294e
Simulate Using Simulink®
To run this example, Simulink® is required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
Simulate with traditional MPC controller in Simulink. Controller "mpcobj" is specified in
the block dialog.
mdl = 'mpc_doubleint';
open_system(mdl);
sim(mdl);
6-15
6
Explicit MPC Design
6-16
Explicit MPC Control of a Single-Input-Single-Output Plant
Simulate with Explicit MPC controller in Simulink. Controller
"mpcobjExplicitSimplified" is specified in the block dialog.
mdlExplicit = 'empc_doubleint';
open_system(mdlExplicit);
sim(mdlExplicit);
6-17
6
Explicit MPC Design
6-18
Explicit MPC Control of a Single-Input-Single-Output Plant
6-19
6
Explicit MPC Design
The closed-loop responses are identical.
fprintf('\nDifference between traditional and Explicit MPC responses in Simulink is %g\
Difference between traditional and Explicit MPC responses in Simulink is 1.56789e-13
bdclose(mdl)
bdclose(mdlExplicit)
Related Examples
•
“Explicit MPC Control of an Aircraft with Unstable Poles”
•
“Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output”
More About
•
6-20
“Explicit MPC” on page 6-2
Explicit MPC Control of an Aircraft with Unstable Poles
Explicit MPC Control of an Aircraft with Unstable Poles
This example shows how to control an unstable aircraft with saturating actuators using
Explicit MPC.
Reference:
[1] P. Kapasouris, M. Athans and G. Stein, "Design of feedback control systems for
unstable plants with saturating actuators", Proc. IFAC Symp. on Nonlinear Control
System Design, Pergamon Press, pp.302--307, 1990
[2] A. Bemporad, A. Casavola, and E. Mosca, "Nonlinear control of constrained linear
systems via predictive reference management", IEEE® Trans. Automatic Control, vol.
AC-42, no. 3, pp. 340-349, 1997.
See also MPCAIRCRAFT.
Define Aircraft Model
The linear open-loop dynamic model is as follows:
A = [-0.0151 -60.5651 0 -32.174;
-0.0001 -1.3411 0.9929 0;
0.00018 43.2541 -0.86939 0;
0
0
1
0];
B = [-2.516 -13.136;
-0.1689 -0.2514;
-17.251 -1.5766;
0
0];
C = [0 1 0 0;
0 0 0 1];
D = [0 0;
0 0];
plant = ss(A,B,C,D);
x0 = zeros(4,1);
The manipulated variables are the elevator and flaperon angles, the attack and pitch
angles are measured outputs to be regulated.
The open-loop response of the system is unstable.
pole(plant)
6-21
6
Explicit MPC Design
ans =
-7.6636
5.4530
-0.0075
-0.0075
+
+
+
-
0.0000i
0.0000i
0.0556i
0.0556i
Design MPC Controller
To obtain an Explicit MPC controller, you must first design a traditional MPC (also
referred as Implicit MPC) that is able to achieves your control objectives.
% *MV Constraints*
Both manipulated variables are constrained between +/- 25 degrees. Since the plant
inputs and outputs are of different orders of magnitude, you also use scale factors to
faciliate MPC tuning. Typical choices of scale factor are the upper/lower limit or the
operating range.
MV = struct('Min',{-25,-25},'Max',{25,25},'ScaleFactor',{50,50});
OV Constraints
Both plant outputs have constraints to limit undershoots at the first prediction horizon.
You also specify scale factors for outputs.
OV = struct('Min',{[-0.5;-Inf],[-100;-Inf]},'Max',{[0.5;Inf],[100;Inf]},'ScaleFactor',{
Weights
The control task is to get zero offset for piecewise-constant references, while avoiding
instability due to input saturation. Because both MV and OV variables are already scaled
in MPC controller, MPC weights are dimensionless and applied to the scaled MV and OV
values. In this example, you penalize the two outputs equally with the same OV weights.
Weights = struct('MV',[0 0],'MVRate',[0.1 0.1],'OV',[10 10]);
Construct the traditional MPC controller
Create an MPC controller with plant model, sample time and horizons.
Ts = 0.05;
% Sampling time
p = 10;
% Prediction horizon
m = 2;
% Control horizon
mpcobj = mpc(plant,Ts,p,m,Weights,MV,OV);
6-22
Explicit MPC Control of an Aircraft with Unstable Poles
Generate Explicit MPC Controller
Explicit MPC executes the equivalent explicit piecewise affine version of the MPC control
law defined by the traditional MPC. To generate an Explicit MPC from a traditional
MPC, you must specify range for each controller state, reference signal, manipulated
variable and measured disturbance so that the multi-parametric quadratic programming
problem is solved in the parameter space defined by these ranges.
Obtain a range structure for initialization
Use generateExplicitRange command to obtain a range structure where you can
specify range for each parameter afterwards.
range = generateExplicitRange(mpcobj);
-->Converting model to discrete time.
-->Integrated white noise added on measured output channel #1.
-->Integrated white noise added on measured output channel #2.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
Specify ranges for controller states
MPC controller states include states from plant model, disturbance model and noise
model in that order. Setting the range of a state variable is sometimes difficult when the
state does not correspond to a physical parameter. In that case, multiple runs of openloop plant simulation with typical reference and disturbance signals are recommended in
order to collect data that reflect the ranges of states.
range.State.Min(:) = -10000;
range.State.Max(:) = 10000;
Specify ranges for reference signals
Usually you know the practical range of the reference signals being used at the nominal
operating point in the plant. The ranges used to generate Explicit MPC must be at least
as large as the practical range.
range.Reference.Min = [-1;-11];
range.Reference.Max = [1;11];
Specify ranges for manipulated variables
If manipulated variables are constrained, the ranges used to generate Explicit MPC must
be at least as large as these limits.
6-23
6
Explicit MPC Design
range.ManipulatedVariable.Min = [MV(1).Min; MV(2).Min] - 1;
range.ManipulatedVariable.Max = [MV(1).Max; MV(2).Max] + 1;
Construct the Explicit MPC controller
Use generateExplicitMPC command to obtain the Explicit MPC controller with the
parameter ranges previously specified.
mpcobjExplicit = generateExplicitMPC(mpcobj, range);
display(mpcobjExplicit);
Regions found / unexplored:
483/
0
Explicit MPC Controller
--------------------------------------------Controller sample time:
0.05 (seconds)
Polyhedral regions:
483
Number of parameters:
10
Is solution simplified:
No
State Estimation:
Default Kalman gain
--------------------------------------------Type 'mpcobjExplicit.MPC' for the original implicit MPC design.
Type 'mpcobjExplicit.Range' for the valid range of parameters.
Type 'mpcobjExplicit.OptimizationOptions' for the options used in multi-parametirc QP c
Type 'mpcobjExplicit.PiecewiseAffineSolution' for regions and gain in each solution.
Use simplify command with the "exact" method to join pairs of regions whose
corresponding gains are the same and whose union is a convex set. This practice
can reduce memory footprint of the Explicit MPC controller without sacrifice any
performance.
mpcobjExplicitSimplified = simplify(mpcobjExplicit, 'exact');
display(mpcobjExplicitSimplified);
Regions to analyze:
471/
471
Explicit MPC Controller
--------------------------------------------Controller sample time:
0.05 (seconds)
6-24
Explicit MPC Control of an Aircraft with Unstable Poles
Polyhedral regions:
471
Number of parameters:
10
Is solution simplified:
Yes
State Estimation:
Default Kalman gain
--------------------------------------------Type 'mpcobjExplicitSimplified.MPC' for the original implicit MPC design.
Type 'mpcobjExplicitSimplified.Range' for the valid range of parameters.
Type 'mpcobjExplicitSimplified.OptimizationOptions' for the options used in multi-param
Type 'mpcobjExplicitSimplified.PiecewiseAffineSolution' for regions and gain in each so
The number of piecewise affine region has been reduced.
Plot Piecewise Affine Partition
You can review any 2-D section of the piecewise affine partition defined by the Explicit
MPC control law.
Obtain a plot parameter structure for initialization
Use generatePlotParameters command to obtain a parameter structure where you
can specify which 2-D section to plot afterwards.
params = generatePlotParameters(mpcobjExplicitSimplified);
Specify parameters for a 2-D plot
In this example, you plot the pitch angle (the 4th state variable) vs. its reference (the 2nd
reference signal). All the other parameters must be fixed at a value within its range.
Fix other state variables
params.State.Index = [1 2 3 5 6];
params.State.Value = [0 0 0 0 0];
Fix other reference signals
params.Reference.Index = 1;
params.Reference.Value = 0;
Fix manipulated variables
params.ManipulatedVariable.Index = [1 2];
params.ManipulatedVariable.Value = [0 0];
Plot the 2-D section
Use plotSection command to plot the 2-D section defined previously.
6-25
6
Explicit MPC Design
plotSection(mpcobjExplicitSimplified, params);
axis([-10 10 -10 10]);
grid;
xlabel('Pitch angle (x_4)');
ylabel('Reference on pitch angle (r_2)');
Simulate Using Simulink®
To run this example, Simulink® is required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
6-26
Explicit MPC Control of an Aircraft with Unstable Poles
Simulate closed-loop control of the linear plant model in Simulink, using the Explicit
MPC Controller block. Controller "mpcobjExplicitSimplified" is specified in the block
dialog.
mdl = 'empc_aircraft';
open_system(mdl)
sim(mdl)
6-27
6
Explicit MPC Design
6-28
Explicit MPC Control of an Aircraft with Unstable Poles
The closed-loop response is identical to the traditional MPC controller designed in the
"mpcaircraft" example.
bdclose(mdl)
Related Examples
•
“Explicit MPC Control of a Single-Input-Single-Output Plant”
•
“Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output”
More About
•
“Explicit MPC” on page 6-2
6-29
6
Explicit MPC Design
Explicit MPC Control of DC Servomotor with Constraint on
Unmeasured Output
This example shows how to use Explicit MPC to control DC servomechanism under
voltage and shaft torque constraints.
Reference
[1] A. Bemporad and E. Mosca, ''Fulfilling hard constraints in uncertain linear systems
by reference managing,'' Automatica, vol. 34, no. 4, pp. 451-461, 1998.
See also MPCMOTOR.
Define DC-Servo Motor Model
The linear open-loop dynamic model is defined in "plant". Variable "tau" is the maximum
admissible torque to be used as an output constraint.
[plant, tau] = mpcmotormodel;
Design MPC Controller
Specify input and output signal types for the MPC controller. The second output, torque,
is unmeasurable.
plant = setmpcsignals(plant,'MV',1,'MO',1,'UO',2);
MV Constraints
The manipulated variable is constrained between +/- 220 volts. Since the plant inputs
and outputs are of different orders of magnitude, you also use scale factors to faciliate
MPC tuning. Typical choices of scale factor are the upper/lower limit or the operating
range.
MV = struct('Min',-220,'Max',220,'ScaleFactor',440);
OV Constraints
Torque constraints are only imposed during the first three prediction steps to limit the
complexity of the explicit MPC design.
6-30
Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output
OV = struct('Min',{Inf, [-tau;-tau;-tau;-Inf]},'Max',{Inf, [tau;tau;tau;Inf]},'ScaleFac
Weights
The control task is to get zero tracking offset for the angular position. Since you only
have one manipulated variable, the shaft torque is allowed to float within its constraint
by setting its weight to zero.
Weights = struct('MV',0,'MVRate',0.1,'OV',[0.1 0]);
Construct MPC controller
Create an MPC controller with plant model, sample time and horizons.
Ts = 0.1;
% Sampling time
p = 10;
% Prediction horizon
m = 2;
% Control horizon
mpcobj = mpc(plant,Ts,p,m,Weights,MV,OV);
Generate Explicit MPC Controller
Explicit MPC executes the equivalent explicit piecewise affine version of the MPC control
law defined by the traditional MPC. To generate an Explicit MPC from a traditional
MPC, you must specify the range for each controller state, reference signal, manipulated
variable and measured disturbance so that the multi-parametric quadratic programming
problem is solved in the parameter sets defined by these ranges.
Obtain a range structure for initialization
Use generateExplicitRange command to obtain a range structure where you can
specify the range for each parameter afterwards.
range = generateExplicitRange(mpcobj);
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
Specify ranges for controller states
MPC controller states include states from plant model, disturbance model and noise
model in that order. Setting the range of a state variable is sometimes difficult when the
6-31
6
Explicit MPC Design
state does not correspond to a physical parameter. In that case, multiple runs of openloop plant simulation with typical reference and disturbance signals are recommended in
order to collect data that reflect the ranges of states.
range.State.Min(:) = -1000;
range.State.Max(:) = 1000;
Specify ranges for reference signals
Usually you know the practical range of the reference signals being used at the nominal
operating point in the plant. The ranges used to generate Explicit MPC must be at least
as large as the practical range. Note that the range for torque reference is fixed at 0
because it has zero weight.
range.Reference.Min = [-5;0];
range.Reference.Max = [5;0];
Specify ranges for manipulated variables
If manipulated variables are constrained, the ranges used to generate Explicit MPC must
be at least as large as these limits.
range.ManipulatedVariable.Min = MV.Min - 1;
range.ManipulatedVariable.Max = MV.Max + 1;
Construct the Explicit MPC controller
Use generateExplicitMPC command to obtain the Explicit MPC controller with the
parameter ranges previously specified.
mpcobjExplicit = generateExplicitMPC(mpcobj, range);
display(mpcobjExplicit);
Regions found / unexplored:
75/
0
Explicit MPC Controller
--------------------------------------------Controller sample time:
0.1 (seconds)
Polyhedral regions:
75
Number of parameters:
6
6-32
Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output
Is solution simplified:
No
State Estimation:
Default Kalman gain
--------------------------------------------Type 'mpcobjExplicit.MPC' for the original implicit MPC design.
Type 'mpcobjExplicit.Range' for the valid range of parameters.
Type 'mpcobjExplicit.OptimizationOptions' for the options used in multi-parametirc QP c
Type 'mpcobjExplicit.PiecewiseAffineSolution' for regions and gain in each solution.
Plot Piecewise Affine Partition
You can review any 2-D section of the piecewise affine partition defined by the Explicit
MPC control law.
Obtain a plot parameter structure for initialization
Use generatePlotParameters command to obtain a parameter structure where you
can specify which 2-D section to plot afterwards.
params = generatePlotParameters(mpcobjExplicit);
Specify parameters for a 2-D plot
In this example, you plot the 1th state variable vs. the 2nd state variable. All the other
parameters must be fixed at a value within its range.
Fix other state variables
params.State.Index = [3 4];
params.State.Value = [0 0];
Fix reference signals
params.Reference.Index = [1 2];
params.Reference.Value = [pi 0];
Fix manipulated variables
params.ManipulatedVariable.Index = 1;
params.ManipulatedVariable.Value = 0;
Plot the 2-D section
Use plotSection command to plot the 2-D section defined previously.
6-33
6
Explicit MPC Design
plotSection(mpcobjExplicit, params);
axis([-.3 .3 -2 2]);
grid
title('Section of partition [x3(t)=0, x4(t)=0, u(t-1)=0, r(t)=pi]')
xlabel('x1(t)');
ylabel('x2(t)');
Simulate Using SIM Command
Compare closed-loop simulation between traditional MPC (as referred as Implicit MPC)
and Explicit MPC
Tstop = 8;
6-34
% seconds
Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output
Tf = round(Tstop/Ts);
% simulation iterations
r = [pi 0];
% reference signal
[y1,t1,u1] = sim(mpcobj,Tf,r); % simulation with traditional MPC
[y2,t2,u2] = sim(mpcobjExplicit,Tf,r);
% simulation with Explicit MPC
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white
-->The "Model.Noise" property of the "mpc" object is
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white
-->The "Model.Noise" property of the "mpc" object is
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white
-->The "Model.Noise" property of the "mpc" object is
noise.
empty. Assuming white noise on eac
noise.
empty. Assuming white noise on eac
noise.
empty. Assuming white noise on eac
The simulation results are identical.
fprintf('SIM command: Difference between QP-based and Explicit MPC trajectories = %g\n'
SIM command: Difference between QP-based and Explicit MPC trajectories = 6.88112e-12
Simulate Using Simulink®
To run this example, Simulink® is required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
Simulate closed-loop control of the linear plant model in Simulink, using the Explicit
MPC Controller block. Controller "mpcobjExplicit" is specified in the block dialog.
mdl = 'empc_motor';
open_system(mdl)
sim(mdl);
6-35
6
Explicit MPC Design
6-36
Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output
6-37
6
Explicit MPC Design
The closed-loop response is identical to the traditional MPC controller designed in the
"mpcmotor" example.
Control Using Sub-optimal Explicit MPC
To reduce the memory footprint, you can use simplify command to reduce the number
of piecewise affine solution regions. For example, you can remove regions whose
Chebychev radius is smaller than .08. However, the price you pay is that the controler
performance now becomes sub-optimal.
Use simplify command to generate Explicit MPC with sub-optimal solutions.
mpcobjExplicitSimplified = simplify(mpcobjExplicit, 'radius', 0.08);
disp(mpcobjExplicitSimplified);
Regions to analyze:
75/
75 --> 37 regions deleted.
explicitMPC with properties:
MPC:
Range:
OptimizationOptions:
PiecewiseAffineSolution:
6-38
[1x1 mpc]
[1x1 struct]
[1x1 struct]
[1x38 struct]
Explicit MPC Control of DC Servomotor with Constraint on Unmeasured Output
IsSimplified: 1
The number of piecewise affine regions has been reduced.
Compare closed-loop simulation between sub-optimal Explicit MPC and Explicit MPC.
[y3,t3,u3] = sim(mpcobjExplicitSimplified, Tf, r);
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white
-->The "Model.Noise" property of the "mpc" object is
-->Converting model to discrete time.
Assuming unmeasured input disturbance #1 is white
-->The "Model.Noise" property of the "mpc" object is
noise.
empty. Assuming white noise on eac
noise.
empty. Assuming white noise on eac
The simulation results are not the same.
fprintf('SIM command: Difference between exact and suboptimal MPC trajectories = %g\n',
SIM command: Difference between exact and suboptimal MPC trajectories = 439.399
Plot results.
figure;
subplot(3,1,1)
plot(t1,y1(:,1),t3,y3(:,1),'o');
grid
title('Angle (rad)')
legend('Explicit','sub-optimal Explicit')
subplot(3,1,2)
plot(t1,y1(:,2),t3,y3(:,2),'o');
grid
title('Torque (Nm)')
legend('Explicit','sub-optimal Explicit')
subplot(3,1,3)
plot(t1,u1,t3,u3,'o');
grid
title('Voltage (V)')
legend('Explicit','sub-optimal Explicit')
6-39
6
Explicit MPC Design
The simulation result with the sub-optimal Explicit MPC is slightly worse.
bdclose(mdl)
Related Examples
•
“Explicit MPC Control of a Single-Input-Single-Output Plant”
•
“Explicit MPC Control of an Aircraft with Unstable Poles”
More About
•
6-40
“Explicit MPC” on page 6-2
7
Gain Scheduling MPC Design
• “Gain-Scheduled MPC” on page 7-2
• “Design Workflow for Gain Scheduling” on page 7-3
• “Gain Scheduled MPC Control of Nonlinear Chemical Reactor” on page 7-5
• “Gain Scheduled MPC Control of Mass-Spring System” on page 7-26
7
Gain Scheduling MPC Design
Gain-Scheduled MPC
The Multiple MPC Controllers block for Simulink allows you to switch between a defined
set of MPC Controllers. You might need this feature if the plant operating characteristics
change in a predictable way, and the change is such that a single prediction model cannot
provide adequate accuracy. This approach is comparable to the use of gain scheduling in
conventional feedback control.
The individual MPC controllers coordinate to make switching from one to another
bumpless, avoiding a sudden change in the manipulated variables when the switch
occurs.
You can perform command-line simulations using the mpcmoveMultiple command.
More About
7-2
•
“Design Workflow for Gain Scheduling”
•
“Relationship of Multiple MPC Controllers to MPC Controller Block”
Design Workflow for Gain Scheduling
Design Workflow for Gain Scheduling
In this section...
“General Design Steps” on page 7-3
“Tips” on page 7-3
General Design Steps
• Define and tune a nominal MPC controller for the most likely (or average) operating
conditions. (See “MPC Design”.)
• Use simulations to determine an operating condition at which the nominal controller
loses robustness. See “Simulation”.
• Identify a measurement (or combination of measurements) signaling when the
nominal controller should be replaced.
• Determine a plant prediction model to be used at the new condition. Its input and
output variables must be the same as in the nominal case.
• Define a new MPC controller based on the new prediction model. Use the nominal
controller settings as a starting point, and test and retune controller settings if
necessary.
• If two controllers are inadequate to provide robustness over the full operational range,
consider adding another. If it appears that you need more than three controllers to
provide robustness over the full range, consider using adaptive MPC instead. See
“Adaptive MPC Design”.
• In your Simulink model, configure the Multiple MPC Controllers block. Specify the set
of MPC controllers to be used, and specify the switching criterion.
• Test in closed-loop simulation over the full operating range to verify robustness and
bumpless switching.
Tips
• Recommended MPC start-up practice is a warm-up period in which the plant operates
under manual control while the controller initializes its state estimate. This typically
requires 10-20 control intervals. A warm-up is especially important for the Multiple
MPC Controllers block. Otherwise, switching between MPC controllers might upset
the manipulated variables.
7-3
7
Gain Scheduling MPC Design
• If you select the Multiple MPC Controllers block’s custom state estimation option, all
MPC controllers in the set must have the same state dimension. This places implicit
restrictions on plant and disturbance models.
See Also
mpcmoveMultiple | Multiple MPC Controllers
Related Examples
•
“Schedule Controllers at Multiple Operating Points”
•
“Coordinate Multiple Controllers at Different Operating Points”
•
“Gain Scheduled MPC Control of Nonlinear Chemical Reactor”
•
“Gain Scheduled MPC Control of Mass-Spring System”
More About
•
7-4
“Relationship of Multiple MPC Controllers to MPC Controller Block”
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
This example shows how to use multiple MPC controllers to control a nonlinear
continuous stirred tank reactor (CSTR) as it transitions from low conversion rate to high
conversion rate.
Multiple MPC Controllers are designed at different operating conditions and then
implemented with the Multiple MPC Controller block in Simulink. At run time, a
scheduling signal is used to switch controller from one to another.
About the Continuous Stirred Tank Reactor
A Continuously Stirred Tank Reactor (CSTR) is a common chemical system in the
process industry. A schematic of the CSTR system is:
This is a jacketed non-adiabatic tank reactor described extensively in Seborg's book,
"Process Dynamics and Control", published by Wiley, 2004. The vessel is assumed to be
7-5
7
Gain Scheduling MPC Design
perfectly mixed, and a single first-order exothermic and irreversible reaction, A --> B,
takes place. The inlet stream of reagent A is fed to the tank at a constant volumetric rate.
The product stream exits continuously at the same volumetric rate and liquid density is
constant. Thus the volume of reacting liquid is constant.
The inputs of the CSTR model are:
and the outputs (y(t)), which are also the states of the model (x(t)), are:
The control objective is to maintain the concentration of reagent A,
at its desired
setpoint, which changes over time when reactor transitions from low conversion rate
to high conversion rate. The coolant temperature
is the manipulated variable used
by the MPC controller to track the reference. The inlet feed stream concentration
and temperature are assumed to be constant. The Simulink model mpc_cstr_plant
implements the nonlinear CSTR plant.
About Gain Scheduled Model Predictive Control
It is well known that the CSTR dynamics are strongly nonlinear with respect to reactor
temperature variations and can be open-loop unstable during the transition from
one operating condition to another. A single MPC controller designed at a particular
operating condition cannot give satisfactory control performance over a wide operating
range.
To control the nonlinear CSTR plant with linear MPC control technique, you have a few
options:
• If a linear plant model cannot be obtained at run time, first you need to obtain several
linear plant models offline at different operating conditions that cover the typical
operating range. Next you can choose one of the two approaches to implement MPC
control strategy:
7-6
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
(1) Design several MPC controllers offline, one for each plant model. At run time, use
Multiple MPC Controller block that switches MPC controllers from one to another based
on a desired scheduling strategy, as discussed in this example. Use this approach when
the plant models have different orders or time delays.
(2) Design one MPC controller offline at a nominal operating point. At run time, use
Adaptive MPC Controller block (updating predictive model at each control interval)
together with Linear Parameter Varying (LPV) System block (supplying linear plant
model with a scheduling strategy). See "Adaptive MPC Control of Nonlinear Chemical
Reactor Using Linear Parameter Varying System" for more details. Use this approach
when all the plant models have the same order and time delay.
• If a linear plant model can be obtained at run time, you should use Adaptive MPC
Controller block to achieve nonlinear control. There are two typical ways to obtain a
linear plant model online:
(1) Use successive linearization. See "Adaptive MPC Control of Nonlinear Chemical
Reactor Using Successive Linearization" for more details. Use this approach when a
nonlinear plant model is available and can be linearized at run time.
(2) Use online estimation to identify a linear model when loop is closed. See "Adaptive
MPC Control of Nonlinear Chemical Reactor Using Online Model Estimation" for more
details. Use this approach when linear plant model cannot be obtained from either an
LPV system or successive linearization.
Obtain Linear Plant Model at Initial Operating Condition
To run this example, Simulink® and Simulink Control Design® are required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
if ~mpcchecktoolboxinstalled('slcontrol')
disp('Simulink Control Design(R) is required to run this example.')
return
end
First, a linear plant model is obtained at the initial operating condition, CAi is 10
kgmol/m^3, Ti and Tc are 298.15 K. Functions from Simulink Control Design such as
"operspec", "findop", "linearize", are used to generate the linear state space system from
the Simulink model.
7-7
7
Gain Scheduling MPC Design
Create operating point specification.
plant_mdl = 'mpc_cstr_plant';
op = operspec(plant_mdl);
Feed concentration is known at the initial condition.
op.Inputs(1).u = 10;
op.Inputs(1).Known = true;
Feed temperature is known at the initial condition.
op.Inputs(2).u = 298.15;
op.Inputs(2).Known = true;
Coolant temperature is known at the initial condition.
op.Inputs(3).u = 298.15;
op.Inputs(3).Known = true;
Compute initial condition.
[op_point, op_report] = findop(plant_mdl,op);
% Obtain nominal values of x, y and u.
x0 = [op_report.States(1).x;op_report.States(2).x];
y0 = [op_report.Outputs(1).y;op_report.Outputs(2).y];
u0 = [op_report.Inputs(1).u;op_report.Inputs(2).u;op_report.Inputs(3).u];
Operating Point Search Report:
--------------------------------Operating Report for the Model mpc_cstr_plant.
(Time-Varying Components Evaluated at time t=0)
Operating point specifications were successfully met.
States:
---------(1.) mpc_cstr_plant/CSTR/Integrator
x:
311
dx:
8.12e-11 (0)
(2.) mpc_cstr_plant/CSTR/Integrator1
x:
8.57
dx:
-6.87e-12 (0)
Inputs:
7-8
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
---------(1.) mpc_cstr_plant/CAi
u:
10
(2.) mpc_cstr_plant/Ti
u:
298
(3.) mpc_cstr_plant/Tc
u:
298
Outputs:
---------(1.) mpc_cstr_plant/T
y:
311
(2.) mpc_cstr_plant/CA
y:
8.57
[-Inf Inf]
[-Inf Inf]
Obtain linear model at the initial condition.
plant = linearize(plant_mdl, op_point);
Verify that the linear model is open-loop stable at this condition.
eig(plant)
ans =
-0.5223
-0.8952
Design MPC Controller for Initial Operating Condition
You design an MPC at the initial operating condition.
Ts = 0.5;
Specify signal types used in MPC. Assume both reactor temperature and concentration
are measurable.
plant.InputGroup.UnmeasuredDisturbances = [1 2];
plant.InputGroup.ManipulatedVariables = 3;
plant.OutputGroup.Measured = [1 2];
plant.InputName = {'CAi','Ti','Tc'};
7-9
7
Gain Scheduling MPC Design
plant.OutputName = {'T','CA'};
Create MPC controller with default prediction and control horizons
mpcobj = mpc(plant, Ts);
-->The
-->The
-->The
-->The
-->The
for
"PredictionHorizon" property of "mpc" object is empty. Trying PredictionHorizon
"ControlHorizon" property of the "mpc" object is empty. Assuming 2.
"Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
"Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
"Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
output(s) y1 and zero weight for output(s) y2
Set nominal values in the controller. Note that nominal values for unmeasured
disturbance must be zero.
mpcobj.Model.Nominal = struct('X', x0, 'U', [0;0;u0(3)], 'Y', y0, 'DX', [0 0]);
Set scale factors because plant input and output signals have different orders of
magnitude
Uscale = [10;30;50];
Yscale = [50;10];
mpcobj.DV(1).ScaleFactor = Uscale(1);
mpcobj.DV(2).ScaleFactor = Uscale(2);
mpcobj.MV.ScaleFactor = Uscale(3);
mpcobj.OV(1).ScaleFactor = Yscale(1);
mpcobj.OV(2).ScaleFactor = Yscale(2);
The goal will be to track a specified transition in the reactor concentration. The reactor
temperature will be measured and used in state estimation but the controller will not
attempt to regulate it directly. It will vary as needed to regulate the concentration. Thus,
set its MPC weight to zero.
mpcobj.Weights.OV = [0 1];
Plant inputs 1 and 2 are unmeasured disturbances. By default, the controller assumes
integrated white noise with unit magnitude at these inputs when configuring the state
estimator. Try increasing the state estimator signal-to-noise by a factor of 10 to improve
disturbance rejection performance.
D = ss(getindist(mpcobj));
D.b = eye(2)*10;
7-10
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
setindist(mpcobj, 'model', D);
-->Converting model to discrete time.
-->The "Model.Disturbance" property of "mpc" object is empty:
Assuming unmeasured input disturbance #1 is integrated white noise.
Assuming unmeasured input disturbance #2 is integrated white noise.
Assuming unmeasured input disturbance #2 is white noise.
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
All other MPC parameters are at their default values.
Test the Controller With a Step Disturbance in Feed Concentration
"mpc_cstr_single" contains a Simulink® model with CSTR and MPC Controller blocks in
a feedback configuration.
mpc_mdl = 'mpc_cstr_single';
open_system(mpc_mdl)
7-11
7
Gain Scheduling MPC Design
Note that the MPC Controller block is configured to look ahead (preview) the setpoint
changes in the future, i.e., anticipating the setpoint transition. This generally improves
setpoint tracking.
Define a constant setpoint for the output.
CSTR_Setpoints.time = [0; 60];
CSTR_Setpoints.signals.values = [y0 y0]';
Test the response to a 5% increase in feed concentration.
set_param([mpc_mdl '/Feed Concentration'], 'Value', '10.5');
Set plot scales and simulate the response.
open_system([mpc_mdl '/Measurements'])
open_system([mpc_mdl '/Coolant Temperature'])
set_param([mpc_mdl '/Measurements'], 'Ymin', '305~8', 'Ymax', '320~9')
set_param([mpc_mdl '/Coolant Temperature'], 'Ymin', '295', 'Ymax', '305')
sim(mpc_mdl, 10);
-->Converting model to discrete time.
Assuming unmeasured input disturbance #2 is white noise.
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
7-12
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
7-13
7
Gain Scheduling MPC Design
The closed-loop response is satisfactory.
Simulate Designed MPC Controller Using Full Transition
First, define the desired setpoint transition. After a 10-minute warm-up period, ramp the
concentration setpoint downward at a rate of 0.25 per minute until it reaches 2.0 kmol/
m^3.
CSTR_Setpoints.time = [0 10 11:39]';
CSTR_Setpoints.signals.values = [y0(1)*ones(31,1),[y0(2);y0(2);(y0(2):-0.25:2)';2;2]];
Remove the 5% increase in feed concentration used previously.
set_param([mpc_mdl '/Feed Concentration'], 'Value', '10')
Set plot scales and simulate the response.
set_param([mpc_mdl '/Measurements'], 'Ymin', '300~0', 'Ymax', '400~10')
set_param([mpc_mdl '/Coolant Temperature'], 'Ymin', '240', 'Ymax', '360')
7-14
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
Simulate model.
sim(mpc_mdl, 60)
7-15
7
Gain Scheduling MPC Design
The closed-loop response is unacceptable. Performance along the full transition can be
improved if other MPC controllers are designed at different operating conditions along
the transition path. In the next two section, two additional MPC controllers are design at
intermediate and final transition stages respectively.
Design MPC Controller for Intermediate Operating Condition
Create operating point specification.
op = operspec(plant_mdl);
Feed concentration is known.
op.Inputs(1).u = 10;
op.Inputs(1).Known = true;
Feed temperature is known.
op.Inputs(2).u = 298.15;
7-16
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
op.Inputs(2).Known = true;
Reactor concentration is known
op.Outputs(2).y = 5.5;
op.Outputs(2).Known = true;
Find steady state operating condition.
[op_point, op_report] = findop(plant_mdl,op);
% Obtain nominal values of x, y and u.
x0 = [op_report.States(1).x;op_report.States(2).x];
y0 = [op_report.Outputs(1).y;op_report.Outputs(2).y];
u0 = [op_report.Inputs(1).u;op_report.Inputs(2).u;op_report.Inputs(3).u];
Operating Point Search Report:
--------------------------------Operating Report for the Model mpc_cstr_plant.
(Time-Varying Components Evaluated at time t=0)
Operating point specifications were successfully met.
States:
---------(1.) mpc_cstr_plant/CSTR/Integrator
x:
339
dx:
3.42e-08 (0)
(2.) mpc_cstr_plant/CSTR/Integrator1
x:
5.5
dx:
-2.87e-09 (0)
Inputs:
---------(1.) mpc_cstr_plant/CAi
u:
10
(2.) mpc_cstr_plant/Ti
u:
298
(3.) mpc_cstr_plant/Tc
u:
298
Outputs:
---------(1.) mpc_cstr_plant/T
y:
339
(2.) mpc_cstr_plant/CA
y:
5.5
[-Inf Inf]
[-Inf Inf]
(5.5)
7-17
7
Gain Scheduling MPC Design
Obtain linear model at the initial condition.
plant_intermediate = linearize(plant_mdl, op_point);
Verify that the linear model is open-loop unstable at this condition.
eig(plant_intermediate)
ans =
0.4941
-0.8357
Specify signal types used in MPC. Assume both reactor temperature and concentration
are measurable.
plant_intermediate.InputGroup.UnmeasuredDisturbances = [1 2];
plant_intermediate.InputGroup.ManipulatedVariables = 3;
plant_intermediate.OutputGroup.Measured = [1 2];
plant_intermediate.InputName = {'CAi','Ti','Tc'};
plant_intermediate.OutputName = {'T','CA'};
Create MPC controller with default prediction and control horizons
mpcobj_intermediate = mpc(plant_intermediate, Ts);
-->The
-->The
-->The
-->The
-->The
for
"PredictionHorizon" property of "mpc" object is empty. Trying PredictionHorizon
"ControlHorizon" property of the "mpc" object is empty. Assuming 2.
"Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
"Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
"Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
output(s) y1 and zero weight for output(s) y2
Set nominal values, scale factors and weights in the controller
mpcobj_intermediate.Model.Nominal = struct('X', x0, 'U', [0;0;u0(3)], 'Y', y0, 'DX', [0
Uscale = [10;30;50];
Yscale = [50;10];
mpcobj_intermediate.DV(1).ScaleFactor = Uscale(1);
mpcobj_intermediate.DV(2).ScaleFactor = Uscale(2);
7-18
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
mpcobj_intermediate.MV.ScaleFactor = Uscale(3);
mpcobj_intermediate.OV(1).ScaleFactor = Yscale(1);
mpcobj_intermediate.OV(2).ScaleFactor = Yscale(2);
mpcobj_intermediate.Weights.OV = [0 1];
D = ss(getindist(mpcobj_intermediate));
D.b = eye(2)*10;
setindist(mpcobj_intermediate, 'model', D);
-->Converting model to discrete time.
-->The "Model.Disturbance" property of "mpc" object is empty:
Assuming unmeasured input disturbance #1 is integrated white noise.
Assuming unmeasured input disturbance #2 is integrated white noise.
Assuming unmeasured input disturbance #2 is white noise.
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
Design MPC Controller for Final Operating Condition
Create operating point specification.
op = operspec(plant_mdl);
Feed concentration is known.
op.Inputs(1).u = 10;
op.Inputs(1).Known = true;
Feed temperature is known.
op.Inputs(2).u = 298.15;
op.Inputs(2).Known = true;
Reactor concentration is known
op.Outputs(2).y = 2;
op.Outputs(2).Known = true;
Find steady state operating condition.
[op_point, op_report] = findop(plant_mdl,op);
% Obtain nominal values of x, y and u.
x0 = [op_report.States(1).x;op_report.States(2).x];
y0 = [op_report.Outputs(1).y;op_report.Outputs(2).y];
u0 = [op_report.Inputs(1).u;op_report.Inputs(2).u;op_report.Inputs(3).u];
7-19
7
Gain Scheduling MPC Design
Operating Point Search Report:
--------------------------------Operating Report for the Model mpc_cstr_plant.
(Time-Varying Components Evaluated at time t=0)
Operating point specifications were successfully met.
States:
---------(1.) mpc_cstr_plant/CSTR/Integrator
x:
373
dx:
5.57e-11 (0)
(2.) mpc_cstr_plant/CSTR/Integrator1
x:
2
dx:
-4.6e-12 (0)
Inputs:
---------(1.) mpc_cstr_plant/CAi
u:
10
(2.) mpc_cstr_plant/Ti
u:
298
(3.) mpc_cstr_plant/Tc
u:
305
Outputs:
---------(1.) mpc_cstr_plant/T
y:
373
(2.) mpc_cstr_plant/CA
y:
2
[-Inf Inf]
[-Inf Inf]
(2)
Obtain linear model at the initial condition.
plant_final = linearize(plant_mdl, op_point);
Verify that the linear model is again open-loop stable at this condition.
eig(plant_final)
ans =
-1.1077 + 1.0901i
-1.1077 - 1.0901i
7-20
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
Specify signal types used in MPC. Assume both reactor temperature and concentration
are measurable.
plant_final.InputGroup.UnmeasuredDisturbances = [1 2];
plant_final.InputGroup.ManipulatedVariables = 3;
plant_final.OutputGroup.Measured = [1 2];
plant_final.InputName = {'CAi','Ti','Tc'};
plant_final.OutputName = {'T','CA'};
Create MPC controller with default prediction and control horizons
mpcobj_final = mpc(plant_final, Ts);
-->The
-->The
-->The
-->The
-->The
for
"PredictionHorizon" property of "mpc" object is empty. Trying PredictionHorizon
"ControlHorizon" property of the "mpc" object is empty. Assuming 2.
"Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
"Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
"Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
output(s) y1 and zero weight for output(s) y2
Set nominal values, scale factors and weights in the controller
mpcobj_final.Model.Nominal = struct('X', x0, 'U', [0;0;u0(3)], 'Y', y0, 'DX', [0 0]);
Uscale = [10;30;50];
Yscale = [50;10];
mpcobj_final.DV(1).ScaleFactor = Uscale(1);
mpcobj_final.DV(2).ScaleFactor = Uscale(2);
mpcobj_final.MV.ScaleFactor = Uscale(3);
mpcobj_final.OV(1).ScaleFactor = Yscale(1);
mpcobj_final.OV(2).ScaleFactor = Yscale(2);
mpcobj_final.Weights.OV = [0 1];
D = ss(getindist(mpcobj_final));
D.b = eye(2)*10;
setindist(mpcobj_final, 'model', D);
-->Converting model to discrete time.
-->The "Model.Disturbance" property of "mpc" object is empty:
Assuming unmeasured input disturbance #1 is integrated white noise.
Assuming unmeasured input disturbance #2 is integrated white noise.
Assuming unmeasured input disturbance #2 is white noise.
Assuming unmeasured input disturbance #1 is white noise.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
7-21
7
Gain Scheduling MPC Design
Control the CSTR Plant With the Multiple MPC Controllers Block
The following model uses the Multiple MPC Controllers block to implement three MPC
controllers across the operating range.
mmpc_mdl = 'mpc_cstr_multiple';
open_system(mmpc_mdl);
Note that it has been configured to use the three controllers in a sequence: mpcobj,
mpcobj_intermediate and mpcobj_final.
open_system([mmpc_mdl '/Multiple MPC Controllers']);
Note also that the two switches specify when to switch from one controller to another.
The rules are: 1. If CSTR concentration >= 8, use "mpcobj" 2. If 3 <= CSTR concentration
< 8, use "mpcobj_intermediate" 3. If CSTR concentration < 3, use "mpcobj_final"
Simulate with the Multiple MPC Controllers block
open_system([mmpc_mdl '/Measurements']);
open_system([mmpc_mdl '/MV']);
7-22
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
sim(mmpc_mdl)
-->Converting model to discrete time.
Assuming unmeasured input disturbance #2 is white
Assuming unmeasured input disturbance #1 is white
-->The "Model.Noise" property of the "mpc" object is
-->Converting model to discrete time.
Assuming unmeasured input disturbance #2 is white
Assuming unmeasured input disturbance #1 is white
-->The "Model.Noise" property of the "mpc" object is
noise.
noise.
empty. Assuming white noise on eac
noise.
noise.
empty. Assuming white noise on eac
7-23
7
Gain Scheduling MPC Design
The transition is now well controlled. The major improvement is in the transition
through the open-loop unstable region. The plot of the switching signal shows when
controller transitions occur. The MV character changes at these times because of the
change in dynamic characteristics introduced by the new prediction model.
bdclose(plant_mdl)
bdclose(mpc_mdl)
7-24
Gain Scheduled MPC Control of Nonlinear Chemical Reactor
bdclose(mmpc_mdl)
Related Examples
•
“Schedule Controllers at Multiple Operating Points”
•
“Coordinate Multiple Controllers at Different Operating Points”
•
“Gain Scheduled MPC Control of Mass-Spring System”
More About
•
“Design Workflow for Gain Scheduling”
7-25
7
Gain Scheduling MPC Design
Gain Scheduled MPC Control of Mass-Spring System
This example shows how to use an Multiple MPC Controllers block to implement gain
scheduled MPC control of a nonlinear plant.
System Description
The system is composed by two masses M1 and M2 connected to two springs k1 and k2
respectively. The collision is assumed completely inelastic. Mass M1 is pulled by a force
F, which is the manipulated variable. The objective is to make mass M1's position y1
track a given reference r.
The dynamics are twofold: when the masses are detached, M1 moves freely. Otherwise,
M1+M2 move together. We assume that only M1 position and a contact sensor are
available for feedback. The latter is used to trigger switching the MPC controllers. Note
that position and velocity of mass M2 are not controllable.
/-----\
k1
||
F <--- | M1 |----/\/\/\-------------[|| wall
||
| |---/
||
||
k2
\-/
/----\
||
wall||]--/\/\/\-------------------| M2 |
||
||
\----/
||
||
||
----yeq2------------------ y1 ------ y2 ----------------yeq1----> y axis
The model is a simplified version of the model proposed in the following reference:
A. Bemporad, S. Di Cairano, I. V. Kolmanovsky, and D. Hrovat, "Hybrid modeling and
control of a multibody magnetic actuator for automotive applications," in Proc. 46th
IEEE® Conf. on Decision and Control, New Orleans, LA, 2007.
Model Parameters
M1=1;
M2=5;
k1=1;
k2=0.1;
b1=0.3;
b2=0.8;
yeq1=10;
yeq2=-10;
7-26
%
%
%
%
%
%
%
%
mass
mass
spring constant
spring constant
friction coefficient
friction coefficient
wall mount position
wall mount position
Gain Scheduled MPC Control of Mass-Spring System
State Space Models
states: position and velocity of mass M1; manipulated variable: pull force F measured
disturbance: a constant value of 1 which provides calibrates spring force to the right
value measured output: position of mass M1
State-space model of M1 when masses are not in contact.
A1=[0 1;-k1/M1 -b1/M1];
B1=[0 0;-1/M1 k1*yeq1/M1];
C1=[1 0];
D1=[0 0];
sys1=ss(A1,B1,C1,D1);
sys1=setmpcsignals(sys1,'MD',2);
-->Assuming unspecified input signals are manipulated variables.
State-space model when the two masses are in contact.
A2=[0 1;-(k1+k2)/(M1+M2) -(b1+b2)/(M1+M2)];
B2=[0 0;-1/(M1+M2) (k1*yeq1+k2*yeq2)/(M1+M2)];
C2=[1 0];
D2=[0 0];
sys2=ss(A2,B2,C2,D2);
sys2=setmpcsignals(sys2,'MD',2);
-->Assuming unspecified input signals are manipulated variables.
Design MPC Controllers
Common parameters
Ts=0.2;
p=20;
m=1;
% sampling time
% prediction horizon
% control horizon
Define MPC object for mass M1 detached from M2.
MPC1=mpc(sys1,Ts,p,m);
MPC1.Weights.OV=1;
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
7-27
7
Gain Scheduling MPC Design
Define constraints on the manipulated variable.
MPC1.MV=struct('Min',0,'Max',Inf,'RateMin',-1e3,'RateMax',1e3);
Define MPC object for mass M1 and M2 stuck together.
MPC2=mpc(sys2,Ts,p,m);
MPC2.Weights.OV=1;
-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming defau
-->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming d
-->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.
Define constraints on the manipulated variable.
MPC2.MV=MPC1.MV;
Simulate Gain Scheduled MPC in Simulink®
To run this example, Simulink® is required.
if ~mpcchecktoolboxinstalled('simulink')
disp('Simulink(R) is required to run this example.')
return
end
mdl = 'mpc_switching';
Simulate gain scheduled MPC control with Multiple MPC Controllers block.
y1initial=0;
% Initial positions
y2initial=10;
open_system(mdl);
if exist('animationmpc_switchoff','var') && animationmpc_switchoff
close_system([mdl '/Animation']);
clear animationmpc_switchoff
end
7-28
Gain Scheduled MPC Control of Mass-Spring System
disp('Start simulation by switching control between MPC1 and MPC2 ...');
disp('Control performance is satisfactory.');
open_system([mdl '/signals']);
sim(mdl);
Start simulation by switching control between MPC1 and MPC2 ...
7-29
7
Gain Scheduling MPC Design
Control performance is satisfactory.
-->Converting model to discrete time.
-->Integrated white noise added on measured output channel #1.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
-->Converting model to discrete time.
-->Integrated white noise added on measured output channel #1.
-->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on eac
7-30
Gain Scheduled MPC Control of Mass-Spring System
7-31
7
Gain Scheduling MPC Design
Use of two controllers provides good performance under all conditions.
Repeat Simulation Using MPC1 Only (Assumes Masses Never in Contact)
disp('Now repeat simulation by using only MPC1 ...');
disp('When two masses stick together, control performance deteriorates.');
MPC2save=MPC2;
MPC2=MPC1; %#ok<*NASGU>
sim(mdl);
Now repeat simulation by using only MPC1 ...
When two masses stick together, control performance deteriorates.
7-32
Gain Scheduled MPC Control of Mass-Spring System
7-33
7
Gain Scheduling MPC Design
In this case, performance degrades whenever the two masses join.
Repeat Simulation Using MPC2 Only (Assumes Masses Always in Contact)
disp('Now repeat simulation by using only MPC2 ...');
disp('When two masses are detached, control performance deteriorates.');
MPC1=MPC2save;
MPC2=MPC1;
sim(mdl);
Now repeat simulation by using only MPC2 ...
When two masses are detached, control performance deteriorates.
7-34
Gain Scheduled MPC Control of Mass-Spring System
7-35
7
Gain Scheduling MPC Design
In this case, performance degrades when the masses separate, causing the controller to
apply excessive force.
bdclose(mdl)
close(findobj('Tag','mpc_switching_demo'))
Related Examples
•
“Schedule Controllers at Multiple Operating Points”
•
“Coordinate Multiple Controllers at Different Operating Points”
•
“Gain Scheduled MPC Control of Nonlinear Chemical Reactor”
More About
•
7-36
“Design Workflow for Gain Scheduling”
8
Reference for the Design Tool GUI
This chapter is the reference manual for the Model Predictive Control Toolbox design tool
(graphical user interface).
• “Working with the Design Tool” on page 8-2
• “Weight Sensitivity Analysis” on page 8-57
• “Customize Response Plots” on page 8-69
8
Reference for the Design Tool GUI
Working with the Design Tool
In this section...
“Opening the MPC Design Tool” on page 8-2
“Creating a New MPC Design Task” on page 8-3
“Menu Bar” on page 8-4
“Toolbar” on page 8-6
“Tree View” on page 8-6
“Importing a Plant Model” on page 8-8
“Importing a Controller” on page 8-12
“Exporting a Controller” on page 8-15
“Signal Definition View” on page 8-16
“Plant Models View” on page 8-20
“Controllers View” on page 8-23
“Simulation Scenarios List” on page 8-26
“Controller Specifications View” on page 8-29
“Simulation Scenario View” on page 8-49
Opening the MPC Design Tool
To open the Design Tool in MATLAB, type
mpctool
The design tool is part of the Control and Estimation Tools Manager. When invoked as
shown above, the design tool opens and creates a new project named MPC Design Task.
If you started the tool previously, the above command makes the tool visible but does not
create a new project.
Alternatively, if your Simulink model contains a Model Predictive Controller block, you
can double-click the block to obtain its mask (see example below) and click the Design
button. If the MPC controller field is empty, the design tool will create a default
controller. Otherwise, it will load the named controller object, which must be in your base
workspace. You can then view and modify the controller design.
8-2
Working with the Design Tool
Creating a New MPC Design Task
To create a new MPC Design Task:
1
Select the Workspace node in the Control and Estimation Tools Manager.
2
Click File > New > Task.
3
In the New Project dialog box, click the Select check box for Model Predictive
Control Task: MPC Controller or Model Predictive Control Task: Multiple
MPC Controllers.
Click OK.
8-3
8
Reference for the Design Tool GUI
Menu Bar
The design tool's menu bar appears whenever you've selected a Model Predictive Control
Toolbox project or task in the tree (see “Tree View” on page 8-6). The menu bar's
MPC option distinguishes it from other control and estimation tools. See the example
below. The following sections describe each menu option.
File Menu
• “Load” on page 8-4
• “Save” on page 8-4
• “Close” on page 8-5
Load
Loads a saved design. A dialog box asks you to specify the MAT-file containing the saved
design. If the MAT-file contains multiple projects, you must select the one(s) to be loaded
(see example below).
You can also load a design using the toolbar (see “Toolbar” on page 8-6).
Save
Saves a design so you can use it later. The data are saved in a MAT-file. A dialog allows
you to specify the file name (see below). If you are working on multiple projects, you can
select those to be saved.
8-4
Working with the Design Tool
You can also select the Save option using the toolbar (see “Toolbar” on page 8-6).
Close
Closes the design tool. If you've modified the design, you'll be asked whether or not you
want to save it before closing.
MPC Menu
• “Import” on page 8-5
• “Export” on page 8-5
• “Simulate” on page 8-6
Import
You have the following options:
• Plant model – Import a plant model using the model import dialog box (see
“Importing a Plant Model” on page 8-8).
• Controller – Import a controller using the controller import dialog box (see
“Importing a Controller” on page 8-12).
Export
Export a controller using the export dialog box (see “Exporting a Controller” on page
8-15). This option is disabled until your project includes at least one controller.
8-5
8
Reference for the Design Tool GUI
Simulate
Simulate the current scenario, i.e., the one most recently simulated or selected in the
tree (see “Tree View” on page 8-6). You can select this option from the keyboard by
pressing Ctrl+R, or using the toolbar icon (see “Toolbar” on page 8-6).
The Simulate option is disabled until your project includes at least one valid simulation
scenario.
Toolbar
The toolbar, shown in the following figure, lets you perform the following tasks:
• Load a saved design
• Simulate a current scenario
• Save the current design
• Toggle the output area
For more information on the first three tasks, see the following:
• “Load” on page 8-4
• “Save” on page 8-4
• “Simulate” on page 8-6
The text output area is at the bottom of the tool. It displays progress messages and
diagnostics. In the above view, the toggle button is pushed in, so the text display area
appears. If you are working on a small screen, you might use the toggle button to hide
the text area, allowing more room to display the controller design.
Tree View
The tree view appears in a frame on the design tool's left-hand side (see example below).
When you select one of the tree's nodes (by clicking its name or icon) the larger frame
to its right shows a dialog pane that allows you to view and edit the specifications
associated with that item.
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Node Types
Plant models list
Controllers list
MPC project
nodes
Controller specifications
Simulation scenarios list
Scenario
specifications
The above example shows two Model Predictive Control Toolbox design project nodes,
Distillation Control and CSTR Control, and their subnodes. For more details on each
node type, see the following:
• MPC design project/task – See “Signal Definition View” on page 8-16.
• Plant models list – See “Plant Models View” on page 8-20.
• Controllers list – See “Controllers View” on page 8-23.
• Controller specifications – See “Simulation Scenarios List” on page 8-26.
• Scenarios list – See “Simulation Scenario View” on page 8-49.
• Scenario specifications – See “Controller Specifications View” on page 8-29.
Renaming a Node
You can rename following node types:
• MPC design project/task
• Controller specifications
• Scenario specifications
To rename a node, do one of the following:
• Click the name, wait for an edit box to appear, type the desired name, and press the
Enter key to finalize your choice.
• Right-click the name, select the Rename menu option, and enter the desired name in
the dialog box.
• To rename a controller, select Controllers and edit the controller name in the table.
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• To rename a scenario, select Scenarios and edit the scenario name in the table.
Importing a Plant Model
To import a plant model, do one of the following:
• Select the MPC/Import/Plant Model menu option.
• Select the MPC project/task node in the tree (see “Tree View” on page 8-6), and
then click the Import Plant button.
• Right-click the MPC project/task node and select the Import Plant Model menu
option.
• If you've already imported a model, select the Plant models node, and then click the
Import button, or right-click the Plant models node and select the Import Model
menu option.
The Plant Model Importer dialog box opens (see the dialog box in “Import from” on
page 8-13 for an example). Within the dialog box you can import an LTI model from
the workspace or, when you have Simulink Control Design software, you can import a
linearized plant model from the Simulink model.
Note Once you have imported a model, any additional models you import to the same
MPC project or task must have the identical structure, i.e., the same number of input
and output signals, each appearing in the same sequence and having the same signal
type designations. If you attempt to import a model that violates one of these conditions,
the design tool issues a warning message. If you persist, all previously loaded models will
be deleted and the controller design will be re-initialized using the latest model.
The following sections describe model import options:
• “Import from” on page 8-13
• “Import to” on page 8-14
• “Buttons” on page 8-14
• “Importing a Linearized Plant Model” on page 8-10
Import from
Use these options to set the location from which the model will be imported.
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MATLAB workspace
This is the default option and is the case shown in the above example. The Items in
your workspace area in the upper-right corner lists all candidate models in your
MATLAB workspace. Select one by clicking it. The Properties area lists the selected
model's properties (the DC model in the above example).
MAT-file
The upper part of the dialog box changes as shown below.
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The MAT-file name edit box becomes active. Type the desired MAT-file name (if it's not
on the MATLAB path, enter the complete file path). You can also use the Browse button
which opens a file chooser window.
In the above example, file DCmodels.mat contains two models. Their names appear
in the Items in your MAT-file area in the upper-right corner. As with the workspace
option, the selected model's properties appear in the Properties area.
Import to
The combo box at the bottom of the dialog box allows you to specify the MPC project/task
into which the plant model will be imported (see example below). It defaults to the active
project.
Buttons
Import
Select the model you want to import from the Items list in the upper-right corner of the
dialog box. Verify that the Import to option designates the correct project/task. Click the
Import button to import the model.
To verify that the model has been loaded, select Plant models in the tree. (See “Tree
View” on page 8-6, and “Plant Models View” on page 8-20.)
The import dialog box remains visible until you close it so you can import additional
models.
Close
Click Close to close the dialog box. You can also click the Close icon on the title bar.
Importing a Linearized Plant Model
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1
Open the design tool from within a Simulink model as discussed in “Opening the
MPC Design Tool” on page 8-2.
2
Open the Plant Model Importer dialog box (see “Importing a Plant Model” on page
8-8).
Working with the Design Tool
3
Click the Linearized Plant from Simulink tab (see the following example).
Note If you haven't activated the design tool within Simulink, the Linearized
Plant from Simulink tab is unavailable.
Linearization Process
If you click OK, the design tool uses Simulink Control Design software to create a
linearized plant model. It performs the following tasks automatically:
1
Configures the Control and Estimation Tools Manager.
2
Temporarily inserts linearization input and output points in the Simulink model at
the inputs and outputs of the MPC block.
3
When the Create a new operating condition from MPC I/O values option is
selected, the Model Predictive Control Toolbox software temporarily inserts output
constraints at the inputs/outputs of the MPC block.
4
Finds a steady state operating condition based on the constraints or uses the
specified operating condition.
5
Linearizes the plant model about the operating point.
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The linearized plant model appears as a new node under Plant Models. For details of
the linearization process, refer to the Simulink Control Design documentation.
Linearization Options
You can customize the linearization process in several ways:
• To specify a name for the linearized plant model, enter the name in the
Linearization model name edit box.
• To use an alternative operating condition, you can:
• Select one from the menu next to Use the previously computed operating
condition. This list contains all operating conditions that exist within the current
project.
• Select Create a new operating condition from MPC I/O values to compute an
operating condition by optimization, using the nominal plant values as constraints.
See “Linearize Simulink Models” for an example involving a nonlinear chemical
reactor.
• To replace the nominal plant values with the operating point used in the
linearization, select the check box next to Replace the MPC nominal I/O values
with those derived from the operating condition.
• When there are multiple MPC blocks in the Simulink diagram, use the Import to
menu to select the one that will receive the plant model.
Note The above linearization process automatically identifies the plant's input
and output variables according your signal connections in the Simulink model. The
controller block does not allow signals corresponding to unmeasured disturbance
or unmeasured output variables. Consequently, such variables cannot be included
in a model created via the above linearization procedure. If you must include such
variables in your controller design, use the Simulink Control Design tool to designate
the signals to be used, linearize the plant, and then import this linearized model
into the MPC design tool. See “Linearize Simulink Models” for an example of this
procedure.
Importing a Controller
To import a controller, do one of the following:
• Select the MPC/Import/Controller menu option.
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• Select the MPC project/task node in the tree (see “Tree View” on page 8-6), and
then click the Import Controller button.
• Right-click the MPC project/task node and select the Import Controller menu
option.
• If you've already designed a controller, select the Controllers node and then click
the Import button, or right-click the Controllers node and select the Import
Controller menu option.
The MPC Controller Importer dialog box opens. The following sections describe its
options:
• “Import from” on page 8-13
• “Import to” on page 8-14
• “Buttons” on page 8-14
Import from
Use these options to set the location from which the controller will be imported.
MATLAB Workspace
This is the default option and is the case shown in the above example. The Items in
your workspace area in the upper-right corner lists all MPC objects in your workspace.
Select one by clicking it. The Properties area lists the properties of the selected model.
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MAT-File
The upper part of the dialog box changes as shown below.
The MAT-file name edit box becomes active. Type the desired MAT-file name here (if
it's not on the MATLAB path, enter the complete file path). You can also use the Browse
button which opens a standard file chooser dialog box.
In the above example, file Controllers.mat contains two MPC objects. Their names
appear in the Items in your MAT-file area in the upper-right corner.
Import to
This allows you to specify the MPC task into which the controller will be imported (see
example below). It defaults to that most recently active.
Buttons
Import
Select the controller you want to import from the Items list in the upper-right corner.
Verify that the Import to option designates the correct project/task. Click the Import
button to import the controller.
The new controller should appear in the tree as a subnode of Controllers. (See “Tree
View” on page 8-6.)
The imported controller contains a plant model, which appears in the Plant models list.
(See “Plant Models View” on page 8-20.)
Note If the selected controller is incompatible with any others in the designated project,
the design tool will not import it.
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Close
Click Close to close the dialog box. You can also click the Close icon on the title bar.
Exporting a Controller
To export a controller, do one of the following:
• Select the MPC/Export menu option.
• Select Controllers in the tree and click its Export button.
• In the tree, right-click Controllers and select the Export Controller menu option.
• In the tree, right-click the controller you want to export and select the Export
Controller menu option.
The MPC Controller Exporter dialog box opens (see example below). The following
sections describe its options:
• “Dialog Box Options” on page 8-15
• “Buttons” on page 8-16
Dialog Box Options
The following sections describe the dialog box options.
Controller source
Use this to select the project/task containing the controller to be exported. It defaults to
the project/task most recently active.
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Controller to export
Use this to specify the controller to be exported. It defaults to the controller most recently
selected in the tree.
Name to assign
Use this to assign a valid MATLAB variable name (no spaces). It defaults to the selected
controller's name (with spaces removed, if any).
Export to MATLAB workspace
Select this option if you want the controller to be exported to the MATLAB workspace.
Export to MAT-file
Select this option if you want the controller to be exported to a MAT-file.
Buttons
Export
If you've selected the Export to MATLAB workspace option, clicking Export causes
a new MPC object to be created in your MATLAB workspace. (If one having the assigned
name already exists, you'll be asked if you want to overwrite it.) You can use the
MATLAB whos command to verify that the controller has been exported.
If you've selected the Export to MAT-file option, clicking Export opens a standard file
chooser that allows you to specify the file.
In either case, the dialog box remains visible, allowing you to export additional
controllers.
Close
Click Close to close the dialog box. You can also click the Close icon on the title bar.
Signal Definition View
The signal definition view appears whenever you select a Model Predictive Control
Toolbox project or task node in the tree (see “Tree View” on page 8-6). You'll see this
view when you open the design tool for the first time. An example appears below.
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The following sections describe the view's main features:
• “MPC Structure Overview” on page 8-17
• “Buttons” on page 8-18
• “Signal Properties Tables” on page 8-18
• “Right-Click Menu Options” on page 8-20
MPC Structure Overview
This upper section is a noneditable display of your application's structure. Once you've
imported a plant model (or controller), tool counts and displays the five possible signal
types, as in the example below.
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The counts change if you edit the signal types.
Buttons
Import Plant
Clicking this opens the Plant Model Importer dialog box (see “Importing a Plant Model”
on page 8-8).
Import Controller
Clicking this opens the MPC Controller Importer dialog box (see “Importing a Controller”
on page 8-12).
Note You won't be allowed to proceed with your design until you import a plant model.
You can do so indirectly by importing a controller or loading a saved project.
Signal Properties Tables
Two tables display the properties of each signal in your design.
Input Signal Properties
The plant's input signals appear as table rows (see example below).
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The entries are editable and have the following significance:
• Name – The signal name, an alphanumeric string used to label other tables, graphics,
etc. Each name must be unique. The design tool assigns a default name if your
imported model doesn't specify one.
• Type – One of the three valid Model Predictive Control Toolbox input signal types.
The above example shows one of each. To change a signal's type, click the table cell
and select the desired type. The options are as follows:
Manipulated – A signal that will be manipulated by the controller, i.e., an actuator
(valve, motor, etc.).
Measured Disturbance – An independent input whose value is measured and used
as a controller input for feedforward compensation.
Unmeasured Disturbance – An independent input representing an unknown,
unpredictable disturbance.
• Description – An optional descriptive string.
• Units – Optional units (dimensions), a string. Used to label other dialog boxes,
simulation plots, etc.
• Nominal – The signal's nominal value. The design tool defaults this to zero. Any
value you assign here will be the default initial condition in simulations.
Note Your design must include at least one manipulated variable. The other input
signal types need not be included.
Output Signal Properties
The plant's output signals appear as table rows (see example below).
The entries are editable and have the following significance:
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• Name – The signal name, an alphanumeric string used to label other tables, graphics,
etc. Each name must be unique. The design tool assigns a default name if your
imported model doesn't specify one.
• Type – One of the two valid Model Predictive Control Toolbox output signal types.
The above example shows one of each. To change a signal's type, click the table cell
and select the desired type. The options are as follows:
Measured – A signal the controller can use for feedback.
Unmeasured – Predicted by the plant model but unmeasured. Can be used as an
indicator. Can also be assigned a setpoint or constrained.
• Description – An optional descriptive string.
• Units – Optional units (dimensions), a string. Used to label other dialog boxes,
simulation plots, etc.
• Nominal – The signal's nominal value. The design tool defaults this to zero. Any
value you assign here will be the default initial condition in simulations.
Note Your design must include at least one measured output. Inclusion of
unmeasured outputs is optional.
Right-Click Menu Options
Right-clicking on an MPC project/task node allows you to choose one of the following
menu items:
• Import Plant Model – Opens the Plant Model Importer dialog box (see “Importing a
Plant Model” on page 8-8).
• Import Controller – Opens the MPC Controller Importer dialog box (see “Importing
a Controller” on page 8-12).
• Clear Project – Erases all plant models, controllers, and scenarios in your design,
returning the project to its initial empty state.
• Delete Project – Deletes the selected project node.
Plant Models View
Selecting Plant models in the tree displays this view (see example below).
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The following sections describe the view's main features:
• “Plant Models List” on page 8-21
• “Model Details” on page 8-22
• “Additional Notes” on page 8-22
• “Buttons” on page 8-22
• “Right-Click Options” on page 8-23
Plant Models List
This table lists all the plant models you've imported and/or plant models contained in
controllers that you've imported. The example below lists two imported models, DC and
DCp .
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The Name field is editable. Each model must have a unique name. The name you assign
here will be used within the design tool only.
The Type field is noneditable and indicates the model's LTI object type (see the Control
System Toolbox documentation for a detailed discussion of LTI models).
The Sampling Period field is zero for continuous-time models, and a positive real value
for discrete-time models.
The Imported on field gives the date and time the model was imported.
Model Details
This scrollable viewport shows details of the model currently selected in the plant models
list (see “Plant Models List” on page 8-21). An example appears below.
Additional Notes
You can use this editable text area to enter comments, distinguishing model features, etc.
Buttons
Import
Opens the Plant Model Importer dialog box (see “Importing a Plant Model” on page
8-8).
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Delete
Deletes the selected model. If the model is being used elsewhere (i.e., in a controller or
scenario), the first model in the list replaces it and a warning message appears.
Right-Click Options
Right-clicking the Plant models node causes the following menu option to appear.
Import Model
Opens the Plant Model Importer dialog box (see “Importing a Plant Model” on page
8-8).
Controllers View
Selecting Controllers in the tree displays this view (see example below).
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The following sections describe the view's main features:
• “Controllers List” on page 8-24
• “Controller Details” on page 8-25
• “Additional Notes” on page 8-25
• “Buttons” on page 8-25
• “Right-Click Options” on page 8-26
Controllers List
This table lists all the controllers in your project. The example below lists two controllers,
MPC1 and MPC2 .
The Name field is editable. The name you assign here must be unique. You will refer to
it elsewhere in the design tool, e.g., when you use the controller in a simulation scenario.
Each listed controller corresponds to a subnode of Controllers (see “Tree View” on page
8-6). Editing the name in the table will rename the corresponding subnode.
The Plant Model field is editable. To change the selection, click the cell and choose one
of your models from the list. (All models appearing in the Plant Models view are valid
choices. See “Plant Models View” on page 8-20.)
The Control Interval field is editable and must be a positive real number. You can
also set it in the Controller Specifications view (see “Model and Horizons Tab” on page
8-30 for more details).
The Prediction Horizon field is editable and must be a positive, finite integer. You
can also set in the Controller Specifications view (see “Model and Horizons Tab” on page
8-30 for more details).
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The noneditable Last Update field gives the date and time the controller was most
recently modified.
Controller Details
This scrollable viewport shows details of the controller currently selected in the
controllers list (see “Controllers List” on page 8-24). An example appears below.
Note This view shows controller details once you have used the controller in a
simulation. Prior to that, it is empty. If necessary, you can use the Display button to
force the details to appear.
Additional Notes
You can use this editable text area to enter comments, distinguishing controller features,
etc.
Buttons
Import
Opens the MPC Controller Importer dialog box (see “Importing a Controller” on page
8-12).
Export
Opens the MPC Controller Exporter dialog box (see “Exporting a Controller” on page
8-15).
New
Creates a new controller specification subnode containing the default Model Predictive
Control Toolbox settings, and assigns it a default name.
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Copy
Copies the selected controller, creating a controller specification subnode containing the
same controller settings, and assigning it a default name.
Display
Calculates and displays details for the selected controller.
Delete
Deletes the selected controller. If the controller is being used elsewhere (i.e., in a
simulation scenario), the first controller in the list replaces it (and a warning message
appears).
Right-Click Options
Right-clicking the Controllers node causes the following menu options to appear.
New Controller
Creates a new controller specification subnode containing the default Model Predictive
Control Toolbox settings, and assigns it a default name.
Import Controller
Opens the MPC Controller Importer dialog box (see “Importing a Controller” on page
8-12).
Export Controller
Opens the MPC Controller Exporter dialog box (see “Exporting a Controller” on page
8-15).
Simulation Scenarios List
Selecting Scenarios in the tree causes this view to appear (see example below).
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The following sections describe the view's main features:
• “Scenarios List” on page 8-27
• “Scenario Details” on page 8-28
• “Additional Notes” on page 8-29
• “Buttons” on page 8-29
• “Right-Click Options” on page 8-29
Scenarios List
This table lists all the scenarios in your project. The example below lists two, Scenario1
and Scenario2 .
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The Name field is editable. The assigned name must be unique. Each listed scenario
corresponds to a subnode of Scenarios (see “Tree View” on page 8-6). Editing the
name in the table will rename the corresponding subnode.
The Controller field is editable. To change the selection, click the cell and select one
of your controllers from the list. (All controllers appearing in the Controllers view are
valid choices. See “Controllers View” on page 8-23.) You can also set this using the
Scenario Specifications view (for more discussion, see “Simulation Scenario View” on
page 8-49).
The Plant field is editable. To change the selection, click the cell and select one of your
plant models from the list. (All models appearing in the Plant Models view are valid
choices. See “Plant Models View” on page 8-20.) You can also set this in the scenario
specifications (for more discussion, see “Simulation Scenario View” on page 8-49).
The Closed Loop field is an editable check box. If cleared, the simulation will be
open loop. You can also set it in the scenario specifications (for more discussion see
“Simulation Scenario View” on page 8-49).
The Constrained field is an editable check box. If cleared, the simulation will ignore
all constraints specified in the controller design. You can also set it in the scenario
specifications (for more discussion see “Simulation Scenario View” on page 8-49).
The Duration field is editable and must be a positive, finite real number. It sets
the simulation duration. You can also set it in the scenario specifications (for more
discussion, see “Simulation Scenario View” on page 8-49).
Scenario Details
This area is blank at all times.
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Additional Notes
You can use this editable text area to enter comments, distinguishing scenario features,
etc.
Buttons
New
Creates a new scenario specification subnode containing the default Model Predictive
Control Toolbox settings, and assigns it a default name.
Copy
Copies the selected scenario, creating a scenario specification subnode containing the
same settings, and assigning it a default name.
Delete
Deletes the selected scenario.
Right-Click Options
Right-clicking the Scenarios node causes the following menu option to appear
New Scenario
Creates a new scenario specification subnode containing the default Model Predictive
Control Toolbox settings, and assigns it a default name.
Controller Specifications View
This view appears whenever you select one of your controller nodes (see “Tree View” on
page 8-6). It allows you to review and edit controller settings. It consists of four tabs,
each devoted to a particular design aspect. All settings have default values.
The following sections describe the view's main features:
• “Model and Horizons Tab” on page 8-30
• “Constraints Tab” on page 8-32
• “Constraint Softening” on page 8-35
• “Weight Tuning Tab” on page 8-38
• “Estimation Tab” on page 8-41
• “Right-Click Menus” on page 8-48
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Model and Horizons Tab
Plant Model
This combo box allows you to specify the plant model the controller uses for its
predictions. You can choose any of the plant models you've imported. (See “Importing a
Plant Model” on page 8-8.)
Horizons
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The Control interval option sets the elapsed time between successive controller moves.
It must be a positive, finite real number. The calculations assume a zero-order hold on
the manipulated variables (the signals adjusted by the controller). Thus, these signals
are constant between moves.
The Prediction horizon option sets the number of control intervals over which the
controller predicts its outputs when computing controller moves. It must be a positive,
finite integer.
The Control horizon option sets the number of moves computed. It must be a positive,
finite integer, and must not exceed the prediction horizon. If less than the prediction
horizon, the final computed move fills the remainder of the prediction horizon.
For more discussion, see “Choosing Sample Time and Horizons” on page 1-5.
Blocking
By default, the Blocking option is cleared (off). When selected as shown above, the
design tool replaces the Control horizon specification (see “Horizons” on page 8-30)
with a move pattern determined by the following settings:
• Blocking allocation within prediction horizon – Choices are:
Beginning – Successive moves at the beginning of the prediction horizon, each with a
duration of one control interval.
Uniform – The prediction horizon is divided by the number of moves and rounded to
obtain an integer duration, and each computed move has this duration (the last move
extends to fill the prediction horizon).
End – Successive moves at the end of the prediction horizon, each with a duration of
one control interval.
Custom – You specify the duration of each computed move.
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• Number of moves computed per step – The number of moves computed when the
allocation setting is Beginning, Uniform, or End. Must be a positive integer not
exceeding the prediction horizon.
• Custom move allocation vector – The duration of each computed move, specified
as a row vector. In the example below, there are four moves, the first lasting 1 control
interval, the next two lasting 3, and the final lasting 8 for a total of 15. The Number
of moves computed per step setting is disabled (ignored).
The sum of the vector elements should equal the prediction horizon (15 in this case). If
not, the last move is extended or truncated automatically.
Note When Blocking is off, the controller uses the Beginning allocation with
Number of moves computed per step equal to the Control horizon.
For more discussion, see “Manipulated Variable Blocking” on page 2-26.
Constraints Tab
This tab allows you to specify constraints (bounds) on manipulated variables and outputs.
Constraints can be hard or soft. By default, all variables are unconstrained, as shown in
the view below.
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Note If you specify constraints, manipulated variable constraints are hard by default,
whereas output variable constraints are soft by default. You can customize this behavior,
as discussed in the following sections. For additional information on constraints, see
“Constraints” on page 2-7.
Each table entry may be a scalar or a vector. A scalar entry defines a constraint that
is constant for the entire prediction horizon. A vector entry defines a time-varying
constraint. See “Entering Vectors in Table Cells” on page 8-41 for the required
format.
An entry may also be any valid MATLAB expression provided that it evaluates to yield
an appropriate scalar or vector quantity.
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Constraints on Manipulated Variables
The example below is for an application with two manipulated variables (MVs), each
represented by a table row.
The Name and Units columns are noneditable. To change them, use the signal definition
view. (See “Signal Definition View” on page 8-16. Any changes there apply to the
entire design.)
The remaining entries are editable. If you leave a cell blank, the controller ignores that
constraint. You can achieve the same effect by entering -Inf or Inf (for a minimum or
maximum, respectively).
The Minimum and Maximum values set each MV's range.
The Max down rate and Max up rate values set the amount the MV can change in a
single control interval. The Max down rate must be negative or zero. The Max up rate
must be positive or zero.
Constraint values must be consistent with your nominal values (see “Input Signal
Properties” on page 8-18). In other words, each MV's nominal value must satisfy the
constraints.
Constraint values must also be self-consistent. For example, an MV's lower bound must
not exceed its upper bound.
Constraints on Output Variables
The example below is for an application with two output variables, each represented by a
table row.
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Working with the Design Tool
The Name and Units columns are noneditable. To change them, use the signal definition
view. (See “Signal Definition View” on page 8-16. Any changes there apply to the
entire design.)
The remaining entries are editable. If you leave a cell blank (as above), the controller
ignores that constraint. You can achieve the same effect by entering -Inf (for a
Minimum) or Inf (for a Maximum).
Constraint values must be consistent with your nominal values (see “Output Signal
Properties” on page 8-19). In other words, each output's nominal value must satisfy
the constraints.
Constraint values must also be self-consistent. For example, an output's lower bound
must not exceed its upper bound.
Note Don't constrain outputs unless this is an essential aspect of your application. It is
usually better to define output setpoints (reference values) rather than constraints.
Constraint Softening
A hard constraint cannot be violated. Hard constraints are risky, especially for outputs,
because the controller will ignore its other objectives in order to satisfy them. Also,
the constraints might be impossible to satisfy in certain situations, in which case the
calculations are mathematically infeasible.
Model Predictive Control Toolbox software allows you to specify soft constraints. These
can be violated, but you specify a violation tolerance for each (the relaxation band). See
the example specifications below.
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To open this dialog box, click the Constraint softening button at the bottom of the
Constraints tab in the Controller Specification view (see “Constraints Tab” on page
8-32).
As for the constraints themselves, an entry can be a scalar or a vector. The latter defines
a time-varying relaxation band. See “Entering Vectors in Table Cells” on page 8-41
for the required format.
Input Constraints
An example input constraint softening specification appears below.
The Name and Units columns are noneditable. To change them, use the signal definition
view. (See “Signal Definition View” on page 8-16. Any changes there apply to the
entire design.)
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Working with the Design Tool
The Minimum, Maximum, Max down rate, and Max up rate columns are editable.
Their values are the same as on the main Constraints tab (see “Constraints on
Manipulated Variables” on page 8-34). You can specify them in either location.
The remaining columns specify the relaxation band for each constraint. An empty cell is
equivalent to a zero, i.e., a hard constraint.
Entries must be zero or positive real numbers. To soften a constraint, increase its
relaxation band.
The example above shows a relaxation band of 2 moles/min for the steam flow rate's
lower and upper bounds. The lack of a relaxation band setting for the reflux flow rate's
constraints means that these will be hard.
Note The relaxation band is a relative tolerance, not a strict bound. In other words, the
actual constraint violation can exceed the relaxation band.
Output Constraints
An example output constraint specification appears below.
The Name and Units columns are noneditable. To change them, use the signal definition
view. (See “Signal Definition View” on page 8-16. Any changes there apply to the
entire design.)
The Minimum and Maximum columns are editable. Their values are the same as on
the main Constraints tab (see “Constraints on Output Variables” on page 8-34). You
can specify them in either location.
The remaining columns specify the relaxation band for each constraint. An empty cell is
equivalent to 1.0, i.e., a soft constraint.
Entries must be zero or positive real numbers. To soften a constraint, increase its
relaxation band.
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The example above shows a relaxation band of 0.5 mole % for the distillate purity lower
bound, and a relaxation band of 2 mole % for the bottoms purity lower bound (the softer
of the two constraints).
Note The relaxation band is a relative tolerance, not a strict bound. In other words, the
actual constraint violation can exceed the relaxation band.
Overall Constraint Softness
The relaxation band settings allow you to adjust the hardness/softness of each constraint.
You can also soften/harden all constraints simultaneously using the slider at the bottom
of the dialog box pane.
You can move the slider or edit the value in the edit box, which must be between 0 and 1.
Buttons
OK – Closes the constraint softening dialog box, implementing changes to the tabular
entries or the slider setting.
Cancel – Closes the constraint softening dialog box without changing anything.
Weight Tuning Tab
The example below shows the Model Predictive Control Toolbox default tuning weights
for an application with two manipulated variables and two outputs.
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Working with the Design Tool
The following sections discuss the three tab areas in more detail. For additional
information, see “Optimization Problem” on page 2-2.
Each table entry may be a scalar or a vector. A scalar entry defines a weight that is
constant for the entire prediction horizon. A vector entry defines a time-varying weight.
See “Entering Vectors in Table Cells” on page 8-41 for the required format.
Input Weights
The Name, Description, and Units columns are noneditable. To change them, use the
signal definition view. (See “Signal Definition View” on page 8-16. Any changes there
apply to the entire design.)
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The Weight column sets a penalty on deviations of each manipulated variable (MV)
from its nominal value. The weight must be zero or a positive real number. The default
is zero, meaning that the corresponding MV can vary freely provided that it satisfies its
constraints (see “Constraints on Manipulated Variables” on page 8-34).
A large Weight discourages the corresponding MV from moving away from its nominal
value. This can cause steady state error (offset) in the output variables unless you have
extra MVs at your disposal.
Note To set the nominal values, use the signal definition view. (See “Signal Definition
View” on page 8-16. Any changes there apply to the entire design.)
The Rate Weight value sets a penalty on MV changes, i.e., on the magnitude of each MV
move. Increasing the penalty on a particular MV causes the controller to change it more
slowly. The table entries must be zero or positive real numbers. These values have no
effect in steady state.
Output Weights
The Name, Description, and Units columns are noneditable. To change them, use the
signal definition view. (See “Signal Definition View” on page 8-16. Any changes there
apply to the entire design.)
The Weight column sets a penalty on deviations of each output variable from its setpoint
(or reference) value. The weight must be zero or a positive real number.
A large Weight discourages the corresponding output from moving away from its
setpoint.
If you don't need to hold a particular output at a setpoint, set its Weight to zero. This
may be the case, for example, when an output doesn't have a target value and is being
used as an indicator variable only.
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Working with the Design Tool
Overall (Slider Control)
The slider adjusts the weights on all variables simultaneously. Moving the slider to the
left increases rate penalties relative to setpoint penalties, which often (but not always!)
increases controller robustness. The disadvantage is that disturbance rejection and
setpoint tracking become more sluggish.
You can also change the value in the edit box. It must be a real number between 0 and 1.
The actual effect is nonlinear. You will generally need to run trials to determine the best
setting.
Entering Vectors in Table Cells
In the above examples all constraints and weights were entered as scalars. A scalar entry
defines a value that is constant for the entire prediction horizon.
You can also define a constraint or weight that varies with time by entering a vector.
For the rationale and theoretical basis, see “View and Alter Controller Properties” and
“Optimization Problem” on page 2-2.
Enter vectors using the standard MATLAB syntax. For example, [1, 2, 3] defines a
vector containing three elements, the values 1, 2, and 3.
Entries can be either row or column vectors. A MATLAB expression that produces a
vector works too. For example, 4*ones(3,1) would be a valid entry.
If a vector contains fewer than P elements, where P is the horizon length, the controller
automatically extends the vector using its last element. For example, if you entered [1,
2, 3] and P = 5, the vector used in controller calculations would be [1, 2, 3, 3, 3] .
Estimation Tab
Use these specifications to shape the controller's response to unmeasured disturbances
and measurement noise.
The example below shows Model Predictive Control Toolbox default settings for an
application with two output variables and no unmeasured disturbance inputs.
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The following sections cover each estimation feature in detail. For additional information,
see“Controller State Estimation” on page 2-29.
Button (MPC Default Settings)
If you edit any of the Estimation tab settings, the display near the top will appear as
follows.
To return the settings to the default state, click the Use MPC Defaults button, causing
the display to revert to the default condition shown below.
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Working with the Design Tool
Overall Estimator Gain
This slider determines the controller's overall disturbance response. As you move the
slider to the left, the controller responds less aggressively to unexpected changes in the
outputs, i.e., it assumes that such changes are more likely to be caused by measurement
noise rather than a real disturbance.
You can also change the value in the edit box. It must be between zero and 1. The effect
is nonlinear, and you might need to run trial simulations to achieve the desired result.
Output Disturbances
Use these settings to model unmeasured disturbances adding to the plant outputs.
The example below shows the tab's appearance with the Signal-by-signal option
selected for an application having two plant outputs.
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The graphic shows the disturbance location.
Use the table to specify the disturbance character for each output.
The Name and Units columns are noneditable. To change them, use the signal definition
view. (See “Signal Definition View” on page 8-16. Any changes there apply to the
entire design.)
The Type column sets the disturbance character. To edit this, click the cell and select
from the resulting menu. You have the following options:
• Steps – Simulates random step-like disturbances (integrated white noise).
• Ramps – Simulates a random drifting disturbance (doubly-integrated white noise).
• White – White noise.
The Magnitude column specifies the standard deviation of the white noise assumed to
create the disturbance. Set it to zero if you want to turn off a particular disturbance.
For example, if Type is Steps and Magnitude is 2, the disturbance model is integrated
white noise, where the white noise has a standard deviation of 2.
If these options are too restrictive, select the LTI model in workspace option. The tab
appearance changes to the view shown below.
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Working with the Design Tool
You must specify an LTI output disturbance model residing in your workspace. The
Browse button opens a dialog box listing all LTI models in your workspace, and allows
you to choose one. You can also type the model name in the edit box, as shown above.
The model must have the same number of outputs as the plant.
The white noise entering the model is assumed to have unity standard deviation.
Input Disturbances
Use these settings to model disturbances affecting the plant's unmeasured disturbance
inputs.
Note This option is available only if your plant model includes unmeasured disturbance
inputs.
The example below shows the tab's appearance with the Signal-by-signal option
selected for a plant having one unmeasured disturbance input. The graphic shows the
disturbance location.
Use the table to specify the character of each unmeasured disturbance input.
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The Name and Units columns are noneditable. To change them, use the signal definition
view. (See “Signal Definition View” on page 8-16. Any changes there apply to the
entire design.)
The Type column sets the disturbance character. To edit this, click the cell and select
from the resulting menu. You have the following options:
• Steps – Simulates random step-like disturbances (integrated white noise).
• Ramps – Simulates a random drifting disturbance (doubly-integrated white noise).
• White – White noise.
The Magnitude column specifies the standard deviation of the white noise assumed to
create the disturbance. Set it to zero if you want to turn off a particular disturbance.
For example, if Type is Steps and Magnitude is 2, the disturbance model is integrated
white noise, where the white noise has a standard deviation of 2.
If the above options are too restrictive, select the LTI model in workspace option. The
tab appearance changes to the view shown below.
You must specify an LTI disturbance model residing in your workspace. The Browse
button opens a dialog box listing all LTI models in your workspace, and allows you to
choose one. You can also type the model name in the edit box, as shown above.
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Working with the Design Tool
The number of model outputs must equal the number of plant unmeasured disturbance
inputs. The white noise entering the model is assumed to have unity standard deviation.
Noise
Use these settings to model noise in the plant's measured outputs.
The example below shows the tab's appearance with the Signal-by-signal option
selected for a plant having two measured outputs. The graphic shows the noise location.
Use the table to specify the character of each noise input.
The Name and Units columns are noneditable. To change them, use the signal definition
view. (See “Signal Definition View” on page 8-16. Any changes there apply to the
entire design.)
The Type column sets the noise character. To edit this, click the cell and select from the
resulting menu. You have the following options:
• White – White noise.
• Steps – Simulates random step-like disturbances (integrated white noise).
The Magnitude column specifies the standard deviation of the white noise assumed to
create the noise. Set it to zero if you want to specify that an output is noise-free.
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For example, if Type is Steps and Magnitude is 2, the noise model is integrated white
noise, where the white noise has a standard deviation of 2.
If the above options are too restrictive, select the LTI model in workspace option. The
tab appearance changes as follows.
You must specify an LTI model residing in your workspace. The Browse button opens
a dialog box listing all LTI models in your workspace, and allows you to choose one. You
can also type the model name in the edit box, as shown above.
The number of noise model outputs must equal the number of plant measured outputs.
The white noise entering the model is assumed to have unity standard deviation.
Right-Click Menus
Copy Controller
Creates a new controller having the same settings and a default name.
Delete Controller
Deletes the controller. If the controller is being used in a simulation scenario, the design
tool replaces it with the first controller in your list, and displays a warning message.
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Working with the Design Tool
Rename Controller
Opens a dialog box allowing you to rename the controller.
Note Each controller in a design project/task must have a unique name.
Export Controller
Opens the MPC Controller Exporter dialog box (see “Exporting a Controller” on page
8-15).
Simulation Scenario View
This view appears whenever you select one of your scenario specification nodes (see “Tree
View” on page 8-6). It allows you to specify simulation settings and independent
variables. All have default values, but you will want to change at least some of them
(otherwise all independent variables will be constant). Defaults for a plant with three
inputs and two outputs appears below.
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The middle table won't appear unless you have designated at least one input signal to be
a measured disturbance.
The following sections describe the view's main features:
• “Model and Horizons Tab” on page 8-30
• “Simulation Settings” on page 8-51
• “Setpoints” on page 8-51
• “Measured Disturbances” on page 8-52
• “Unmeasured Disturbances” on page 8-53
• “Signal Type Settings” on page 8-54
• “Simulation Button” on page 8-56
• “Tuning Advisor Button” on page 8-56
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Working with the Design Tool
• “Right-Click Menus” on page 8-56
Simulation Settings
Use this section to set the following:
• Controller – Select one of your controllers,
• Plant – Select the plant model that will act as the “real” plant in the simulation, i.e.,
it need not be the same as that used for controller predictions.
• Duration – The simulation duration in time units.
• Close loops – If cleared, the simulation will be open-loop.
• Enforce Constraints – If cleared, all controller constraints will be ignored.
The Control interval field is display-only, and reflects the setting in your Controller
selection. You can change it there if necessary (see “Model and Horizons Tab” on page
8-30).
Setpoints
Note Setpoint specifications affect closed-loop simulations only.
Use this table to specify the setpoint for each output. In the example below, which is
for an application having two plant outputs, the first would be constant at 0.0, and the
second would change step-wise.
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The Name and Units columns are display-only. To change them, use the signal
definition view. (See “Signal Definition View” on page 8-16. Any changes apply to the
entire design.)
The Type column specifies the setpoint variation. To change this, click the cell and select
a choice from the resulting menu.
The significance of the Initial Value, Size, Time, and Period columns depends on the
Type. If a cell is gray (noneditable), it doesn't apply to the Type you've chosen.
For details on the signal types, see “Signal Type Settings” on page 8-54.
If the Look Ahead option is selected (i.e., on), the controller will use future values of the
setpoints in its calculations. This improves setpoint tracking, but knowledge of future
setpoint changes is unusual in practice.
Note In the current implementation, selecting or clearing the Look ahead option for one
output will set the others to the same state. Model Predictive Control Toolbox code does
not allow you to Look ahead for some outputs but not for others.
Measured Disturbances
Use this table to specify the variation of each measured disturbance. In the example
below, which is for an application having a single measured disturbance, the “Steam
Rate” input would be constant at 0.0.
The Name and Units columns are display-only. To change them, use the signal
definition view. (See “Signal Definition View” on page 8-16. Any changes apply to the
entire design.)
The Type column specifies the disturbance variation. To change this, click the cell and
select a choice from the resulting menu.
The significance of the Initial Value, Size, Time, and Period columns depends on the
Type. If a cell is gray (noneditable), it doesn't apply to the Type you've chosen.
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Working with the Design Tool
For details on the signal types, see “Signal Type Settings” on page 8-54.
If the Look Ahead option is selected (i.e., on), the controller will use future values of
the measured disturbance(s) in its calculations. This improves disturbance rejection, but
knowledge of future disturbances is unusual in practice. It has no effect in an open-loop
simulation.
Note In the current implementation, selecting or clearing the Look ahead option for one
input will set the others to the same state. Model Predictive Control Toolbox code does
not allow you to Look ahead for some inputs but not for others.
Unmeasured Disturbances
Use this table to specify the variation of each measured unmeasured disturbance. In the
example below, all would be constant at 0.0.
Unmeasured Disturbance Locations
You can simulate an unmeasured disturbance in any of the following locations:
• The plant's unmeasured disturbance (UD) inputs (if any)
• The plant's measured outputs (MO)
• The plant's manipulated variable (MV) inputs
All of the above will appear as rows in the table. In the case of a measured output or
manipulated variable, the disturbance is an additive bias.
The Name and Units columns are display-only. To change them, use the signal
definition view. (See “Signal Definition View” on page 8-16. Any changes apply to the
entire design.)
The Type column specifies the disturbance variation. To change this, click the cell and
select a choice from the resulting menu.
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The significance of the Initial Value, Size, Time, and Period columns depends on the
Type. If a cell is gray (noneditable), it doesn't apply to the Type you've chosen.
For details on the signal types, see “Signal Type Settings” on page 8-54.
Open-Loop Simulations
For open-loop simulations, you can vary the MV unmeasured disturbance to simulate the
plant's response to a particular MV. The MV signal coming from the controller stays at
its nominal value, and the MV unmeasured disturbance adds to it.
For example, suppose Reflux Rate is an MV, and the corresponding row in the table
below represents an unmeasured disturbance in this MV.
You could set it to a constant value of 1 to simulate the plant's open-loop unit-step
response to the Reflux Rate input. (In a closed-loop simulation, controller adjustments
would also contribute, changing the response.)
Similarly, an unmeasured disturbance in an MO adds to the output signal coming from
the plant. If there are no changes at the plant input, the plant outputs are constant, and
you see only the change due to the disturbance. This allows you to check the disturbance
character before running a closed-loop simulation.
Signal Type Settings
The table below is an example that uses five of the six available signal types (the
Constant option has been illustrated above). The cells with white backgrounds are the
entries you must supply. All have defaults.
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Working with the Design Tool
Constant
The signal will be held at the specified Initial Value for the entire simulation.
y = y0 for t ≥ 0.
Step
Prior to Time, the signal = Initial Value. At Time, the signal changes step-wise by Size
units. Its value thereafter = Initial Value + Size.
y = y0 for 0 ≤ t < t0, where y0 = Initial Value, t0 = Time
y = y0+M for t > 0, where M = Size.
Ramp
Prior to Time, the signal = Initial Value. At Time, the signal begins to vary linearly
with slope Size.
y = y0 for 0 ≤ t < t0, where y0 = Initial Value, t0 = Time
y = y0+M(t – t0) for t ≥ t0, where M = Size.
Sine
Prior to Time, the signal = Initial Value. At Time, the signal begins to vary
sinusoidally with amplitude Size and period Period.
y = y0 for 0 ≤ t < t0, where y0 = Initial Value, t0 = Time
y = y0+Msin[ω(t – t0)] for t ≥ t0, where M = Size, ω = 2π/Period.
Pulse
Prior to Time, the signal = Initial Value. At Time, a square pulse of duration Period
and magnitude Size occurs.
y = y0 for 0 ≤ t < t0, where y0 = Initial Value, t0 = Time
y = y0+M for t0 ≤ t ≤ t0, where M = Size, T = Period
y = y0 for t ≥ t0 + T
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Gaussian
Prior to Time, the signal = Initial Value. At Time, the signal begins to vary randomly
about Initial Value with standard deviation Size.
y = y0 for 0 ≤ t < t0, where y0 = Initial Value, t0 = Time
y = y0+Mrandn for t ≥ t0, where M = Size.
randn is the MATLAB random-normal function, which generates random numbers
having zero mean and unit variance.
Simulation Button
Click the Simulate button to simulate the scenario. You can also press Ctrl+R, use the
toolbar icon (see “Toolbar” on page 8-6), or use the MPC/Simulate menu option (see
“Menu Bar” on page 8-4).
Tuning Advisor Button
Click Tuning Advisor to open a window that helps you improve your controller's
performance. See “Weight Sensitivity Analysis” on page 8-57.
Right-Click Menus
Copy Scenario
Creates a new simulation scenario having the same settings and a default name.
Delete Scenario
Deletes the scenario.
Rename Scenario
Opens a dialog box allowing you to rename the scenario.
Note Each scenario in a design project/task must have a unique name.
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Weight Sensitivity Analysis
Weight Sensitivity Analysis
In this section...
“Defining the Performance Metric” on page 8-59
“Baseline Performance” on page 8-60
“Sensitivities and Tuning Advice” on page 8-61
“Refine Controller Tuning Weights” on page 8-63
“Updating the Controller” on page 8-67
“Restoring Baseline Tuning” on page 8-67
“Modal Dialog Behavior” on page 8-68
“Scenarios for Performance Measurement” on page 8-68
When you design MPC controllers, you can use the Tuning Advisor to help you determine
which weight has the most influence on the closed-loop performance. The Tuning
Advisor also helps you determine in which direction to change the weight to improve
performance. Using the Advisor, you can know numerically how each weight impacts the
closed-loop performance, which makes designing MPC controllers easier when the closedloop responses does not depend intuitively on the weights.
To start the Tuning Advisor, click Tuning Advisor in a simulation scenario view
(see “Tuning Advisor Button” on page 8-56). The next figure shows the default Tuning
Advisor window for a distillation process in which there are two controlled outputs,
two manipulated variables, and one measured disturbance (which the Tuning Advisor
ignores). In this case, the originating scenario is Scenario1.
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The Tuning Advisor populates the Current Tuning column with the most recent tuning
weights of the controller displayed in the Controller in Design. In this case, Obj is
the controller. The Advisor also initializes the Performance Weight column to the
same values. The Scenario in Design displays the scenario from which you started the
Tuning Advisor. The Advisor uses this scenario to evaluate the controller's performance.
The columns highlighted in grey are Tuning Advisor displays and are read-only. For
example, signal names come from the “Signal Definition View” on page 8-16 and are
blank unless you defined them there.
To tune the weights using the Tuning Advisor:
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1
Specify the performance metric.
2
Compute the baseline performance.
3
Adjust the weights based on the computed sensitivities.
4
Recompute the performance metric.
5
Update the controller
Weight Sensitivity Analysis
Defining the Performance Metric
In order to obtain tuning advice, you must first provide a quantitative scalar performance
measure, J.
Select the Performance Function
Select a performance metric from the Select a performance function drop-down list in
the upper right-hand corner of the Advisor. You can choose one of four standard ways to
compute the performance measure, J. In each case, the goal is to minimize J.
• ISE (Integral of Squared Error, the default). This is the standard linear quadratic
weighting of setpoint tracking errors, manipulated variable movements, and
deviations of manipulated variables from targets (if any). The formula is
J=
Tstop Ê n y
nu
ˆ
Á
(w yj eyij ) 2 +
[(wuj euij ) 2 + (w Dj u Duij ) 2 ]˜
Á
˜
i =1 Ë j =1
j =1
¯
 Â
Â
where Tstop is the number of controller sampling intervals in the scenario, eyij is the
deviation of output j from its setpoint (reference) at time step i, euij is the deviation of
manipulated variable j from its target at time step i, Δuij is the change in manipulated
variable j at time step i (i.e., Δuij = uij – ui–1, j), and w yj , wuj , and wDj u are nonnegative
performance weights.
• IAE (Integral of Absolute Error). Similar to the ISE but with squared terms replaced
by absolute values
J=
Tstop Ê n y
nu
ˆ
Á | w y e yij | + (| wuj euij | + | wDj u Duij |) ˜
j
Á
˜
i =1 Ë j =1
j =1
¯
 Â
Â
The IAE gives less emphasis to any large deviations.
• ITSE (time-weighted Integral of Squared Errors)
nu
Ê ny
ˆ
J=
iDt Á
(w yj eyij ) 2 +
[( wuj euij ) 2 + (w Dj u Duij ) 2 ] ˜
Á j =1
˜
i =1
j =1
Ë
¯
Tstop
Â
Â
Â
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which penalizes deviations at long times more heavily than the ISE, i.e., it favors
controllers that rapidly eliminate steady-state offset.
• ITAE (time-weighted Integral of Absolute Errors)
nu
Ê ny
ˆ
iDt Á | w yj e yij | + (| wuj euij | + | wDj u D uij |) ˜
Á j =1
˜
i =1
j =1
Ë
¯
Tstop
J=
Â
Â
Â
which is like the ITSE but with less emphasis on large deviations.
Specify Performance Weights
Each of the above formulae use the same three performance weights, w yj , wuj , and wDj u .
All must be non-negative real numbers. Use the weights to:
• Eliminate a term by setting its weight to zero. For example, a manipulated variable
rarely has a target value, in which case you should set its wuj to zero. Similarly if a
plant output is monitored but doesn't have a setpoint, set its w yj to zero.
• Scale the variables so their absolute or squared errors influence J appropriately.
For example, an eyij of 0.01 in one output might be as important as a value of 100 in
another. If you have chosen the ISE, the first should have a weight of 100 and the
second 0.01. In other words, scale all equally important expected errors to be of order
unity.
A Model Predictive Controller uses weights internally as tuning devices. Although
there is some common ground, the performance weights and tuning weights should
differ in most cases. Choose performance weights to define good performance and then
tune the controller weights to achieve it. The Tuning Advisor's main purpose is to
make this task easier.
Baseline Performance
After you define the performance metric and specify the performance weights, compute
a baseline J for the scenario by clicking Baseline. The next figure shows how this
transforms the above example (the two wDj u performance weights have also been set
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Weight Sensitivity Analysis
to zero because manipulated variable changes are acceptable if needed to achieve good
setpoint tracking for the two (equally weighted) outputs. The computed J = 3.435 is
displayed in Baseline Performance, to the right of the Baseline button.
The Tuning Advisor also displays response plots for the scenario with the baseline
controller (not shown but discussed in “Customize Response Plots” on page 8-69).
Sensitivities and Tuning Advice
Click Analyze to compute the sensitivities, as shown in the next figure. The columns
labeled Sensitivity and Tuning Direction now contain advice.
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Each sensitivity value is the partial derivative of J with respect to the controller tuning
weight in the last entry of the same row. For example, the first output has a sensitivity of
0.08663. If we could assume linearity, a 1-unit increase in this tuning weight, currently
equal to 1, would increase J by 0.08663 units. Since we want to minimize J, we should
decrease the tuning weight, as suggested by the Tuning Direction entry.
The challenge is to choose an adjustment magnitude. The behavior is nonlinear so the
sensitivity value is just a rough indication of the likely impact.
You must also consider the tuning weight's current magnitude. For example, if the
current value were 0.01, a 1-unit increase would be extreme and a 1-unit decrease
impossible, whereas if it were 1000, a 1-unit change would be insignificant.
It's best to focus on a small subset of the tuning weights for which the sensitivities
suggest good possibilities for improvement.
In the above example, the wDj u are poor candidates. The maximum possible change in the
suggested direction (decrease) is 0.1, and the sensitivities indicate that this would have a
negligible impact on J. The wuj are already zero and can't be decreased.
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Weight Sensitivity Analysis
The w yj are the only tuning weights worth considering. Again, it seems unlikely that a
change will help much. The display below shows the effect of doubling the tuning weight
on the bottoms purity (second) output. Note the 2 in the last column of this row. After
you click Analyze, the response plots (not shown) make it clear that this output tracks
its setpoint more accurately but at the expense of the other, and the overall J actually
increases.
Notice also that the sensitivities have been recomputed with respect to the revised
controller tuning weights. Again, there are no obvious opportunities for improved
performance.
Thus, we have quickly determined that the default controller tuning weights are nearoptimal in this case, and further tuning is not worth the effort.
Refine Controller Tuning Weights
The Tuning Advisor can help you to refine controller tuning weights for better
performance. It also provides a quantitative performance measurement.
You can access the Tuning Advisor from the Scenarios node in the Control and
Estimation Tools Manager. Before you use the Advisor, choose the controller horizons
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and sampling period, specify constraints, and select a disturbance estimator (if the
default estimator is inappropriate). The Advisor does not provide help with these
parameters.
The example considered here is a plant with four controlled outputs and four
manipulated variables. There are no measured disturbances and the unmeasured
disturbances are unmodeled.
After starting the design tool and importing the plant model, G, which becomes the
controller design basis, we accept the default values for all controller parameters. We
also load a second plant model, Gp, in which all parameters of G have been perturbed
randomly with a standard deviation of 5%.
The scenario shown in the previous figure specifies the controller based on G and the
plant Gp. In other words, it tests the controllers robustness with respect to plant-model
mismatch. It also defines a series of setpoint changes and disturbances.
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Weight Sensitivity Analysis
Clicking Tuning Advisor opens the MPC Tuning Advisor window. In the Tuning
Advisor window, we specify the following settings:
• Select the IAE performance function (an arbitrary choice for illustration only).
• Set all input performance weights to zero because the application does not have input
targets.
• Set all input rate performance weights to zero because the application has no cost for
manipulated variable movement.
• Leave the output performance weights at their default values (unity) because all
controller outputs are of roughly equal magnitude and the application gives equal
priority to the tracking of all four setpoints.
• Click Baseline.
• Click Analyze.
The Tuning Advisor resembles the previous figure. The sensitivity values indicate that
a decrease in the Out4 weight or an increase in the Out2 weight would have the most
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impact. In general, however, the output tuning weights should reflect the setpoint
tracking priorities and it's preferable to adjust the input rate tuning weights.
Sensitivities for Input Rate Weights In1 and In4 are of roughly equal magnitude but
the In4 suggestion is a decrease and this weight is already near its lower bound of zero.
Thus, we focus on the In1 weight.
The next figure shows the Advisor after the In1 weight has been increased in several
steps from 0.1 to 4. Performance has improved by nearly 20% relative to the baseline.
Sensitivities indicate that further adjustments to in input rate tuning weights will have
little impact.
At this point, we can consider adjusting the output tuning weights. It is possible that an
attempt to control a particular output might be causing upsets in other outputs (because
of model error).
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Weight Sensitivity Analysis
The next figure shows the Tuning Advisor after additional adjustments. At this point,
some sensitivities are still rather large, but a small change in the indicated tuning
weight causes the sensitivity to change sign. Therefore, further progress will be difficult.
Overall, we have improved the performance by (26.69 − 20.14)/26.69 which is more than
20%.
Updating the Controller
If you decide a set of modified tuning weights is significantly better than the baseline
set, click Update Controller in MPC Tool. The tuning weights in the Advisor's last
column permanently replace those stored in the Controller in Design and become the
new baseline. All displays update accordingly.
Restoring Baseline Tuning
If you click Restore Baseline Weights, the Advisor will revert to the most recent
baseline condition.
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Modal Dialog Behavior
By default, the Advisor window is modal, meaning that you won't be able to access any
other MATLAB windows while the Advisor is active. You can disable Tuning Advisor is
Modal, as shown in the above example. This is not recommended, however. In particular,
if you return to the Design Tool and modify your controller, your changes won't be
communicated to the Advisor. Instead, close the Advisor, modify the controller and then
reopen the Advisor.
Scenarios for Performance Measurement
The scenario used with the Advisor should be a true test of controller performance. It
should include a sequence of typical setpoint changes and disturbances. It is also good
practice to test controller robustness with respect to prediction model error. The usual
approach is to define a scenario in which the plant being controlled differs from the
controller's prediction model.
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Customize Response Plots
Customize Response Plots
Each time you simulate a scenario, the design tool plots the corresponding plant input
and output responses. The graphic below shows such a response plot for a plant having
two outputs (the corresponding input response plot is not shown).
By default, each plant signal plots in its own graph area (as shown above). If the
simulation is closed loop, each output signal plot include the corresponding setpoint.
The following sections describe response plot customization options:
• “Data Markers” on page 8-69
• “Displaying Multiple Scenarios” on page 8-71
• “Viewing Selected Variables” on page 8-72
• “Grouping Variables in a Single Plot” on page 8-72
• “Normalizing Response Amplitudes” on page 8-73
Data Markers
You can use data markers to label a curve or to display numerical details.
Adding a Data Marker
To add a data marker, click the desired curve at the location you want to mark. The
following graph shows a marker added to each output response and its corresponding
setpoint.
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Data Marker Contents
Each data marker provides information about the selected point, as follows:
• Response – The scenario that generated the curve.
• Time – The time value at the data marker location.
• Amplitude – The signal value at the data marker location.
• Output – The plant variable name (plant outputs only).
• Input – Variable name for plant inputs and setpoints.
Changing a Data Marker's Alignment
To relocate the data marker's label (without moving the marker), right-click the marker,
and select one of the four Alignment menu options. The above example shows three of
the possible four alignment options.
Relocating a Data Marker
To move a marker, left-click it (holding down the mouse key) and drag it along its curve
to the desired location.
Deleting Data Markers
To delete all data markers in a plot, click in the plot's white space.
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Customize Response Plots
To delete a single data marker, right-click it and select the Delete option.
Right-Click Options
Right-click a data marker to use one of the following options:
• Alignment – Relocate the marker's label.
• Font Size – Change the label's font size.
• Movable – On/off option that makes the marker movable or fixed.
• Delete – Deletes the selected marker.
• Interpolation – Interpolate linearly between the curve's data points, or locate at the
nearest data point.
• Track Mode – Changes the way the marker responds when you drag it.
Displaying Multiple Scenarios
By default the response plots include all the scenarios you've simulated. The example
below shows a response plot for a plant with two outputs. The data markers indicate the
two scenarios being plotted: “Accurate Model” and “Perturbed Model”. Both scenarios use
the same setpoints (not marked—the lighter solid lines).
Viewing Selected Scenarios
If your plots are too cluttered, you can hide selected scenarios. To do so:
• Right-click in the plot's white space.
• Select Responses from the resulting context menu.
• Toggle a response on or off using the submenu.
Note This selection affects all variables being plotted.
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Revising a Scenario
If you modify and recalculate a scenario, its data are replotted, replacing the original
curves.
Viewing Selected Variables
By default, the design tool plots all plant inputs in a single window, and plots all plant
outputs in another. If your application involves many signals, the plots of each may be
too small to view comfortably.
Therefore, you can control the variables being plotted. To do so, right-click in a plot's
white space and select Channel Selector from the resulting menu. A dialog box
appears, on which you can opt to show or hide each variable.
Grouping Variables in a Single Plot
By default, each variable appears in its own plot area. You can instead choose to display
variables together in a single plot. To do so, right-click in a plot's white space, and select
Channel Grouping, and then select All.
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Customize Response Plots
To return to the default mode, use the Channel Grouping: None option.
Normalizing Response Amplitudes
When you're using the Channel Grouping: All option, you might find that the variables
have very different scales, making it difficult to view them together. You can choose to
normalize the curves, so that each expands or contracts to fill the available plot area.
For example, the plot below shows two plant outputs together (Channel Grouping: All
option). The outputs have very different magnitudes. When plotted together, it's hard to
see much detail in the smaller response.
The plot below shows the normalized version, which displays each curve's variations
clearly.
The y-axis scale is no longer meaningful, however. If you want to know a normalized
signal's amplitude, use a data marker (see “Adding a Data Marker” on page 8-69).
Note that the two data markers on the plot below are at the same normalized y-axis
location, but correspond to very different amplitudes in the original (unnormalized)
coordinates.
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