http://dspace.nitrkl.ac.in/dspace D E S I G N, D E V E L O P M E N T A N D P E R F O R M A N C E
EVALUATION OF INTELLIGENT SENSORS
Sar oj K umar Mishra
Department of Electronics & Communication Engineering
National Institute of Technology Rourkela
Des ign , D evelo pme n t an d Perf orma nce
Evalu atio n of In tellig ent S ens o rs
A thesis submitted in partial fulfillment of the requirements for the degree of
Master of Technology (Research)
in
Electronics & Communication Engineering
2004-2006
By
Saroj Kumar Mishra
Roll No: 60407003
Under the supervision of
Dr. Ganapati Panda FNAE, FNASc.
Professor
Department of Electronics & Communication Engineering
National Institute of Technology Rourkela
May, 2006
Contents
Certificate
Bio-data of the Candidate
Acknowledgement
i
Model of CPS
39
Switched Capacitor Circuit (SCC)
43
Development of Intelligent Models
of the CPS
43
Abstract
ii
Direct Modeling
44
List of Abbreviations
iv
Inverse Modeling
44
v
Simulation Studies
45
List of Figures
The MLP based Direct Modeling
45
CHAPTER 1
The MLP based Inverse Modeling
46
Introduction
The FLANN based Direct Modeling
48
The FLANN based Inverse Modeling
49
The RBFNN based Direct Modeling
51
The RBFNN based Inverse Modeling
52
Brief Introduction
1
Sensor Fundamentals
2
Characteristics of Sensor
Literature Survey
3
8
Problem Formulation
11
Chapter-wise Organization
13
References
15
Summary and Discussion
54
References
54
CHAPTER 4
Performance Evaluation of Fixed-Point
FLANN based CPS model
CHAPTER 2
Intelligent Techniques and
Algorithms
Brief Introduction
20
Adaptive Techniques
21
Least Mean Square (LMS) Algorithm
21
Recursive Least Square (RLS)
Algorithm
Artificial Neural Network (ANN)
Brief Introduction
56
Fixed-Point Analysis
57
Fixed-point FLANN algorithm for
Direct Modeling
57
Fixed-point FLANN algorithm for
Inverse Modeling
60
Simulation Studies
63
23
Summary and Discussion
68
23
References
68
Single Neuron Structure
24
Multi-Layer Perceptron (MLP)
26
CHAPTER 5
Functional-Link Artificial Neural
Network (FLANN)
30
Nonlinear Compensation of LVDT using
Different ANN Techniques
Radial Basis Function based NN
(RBFNN) Technique
33
Brief Introduction
69
Summary and Discussion
35
Linear Variable Differential
Transformer (LVDT)
70
References
36
Nonlinearity Compensation of LVDT
73
Simulation Studies
74
CHAPTER 3
Direct and Inverse modeling of
Capacitive Pressure Sensor using
Intelligent Techniques
Brief Introduction
38
Capacitive Pressure Sensor (CPS)
39
MLP based Nonlinearity
Compensator
75
FLANN based Nonlinearity
Compensator
76
CFLANN based Nonlinearity
Compensator
78
Computational Complexity
83
Summary and Discussion
83
References
84
CHAPTER 6
Adaptive Nonlinear Compensation
of Control Valve Actuator using
Soft Computing Techniques
Brief Introduction
86
Control Valve Actuator
87
Adaptive Inverse Models
91
LMS based Inverse Model
91
MLP based Inverse Model
92
FLANN based Inverse Model
93
Summary and Discussion
93
References
94
CHAPTER 7
Conclusions
Brief Introduction
96
Scope for Further Research Work
98
Publications out of the Research Work
99
Department of Electronics & Communication Engineering
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
ORISSA, INDIA – 769 008
CERTIFICATE
This is to certify that the thesis titled “Design, Development and Performance Evaluation of Intelligent Sensors”, submitted to the National Institute of Technology, Rourkela by Mr. Saroj Kumar Mishra, Roll No. 60407003 for the award of the degree of Master of Technology (Research) in Electronics & Communication Engineering, is a bona fide record of research work carried out by him under my supervision and guidance. The candidate has fulfilled all the prescribed requirements. The thesis, which is based on candidate’s own work, has not been submitted elsewhere for a degree/diploma. In my opinion, the thesis is of standard required for the award of a Master of Technology (Research) degree in Electronics & Communication Engineering. To the best of my knowledge, he bears a good moral character and decent behavior. Prof. G. Panda, FNAE, FNASc. Department of Electronics & Communication Engineering National Institute of Technology Rourkela‐769 008 (INDIA) Email: [email protected]
BIO-DATA OF THE CANDIDATE
Name of the Candidate
:
Saroj Kumar Mishra
Father’s Name
:
Bijaya Kumar Mishra
Permanent Address
:
C/O- B.K. Mishra
Bapuji Nagar-3rd lane
Medical Bank colony
Berhampur-760 004.
Orissa, India.
Email ID
:
[email protected]
ACADEMIC QUALIFICATION
•
Continuing M. Tech. (Research) in Electronics & Communication
Engineering, National Institute of Technology Rourkela, Orissa
(INDIA).
•
M.Sc. in Electronics, Berhampur University, Orissa (INDIA).
•
B. Sc. Physics (Honours), Aska Science College, Berhampur University,
Orissa (INDIA).
PUBLICATION
ƒ Published/Accepted 06 papers in National
Conferences;
ƒ Communicated 03 papers to International Journals.
and
International
ACKNOWLEDGEMENT
I would like to take this opportunity to extend my deepest gratitude to my teacher and
supervisor, Prof. Ganapati Panda, for his continuous encouragement and active guidance. I
am indebted to him for the valuable time he has spared for me during this work. He is always
there to meet and talk about my ideas, to proofread and mark up my research papers and
chapters, and to ask me good questions to help me think through my problems.
I am very much thankful to Prof. K.K. Mahapatra, Head, Department of Electronics and
Communication Engineering, for his continuous encouragement. Also, I am indebted to
him who provided me all official and laboratory facilities.
I am grateful to Dr. S. Meher and Prof. T.K. Dan for his valuable suggestions and comments
during this research period.
My sincere thanks go to Prof. G.S. Rath, Dr. S.K. Patra, Prof. S. Pramanik, Dr. B. Majhi, and
Prof. B.D. Sahoo whose valuable suggestions helped me a lot in completing this thesis.
I would like to thank Dr. Debi Prasad Das, Scientist, CEERI, Pilani for his kind support
during the on period of my thesis work. In addition, let me thank all my friends Sunil, Manas,
Sushant, Pankaj, Rana Babu, Vamsi, Alekhika, Babita Madam, Sabuj, Debi, Nilamani sir,
Jitendra sir and Peter for their great support and encouragement during the research period.
Also, I am thankful to all the non-teaching staffs of ECE Department for their kind
cooperation.
During the course of this work, I am supported in part by the research project of ISRO and
VLSI-SMDP sponsored by DIT, Govt. of India. I am really thankful to them.
Last but not the least, I take this opportunity to express my regards and obligation to my
parents and family members for educating me in all aspects. I can never forget for their
unconditional support and encouragement to pursue my interests.
Saroj Kumar Mishra
i
ABSTRACT
Many electronic devices, instruments and sensors exhibit inherent nonlinear input-output
characteristics. Nonlinearity also creeps in due to change in environmental conditions such as
temperature and humidity. In addition, aging of the sensors also introduce nonlinearity. Due to
such nonlinearities direct digital readout is not possible. As a result the devices or sensors are
used only in the linear region of their characteristics. In other words the usable range of these
devices gets restricted due to nonlinearity problem. The accuracy of measurement is also
affected if the full range of the instrument is used. The nonlinearity present in the
characteristics is usually time-varying and unpredictable as it depends on many uncertain
factors stated above. Hence the prime objective of the thesis is to study the nonlinearity
problem associated with these devices and suggest novel methods of circumventing these
effects by suitably designing intelligent systems. In the present investigation, sensors such as,
capacitive pressure sensor (CPS), linear variable differential transformer (LVDT) and control
valve actuator are selected for adaptive linearization. In corporation of the intelligent inverse
model is in series with the nonlinear device alleviates the associated nonlinearity and permits
accurate measurement for the full dynamic range. The existing fixed type nonlinearity
compensators are not be effective for time varying nonlinear devices. However, adaptive
techniques using Artificial Neural Network (ANN) are better candidates for solving such
problems. Therefore, in the present thesis, the following major tasks have been carried out.
(i)
Development of adaptive system identification model (direct model) of the CPS for
analyzing the inherent time varying nonlinearity present in their characteristics mainly
due to the variation in temperatures,
(ii)
Development of an inverse model of the CPS for the full range of operations and
placing in series with the device for achieving time varying nonlinearity compensation.
These models (direct and inverse) have been developed using:
(a)
Adaptive tap-delay filter
(b)
The Multi-Layer Perceptron (MLP)
(c)
The Radial Basis Function based Neural Network (RBFNN)
(d)
The Functional-Link Artificial Neural Network (FLANN)
ii
(iii)
To assess the performance of the direct and inverse models in real-time
implementation, fixed-point analysis is required. Hence, the effect of fixed-point
implementation of the low complexity FLANN based direct and inverse model of the
CPS has been analyzed. The performance of fixed-point FLANN model at different
word lengths has been evaluated.
(iv)
An adaptive nonlinear compensator (or inverse model) of the LVDT by taking the
experimental data has been developed. Different ANN models used to develop the
adaptive nonlinear compensators are:
(a)
The MLP
(b)
The FLANN
(c)
The Cascaded FLANN (CFLANN)
This models when connected in series with the LVDT offers extended linearity
characteristics. The overall system provides accurate measurement for the whole range.
The performance comparison in terms of computational complexity and accuracy of
measurement of various models has been made.
(v)
Nonlinearity compensator of equal percentage control valve actuator have also been
developed using the following adaptive techniques:
(a)
The LMS filter
(b)
The MLP
(c)
The FLANN
The performance of various adaptive models and their comparisons has been assessed using
three typical sensors. It is demonstrated that the FLANN model provides quite satisfactory
performance and involves less complexity. Hence the FLANN based intelligent models are
preferable compared to the MLP and RBFNN based compensators.
iii
LIST OF ABBREVIATIONS
ADC
Analog-to-Digital Converter
AKF
Adaptive Kalman Filtering
ANN
Artificial Neural Network
ASP
Adaptive Signal Processing
BP
Back-Propagation algorithm
CFLANN
Cascaded Functional-Link Artificial Neural Network
CPS
Capacitive Pressure Sensor
DAC
Digital-to-Analog Converter
FL
Fuzzy Logic
FLANN
Functional-Link Artificial Neural Network
FPGA
Field Programmable Gate Array
FSO
Full-Scale Output
LMS
Least Mean Square
LVDT
Linear Variable Differential Transformer
MDS
Minimum Detectable Signal
MLP
Multi-Layer Perceptron
MR
Measured Range
MSE
Mean Square Error
PEL
Percentage of Linearity
PIM
Plug-In-Module
RBFNN
Radial Basis Function based Neural Network
RLS
Recursive Least Square
SCC
Switched Capacitor Circuit
VLSI
Very Large Scale Integration
iv
LIST OF FIGURES
1.1
General structure of a sensor
2
1.2
Repeatability in y-x coordinates
4
1.3 (a), (b)
Nonlinearity with (a) best-fit characteristics, (b) end-points characteristics
5
1.4
Hysteresis Curve
6
2.1
Adaptive filter using LMS algorithm
22
2.2
Structure of a Single Neuron
24
2.3
Different types of nonlinear activation function
25
2.4
Structure of multilayer perceptron
27
2.5
Neural network using BP algorithm
28
2.6
Structure of the FLANN model
32
2.7
Structure of radial basis function based neural network
33
3.1
Typical structure of a CPS
39
3.2
Range of linear region of the CPS
41
3.3
Range of linear region of the CPS at different temperature
42
3.4
Switched Capacitor Interface circuit
43
3.5
A scheme of direct modeling of a CPS and SCC using ANN based
model
Development of an inverse ANN model of CPS
44
3.7
Plots of true and estimated forward characteristics of the CPS at -200 C,
300 C and 700 C by MLP
46
3.8
Plots of forward, inverse and overall characteristics of the CPS by MLP
47
3.6
44
0
3.9
Plots of true and estimated forward characteristics of the CPS at -20 C,
300 C and 1000 C by FLANN
48
3.10
Plots of forward, inverse and overall characteristics of the CPS by power
series FLANN
50
3.11
Plots of forward, inverse and overall characteristics of the CPS by
trigonometric FLANN
51
3.12
Plots of true and estimated forward characteristics of the CPS at -500 C,
100 C and 800 C by RBFNN
52
3.13
Plots of forward, inverse and overall characteristics of the CPS by
RBFNN
53
v
4.1
A scheme of direct modeling of a CPS using fixed-point FLANN model
58
4.2
Direct modeling of CPS using fixed-point FLANN used for simulation
study
60
4.3
Fixed-point inverse modeling of CPS by FLANN
60
4.4
Inverse modeling of CPS by fixed-point FLANN used for simulation
study
63
4.5
Simulation results of fixed-point FLANN direct model at 300C, of the
word lengths t = 4, 8, 12 and 16-bits
64
4.6
Plots of forward, inverse and overall characteristics of the CPS by fixedpoint FLANN at t = 8-bits
65
4.7
Plots of forward, inverse and overall characteristics of the CPS by fixedpoint FLANN at t = 12-bits
66
4.8
Plots of forward, inverse and overall characteristics of the CPS by fixedpoint FLANN at t = 16-bits
67
5.1
Cross-sectional view of an LVDT
70
5.2
Range of linear region of LVDT
73
5.3
Scheme of an intelligent nonlinearity compensator of LVDT
73
5.4
Practical set-up of LVDT after training
74
5.5
Response due to MLP (1-100-50-1) network
76
5.6
Response due to trigonometric series FLANN with P = 10
77
5.7
Response due to trigonometric series FLANN with P = 50
77
5.8
Structure of CFLANN model
78
5.9
Plots of response of the CFLANN model
82
6.1
Cross-sectional view of a control valve actuator
87
6.2
Set-up for compensating nonlinearity of control valve actuator
91
6.3
Plots of response of the LMS based inverse model
92
6.4
Plots of response of the MLP based inverse model
92
6.5
Plots of response of the FLANN based inverse model
93
vi
1
Chapter
Introduction
T
HE SENSORS are devices which, for the purpose of measurement, turn physical
input quantities into electrical output signals, their output-input and output-time
relationship being predictable to a known degree of accuracy at specified environmental
conditions. The definition of sensors or transducers according to The Instrument Society of
America is “a device which provides a usable output in response to a specified measurand”. Here the output is
an ‘electrical quantity’ and measurand is a ‘physical quantity’ [1.1-1.3]. It can also be defined as an
element that senses a variation in input energy to produce a variation in another or same form of
energy is called a sensor, whereas, transducer involves a transduction principle which converts a
specified measurand into an usable output.
This Chapter deals with the fundamental of the sensors and their characteristics. Section 1.1
deals with the sensor fundamental along with its characteristics. The literature survey is discussed
in Section 1.2. Problem formulation of the thesis is depicted in Section 1.3. Finally the Chapterwise organization of the thesis is presented in Section 1.4.
1
I N T R O D U C T I O N
1.1 Sensor Fundamentals
The sensor consists of several elements or blocks such as sensing element, signal conditioning
element, signal processing element and data presentation element [1.2].
Sensing element: This is in contact with the process and gives an output which depends in some
way on the variable to be measured. Examples are: thermocouple where millivolt emf depends
on temperature, capacitive pressure sensor where capacitance of a chamber depends on
pressure, linear variable differential transformer where emf at the secondary coil depends on
displacement, etc.
If there is more than one sensing element in a system, the element in contact with the process is
termed the primary sensing element, the others secondary sensing elements.
Signal conditioning element: This takes the output of the sensing element and converts it into a form
more suitable for further processing, usually a dc voltage, dc current or frequency signal.
Examples are: deflection bridge which converts an impedance change into a voltage change,
amplifier which amplifies millivolts to volts, etc.
Signal processing element: This takes the output of the conditioning element and converts it into a
form more suitable for presentation. Examples are: analog-to-digital converter (ADC) which
converts a voltage into a digital form for input to a computer; a microcontroller which calculates
the measured value of the variable from the incoming digital data.
Data presentation element: This presents the measured value in a form which can be easily
recognized by the observer. Examples are: a simple pointer-scale indicator, chart recorder, and
alphanumeric display etc.
Input
True
value
Sensing
element
Signal
conditioning
element
Signal
processing
element
Fig. 1.1 General structure of a sensor
2
Data
presentation
element
Output
Measured
value
I N T R O D U C T I O N
1.1.1 Characteristics of Sensor
Sensors or all measurement systems, have two general characteristics [1.3], i.e. (i) static
characteristics, and (ii) dynamic characteristics.
The Static characteristics comprise of:
(a) Accuracy: These characteristics specified by error. And it is given by,
ε a % = ( xm − xt ) xt ×100
(1.1)
where t is for true value, m for measured value and variable x stands for measurand. This is often
expressed for the full scale output and is given by,
ε fso % =
( xm − xt ) ⋅100
(1.2)
x fso
Obviously,
ε fso ≤ ε a
For multi error systems the overall performance in terms of error can be assessed either through
(i) the worst case approach which assumes that all errors add in the same direction so that the
overall error is very high being the linear sum of all the performance errors, or through (ii) the
root mean square approach which is optimistic as well as practical, when the total performance
error can be assessed as,
⎧
⎫
ε o = ⎨∑ ε i2 ⎬
⎩
i
1
2
(1.3)
⎭
(b) Precision: It describes how far a measured quantity is reproducible as also how close it is to
the true value.
3
I N T R O D U C T I O N
A term repeatability is similar to precision
9
which is the difference in output y at a given
Repeatabilty
8
7
value of the input x when this is obtained in
6
y (% FSO)
two consecutive measurements. It may be
expressed as % Full-scale Output (FSO). Fig.
1.2 shows the repeatability.
5
4
3
2
(c) Resolution: It is defined as the smallest
1
incremental change in the input that would
0
0
produce a detectable change in the output.
1
2
3
4
5
6
x (% range)
7
8
xm
9
10
Fig. 1.2 Repeatability in y-x coordinates
This is often expressed as percentage of the
measured range, MR. The measured range is defined as the difference of the maximum input
and the minimum input, xmax − xmin = MR . For a detectable output ∆y , if the minimum
change in x is ( ∆x )min , then the maximum resolution is
Rmax ( % ) = 100 ( ∆x )min MR
(1.4)
Over the range of operation an average resolution has also been defined as
N
Rav ( % ) = 100∑ ∆xi
( n ⋅ MR )
(1.5)
i =1
(d) Minimum Detectable Signal (MDS): Noise in a sensor occurs due to many reasons internal sources, or, fluctuations due to externally generated mechanical and electromagnetic
influences. Details of noise are considered on individual merits and often an equivalent noise
source is considered for test purposes.
If the input does not contain any noise the minimum signal level that produces a detectable
output from the sensor is determined by its noise performance or noise characteristics. For this,
the equivalent noise source is connected to the input side of the ideal noiseless sensor to yield an
output which is the actual output of the sensor. The MDS is then taken as the RMS equivalent
input noise. When a signal exceeds this value it would be detectable signal.
(e)Threshold: This is the smallest input change that produces a detectable output at zero value
condition of the measurand.
4
I N T R O D U C T I O N
(f) Sensitivity (S): This is defined as the ratio of the incremental output ( ∆y ) to incremental
input ( ∆x ), i.e.
S = ∆y ∆x
(1.6)
In normalized form this is written as
Sn = ( ∆y ∆x ) ( y x )
(1.7)
If sensitivity changes or the output level changes with time, temperature and/or any other
parameters without any change in input level, drift is said to occur in the system which often
leads to instability.
(g) Selectivity and Specificity: The output of a sensor may change when affected by
environmental parameters or other variables and this may appear as an unwanted signal. The
sensor is then said to be non-selective. It is customary to define selectivity or specificity by
considering a system of n sensors each with output yk (k=1, 2, …, n). The partial sensitivity Sjk is
defined as the measure of the kth sensor to these other interfering quantities or variables xj as
S jk = ∆yk ∆x j
(1.8)
A selectivity matrix would thus be obtained with Sjk as jkth entry. Obviously, an ideal selective
system will have only the diagonal entries Sjj in the selectivity matrix. An ideally specific system is
characterized by having a matrix with a single element in the diagonal. The selectivity λ is:
⎡
⎤
⎢
⎥
S jj
⎥ , j = 1, 2,..., n
λ = Min ⎢ n
⎢ S −S ⎥
jk
jj
⎢⎣ ∑
⎥⎦
k =1
(1.9)
Thus, for a selective group the denominator tends to zero and λ → ∞ . Also, specificity is a
special case of selectivity.
5
I N T R O D U C T I O N
(h) Nonlinearity: The deviation from linearity, which itself is defined in terms of superposition
principles, is expressed as a percentage of the full scale output at a given value of the input.
Nonlinearity can, however, be specified in two different ways:
•
deviation from best fit straight line obtained by regression analysis, and
•
deviation from a straight line joining the end points of the scale.
These are shown in Figs. 1.3 (a) and (b). The maximum nonlinearity in the first method is always
10
10
9
9
8
8
7
7
6
6
5
Output
Output
less than the maximum nonlinearity in the second method.
M ax linearity
4
5
Best fit
straight line
3
3
2
2
1
1
0
M ax Nonlinearity
4
0
a
1
2
b
3
4
5
Input
6
7
8
9
0
10
End points
based line
0
1
2
3
4
5
Input
6
7
8
9
10
Fig. 1.3 Nonlinearity with (a) best-fit characteristics, (b) end-points characteristics
A consequence of nonlinearity is distortion, which is defined as the deviation from an expected
output of the sensor or transducer. It also occurs due to presence of additional input
components.
If deviation at each point of the experimental curve is negligibly small from the corresponding
point in the theoretical curve or a curve using least square or other standard fits, the sensor is
said to have conformance which is quantitatively expressed in % FSO at any given value of the
input.
(i) Hysteresis: It is the difference in the output of the sensor y for a given input x when x
reaches the value in upscale and downscale directions as shown in Fig. 1.4. The causes are
different for different types of sensors. In magnetic types, it is lag in alignment of the dipoles,
while in semiconductor types it is the injection type slow traps producing the effect, and so on.
6
I N T R O D U C T I O N
10
Downward reading
Upward reading
9
8
7
Output
6
5
4
3
2
1
0
0
1
2
3
4
5
Input
6
7
8
9
10
Fig. 1.4 Hysteresis curve
(j) Output Impedance: It is a characteristic, which is to be considered on individual merit. It
puts great restriction in interfacing, specifically in the choice of the succeeding stage.
(k) Isolation and Grounding: Isolation is necessary to eliminate or at least reduce undesirable
electrical, magnetic, electromagnetic and mechanical coupling among various parts of the system
and between the system and the environment. Similarly, grounding is necessary to establish a
common node among different parts of the system with respect to which potential of any point
in the system remains constant.
The dynamic characteristics involve determination of transfer function, frequency response,
impulse response and step response and then evaluation of the time-dependent outputs. The
two important parameters in these connections are:
•
Fidelity determined by dynamic error, and
•
Speed of response determined by lag.
For determining the dynamic characteristics different inputs are given to the sensor and the response
characteristics are to be studied. With step input, the specifications in terms of the time constant
of the sensor are to be made. Impulse response as well as its Fourier transform are also to be
considered for time domain as well as frequency domain studies.
Environmental Parameters: The external variables such as temperatures, pressure, humidity,
vibration, etc. which affects the performance of the sensor. Aging is also an important parameter
7
I N T R O D U C T I O N
of the sensor. These parameters are not the ones that are to be sensed. For any environmental
parameters, the performance of the sensor can be studied in terms of its effect on the static and
dynamic characteristics. For this study, one environmental parameter at a time is considered variable
while others are fixed.
1.2 Literature Survey
The capacitive pressure sensor (CPS), in which the capacitance of a chamber changes with
application of pressure finds extensive applications because of its low power consumption
and high sensitivity. It is observed that many sensors exhibit nonlinear input-output
characteristics. Due to such nonlinearities direct digital readout is not possible. As a
result we employ the sensors only in the linear region of their characteristics. Generally,
CPSs are known to have no turn-on temperature drift, high sensitivity, robust structure and
is less sensitive to environmental effects [1.4]. However, its output is nonlinear with respect
to input changes and the sensitivity in the near linear region is not high enough to ignore
many stray capacitance effects [1.5]. Several studies have been carried out to fulfill the
demand of low cost, high sensitivity, resolution and mass producibility. The nonlinear
response characteristics of these sensors give rise to several difficulties including on-chip
interface, direct digital readout and calibration. Nonlinearity also creeps in due to change in
environmental conditions such as temperature and humidity. In addition aging of the
sensors also introduces nonlinearity. Hence, to achieve an easy digital interface for direct
readout, the response characteristics of the sensor should be linear and independent of
variations in the environmental conditions such as temperature, humidity and aging. To
compensate the nonlinear response characteristics of the CPS, several techniques have been
suggested. A switched-capacitor charge balancing technique [1.6], a ROM based look-up
table method and a nonlinear encoding scheme have been proposed [1.7]. And also the
demand of low-cost and small range linear output requirements have led to the
development of the piezo-resistance bridge-based integrated pressure sensor. A thindiaphragm diffused piezo-resistive pressure sensor for biomedical instrumentation has been
developed using monolithic IC techniques. Integrated circuit technology offers several
important advantages in the fabrication of pressure sensors for biomedical instrumentation.
In particular, the ability to control geometries precisely on very small dimensions promises
to allow the fabrication of sensors which combine small size with reliability and stability.
8
I N T R O D U C T I O N
Furthermore, the ability to batch fabricate such transducers should make them relatively
inexpensive. Pressure transducers which depend on the piezo-resistive effect have several
advantages over those based on other pressure sensitive effects. Piezo-resistive transducers
operate at low stress levels, exhibit a resistance change which is a linear function of
pressure over a wide range and have high sensitivity to pressure changes. Because of this
high sensitivity, stylus arrangements to achieve pressure multiplication are not needed; this
fact simplifies the assembly into a completed package [1.8]. Also for nonlinearity estimation
and to obtain a direct digital readout of a CPS, an Artificial Neural Network (ANN) based
modeling technique has been proposed with quite satisfactory performance [1.9,1.10];
however without any consideration of change in the ambient temperature. When we take
temperature into consideration, and the ambient temperature changes frequently, the
situation becomes very complicated and in this case the problem becomes two-dimensional
(2-D) and adaptive signal processing techniques are required to make necessary corrections
to obtain a correct digital readout. A scheme of microcomputer based 2-D lookup table
method has been proposed [1.11]. Another method of ∆-Σ demodulator and complex
signal processing techniques for the sensor model have also been reported [1.12].
By taking temperature into consideration, a 3-layer Multi-Layer Perceptron (MLP) based ANN
has been proposed for auto-calibration and nonlinear compensation of a CPS [1.13], under
variation of ambient temperature. In this method switched capacitor circuit (SCC) converts the
change in capacitance into an equivalent voltage. The intelligent behavior is implanted into the
sensor by training the ANN to adapt to a temperature range of -200 C to 700 C. In the direct
modeling, the ANN is trained in a parallel mode to estimate the capacitance of the CPS. This
model may be used for the purpose of on-line fault detection and quality control of the sensor
during its production. In the inverse modeling, the ANN is trained in a series mode to estimate
the applied pressure which is independent of ambient temperature. A plug-in-module (PIM) is
proposed to implement the scheme on-line.
In many control system applications Linear Variable Differential Transformer (LVDT)
plays an important role to measure the displacement [1.1,1.14]. The performance of
the control system depends on the performance of the sensing element. It is observed
that the LVDT exhibits the same nonlinear input-output characteristics. Due to such
nonlinearities direct digital readout is not possible. As a result we employ the LVDTs
9
I N T R O D U C T I O N
only in the linear region of their characteristics. In other words their usable range gets
restricted due to the presence of nonlinearity. If the LVDT is used for full range of its
nonlinear characteristics, accuracy of measurement is severely affected.
The nonlinearity present is usually time-varying and unpredictable as it depends on
many uncertain factors. Attempts have been made by many researchers to increase the
range of linearity of LVDT. In the conventional design of LVDT sophisticated and
precise winding machines are used to compensate the nonlinearity effects. The
nonlinearity compensation can also be achieved by square coil method in which the
core moves perpendicular to the axis instead of along the axis [1.14]. A selfcompensated LVDT has been modeled using a dual secondary coil which is insensitive
to the variation in excitation current and frequency [1.15]. Some DSP techniques have
been employed in LVDT to achieve better sensitivity and to implement the signal
conditioning circuits [1.16-1.18].
The control valve actuators are the most common actuators in the process industries. They
are found in process plants, manipulating flows to maintain controlled variable at their set
points. A control valve acts as a variable restriction in a process pipe. By changing its opening
it changes the resistance to flow and, thus, the flow itself [1.19]. As discussed earlier
sections, many instruments or sensors exhibit inherent nonlinear input-output
characteristics. The same effect is also observed in case of the control valve actuators. The
equal percentage control valve actuator devices have nonlinearities in their inherent flow
characteristics. But when it is put in a pipe line, it acts as nearly linear control valve, called
installed characteristics. But this linearity depends on the pressure difference across the valve.
As the pressure difference across the valve varies, it is always required to compensate to get
very accurate linear characteristics. These nonlinearities cannot be compensated using
feedback linearization techniques because they do not appear in the feedback path but
rather in the feed-forward path. Some of other fundamental actuator nonlinearities include
friction, deadzone, saturation, backlash and hysteresis. Amongst these the friction, the
deadzone and the saturation are static nonlinearities, whereas the backlash and the hysteresis
are dynamic nonlinearities.
Attempts have been made by many researchers to compensate the static and dynamic nonlinearities
present in the actuators. The nonlinear effect due to friction is common in all mechanical
10
I N T R O D U C T I O N
systems. So it cannot be avoided in control systems. Different friction models and their
compensation techniques are proposed in [1.20,1.21]. [1.22] described an observer based friction
compensation for a dynamical model. Robust adaptive friction compensation of the static
friction model is given in [1.23], and using reinforcement adaptive learning neural networks in
[1.24]. The deadzone effect is present in many electro-mechanical systems. Standard techniques
for overcoming deadzone are variable structure control and dithering [1.25]. The deadzone
compensation in nonlinear systems using artificial neural networks (ANN) is proposed in [1.26].
Backlash and hysteresis nonlinearities usually appear in the feedforward path, and their
compensators are based on applications of dynamic inversion of nonlinear systems. Tao and
Kokotovic [1.27], developed adaptive backlash and hysteresis compensation. The dynamic
inversion is well known technique in aerospace applications. A compensated inverse dynamics
approach using adaptive and robust control techniques is presented in [1.28]. The magnitude and
the rate of actuator inputs exhibit the saturation nonlinearities. When an actuator has reached
such an input limit, it is said to be “saturated”, since efforts to further increase the actuator
output would not result in any variation of the output. To overcome this problem, the general
actuator saturation compensator scheme is developed in [1.29]; Design of a robust anti-windup
controller based on the Lyapunov approach to accommodate the constraints and disturbance is
described in [1.30]. Both static and dynamic compensators using neural networks are considered
in [1.31].
1.3 Problem Formulation
The sensors exhibit nonlinear characteristics which limit the dynamic range of these devices. As
a result the direct digital readout of the output is not possible for the whole input range of the
sensors. In addition the full potential of the sensors cannot be utilized. Thus there is a real
challenge for designing and implementing novel sensors which circumvent the nonlinearity
problems associated with them. In recent past few attempts have been made but not much has
been achieved in this direction. The second problem is the accuracy of measurements in these
sensors which is greatly affected by aging of the sensor, temperature and humidity variations.
Both these issues detoriate the accuracy of measurement. Therefore nonlinearity compensation
is required to achieve accurate measurement under adverse conditions. Most of the existing
nonlinearity compensation techniques work well for fixed and known type of nonlinearities. But
11
I N T R O D U C T I O N
in actual practice the nonlinearity behavior of the sensors changes with time and is usually
unknown. The adaptive inverse model of the CPS can be designed and connected in series with
these devices, so that the associated nonlinearity can be compensated and direct digital readout is
possible for the whole input range. A scheme of direct modeling and inverse modeling of the
CPS using different ANN structure has been developed. The direct modeling is proposed for
calibration of inputs and estimation of internal parameters of the CPS. The purpose of the direct
model is to obtain an ANN model of the CPS in such a way that the outputs of the CPS and the
ANN match closely. Once a model of the CPS is available, it may be used for fault detection of
the sensor. The CPS (interfaced with SCC) output provides a voltage signal proportional to the
capacitance change due to the applied input pressure. And the inverse modeling is proposed for
estimation of applied input pressure. The adaptive inverse model can be developed using Least
Mean Square (LMS) algorithm, Recursive Least Square (RLS) algorithm [1.32,1.33], and
Adaptive Kalman Filtering (AKF) [1.32]. The intelligent techniques such as Artificial Neural
Networks (ANN) [1.34-1.36], Fuzzy Logic (FL) and Neuro-Fuzzy algorithms [1.37] are also
potential candidates for developing adaptive inverse model of these devices.
In this thesis the following burning issues relating CPS, LVDT, and control valve actuators are
analyzed and solution to these problems has been suggested. The adaptive modelings of these
instruments have been carried out using LMS and different ANN techniques. The direct modeling
of these devices provides information about the degree of nonlinearity and how it changes with
the change in environment. The adaptive inverse model of these nonlinear instruments have also
been developed taking environmental conditions into account. Different efficient ANN
structures such as Multi-Layer Perceptron (MLP) [1.34,1.38], Functional-Link ANN (FLANN)
[1.39-1.41], and Radial Basis Function based NN (RBFNN) [1.34-1.37,1.42] have been
designed for nonlinear compensation and their performance has been assessed through fictitious
data of the CPS. The proposed techniques are novel ones and are shown to yield excellent
performance. The simulation results for CPS, LVDT and control valve actuator for achieving
adaptive linearization under different conditions are also presented in this thesis.
The LVDT exhibits nonlinearity behaviour similar to the CPS when the core is moved
towards any one of the secondary coils. In the primary coil region (middle) of the
characteristics, the core movement is almost linear. Because of that, the range of
operation is limited to the primary coil region only. Different attempts have been made
12
I N T R O D U C T I O N
for the linearity enhancement of LVDT. In this thesis, adaptive nonlinearity
compensators have been designed using different ANN structures such as MLP,
FLANN and cascaded FLANN (CFLANN) based model.
The control valve plays an important role in process industries, the linear type is the most
suitable control valve due to its linear relationship between input and output. But the linear type
is associated with constant sensitivity for which the pressure is also remains constant. So the
application gets restricted. To compensate the nonlinearity present in the equal percentage
control valve, an adaptive inverse model have been proposed by using different adaptive
algorithms such as LMS, MLP and FLANN techniques. A comparative study has been made
between these three models, and observed that the LMS based inverse model is more efficient
and computationally less complex.
1.4 Chapter-wise Organization
The chapter-wise organization of the thesis is outlined below.
Chapter-1 Introduction
Brief Introduction
1.1
Sensor Fundamentals
1.1.1
Characteristics of Sensors
1.2
Literature Survey
1.3
Problem Formulation
1.4
Chapter-wise Organization
References
Chapter-2 Intelligent Techniques and Algorithms
Brief Introduction
2.1
2.2
Adaptive Techniques
2.1.1
Least Mean Square (LMS) Algorithm
2.1.2
Recursive Least Square (RLS) Algorithm
Artificial Neural Network (ANN)
2.2.1
Single Neuron Structure
2.2.2
Multilayer Perceptron (MLP)
2.2.3
Functional-Link Artificial Neural Network (FLANN)
13
I N T R O D U C T I O N
2.2.4
2.3
Radial Basis Function based NN (RBFNN) Technique
Summary and Discussion
References
Chapter-3 Direct and Inverse Modeling of Capacitive Pressure Sensor using
Intelligent Techniques
Brief Introduction
3.1
3.2
Capacitive Pressure Sensor (CPS)
3.1.1
Model of CPS
3.1.2
Switched Capacitor Circuit (SCC)
Development of Intelligent Models of the CPS
A scheme of direct modeling and inverse modeling of the CPS using different
ANN structure has been proposed in this section.
3.3
3.4
3.2.1
Direct Modeling
3.2.2
Inverse Modeling
Simulation Studies
3.3.1
The MLP based Direct Modeling
3.3.2
The MLP based Inverse Modeling
3.3.3
The FLANN based Direct Modeling
3.3.4
The FLANN based Inverse Modeling
3.3.5
The RBFNN based Direct Modeling
3.3.6
The RBFNN based Inverse Modeling
Summary and Discussion
References
Chapter-4 Performance Evaluation of Fixed-Point FLANN based CPS model
Brief Introduction
4.1
Fixed-point Analysis
4.1.1
Fixed-point FLANN algorithm for Direct Modeling
4.1.2
Fixed-point FLANN algorithm for Inverse Modeling
4.2
Simulation Studies
4.3
Summary and Discussion
References
14
I N T R O D U C T I O N
Chapter-5 Nonlinear Compensation of LVDT using Different ANN Techniques
Brief Introduction
5.1
Linear Variable Differential Transformer
5.2
Nonlinearity Compensation of LVDT
5.3
Simulation Studies
5.3.1
MLP based Nonlinearity Compensator
5.3.2
FLANN based Nonlinearity Compensator
5.3.3
CFLANN based Nonlinearity Compensator
5.4
Computational Complexity
5.5
Summary and Discussion
References
Chapter-6 Adaptive Nonlinear Compensation of Control Valve Actuator using Soft
Computing Techniques
Brief Introduction
6.1
Control Valve Actuator
6.2
Adaptive Inverse Models
6.3
6.2.1
LMS based Inverse Model
6.2.2
MLP based Inverse Model
6.2.3
FLANN based Inverse Model
Summary and Discussion
References
Chapter-7 Conclusions
Brief Introduction
7.1
Scope for Further Research Work
Publications out of Research Work
REFERENCES
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Hermann K.P. Neubert, “Instrument Transducers: An Introduction to their performance &
design”, 2nd edition, Oxford University Press.
[1.2]
John P. Bentley, “Principles of Measurement Systems”, 3rd edition, Pearson Education.
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D Patranabis, “Sensors and Transducers”, 1st edition, Wheeler Publishing, 1997.
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H. Matsumoto, H. Shimizu and K. Watanabe, “A switched capacitor charge
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M. Yamada, T. Takabayashi, S. I. Notoyama and K. Watanabe, “A switched
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S. Samaun, K. D. Wise, E. D. Nielsen and J. Angel, “An IC piezoresistive pressure
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Aug.2000.
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[1.14] Y. Kano, S. Hasebe, and H. Miyaji, “New linear variable differential transformer
with square coils,” IEEE Transactions on Magnetics, Vol. 26, Issue: 5, pp.2020 – 2022,
Sep 1990.
[1.15] S.C. Saxena, and S.B.L. Sexena, “A self-compensated smart LVDT transducer,”
IEEE Trans. Instrumentation &. Measurement, Vol. 38, Issue: 3, pp.748 – 753, June
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[1.16] G. Y. Tian, Z, X. Zhao, R. W. Baines, N. Zhang, “Computational algorithms foe
linear variable differential transformers (LVDTs)” IEE Proc.-Sci. Meas. Technol., Vol.
144, No. 4,pp 189-192, July 1997.
[1.17] D. Crescini, A. Flammini, D. Marioli, and A. Taroni, “Application of an FFT-based
algorithm to signal processing of LVDT position sensors” IEEE Trans.
Instrumentation &. Measurement, Vol. 47, No. 5, pp. 1119-1123 Oct. 1998.
[1.18] R. M. Ford, R. S. Weissbach, and D. R. Loker, “A novel DSP-based LVDT signal
conditioner” IEEE Trans. Instrumentation &. Measurement, Vol. 50, No. 3, pp. 768774, June 2001.
[1.19] D. R. Coughanowr, “Process Systems Analysis and Control”, 2nd edition, McGraw-Hill
international editions, 1991.
[1.20] B. Armstrong-Htlouvry, P. Dupont, and C. Canudas de Wit, “A survey of models,
analysis tools and compensation methods for the control of machines with fiction,”
Automatica, vol.30, no.7, pp.1083-1138, 1994.
[1.21] C. Canudas de Wit, P. Noel, A. Aubin, and B. Brogliato, “Adaptive friction
compensation in robot manipulators: Low velocities,” Int. J. Robot. Res., vol.10, no.3,
Jun. 1991.
[1.22] H. Olsson and K. J. Astrom, “Observer-based fiction compensation,” Proc. IEEE
Conf. Decis. and Control, Kobe, Japan, Dec. 1996, pp. 4345-4350.
[1.23] S. W. Lee and J. H. Kim, “Robust adaptive stick-slip fiction compensation,” IEEE
Trans. Ind. Electron, vol. 42, no.5, pp. 474-479, Oct. 1995.
[1.24] Y. H. Kim and F. L. Lewis, “Reinforcement adaptive learning neural-net-based
friction compensation control for high speed and precision,” IEEE Trans. Control
Sysi. Technology, vol. 8, no. l, pp.118-126, Jan. 2000.
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[1.25] C. A. Desoer and S. M. Shahruz, “Stability of dithered nonlinear systems with
backlash or hysteresis,” Int. J. Contr., vol. 43, no. 4, pp. 1045-1060, 1986.
[1.26] R. R. Selmic, F. L. Lewis, “Deadzone Compensation in Nonlinear Systems using
Neural Network”, Proc. of the 37th IEEE conference of Decision & control, Dec, 1998.
[1.27] G. Tao and P. V. Kokotovic, “Adaptive Control of Systems with Actuator and Sensor
Nonlinearities”, John Wiley & Sons, New York, 1996.
[1.28] Y. D. Song, T. L. Mitchell, and H. Y. Lai, “Control of a class of nonlinear uncertain
systems via compensated inverse dynamics approach,” IEEE Trans. Automat.
Control, vol. 39, no. 9, pp. 1866-1871, Sep. 1994.
[1.29] K. J. Astrom and B. Wittenmark, “Computer-Controlled Systems: Theory and Design”,
Prentice Hall, 3rd edition, 1996.
[1.30] W. Niu and M. Tomizuka, “‘A Robust Anti-Windup Controller Design for
Asymptotic Tracking of Motion Control System Subjected to Actuator Saturation,’’
The 37th IEEE Conference on Decision and Control, Tampa, December 1998.
[1.31] R. R. Selmic, V. V. Phoha and F. L. Lewis, “Intelligent control of Actuator
Nonlinearities”, Proc. of the 42nd IEEE Conf. on Decis. & Contr., Maui, Hawaii USA,
Dec, 2003.
[1.32] S. Haykin, “Adaptive Filter Theory,” 4th Edition, Pearson Education Asia, 2002.
[1.33] B. Widrow and S.D. Sterns, “Adaptive Signal Processing” Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1985.
[1.34] S. Haykin, “Neural Networks: A comprehensive foundation” 2nd Edition, Pearson
Education Asia, 2002.
[1.35] Dayhoff E.J., “Neural Network Architecture – An Introduction” Van Norstand Reilold,
New York, 1990.
[1.36] Bose N.K., and Liang P., “Neural Network Fundamentals with Graphs, Algorithms,
Applications”, TMH Publishing Company Ltd, 1998.
[1.37] Richard O. Duda, Peter E. Hart and David G. Stork, “Pattern Classification”, 2nd
edition, John Wiley & Sons, INC.2001.
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[1.38] Fu L.M., Hsu H.H., and Principe J.C., “Incremental Back Propagation Learning
Networks”, IEEE trans. on Neural Network, vol.7, no.3, pp.757-762, 1996.
[1.39] Carpenter G.A., and Grossberg S., “The Art of Adaptive Pattern Recognition by
Self-Organizing Neural Network”, IEEE Computer Mag., vol.21, no.3, pp.77-88,
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of the Random Vector Function”, Neuro Computation, 6: 163-180, 1994.
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Channel Equalization”, Signal Processing 43(1995) 181-195, vol.43, no.2, May 1995.
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function network”, IEEE International Conference on Neural Networks, Vol. 4, Page(s):19241928, 3-6 June 1996.
19
2
Chapter
Intelligent Techniques and
Algorithms
I
N RECENT YEARS, a growing field of research in “Adaptive Systems” has resulted in a
variety of adaptive automations whose characteristics in limited ways resemble certain
characteristics of living systems and biological adaptive processes. An adaptive automation
is a system whose structure is alterable or adjustable in such a way that its behavior and
performance improves by its environment. A simple example of an adaptive system is the
automatic gain control used in radio and television receiver. The most important factor in
adaptive system is its time-varying and self-adjusting performance. Their characteristic depends
upon the input signal. If a signal is applied to the input of adaptive system to test its response
characteristic, the system adapts to this specific input and thereby changes its parameters. The
adaptation procedure is carried out using different algorithms such as the Least Mean Square
(LMS), Recursive Least Square (RLS) [2.1,2.2] etc. In many real world problems these algorithms
do not perform satisfactorily. Hence, based on the different neural architecture of human brain,
20
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
different Artificial Neural Algorithms are developed [2.3-2.6]. These are capable of mapping the
input and output nonlinearly.
This Chapter deals with different types of adaptive algorithms, which are used as a tool to
compensate nonlinearity problem of different sensors. The present Chapter is organized as
follows: Section 2.1 deals with the adaptive filtering techniques mainly the LMS and RLS
algorithms. Different types of Artificial Neural Network (ANN) such as Multi-Layer Perceptron
(MLP), Functional-Link ANN (FLANN) and Radial Basis Function based Neural Network
(RBFNN) along with their associated training algorithms are discussed in Section.2.2 and its
subsections. Finally the summary and discussion is presented in Section 2.3.
2.1 Adaptive Techniques
Filter is a primary subsystem in any signal processing system. Filters are employed to remove
undesirable signal components from the desired signal. In Adaptive Filters the coefficients can
be changed from time to time depending on the situation. Here the filter updates its coefficients
from the knowledge of the past input and the present error. The error is generated from the
reference input and actual output. The update procedure depends upon the different algorithms
used.
2.1.1 Least Mean Square (LMS) Algorithm
The general architecture of the LMS based adaptive filter is depicted in Fig. 2.1. The X is
Nth input pattern having one unit delay in each instant. This process is also called as
adaptive linear combiner [2.1,2.2]. Let X k = [ xk
xk −1 … xk − L +1 ] form of the L-by-1
T
tap input vector. Where L-1 is the number of delay elements; these inputs span a
multidimensional
Wk = ⎡ w0 k
⎣
space
denoted
by
ℵk
.
Correspondingly,
the
tap
weights
T
… w( L −1)k ⎤ form the elements of the L-by-1 tap weight vector. The
⎦
w1k
output is represented as,
L −1
yk = ∑ wlk xk −l
(2.1)
l =0
21
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
The output can be represented in vector notation as
yk = X Tk Wk = WkT X k
(2.2)
Generally for the adaptive linear combiner the other data include a “desired response” or
“training signal”, d k . This is accomplished by comparing the output with the desired
response to obtain an “error signal” ek and then adjusting or optimizing the weight vector to
minimize this signal. The error signal is,
ek = d k − yk
(2.3)
The weights associated with the network are then updated using the LMS algorithm [2.1].
The weight updates equation for nth instant
wk ( n + 1) = wk ( n ) + ∆wk ( n )
(2.4)
It can be further derived as
wk ( n + 1) = wk ( n ) + 2 ⋅η ⋅ ek ( n ) ⋅ XTk
(2.5)
where η is the learning rate parameter (0 ≤ η ≤ 1). This procedure is repeated till the Mean
Square Error (MSE) of the network approaches a minimum value. The MSE at the time index k
may be defined as, ξ = E ⎡⎣ ek2 ⎤⎦ , where E [.] is the expectation value or average of the signal.
xk-1
Z-1
xk
w0k
xk-2
Z-1
•••
w2k
w1k
•••
Z-1
w(L-1)k
∑
LMS
Algorithm
ek
∑
- yk
+
dk
Fig. 2.1 Adaptive filter using LMS algorithm
22
xk-L+1
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
2.1.2 Recursive Least Square (RLS) Algorithm
The RLS algorithm basically minimizes the sum of squared errors up to last signal sample
[2.2]. The recursive weight update equation is given by:
w(n + 1) = w(n ) + Γxx−1 (n + 1)x(n + 1)e(n + 1)
(2.6)
where,
−1
xx
Γ
(n + 1) = Γ
−1
xx
( )( )
( )()
(n ) − (Γ n x n ) Γ n x n
−1
xx
T
−1
xx
1 + x T (n )Γxx−1 (n )x T (n )
(2.7)
and,
n
Γxx (n ) = ∑ x(k )x T (k )
(2.8)
k =0
Under low noise conditions the convergence is guaranteed within 2N iterations where N is the
filter order.
2.2 Artificial Neural Network (ANN)
Artificial neural network (ANN) takes their name from the network of nerve cells in the
brain. Recently, ANN has been found to be an important technique for classification and
optimization problem [2.3-2.5]. McCulloch and Pitts have developed the neural networks
for different computing machines. There are extensive applications of various types of
ANN in the field of communication, control and instrumentation. The ANN is capable of
performing nonlinear mapping between the input and output space due to its large parallel
interconnection between different layers and the nonlinear processing characteristics. An
artificial neuron basically consists of a computing element that performs the weighted sum
of the input signal and the connecting weight. The sum is added with the bias or threshold
and the resultant signal is then passed through a nonlinear function of sigmoid or
hyperbolic tangent type. Each neuron is associated with three parameters whose learning
can be adjusted; these are the connecting weights, the bias and the slope of the nonlinear
function. For the structural point of view a NN may be single layer or it may be multilayer.
23
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
In multilayer structure, there is one or many artificial neurons in each layer and for a
practical case there may be a number of layers. Each neuron of the one layer is connected
to each and every neuron of the next layer. The functional-link ANN is another type of
single layer NN. In this type of network the input data is allowed to pass through a
functional expansion block where the input data are nonlinearly mapped to more number
of points. This is achieved by using trigonometric functions, tensor products or power
terms of the input. The output of the functional expansion is then passed through a single
neuron.
The learning of the NN may be supervised in the presence of the desired signal or it may be
unsupervised when the desired signal is not accessible. Rumelhart developed the Backpropagation (BP) algorithm, which is central to much work on supervised learning in MLP [2.3].
A feed-forward structure with input, output, hidden layers and nonlinear sigmoid functions are
used in this type of network. In recent years many different types of learning algorithm using the
incremental back-propagation algorithm [2.8], evolutionary learning using the nearest neighbor
MLP [2.9] and a fast learning algorithm based on the layer-by-layer optimization procedure [2.10]
are suggested in literature. In case of unsupervised learning the input vectors are classified into
different clusters such that elements of a cluster are similar to each other in some sense. The
method is called competitive learning [2.11], because during learning sets of hidden units
compete with each other to become active and perform the weight change. The winning unit
increases its weights on those links with high input values and decreases them on those with low
input values. This process allows the winning unit to be selective to some input values. Different
types of NNs and their learning algorithms are discussed in sequel.
2.2.1 Single Neuron Structure
Input
x1
wj ( n)
•••
x2
α ( n)
Activation
Function
ϕ ( .)
∑
y ( n)
Output
xN
Fig. 2.2. Structure of a single neuron
24
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
The basic structure of an artificial neuron is presented in Fig. 2.2. The operation in a neuron
involves the computation of the weighted sum of inputs and threshold [2.3-2.5]. The resultant
signal is then passed through a nonlinear activation function. This is also called as a perceptron,
which is built around a nonlinear neuron; whereas the LMS algorithm described in the preceding
sections is built around a linear neuron. The output of the neuron may be represented as,
⎡N
⎤
y ( n ) = ϕ ⎢∑ w j ( n ) x j ( n ) + α ( n )⎥
⎣ j =1
⎦
(2.9)
where α ( n ) is the threshold to the neurons at the first layer, w j ( n ) is the weight associated
with the j th input, N is the no. of inputs to the neuron and ϕ (.) is the nonlinear activation
function. Different types of nonlinear function are shown in Fig. 2.3.
ϕh ( v )
ϕt ( v )
ϕs ( v )
+1
+1
v
0
ϕ p (v)
+1
+1
0
v
0
v
0
v
-1
(c)
(b)
(a)
(d)
Fig. 2.3. Different types of nonlinear activation function,
(a) Signum function or hard limiter,
(b) Threshold function,
(c) Sigmoid function,
(d) Piecewise Linear
Signum Function: For this type of activation function, we have
⎧ 1 if v > 0
⎪
ϕ ( v ) = ⎨ 0 if v = 0
⎪−1 if v < 0
⎩
(2.10)
Threshold Function: This function is represented as,
⎧1
⎩0
ϕ (v) = ⎨
if v ≥ 0
if v < 0
(2.11)
25
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
Sigmoid Function: This function is S-shaped, is the most common form of the activation
function used in artificial neural network. It is a function that exhibits a graceful balance
between linear and nonlinear behaviour.
ϕ (v) =
1
1 + e − av
(2.12)
where v is the input to the sigmoid function and a is the slope of the sigmoid function.
For the steady convergence a proper choice of a is required.
Piecewise-Linear Function: This function is
1
⎧
v≥+
⎪1,
2
⎪
1
1
⎪
ϕ ( v ) = ⎨v, + > v > −
2
2
⎪
1
⎪
v≤+
⎪ 0,
2
⎩
(2.13)
where the amplification factor inside the linear region of operation is assumed to be unity.
This can be viewed as an approximation to a nonlinear amplifier.
2.2.2 Multi-Layer Perceptron (MLP)
In the multilayer neural network or multilayer perceptron (MLP), the input signal propagates
through the network in a forward direction, on a layer-by-layer basis. This network has been
applied successfully to solve some difficult and diverse problems by training in a supervised
manner with a highly popular algorithm known as the error back-propagation algorithm [2.3,2.4].
The scheme of MLP using four layers is shown in Fig. 2.4. xi ( n ) represent the input to the
network, f j and f k represent the output of the two hidden layers and yl ( n ) represents the
output of the final layer of the neural network. The connecting weights between the input to the
first hidden layer, first to second hidden layer and the second hidden layer to the output layers
are represented by wij , w jk and wkl respectively.
26
I N T E L L I G E N T
Input
layer
(Layer-1)
+1
A N D
Second
Hidden
layer
(Layer-3)
First
Hidden
layer
(Layer-2)
A L G O R I T H M S
wkl
Output
Signal
yl ( n )
•••
w jk
•••
Input
Signal,
xi ( n )
•••
+1
+1
wij
T E C H N I Q U E S
Output
layer
(Layer-4)
Fig. 2.4 Structure of multilayer perceptron
If P1 is the number of neurons in the first hidden layer, each element of the output vector
of first hidden layer may be calculated as,
⎡N
⎤
f j = ϕ j ⎢ ∑ wij xi ( n ) + α j ⎥ , i = 1, 2,3,...N , j = 1, 2,3,...P1
⎣ i =1
⎦
(2.14)
where α j is the threshold to the neurons of the first hidden layer, N is the no. of inputs
and ϕ (.) is the nonlinear activation function in the first hidden layer of (2.12) type. The
time index n has been dropped to make the equations simpler. Let P2 be the number of
neurons in the second hidden layer. The output of this layer is represented as, f k and may
be written as
⎡ P1
⎤
f k = ϕk ⎢ ∑ w jk f j + α k ⎥ , k=1, 2, 3, …, P2
⎣ j =1
⎦
(2.15)
where, α k is the threshold to the neurons of the second hidden layer. The output of the
final output layer can be calculated as
⎡ P2
yl ( n ) = ϕl ⎢ ∑ wkl f k + α l
⎣ k =1
⎤
⎥ , l=1, 2, 3, … , P3
⎦
27
(2.16)
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
where, α l is the threshold to the neuron of the final layer and P3 is the no. of neurons in
the output layer. The output of the MLP may be expressed as
⎡ P2
⎤
⎛ P1
⎞
⎧N
⎫
yl ( n ) = ϕ n ⎢ ∑ wklϕ k ⎜ ∑ w jkϕ j ⎨∑ wij xi ( n ) + α j ⎬ + α k ⎟ + α l ⎥
⎩ i =1
⎭
⎥⎦
⎝ j =1
⎠
⎣⎢ k =1
(2.17)
Back-propagation (BP) Algorithm
x1
x2
Back-Propagation
Algorithm
el ( n )
Σ
yl ( n )
+
d ( n)
Fig. 2.5 Neural network using BP algorithm
An MLP network with 2-3-2-1 neurons (2, 3, 2 and 1 denote the number of neurons in the
input layer, the first hidden layer, the second hidden layer and the output layer respectively)
with the back-propagation (BP) learning algorithm, is depicted in Fig. 2.5. The parameters
of the neural network can be updated in both sequential and batch mode of operation. In
BP algorithm, initially the weights and the thresholds are initialized as very small random
values. The intermediate and the final outputs of the MLP are calculated by using (2.14),
(2.15.), and (2.16.) respectively.
The final output yl ( n ) at the output of neuron l , is compared with the desired output
d ( n ) and the resulting error signal el ( n ) is obtained as
el ( n ) = d ( n ) − yl ( n )
(2.18)
The instantaneous value of the total error energy is obtained by summing all error signals
over all neurons in the output layer, that is
28
I N T E L L I G E N T
ξ ( n) =
T E C H N I Q U E S
A N D
A L G O R I T H M S
1 P3 2
∑ el ( n )
2 l =1
(2.19)
where P3 is the no. of neurons in the output layer.
This error signal is used to update the weights and thresholds of the hidden layers as well as
the output layer. The reflected error components at each of the hidden layers is computed
using the errors of the last layer and the connecting weights between the hidden and the
last layer and error obtained at this stage is used to update the weights between the input
and the hidden layer. The thresholds are also updated in a similar manner as that of the
corresponding connecting weights. The weights and the thresholds are updated in an
iterative method until the error signal becomes minimum. For measuring the degree of
matching, the Mean Square Error (MSE) is taken as a performance measurement.
The updated weights are,
wkl ( n + 1) = wkl ( n ) + ∆wkl ( n )
(2.20)
w jk ( n + 1) = w jk ( n ) + ∆w jk ( n )
(2.21)
wij ( n + 1) = wij ( n ) + ∆wij ( n )
(2.22)
where, ∆wkl ( n ) , ∆w jk ( n ) and ∆wij ( n ) are the change in weights of the second hidden
layer-to-output layer, first hidden layer-to-second hidden layer and input layer-to-first
hidden layer respectively. That is,
∆wkl ( n ) = −2 µ
dξ ( n )
dwkl ( n )
= 2µ e ( n )
dyl ( n )
dwkl ( n )
⎡ P2
⎤
= 2 µ e ( n ) ϕl′ ⎢ ∑ wkl f k + α l ⎥ f k
⎣ k =1
⎦
(2.23)
Where, µ is the convergence coefficient ( 0 ≤ µ ≤ 1 ). Similarly the ∆w jk ( n ) and ∆wij ( n )
can be computed [2.3].
The thresholds of each layer can be updated in a similar manner, i.e.
29
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
α l ( n + 1) = α l ( n ) + ∆α l ( n )
(2.24)
α k ( n + 1) = α k ( n ) + ∆α k ( n )
(2.25)
α j ( n + 1) = α j ( n ) + ∆α j ( n )
(2.26)
where, ∆α l ( n ) , ∆α k ( n ) and ∆α j ( n ) are the change in thresholds of the output, hidden
and input layer respectively. The change in threshold is represented as,
∆ α l ( n ) = −2 µ
dξ ( n )
dα l ( n )
= 2µ e ( n )
dyl ( n )
dα l ( n )
⎡ P2
⎤
′
= 2 µ e ( n ) ϕl ⎢ ∑ wkl f k + α l ⎥
⎣ k =1
⎦
(2.27)
2.2.3 Functional-link Artificial Neural Network (FLANN)
Pao originally proposed FLANN and it is a novel single layer ANN structure capable of
forming arbitrarily complex decision regions by generating nonlinear decision boundaries
[2.7]. Here, the initial representation of a pattern is enhanced by using nonlinear function
and thus the pattern dimension space is increased. The functional link acts on an element
of a pattern or entire pattern itself by generating a set of linearly independent function and
then evaluates these functions with the pattern as the argument. Hence separation of the
patterns becomes possible in the enhanced space. The use of FLANN not only increases
the learning rate but also has less computational complexity [2.13]. Pao et al [2.12] have
investigated the learning and generalization characteristics of a random vector FLANN and
compared with those attainable with MLP structure trained with back propagation
algorithm by taking few functional approximation problems. A FLANN structure with two
inputs is shown in Fig. 2.6.
Mathematical Derivation of FLANN
Let X is the input vector of size N×1 which represents N number of elements; the nth
element is given by:
30
I N T E L L I G E N T
T E C H N I Q U E S
X ( n ) = xn ,1 ≤ n ≤ N
A N D
A L G O R I T H M S
(2.28)
Each element undergoes nonlinear expansion to form M elements such that the resultant
matrix has the dimension of N×M.
The functional expansion of the element xn by power series expansion is carried out using
the equation given in (2.29)
⎧ x for i = 1
si = ⎨ nl
⎩ xn for i = 2,3, 4,… , M
where l = 1, 2,
(2.29)
,M .
For trigonometric expansion, the
⎧
xn
for i = 1
⎪
⎪
si = ⎨ sin ( lπ xn ) for i = 2, 4,… , M
⎪
for i = 3,5,… , M +1
⎪⎩cos ( lπ xn )
Where l = 1, 2,
(2.30)
, M 2 . In matrix notation the expanded elements of the input vector E, is
denoted by S of size N×(M+1).
The bias input is unity. So an extra unity value is padded with the S matrix and the
dimension of the S matrix becomes N×Q, where Q = ( M + 2 ) .
Let the weight vector is represented as W having Q elements. The output y is given as
Q
y = ∑ si wi
(2.31)
i =1
In matrix notation the output can be,
Y = S ⋅ WT
(2.32)
31
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
At kth iteration the error signal e ( k ) can be computed as
e(k ) = d (k ) − y (k )
(2.33)
Let ξ ( k ) denotes the cost function at iteration k and is given by
ξ (k ) =
1 P 2
∑ ej (k )
2 j =1
(2.34)
where P is the number of nodes at the output layer.
The weight vector can be updated by least mean square (LMS) algorithm, as
w(k + 1) = w(k ) −
µ ˆ
∇(k )
(2.35)
2
where ∇ˆ ( k ) is an instantaneous estimate of the gradient of ξ with respect to the weight
vector w(k ) . Now
ˆ (k ) = ∂ξ = −2e(k ) ∂y (k ) = −2e(k ) ∂[ w(k ) s(k )]
∇
∂w
∂w
∂w
(2.36)
= − 2e ( k ) s ( k )
Substituting the values of ∇ˆ ( k ) in (2.35) we get
w ( k + 1) = w ( k ) + µ e ( k ) s ( k )
(2.37)
where µ denotes the step-size ( 0 ≤ µ ≤ 1) , which controls the convergence speed of the
LMS algorithm.
S
W
1
x2
Functional Expansion
x1
d (k )
y (k )
Σ
Adaptive
Algorithm
-
+
Σ
e(k )
Fig. 2.6 Structure of the FLANN model
32
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
2.2.4 Radial Basis Function based NN (RBFNN) Technique
The Radial Basis Function based neural network (RBFNN) consists of an input layer made
up of source nodes and a hidden layer of large dimension [2.3-2.5,2.14]. The number of
input and output nodes is maintained same and while training the same pattern is
simultaneously applied at the input and the output. The nodes within each layer are fully
connected to the previous layer as shown in the Fig. 2.7. The input variables are each
assigned to a node in the input layer and pass directly to the hidden layer without weights.
The hidden nodes contain the radial basis functions (RBFs) which are Gaussian in
characteristics. Each hidden unit in the network has two parameters called a center (µ), and
a width (σ) associated with it. The Gaussian function of the hidden units is radially
symmetric in the input space and the output of each hidden unit depends only on the radial
distance between the input vector x and the center parameter µ for the hidden unit.
αmo
+1
αm1
Input
x1 ( n )
αm2
x2 ( n )
αm3
Σ
Output
xN ( n )
y ( n)
-
Σ
αmk
e (n)
+
d ( n)
RBF
Algorithm
Fig. 2.7 Structure of radial basis function based neural network
The Gaussian function gives the highest output when the incoming variables are closest to
the center position and decreases monotonically as the distance from the center decreases.
The response of each hidden unit is scaled by its connecting weights (αmi’s) to the output
units and then summed to produce the final network output and. The overall network
output y ( n ) at the time index n is therefore
33
I N T E L L I G E N T
K
y ( n ) = α m 0 + ∑ α mkφk ( x j ) ;
T E C H N I Q U E S
j = 1, 2,..., N
A N D
A L G O R I T H M S
(2.38)
k =1
For each input x j , N represents the no. of inputs, K = number of hidden units, αmk =
connecting weight of the kth hidden unit to output layer, αmo = bias term, m is the number
of output.
The value of φk ( x j ) is given by
φk ( x j ) = exp(−
1
σ
2
k
2
x j − µk )
(2.39)
Where µk is the center vector for the kth hidden unit and σk is the width of the Gaussian
function and
denotes the Euclidean norm.
The parameters of the RBFNN are updated using the RBF algorithm. The RBF algorithm
is an exact analytical procedure for evaluation of the first derivative of the output error with
respect to network parameters. In the present paper we apply a three layer RBF network
with nonlinear output units.
Let the error vector at the nth instant is e ( n ) = d ( n ) − y ( n ) , where d ( n ) = desired output
vector and y ( n ) = estimated output vector. Let ξ ( n ) =
1
e 2 ( n ) . The update equations
∑
2 P
for the center and width of the Gaussian function as well as the connecting and bias
weights are derived as
µk ( n + 1) = µk ( n ) + ∆µk ( n )
(2.40)
σ k ( n + 1) = σ k ( n ) + ∆σ k ( n )
(2.41)
α mk ( n + 1) = α mk ( n ) + ∆α mk ( n )
(2.42)
α m0 ( n + 1) = α m0 ( n ) + ∆α m 0 ( n )
(2.43)
34
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
where ∆µk ( n ) , ∆σ k ( n ) are the change of the centers and spread of the Gaussian
functions; the ∆α mk ( n ) and ∆α m 0 ( n ) are the change in weights and the threshold of the
RBFNN. These are computed by taking the partial derivative of ξ ( n ) with respect to
different network parameters. The key equations obtained are stated in (2.44) to (2.47).
∆µk ( n ) = −η
( x j − µk )
∂ξ
= 2ηe ( n ) φk ( x j ) αmk
∂µk
σ k2
( x j − µk )
∂ξ
∆σ k ( n ) = − η
= 2ηe ( n ) φk ( x j ) αmk
∂σ k
σ k3
(2.44)
2
(2.45)
∆αmk ( n ) = − η
∂ξ
= ηe ( n ) φk ( x j )
∂αmk
(2.46)
∆αm0 ( n ) = − η
∂ξ
= ηe ( n )
∂αm0
(2.47)
where η is the learning rate parameter ( 0 ≤ η ≤ 1 ). By applying each input patterns, the
change in center location, width of the Gaussian function as well as the connecting weights
and bias weights are calculated.
2.3
Summary and Discussion
The various adaptive techniques used in the thesis are presented in this Chapter. The single
layer network or perceptron and an adaptive filter using the LMS algorithm are naturally
related, as evidenced by their weights updates. However, the perceptron and LMS
algorithm differ from each other in some fundamental respects:
ƒ
The LMS algorithm uses a linear neuron, whereas the perceptron uses the nonlinear
neuron.
ƒ
The learning process in the perceptron is performed for a finite number of
iterations and then stops. In contrast, continuous learning takes place in the LMS
algorithm.
35
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
For MLP, the back propagation learning is the standard algorithm. The back-propagation
algorithm derives its name from the fact that the partial derivatives of the cost function
(performance measure) with respect to the free parameters (synaptic weights and biases) of
the network are determined by back-propagating the error signals (computed by output
neurons) through the network, layer by layer.
The structure of an RBFNN is unusual in that the constitution of its hidden units is entirely
different from that of its output units. Each hidden node consists of a Gaussian function.
The learning algorithm of RBFNN is more or less equivalent to LMS algorithm.
REFERENCES
[2.1]
B. Widrow and S.D. Sterns, “Adaptive Signal Processing” Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1985.
[2.2]
S. Haykin, “Adaptive Filter Theory”, 4th edition, Pearson Education Asia, 2002.
[2.3]
S. Haykin, “Neural Networks: A comprehensive foundation” 2nd Edition, Pearson
Education Asia, 2002.
[2.4]
Dayhoff E.J., “Neural Network Architecture – An Introduction” Van Norstand Reilold, New
York, 1990.
[2.5]
Bose N.K., and Liang P., “Neural Network Fundamentals with Graphs, Algorithms,
Applications”, TMH Publishing Company Ltd, 1998.
[2.6]
Richard O. Duda, Peter E. Hart and David G. Stork, “Pattern Classification”, 2nd edition,
John Wiley & Sons, INC.2001.
[2.7]
Y.H. Pao, Adaptive Pattern Recognition and Neural Networks, Addison Wesley, Reading,
Massachusetts, 1989.
[2.8]
Fu L.M., Hsu H.H., and Principe J.C., “Incremental Back Propagation Learning
Networks”, IEEE Trans. on Neural Network, vol.7, no.3, pp.757-762, 1996.
[2.9]
Zhao Q. and Higuchi T., “Evolutionary Learning of Nearest-Neighbor MLP”, IEEE
Trans. on Neural Network, vol.7, no.3, pp.762-768, 1996.
36
I N T E L L I G E N T
T E C H N I Q U E S
A N D
A L G O R I T H M S
[2.10] Wang G.J., and Chen C.C., “A Fast Multilayer Neural Network Training Algorithm
based on the Layer-By-Layer Optimization Procedure”, IEEE Trans. on Neural Network,
vol.7, no.3, pp.768-776, 1996.
[2.11] Carpenter G.A., and Grossberg S., “The Art of Adaptive Pattern Recognition by SelfOrganizing Neural Network”, IEEE Computer Mag. vol.21, no.3, pp.77-88, March 1988.
[2.12] Pao Y.H., Park G.H., and Sobjic D.J., “Learning and Generalization Characteristics of
the Random Vector Function”, Neuro Computation, 6: 163-180, 1994.
[2.13] Patra J.C., and Pal R.N., “A Functional Link Artificial Neural Network for Adaptive
Channel Equalization”, Signal Processing 43(1995) 181-195, vol.43, no.2, May 1995.
[2.14] Jungsik Lee, Beach, C.D., Tepedelenlioglu, N, “Channel equalization using radial basis
function network”, IEEE International Conference on Neural Networks, Vol. 4, Page(s):19241928, 3-6 June 1996.
37
3
Chapter
Direct and Inverse modeling of
Capacitive Pressure Sensor using
Intelligent Techniques
T
HE CAPACITIVE PRESSURE SENSOR (CPS), in which the capacitance of a
chamber changes with the application of pressure, finds extensive usage because of its
low power consumption and high sensitivity as discussed in Chapter-1. The CPS
exhibits nonlinearity in its transfer characteristics [3.1-3.3]. This prevents its direct digital readout
and restricts its dynamic range. The second problem is the accuracy of measurements in these
sensors, which is greatly affected specially by aging of the sensor as well as variations in
environmental temperature and humidity. These changes also introduce nonlinearity in the
device characteristics, which is usually time varying and unpredictable as it depends on many
uncertain factors.
38
D I R E C T
A N D
I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
This chapter deals with the design and development of the direct model as well as the inverse
model of the CPS. The direct model estimates the CPS; alternatively it develops the electronic
model of the CPS. This scheme is very similar to a system identification problem in control
systems. This adaptive model of the sensor is developed using different ANN techniques such
that the outputs of the sensor and the model will be identical. And the inverse model
compensates the nonlinearity present in the sensor. It is analogous to the channel equalization
scheme of a digital communication system to cancel the adverse effect of the channel.
Section 3.1 deals with the mathematical modeling of the CPS. In Section 3.2, the direct modeling
of the CPS along with the simulation results of all types of ANN modeling has been discussed.
The inverse modeling along with the simulation results has been depicted in Section 3.3. Finally,
Section 3.4 shows the summary and discussion on different models developed.
3.1 Capacitive Pressure Sensor (CPS)
3.1.1 Model of CPS
P
A commonly used CPS is shown in Fig. 3.1. Here one
plate is a fixed metal disc and the other is a flexible flat
R
circular diaphragm, clamped around its circumference
[3.2,3.3]. The dielectric material is air ( ε ≈ 1 ). The
diaphragm is an elastic sensing element which is bent into
a curve by the applied pressure P (Nm-2). The deflection
3 (1 − ρ
16 Et 3
2
)
(R
r
d(r)
Diaphragm
Chamber
Substrate (Fixed Disc)
Fig. 3.1 Typical structure of a CPS
d ( r ) at any radius r is given by,
d (r ) =
t
d0
Inlet
2
− r2 ) P
2
(m)
(3.1)
where, R = radius of the circular diaphragm (m), t = thickness of diaphragm (m), E =Young’s
modulus or elastic modulus (Nm-2), and ρ =Poisson’s ratio.
The capacitance of the chamber is,
C ( P ) = ε 0ε r ∫
R
0
2π r
dr
d0 − d ( r )
(3.2)
39
D I R E C T
A N D
I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
where, ε 0 is the permittivity of free space (vacuum), and ε r is the relative permittivity or
dielectric constant of the insulating material.
Putting the value of d ( r ) , obtained in (3.1), into (3.2) and integrating it, the capacitance may be
expressed as a function of the fractional pressure x :
C ( x ) = C0
1
tanh −1
x
( x)
(3.3)
where
x=
P
Pmax
Pmax =
( ≤ 1)
(3.4)
16 E t 3 d 0
3 (1 − ρ 2 ) R 4
(3.5)
It is clear that Pmax is the maximum allowed pressure which causes a center deflection equal to
the chamber depth d 0 . The offset capacitance (C0 ) when P = 0 is,
C0 =
π R 2ε 0ε r
(3.6)
d0
Expanding (3.3) by Taylor series, we have
∞
xn
n = 0 2n + 1
C ( x ) = C0 ∑
≅ C0
1 − ( 2 3) x
1− x
+ O ( x2 )
(3.7)
Where O ( x 2 ) denotes the residual term of second-order and is considered to be negligible. It is
seen that some of the practical sensors use stepped or square diaphragms for better sensitivity.
Even though the diaphragm construction is different for the capacitive sensors, their response
characteristics are hyperbolic in nature and can be expressed by the relationships provided in
(3.3) and (3.7):
40
D I R E C T
A N D
C ( x ) = C0
I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
1−α x
= C0 + ∆ C ( x )
1− x
(3.8)
Where
∆ C ( x ) = C0
1−α
x
1− x
(3.9)
The parameters α and Pmax of each sensor are determined by the measured capacitances. Error
between the measured and approximated values is less than 1%. Therefore, a CPS can be
regarded reasonable as a nonlinear capacitor. Plot of (3.8) is shown in Fig. 3.2, and it is observed
that the CPS is quite nonlinear in its full range.
-3
5
x 10
4.5
4
Capacitance
3.5
3
Ideal
2.5
Actual
2
Range of
Linearity
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Applied Pressure
0.35
0.4
0.45
0.5
Fig. 3.2 Range of linear region of the CPS
Input-output relation of CPS taking pressure (P) and temperature (T) as inputs:
The input/output equation for the CPS [3.4-3.8] as a function of pressure (P) and temperature
(T) is,
C N = C ( P, T ) / C0
C ( P, T ) = C0 g1 (T ) + ∆C ( P, T0 ) g 2 (T )
41
(3.10)
D I R E C T
A N D
I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
Where, CN is the normalized capacitance. C0 is the sensor capacitance when P = 0 at the
reference temperature T0; When pressure is applied to the CPS, the change in capacitance,
∆C ( P, T0 ) = C0 ⋅ γ , where,
γ = PN
1−α
1 − PN
(3.11)
where, α = sensitivity parameter of the CPS which varies with the geometrical structure,
PN = normalized pressure = P Pmax ( Pmax is the maximum permissible applied pressure).
Assume that the change in capacitance (or voltage) with respect to temperature is linear.
Hence, the linear functions are,
TN = normalized temperature =
gi ( ⋅) ( i = 1, 2 ) and,
gi (T ) = 1 + βi ⋅ TN , where,
T − T0
, and βi is the coefficients which determines the
Tmax
influence of the temperature on the sensor characteristics. Hence, (3.10) can be rewritten as
(3.12), and the characteristics of the CPS is at different temperatures is shown in Fig. 3.3.
C N = (1 + β1 ⋅ TN ) + PN ⋅
1−α
(1 + β 2 ⋅ TN )
1 − PN
(3.12)
0.9
CPS output at -20 deg C
CPS output at 30 deg C
CPS output at 70 deg C
0.8
Normalised Voltage
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
Normalised Pressure
0.5
0.6
0.7
Fig. 3.3 Range of linear region of the CPS at different temperatures
42
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S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
3.1.2 Switched Capacitor Circuit (SCC)
CS
φ
VR
φ
φ
C ( P)
φ
φ
φ
+
-
V0
Fig. 3.4 Switched Capacitor Interface circuit
Fig. 3.4 shows a switched capacitor circuit (SCC) for interfacing the CPS [3.5,3.6], where C ( P )
denotes the CPS. As defined earlier, the SCC output provides an equivalent voltage signal
proportional to the capacitance change in the CPS due to applied pressure. The circuit operation
can be controlled by a reset signal φ . When φ = 1( +5v ) , C ( P ) charges to the reference
voltage VR while the capacitor CS is discharged to ground. And, when φ = 1 , the total charge
C ( P ) .VR stored in the C ( P ) is transferred to CS and producing an output voltage,
V0 = K ⋅ C ( P ) , where K = − VR CS . It may be noted if ambient temperature changes, then the
SCC output also changes though the applied pressure remains same. By choosing proper values
of CS and VR , the normalized SCC output can be adjusted in such a way that VN = CN . The
SCC does not take into account the effect of temperature variation in the voltage measurement.
The intelligent behavior is implemented in the sensor by training the RBFNN to adapt any
temperature change.
3.2 Development of Intelligent Models of the CPS
A scheme of direct modeling and inverse modeling [3.7] of the CPS using different ANN
structure has been proposed in this section. The direct modeling is proposed for calibration of
inputs and estimation of internal parameters of the CPS. And the inverse modeling is proposed
for estimation of applied input pressure.
43
D I R E C T
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I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
3.2.1 Direct Modeling
PN
CPS
+
SCC
TN
VN
+
Σ
VN′
ANN
Model
e
-
UPDATE
Algorithm
Fig. 3.5 A scheme of direct modeling of a CPS and SCC using ANN based model
The direct modeling is analogous to that of the system identification problem in control system.
The purpose of the direct model is to obtain an ANN model of the CPS in such a way that the
outputs of the CPS and the ANN match closely. Once a model of the CPS is available, it may be
used for fault detection of the sensor. The CPS (interfaced with SCC) output provides an
equivalent voltage signal proportional to the capacitance change due to the applied input
pressure [3.4, 3.8]. By applying pressure to the CPS, the response is the change in capacitance
value with respect to the applied pressure. By connecting the switched capacitor interface circuit
(SCC) [3.9, 3.10] with the CPS to get the equivalent voltage signal proportional to the change in
capacitance is obtained. Fig. 3.5 represents a scheme of direct model of the CPS using ANN
based approach. Since the temperature also affects the output voltage of the sensor, both
normalized pressure ( PN ) and normalized temperature ( TN ) are used as input to the CPS. The
output voltage ( VN ) of the CPS and that of the ANN model ( VN′ ) are compared to produce the
error signal, e . This error information is used to update the ANN model. The model of CPS
has been developed by separately applying all types of neural models such as MLP, FLANN and
RBFNN.
3.2.2 Inverse Modeling
PN
TN
CPS
Model
VN
ANN
Model
Update
Algorithm
PN′
e
PN
-
Σ
+
Fig. 3.6 Development of an inverse ANN model of CPS
44
D I R E C T
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I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
A scheme of inverse modeling of a CPS using different ANN techniques for estimation of
applied pressure is shown in Fig. 3.6. This is analogous to the channel equalization scheme
used in a digital communication receiver to cancel the adverse effects of the channel on the
data being transmitted. To obtain a direct digital readout of the applied pressure, an inverse
model of the CPS may be used in cascade with it to compensate for the adverse effects on
the CPS output due to the nonlinear response characteristics and the variations with
ambient temperature. The generation of training-set and test-set patterns is similar to that
of the direct modeling scheme. However, in the inverse modeling scheme, the normalized
temperature TN and the CPS output VN are taken as input patterns, and the normalized
input pressure PN is taken as the desired output pattern in the ANN model.
3.3 Simulation Studies
Different ANN technique based direct and inverse models have been simulated extensively in
MATLAB 7.0 environment. The SCC output voltage VN is obtained experimentally at the
reference temperature of 250C for different values of normalized pressure chosen between 0.1
and 0.7 with an interval of 0.05. Thus, these 13 pairs of input-output data constitute a set of
patterns at the reference temperature. The functions g1 ( t ) and g 2 ( t ) are generated by taking
the values of β1 and β 2 to -2.0×10-3 and 7.0×10-3. The geometrical structure of the CPS α is
taken as 0.64. From the available CPS pattern set at the reference temperature, i.e., PN ∼ CN , and
with the knowledge of functions g1 ( t ) and g 2 ( t ) , eight sets of patterns (each containing 13
pairs of input-output data) are obtained at an interval of 100C ranging from -200C to 1000C.
Details of each ANN based direct and inverse modeling is described below:
3.3.1 The MLP based Direct Modeling
Simulation studies for MLP network are carried out to obtain a direct model of the CPS.
For the simulation study, a two-layer MLP with a 3-5-1 structure is chosen for direct
modeling of a CPS as shown in Fig. 2.4 of Chapter 2. (3, 5, and 1 denote the number of
nodes including the bias units in the input layer, the first layer, i.e. the hidden layer and the
output layer of the ANN, respectively) [3.8]. The hidden layer and the output layer contains
the tanh(.) type activation function. The back-propagation (BP) algorithm [3.12], in which
45
D I R E C T
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I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
both the learning rate and the momentum rate are chosen as 0.5 and 1 respectively, is used
to adapt the weights of the MLP. The normalized temperature ( TN ) and the normalized
applied pressure ( PN ) are used as input pattern, and the SCC output ( VN = CN ) is used as the
desired pattern to the MLP. After application of each pattern, the ANN weights are
updated using the BP algorithm. Completion of all patterns of all the training sets
constitutes one iteration of training. To make the learning complete and effective, 10,000
iterations are made to train the ANN. Then, the weights of the MLP are frozen and stored
in an EPROM. During the testing phase, the frozen weights are loaded into the MLP
model. Then the inputs, PN and TN from the test set are fed to this model. The model
output is computed and compared with the actual output to verify the effectiveness of the
model. The CPS response characteristics of the chosen values at different temperatures
T = −200 C,300 C and 700 C are plotted in Fig. 3.7.
MLP based direct modeling
0.9
0.8
Normalised Voltage
0.7
Actual sensor output at -20 deg C
Estimated sensor output at -20 deg C
Actual sensor output at 30 deg C
Estimated sensor output at 30 deg C
Actual sensor output at 70 deg C
Estimated sensor output at 70 deg C
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0.2
0.3
0.4
0.5
Normalised Pressure
0.6
0.7
0.8
Fig. 3.7 Plots of true and estimated forward characteristics of the CPS at –200 C, 300 C and 700 C by MLP
3.3.2 The MLP based Inverse Modeling
In the inverse modeling the same MLP with a 3-5-1 structure is chosen for simulation [3.8]. The
MLP is trained in a similar manner as in the case of direct modeling. In this case also, all the 13
patterns corresponding to temperature values -200C, 300C and 700C are applied during training.
The learning rate and the momentum parameter are as same as in the direct modeling i.e. 0.5 and
1 respectively. The MLP is trained for 10,000 iterations using the BP algorithm; the evolved
46
D I R E C T
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I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
weights are frozen, and stored in the memory. In the testing phase, the CPS output VN is
applied as input to the MLP network along with the normalized temperature TN . Then, the
estimated pressure PN′ is obtained from the output of the MLP model. The plots of inverse
model at different temperatures T = −200 C , 300 C and 700 C are shown in Fig. 3.8.
MLP based Inverse modeling
MLP based Inverse modeling
0.8
0.8
Sensor characteristics at -20 deg C
Inverse modeling at -20 deg C
Overall response
0.7
0.7
0.6
VN,PN`,PN` --->
0.6
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.2
0.3
0.4
0.5
PN,VN,PN --->
0.6
0.7
0.1
0.1
0.8
0.2
0.3
0.4
0.5
PN,VN,PN --->
0.6
0.7
MLP based Inverse modeling
0.9
0.8
Sensor characteristics at 70 deg C
Inverse modeling at 70 deg C
Overall response
0.7
VN,PN`,PN` --->
VN,PN`,PN` --->
Sensor characteristics at 30 deg C
Inverse modeling at 30 deg C
Overall response
0.6
0.5
0.4
0.3
0.2
0.1
0.1
a
0.2
0.3
0.4
0.5
0.6
PN,VN,PN --->
0.7
0.8
0.9
b
c
Fig. 3.8 Plots of forward, inverse and overall characteristics of the CPS by MLP; (a) at –200C, (b) at 300C;
and (c) at 700C.
Axes Parameters:
PN = Normalized pressure
VN = Normalized voltage (output of the CPS+SCC)
PN’ = Estimated pressure
47
0.8
D I R E C T
A N D
I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
3.3.3 The FLANN based Direct Modeling
As stated in Chapter 2, the FLANN is a single layered network [3.9,3.10]. Two different
experiments are carried out to obtain the direct model of the CPS. In the first case the power
series FLANN with 10 number of functional expansions of each input ( PN and TN ) and one
product term PN ⋅ TN is taken. Second is the trigonometric expansion (sin(.) and cos(.)
expansions) of input with the product term of PN ⋅ TN are considered. The learning rate is
chosen as 0.035 in both the FLANN update algorithms is used to update the weights of the
FLANN model. The normalized temperature ( TN ) and the normalized applied pressure ( PN ) are
used as the input patterns, and the SCC output ( VN = CN ) is used as the desired pattern to the
FLANN model. 10,000 iterations are used (for both the cases) for training of these FLANN
models completely. Then, the weights of the FLANN models are frozen and stored in the
memory. During the testing phase the inputs, PN and TN are fed to this model. The FLANN
model output is computed and compared with the actual output to verify the effectiveness of
these proposed models. The results of the modeling of CPS using two FLANN structures are
shown in Fig. 3.9 (a) and (b).
Trigonometric FLANN based direct modeling
Power series FLANN based direct modeling
0.9
0.9
Normalized Voltage
0.7
0.6
Actual sensor output at -20 deg C
Estimated sensor output at -20 deg C
Actual sensor output at 30 deg C
Estimated sensor output at 30 deg C
Actual sensor output at 100 deg C
Estimated sensor output at 100 deg C
0.8
0.7
Normalized Voltage
0.8
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0.1
0.2
a
0.3
0.4
0.5
Normalized Pressure
0.6
0.7
0
0.1
0.8
Actual sensor output at -20 deg C
Estimated sensor output at -20 deg C
Actual sensor output at 30 deg C
Estimated sensor output at 30 deg C
Actual sensor output at 100 deg C
Estimated sensor output at 100 deg C
0.2
0.3
0.4
0.5
Normalized Pressure
0.6
b
Fig. 3.9 Plots of true and estimated forward characteristics of the CPS at –200 C, 300 C and 1000 C
(a) power series FLANN (b) trigonometric series FLANN
48
0.7
0.8
D I R E C T
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I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
3.3.4 The FLANN based Inverse Modeling
For the inverse modeling of the CPS the power series and the trigonometric FLANNs are
successfully trained. In case of the power series FLANN, 12 number of expansions of each
input ( VN and TN ) along with the product term of the inputs are computed. The desired signal
to the model is the input applied pressure, PN . To train the network effectively, 50,000 iterations
are chosen. The learning rate is set at 0.025. In trigonometric FLANN model, 10 sets of sin(.)
expansions and 10 sets of cos(.) expansions with the product term are also obtained. In this case
the learning rate is 0.03. Both the FLANN models are trained similar to that of direct modeling.
All 13 patterns of inputs corresponding to temperature values of -200C to 1000C with an interval
of 100C are applied during the training period until the training is complete. Once the ANN
model is successfully trained, the weights are frozen for testing. The plots of the inverse model
of the power series FLANN and the trigonometric FLANN models at different temperature
values of -200C, 300C and 1000C are shown in Fig. 3.10 and Fig. 3.11 respectively. From the
computed results, it is observed that the trigonometric FLANN model yields improved results
compared to the power series counterpart.
Power series FLANN based Inverse modeling
Power series FLANN based Inverse modeling
0.8
0.8
0.7
Actual sensor output at -20 deg C
Estimated sensor output at -20 deg C
Overall response
0.7
0.6
VN,PN`,PN` --->
VN,PN`,PN` --->
0.6
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
Actual sensor output at 30 deg C
Estimated sensor output at 30 deg C
Overall response
0.2
0.3
0.4
0.5
PN,VN,PN --->
0.6
0.7
0.1
0.1
0.8
0.2
0.3
0.4
0.5
PN,VN,PN --->
0.6
0.7
Contd…
49
0.8
D I R E C T
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S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
Power series FLANN based Inverse modeling
0.9
Actual sensor output at 100 deg C
Estimated sensor output at 100 deg C
Overall response
0.8
0.7
VN,PN`,PN` --->
0.6
0.5
0.4
0.3
0.2
0.1
0
a
Page
break
0
0.1
0.2
0.3
b
0.4
0.5
PN,VN,PN --->
0.6
0.7
0.8
0.9
c
Fig. 3.10 Plots of forward, inverse and overall characteristics of the CPS by power series FLANN;
(a) at –200C, (b) at 300C; and (c) at 1000C.
Axes Parameters:
PN = Normalized pressure
VN = Normalized voltage (output of the CPS+SCC)
PN’ = Estimated pressure
Trigonometric FLANN based Inverse modeling
Trigonometric FLANN based Inverse modeling
0.8
0.9
0.8
Actual sensor output at -20 deg C
Estimated sensor output at -20 deg C
Overall response
0.7
0.6
VN,PN`,PN` --->
VN,PN`,PN` --->
0.7
Actual sensor output at 30 deg C
Estimated sensor output at 30 deg C
Overall response
0.6
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.2
0.3
0.4
0.5
0.6
PN,VN,PN --->
0.7
0.8
0.1
0.1
0.9
0.2
0.3
0.4
0.5
PN,VN,PN --->
0.6
0.7
0.8
Contd…
50
D I R E C T
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I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
Trigonometric FLANN based Inverse modeling
0.9
Actual sensor output at 100 deg C
Estimated sensor output at 100 deg C
Overall response
0.8
0.7
VN,PN`,PN` --->
0.6
0.5
0.4
0.3
0.2
0.1
0
Page
break
a
0
0.1
0.2
0.3
0.4
0.5
PN,VN,PN --->
0.6
0.7
0.8
0.9
b
c
Fig. 3.11 Plots of forward, inverse and overall characteristics of the CPS by trigonometric FLANN;
(a) at –200C, (b) at 300C; and (c) at 1000C.
Axes Parameters:
PN = Normalized pressure
VN = Normalized voltage (output of the CPS+SCC)
PN’ = Estimated pressure
3.3.5 The RBFNN based Direct Modeling
Unlike the MLP, the RBFNN is a single layered network [3.12,3.13]. The detailed theory of
RBFNN is depicted in Chapter 2. For the simulation, an RBFNN with a 2-6-1 structure is
chosen for direct modeling of a CPS. (2, 6, and 1 denote the number of nodes in the input layer,
the first layer including the bias units, i.e. the hidden layer and the output layer of the ANN,
respectively) [3.14]. The update algorithm, in which both the learning rate is chosen as 0.05, is
used to adapt the weights of the RBFNN. The normalized temperature ( TN ) and the normalized
applied pressure ( PN ) are used as input pattern, and the SCC output ( VN = CN ) is used as the
desired pattern to the RBFNN. In this case the temperature range is -500C to 800C. After
application of all patterns, the ANN weights are updated using the update algorithm. This
process is repeated till the mean square error (MSE) is minimized. Once the training is complete,
51
D I R E C T
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S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
the RBFNN model will be equivalent to the CPS. To make the training successful, 10,000
iterations are needed. Once the training is complete the weights of the RBFNN can be frozen.
During the testing phase, the frozen weights of RBFNN model are used for testing. Then the
inputs, PN and TN from the test set are fed to this model. The model output is computed and
compared with the actual output to verify the effectiveness of the model. The plot of the direct
modeling at various temperatures, T = −500 C , 100 C and 800 C are shown in Fig. 3.12.
RBFNN based direct modeling
0.9
0.8
Normalised Voltage
0.7
Actual sensor output at -50 deg C
Estimated sensor output at -50 deg C
Actual sensor output at 10 deg C
Estimated sensor output at 10 deg C
Actual sensor output at 80 deg C
Estimated sensor output at 80 deg C
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0.2
0.3
0.4
0.5
Normalised Pressure
0.6
0.7
0.8
Fig. 3.12 Plots of true and estimated forward characteristics of the CPS at –500 C, 100 C and 800 C by
RBFNN
3.3.6 The RBFNN based Inverse Modeling
For inverse model, same 2-6-1 structured RBFNN has been chosen for simulation. Each hidden
node contains a radial basis type function [3.14]. The output of the CPS ( VN ) and the normalized
temperature ( TN ) are the inputs to the RBFNN based inverse model. The normalized input
pressure is then compared with the output of the inverse model PN′ and produces error signal.
Then the weights of the network are trained by the update algorithm to match both the outputs.
In the updation process the learning rate is chosen as 0.001. And 50,000 iterations are needed to
successfully train the network. The plots of inverse model at different temperatures
T = −500 C , 100 C and 800 C are shown in Fig. 3.13.
52
D I R E C T
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S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
RBF based Inverse modeling
RBF based Inverse modeling
0.8
0.7
0.8
Sensor characteristics at -50 deg C
Inverse modeling at -50 deg C
Overall response
0.7
VN,PN`,PN` --->
0.6
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.2
0.3
0.4
0.5
PN,VN,PN --->
0.6
0.7
0.1
0.1
0.8
0.2
0.3
0.4
0.5
PN,VN,PN --->
0.6
0.7
RBF based Inverse modeling
0.9
0.8
Sensor characteristics at 80 deg C
Inverse modeling at 80 deg C
Overall response
0.7
VN,PN`,PN` --->
VN,PN`,PN` --->
0.6
Sensor characteristics at 10 deg C
Inverse modeling at 10 deg C
Overall response
0.6
0.5
0.4
0.3
0.2
0.1
0.1
a
0.2
0.3
0.4
0.5
0.6
PN,VN,PN --->
0.7
0.8
0.9
b
c
Fig. 3.13 Plots of forward, inverse and overall characteristics of the CPS by RBFNN; (a) at –500C, (b) at
100C; and (c) at 800C.
Axes Parameters:
PN = Normalized pressure
VN = Normalized voltage (output of the CPS+SCC)
PN’ = Estimated pressure
53
0.8
D I R E C T
3.4
A N D
I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
Summary and Discussion
This chapter presents the development of direct modeling and the inverse modeling of the CPS
with various ANN techniques. The direct modeling is proposed to estimate the nonlinearity
present in a CPS. It is more or less equivalent to the system identification problem in control
systems. The direct model is also used for fault detection of the sensor. Various ANN (MLP,
FLANN and RBFNN) based direct models are proposed in this chapter.
Inverse models are developed to compensate the nonlinearity present in a CPS. The inverse
modeling is analogous to the channel equalization problem in communication system. The
inverse model is itself a nonlinear system; however its characteristics is the inverse of the sensor
characteristics. It is connected in cascade with the sensor to achieve the overall linearity over a
wide range. It has been developed by using different ANN techniques. It is observed that the
linearity is high in case of MLP and RBFNN. However, if we increase the number of expansions
in FLANN, the percentage of linearity can be high. The FLANN network involves less
computational complexity but provides satisfactory performance. Therefore FLANN based
models are easily implementable and may be preferred for real-time applications.
REFERENCES
[3.1]
Hermann K.P. Neubert, “Instrument Transducers: An Introduction to their performance &
design”, 2nd edition, Oxford University Press.
[3.2]
John P. Bentley, “Principles of Measurement Systems”, 3rd edition, Pearson Education.
[3.3]
G.C. Meijer, “Concepts and focus points for intelligent sensor systems”, Sensor and
Actuators: A Physical, 41-42 (1994), 183-191.
[3.4]
D.D. Bruyker, A. Cozma and Robert Puers, “A combined piezoresistive /
capacitive pressure sensor with self-test function on thermal actuation”, Proc. of
IEEE conference on solid state sensors & actuators, 16th-19th June, 1997, Chicago.
[3.5]
H. Matsumoto, H. Shimizu, and K. Watanabe, “A switched capacitor charge
balancing
analog-to-digital
converters
and
its
application
to
capacitor
measurement”, IEEE Trans. Instrumentation & Measurement, vol. IM-36, pp. 873-878,
Dec. 1987.
54
D I R E C T
[3.6]
A N D
I N V E R S E M O D E L I N G O F C A P A C I T I V E P R E S S U R E
S E N S O R U S I N G I N T E L L I G E N T T E C H N I Q U E S
M. Yamada, T. Takebayashi, S.I. Notoyama, and K. Watanabe, “A switched
capacitor interface for capacitive pressure sensors”, IEEE Trans. Instrumentation &
Measurement, vol. 41, pp. 81-86, Feb. 1992.
[3.7]
M. Yamada and K. Watanabe, “A capacitive pressure sensor interface using
oversampling ∆-Σ demodulation techniques”, IEEE Trans. Instrumentation &
Measurement, vol. 46, pp. 3-7, Feb. 1997.
[3.8]
J.C. Patra, A.C. Kot and G. Panda, “An intelligent pressure sensor using neural
networks”, IEEE Trans. Instrumentation & Measurement, vol. 49, No. 4, Aug.2000.
[3.9]
J.C. Patra, G. Panda, and R. Baliarsingh, “Artificial neural network-based
nonlinearity estimation of pressure sensors”, IEEE Trans. Instrumentation &
Measurement, vol. 43, No. 6, Dec. 1994.
[3.10] S.K. Rath, J.C. Patra, and A.C. Kot, “An intelligent pressure sensor with selfcalibration capability using artificial neural network”, Proc. of IEEE International
Conference on Systems, Man, and Cybernetics, Vol. 4, pp.2563 – 2568, 8th-11th, Oct. 2000.
[3.11] B. Widrow, S.D. Stearns, “Adaptive Signal Processing”, 2nd edition, Pearson Education
Asia.
[3.12] S. Haykin, “Neural Networks-A comprehensive foundation”, Second edition, Pearson
Education Asia, 2002.
[3.13] Y.H. Pao, “Adaptive Pattern Recognition and Neural Networks”, Reading, MA: AddisonWesley, 1989.
[3.14] G. Panda, D.P. Das, S.K. Mishra, S.K. Panda, S. Meher and K.K. Mahapatra,
“Development of an Efficient Intelligent Pressure Sensor using RBF based ANN”,
Proceedings of International Conference on Intelligent Signal Processing & Robotics (ISPR-2004),
(ITS2 in CD), 20th-23rd Feb, 2004, IIIT, Allahabad.
55
4
Chapter
Performance Evaluation of FixedPoint FLANN based CPS model
T
HIS CHAPTER deals with the fixed-point analysis of the FLANN based direct and
inverse model of the CPS. To assess the performance of the direct and the inverse
model in actual implementation a fixed-point analysis is required. In the previous
Chapter it is proposed that the FLANN model involves less computational complexity offering
almost similar performance as that of other types of ANN [4.1-4.2]. Due to involvement of less
computational complexity and better performance, the power series FLANN is considered for
implementation. Computer simulation studies have been carried out to demonstrate the
performance of fixed-point FLANN structures of both the direct and the inverse models of the
CPS. This is primarily contributed by the word lengths used in the implementation. The present
study has been made to assist to choose the desired word length for implementation.
56
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
In Section 4.1, the fixed-point analysis of the CPS for both direct and inverse modelings has
been carried out. Section 4.2 deals with the simulation studies of fixed-point implementation.
The effect of word lengths on the performance of the CPS is demonstrated in Section 4.3.
4.1 Fixed-point Analysis
The input-output relation of the CPS as a function of pressure ( PN ) and temperature ( TN )
is derived in (3.12) of Chapter-3. In this fixed-point analysis, the inputs PN and TN are
represented into ( PN ) F and (TN ) F with t-bits. We denote these variables as,
( PN )F = scale ( PN , t )
and
(TN ) F = scale (TN , t )
(4.1)
where,
scale ( x, t ) is a function, which represents the input x in t-bits.
Then (3.12) (of Chapter-3) can be written as,
(VN )F
{
= add f1 (T ) , mul ( γ , f 2 (T ) , t ) ,t
}
(4.2)
where,
add ( x, y, t ) is the function that calculates fixed-point addition of the inputs x and y, with
t-bits.
Similarly, mul ( x, y, t ) is the function that performs the fixed-point multiplication of inputs x
and y, using finite length of t-bit.
4.1.1 Fixed-point FLANN Algorithm for Direct Modeling
Fig. 4.1 shows a direct modeling scheme of a CPS, where ( PN ) F and (TN ) F stands for fixedpoint representation of PN and TN respectively. Similarly (VN ) F and (VN′ ) F represent the fixedpoint outputs of a CPS and fixed-point FLANN model. In fixed-point mode the error signal is
57
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
also represented by ( e ) F . The magnitude of the fixed-point variables depends on the word
length and the type of arithmetic used to represent the numbers.
PN
TN
Fixed-point
conversion
CPS
Fixed-point
conversion
( PN )F
(TN )F
(VN )F
+
Σ
Fixed-point
FLANN model
(VN′ )F
( e )F
-
Update
Algorithm
Fig. 4.1 A scheme of direct modeling of a CPS using fixed-point FLANN model
Algorithm
The output of the fixed-point FLANN model VN′ is calculated by expanding the input
elements nonlinearly into power series expansion as,
VN ′ = w1 ⋅ PN + w2 ⋅ PN2 + w3 ⋅ PN3 + w4 ⋅ PN4 + w5 ⋅ PN5 +
(4.3)
w6 ⋅ TN + w7 ⋅ TN2 + w8 ⋅ TN3 + w9 ⋅ TN4 + w10 ⋅ TN5
Using fixed-point operations (4.3) may be expressed as,
( ( ))
( ( ))
( ( ))
(
⎛ (w1 ⋅ (PN )F )F + w 2 ⋅ PN2 F F + "
⎜
⎛⎜ V ′ ⎞⎟ = ⎜
+ w5 ⋅ PN5 F F + (w 6 ⋅ (T N )F )F +
⎝ N ⎠F ⎜
⎜
w 7 ⋅ T N2 F F + " + w10 ⋅ T N5 F F
⎝
( ))
⎞
⎟
⎟
⎟
⎟
⎠F
(4.4)
Thus (4.4) can be obtained by summing up the partial fixed-products.
The power terms of inputs can be computed in fixed-point mode as,
(P )
(T )
i +1
N
F
i +1
N
F
(
= mul ( (T )
),
,t)
= mul ( PNi ) , ( PN ) F , t
i
N
F
F
, (TN ) F
for i = 0,1,2,3,4.
58
(4.5)
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
These power terms are multiplied in fixed-point arithmetic with the respective weights to give
fixed point results as,
(
x1 = mul w1 , ( PN ) F , t
(
= mul ( w , ( P )
= mul ( w , ( P )
= mul ( w , ( P )
)
)
,t)
,t)
,t)
x2 = mul w2 , ( PN2 ) , t
x3
x4
x5
F
3
3
N F
4
4
N F
5
5
N F
(
x6 = mul w6 , (TN ) F , t
(4.6)
)
(
)
x = mul ( w , (T ) , t )
x = mul ( w , (T ) , t )
x7 = mul w7 , (TN2 ) , t
F
8
8
3
N F
9
9
4
N F
Similarly the fixed-point additions are performed on these partial products as,
y1 = add ( x1 , x2 , t )
yi +1 = add ( yi , xi + 2 , t ) , for i = 0,1,2, ... , 8.
(4.7)
Finally, the overall fixed-point output of the FLANN is given by,
(V ′ )
N
F
= y9
(4.8)
The FLANN algorithm is used in fixed-point mode to update the weights [4.5,4.6]. It can
be expressed as,
( w ( n + 1) )
i
F
= ( wi ( n ) ) F + ( 2λ e ( n ) x ( n ) ) F
The second term is,
59
(4.9)
P E R F O R M A N C E
( ∆w ( n ) )
F
E V A L U A T I O N
( ((
F I X E D
P O I N T
F L A N N
))
))
(
B A S E D C P S
M O D E L
= mul ⎛⎜ 2, mul mul λ , ( e ( n ) ) F , t , ( x ( n ) ) F , t , t ⎞⎟
⎝
⎠
( w ( n + 1) )
i
O F
F
= add
( ( w ( n ) ) , ( ∆w ( n ) ) , t )
i
F
(4.10)
(4.11)
F
where [x(n)]F is the fixed-point representation of the FLANN input vectors computed as
before,
⎡⎣ x ( n ) ⎤⎦ F = ⎡ ( PN )F
⎣
(P )
2
N F
( P ) (T ) (T )
"
5
N F
N
2
N F
F
" (TN5 ) ⎤
F ⎦
The details of fixed-point FLANN based direct model is depicted in the Fig. 4.2.
PN
TN
Fixed-point
conversion
( PN )F
(TN′ )F
E
X
P
A
N
S
I
O
N
(VN )F
Fixed-point
conversion
CPS
(P )
(P )
(P )
(P )
(P )
(T )
(T )
(T )
(T )
(T )
N
F
2
N
F
3
N
F
4
N
F
5
N
F
N
F
2
N
F
3
N
F
4
N
F
5
N
F
+
wi
Σ
( e )F
-
xi
∑
(VN′ )F
Update
Algorithm
Fig. 4.2 Direct modeling of CPS using fixed-point FLANN used for simulation study
4.1.2 Fixed-point FLANN Algorithm for Inverse Modeling
PN
TN
Fixed-point
CPS Model
Fixed-point
conversion
(VN )F
(TN )F
Fixed-point
inverse
FLANN
model
( PN′ )F
Update
Algorithm
+
( PN )F
Σ
Fig. 4.3 Fixed-point inverse modeling of CPS by FLANN
60
( e )F
(4.12)
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
Algorithm
Referring to Fig. 4.3 and 4.4, the estimated output from the fixed-point power series
FLANN model of 5th order expansion can be computed as,
( )
PN ′
) (
) (
(
F
)
(
)
⎛ W ⋅ (V )
+ W2 ⋅ (VN2 )
+ " + W5 ⋅ (VN5 )
F F
F F
⎜ 1 N F F
=⎜
⎜ W6 ⋅ (TN ) F F + W7 ⋅ (TN2 ) F + " + W10 ⋅ (TN5 ) F
F
⎝
(
)
(
)
⎞
⎟
⎟
⎟
F ⎠F
(4.13)
where (VN ) F is the t-bit representation of output of the CPS, and (TN ) F is the t-bit
representation of normalized temperature.
The other power terms of inputs to the FLANN model can be calculated similar to the direct
modeling and may be represented by:
(V )
(T )
i +1
N
F
i +1
N
F
(
= mul ( (T )
= mul (VNi ) , (VN ) F , t
i
N
F
F
, (TN ) , t
)
) , for i = 0,1,2,3,4.
(4.14)
The partial products of the FLANN model is termed as,
(
X 1 = mul W1 , (VN ) F , t
( (
= mul(W , (V
= mul (W , (V
= mul (W , (V
X 2 = mul W2 , V N2
X3
X4
X5
)
)
)
)
F
3
3
N F
4
4
N F
5
5
N F
)
)
, t)
, t)
, t)
,t
(4.15)
X 6 = mul (W 6 , (T N ) F , t )
( ( ) , t)
X 7 = mul W7 , T N2
F
(
)
= mul (W , (T ) , t )
X 8 = mul W8 , (TN3 ) , t
F
X9
9
4
N F
61
P E R F O R M A N C E
E V A L U A T I O N
(
X 10 = mul W10 , (TN5 ) , t
F
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
)
Now by adding all these fixed-point products in t-bits each gives rise to,
Y1 = add ( X 1 , X 2 , t )
(4.16)
Yi +1 = add (Yi , X i + 2 , t ) , for i = 1, 2, … 8.
Hence the overall output of the fixed-point FLANN model is given by,
(P ′)
N
F
= Y9
(4.17)
The FLANN algorithm used to update the weights [4.3] in fixed-point mode is given by,
(W ( n + 1) )
i
F
= (Wi ( n ) ) F + ( 2λ e ( n ) X ( n ) ) F
(4.18)
The second term is,
( ∆W ( n ) )
F
( ((
))
(
))
= mul ⎛⎜ 2, mul mul λ , ( e ( n ) ) F , t , ( X ( n ) ) F , t , t ⎞⎟
⎝
⎠
(4.19)
Thus, (4.22) can be obtained as,
(W ( n + 1) )
i
F
(
= add (Wi ( n ) ) F , ( ∆W ( n ) ) F , t
)
(4.20)
where, ⎡⎣ X ( n ) ⎤⎦ F is the fixed-point representation of FLANN input vectors computed as
before
⎡⎣ X ( n ) ⎤⎦ = ⎡(VN )F
F
⎣
(V )
2
N F
" (VN5 )
F
(T ) (T )
N
62
F
2
N F
" (TN5 ) ⎤
F⎦
(4.21)
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
The fixed-point FLANN based inverse model for simulation is depicted in the Fig. 4.4.
Fixed-point
conversion
(VN ) F
PN
TN
(VN )F
Fixed-point
CPS Model
(TN ) F
Fixed-point
conversion
E
X
P
A
N
S
I
O
N
(V )
(V )
(V )
(V )
2
N
F
3
N
4 F
N
5 F
N
F
TN
F
TN2
F
TN3
F
TN4
F
TN5
F
( )
(
(
(
(
)
)
)
)
( PN )F
Wi
Xi
+
( PN′ ) F
∑
-
Σ
( e )F
Update
Algorithm
Fig. 4.4 Inverse modeling of CPS by fixed-point FLANN used for simulation study
4.2 Simulation Studies
Both the fixed-point direct and inverse models shown in Figs. 4.3 and 4.4 of the CPS are
simulated in this section. The actual CPS model, which is a function of both pressure and
temperature, is assumed to have the nonlinear form as in (3.12). The direct modeling block
is simulated for the following conditions,
PN = 0.0 to 0.8 at a step of 0.1,
T = 300C,
α = 0.7,
β1 = -2.0×10-3 and β2 = 7.0×10-3
The CPS and the SCC blocks are mathematically represented by (4.2). The output of this sensor
is given by VN. The same input is applied to the FLANN model and the corresponding output
is calculated by (4.4). Both infinite precision as well as finite precision FLANN models are
simulated. In the fixed-point FLANN model, the word length t is varied between 4-bit to 16-bit
63
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
at an interval of 4-bit. The output of the FLANN model is represented as VN′. The two outputs
are compared and the error is used to update the parameter of the FLANN model. After some
iterations of training the minimum mean square error attains a steady state value, at that point of
time the training is stopped and the FLANN parameters are frozen. Now fresh normalized
pressure is applied and the corresponding outputs of the CPS, both floating-point FLANN and
fixed-point FLANN models are obtained. The plots of fixed-point FLANN model outputs are
shown in Fig. 4.5. The plots clearly indicate that at low bit length (e.g. t = 4 and 8-bit) the match
between true and estimated response is observed to be poorer. The agreement improves when
the bit length is increased to 12 and it becomes excellent when the bit length is 16. Thus the bit
length substantially affects the response of the direct model.
Fixed-point analysis of the direct m odel of CPS by FLANN at 30 deg C
1.4
Response of the CPS
Direct m odel by 4-bit FLANN
1.2
Direct m odel by 8-bit FLANN
Direct m odel by 12-bit FLANN
Direct m odel by 16-bit FLANN
Normalized voltage
1
0.8
0.6
0.4
0.2
0
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized pressure
0.7
0.8
0.9
Fig. 4.5 Simulation results of fixed-point FLANN direct model at 300C, of the word lengths t = 4, 8, 12
and 16-bits.
The plot of the fixed-point FLANN based inverse model is shown in Fig. 4.4. In this case
the inverse model is connected in series and is of fixed-point type. The same parameters, as
used for direct model, are used for this simulation also. The results of simulations (direct,
inverse and resultant) at different bit lengths and at different temperatures are shown in
Figs. 4.6 through Fig. 4.8 [4.4,4.5]. These plots clearly show that at low bit length, the
resultant plot between ( PN ) F and ( PN′ ) F are not quite linear. This means that at low bit
length (e.g. t = 8-bit) the linearization offered by fixed-point FLANN model is poor. Fig.
64
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
4.6 shows the performance of the inverse model corresponding 8-bit representation of
data.
8-bit fixed-point analysis of the inverse model of CPS by FLANN
1.4
Response of the CPS at -10 deg C
Inverse response by FLANN at -10 deg C
Overall response
(VN)F, (PN')F, (PN')F ------->
1.2
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
(PN)F, (VN)F, (PN')F ------->
1.2
1.4
8-bit fixed-point analysis of the inverse model of CPS by FLANN
1.4
(VN)F, (PN')F, (PN')F ------->
1.2
1
0.8
0.6
0.4
0.2
Response of the CPS at 100 deg C
Inverse response by FLANN at 100 deg C
Overall response
0
a
0
0.2
0.4
0.6
0.8
1
(PN)F, (VN)F, (PN')F ------->
1.2
1.4
b
Fig. 4.6 Plots of forward, inverse and overall characteristics of the CPS by fixed-point FLANN at t = 8bits; (a) at –100C, (b) at 1000C.
Axes Parameters:
(PN)F = fixed-point normalized pressure
(VN)F = fixed-point normalized voltage (output of the CPS+SCC)
(PN’)F = fixed-point estimated pressure
65
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
When the bit length is increased to 12 the linearization behaviour improves. This is shown
in Fig. 4.7 (a) and (b).
12-bit fixed-point analysis of the inverse modeling of CPS by FLANN
1.4
(VN)F, (PN')F, (PN')F ------->
1.2
1
0.8
0.6
0.4
0.2
0
Response of the CPS at -10 deg C
Inverse response by FLANN at -10 deg C
Overall response
0
0.2
0.4
0.6
0.8
1
(PN)F, (VN)F, (PN')F ------->
1.2
1.4
12-bit fixed-point analysis of the inverse modeling of CPS by FLANN
1.4
(VN)F, (PN')F, (PN')F ------->
1.2
1
0.8
0.6
0.4
0.2
Response of the CPS at 100 deg C
Inverse response by FLANN at 100 deg C
0
(PN)F,Overall
(VN)F,response
(PN')F ------->
a
b
-0.2
-0.2
0
0.2
0.4
0.6
0.8
(PN)F, (VN)F, (PN')F ------->
1
1.2
1.4
Fig. 4.7 Plots of forward, inverse and overall characteristics of the CPS by fixed-point FLANN at t = 12bits; (a) at –100C, (b) at 1000C.
Axes Parameters:
(PN)F = fixed-point normalized pressure
(VN)F = fixed-point normalized voltage (output of the CPS+SCC)
(PN’)F = fixed-point estimated pressure
66
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
At 16-bit word length the fixed-point effect is greatly reduced which is evident from Fig.
4.8.
16-bit fixed-point analysis of the inverse model of CPS by FLANN
1.4
(VN)F, (PN')F, (PN')F ------->
1.2
1
0.8
0.6
0.4
0.2
0
Response of the CPS at -10 deg C
Inverse response by FLANN at -10 deg C
Overall response
0
0.2
0.4
0.6
0.8
1
(PN)F, (VN)F, (PN')F ------->
1.2
1.4
16-bit fixed-point analysis of the inverse model of CPS by FLANN
1.4
(VN)F, (PN')F, (PN')F ------->
1.2
1
0.8
0.6
0.4
0.2
Response of the CPS at 100 deg C
Inverse response by FLANN at 100 deg C
Overall response
0
a
b
-0.2
-0.2
0
0.2
0.4
0.6
0.8
(PN)F, (VN)F, (PN')F ------->
1
1.2
1.4
Fig. 4.8 Plots of forward, inverse and overall characteristics of the CPS by fixed-point FLANN at t = 16bits; (a) at –100C, (b) at 1000C.
Axes Parameters:
(PN)F = fixed-point normalized pressure
(VN)F = fixed-point normalized voltage (output of the CPS+SCC)
(PN’)F = fixed-point estimated pressure
67
P E R F O R M A N C E
E V A L U A T I O N
O F
F I X E D
P O I N T
F L A N N
B A S E D C P S
M O D E L
4.3 Summary and Discussion
In floating-point implementation the performance of the direct and inverse models of various
sensors is quite good. However, it is shown that the word length greatly affects the linearity
performance of these sensors. Thus to achieve an appreciation performance, judicious choosing
of word length is very helpful. The present Chapter provides in-depth information on this
burning issue.
REFERENCES
[4.1]
J.C. Patra, Alex C. Kot, and G. Panda, “An Intelligent Pressure Sensor Using
Neural Networks,” IEEE Trans. on Instrumentation & Measurement, vol.49, No.4,
pp.829-834, Aug.2000.
[4.2]
J.C. Patra and A. van den Bos, “Modeling of an intelligent pressure sensor using
functional link artificial neural networks”, ISA Transactions, 39 (2000), pp. 15-27.
[4.3]
B. Widrow, S.D. Stearns, “Adaptive Signal Processing”, 2nd edition, Pearson Education
Asia.
[4.4]
S.K. Mishra, G. Panda, D.P. Das, S.K. Pattanaik and M.R. Meher, “Performance
Evaluation of Fixed-point Inverse FLANN model of Intelligent Pressure Sensors,”
Proceedings of NCC-2005, pp.647-650, 28th-30th Jan. 2005, IIT, Kharagpur.
[4.5]
G.Panda and S.K.Mishra, “Performance Evaluation of Fixed-point FLANN model of
Intelligent Pressure Sensors”, communicated to IETE Journal of Research, on Oct, 2005.
68
5
Chapter
Nonlinear Compensation of LVDT
using Different ANN Techniques
T
HIS CHAPTER deals with the nonlinearity compensation of Linear Variable
Differential Transformer (LVDT). In many practical control systems LVDT is used as
the sensing element for displacement [5.1,5.2]. The performance of the control system
depends upon the performance of the sensing elements. Many researchers have worked to
design LVDT with high linearity [5.3-5.7]. In its conventional design methodology achieving
high linearity involves complex design task. Sophisticated and precise winding machines are used
to achieve that. It is difficult to have all LVDT fabricated in a factory at a time to be equally
linear. LVDT having different nonlinearity present in a control system malfunctions at times
because of the difference in sensor characteristics.
In Section 5.1, the electrical characteristics of LVDT are presented. Section 5.2 deals with the
nonlinearity compensation of LVDT and its experimental set-up. The simulation studies are
69
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O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
carried out using different ANN models (MLP, FLANN and CFLANN) and the experimental
dataset of a standard LVDT is dealt in Section 5.3. The results obtained from different models
and their computational complexities are presented in Section 5.4. The summary and discussion
of this Chapter is depicted in Section 5.5.
5.1 Linear Variable Differential Transformer (LVDT)
The LVDT consists of the primary coil and the secondary coil. The secondary coil itself has
two coils connected differentially for providing the output. The two secondary coils are
located on the two sides of the primary coil on the bobbin or sleeve and these two output
windings (secondary coils) are connected in opposition to produce zero output at the
middle position of the armature [5.1,5.2]. The lengths of primary and two identical halves
of the secondary coils are b and m respectively. The coils have an inside radius ri and an
outside radius of r0. The spacing between the coils is d. Inside the coils a ferromagnetic
armature of length La and radius ri (neglecting the bobbin thickness) moves in an axial
direction. The number of turns in the primary coil is np and ns is the number of turns in
each secondary coil. The cross-sectional view of LVDT is shown in Fig. 5.1.
m
r0
Primary
d
d
b
m
ri
Armature
Sec. I
S1
Sec. II
S2
L1
La
L2
Fig. 5.1 Cross-sectional view of an LVDT
With a primary sinusoidal excitation voltage, Vp and the current Ip (r.m.s.) of frequency f,
the r.m.s. voltage e1 induced emf in the secondary coil S1 [5.1] is,
e1 =
4π 3 fI p n p ns 2 L2 + b 2
⋅
⋅
x1 ,
107 ln ( r0 ri ) mLa
70
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O F
L V D T
U S I N G
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T E C H N I Q U E S
and in coil S2,
e2 =
4π 3 fI p n p ns 2 L1 + b 2
⋅
⋅
x2 .
107 ln ( r0 ri ) mLa
where
x1 = Distance penetrated by the armature towards the secondary coil S1, and x2 =
Distance penetrated by the armature towards the secondary coil S2,
The differential voltage e = e1 − e2 is thus given by
e = k1 x (1 − k 2 x 2 )
where x =
1
( x1 − x2 ) is the armature displacement and,
2
k1 =
with
(5.1)
x0 =
16π 3 fI p n p ns ( b + 2d + x0 ) x0
10 ln ( r0 ri ) mLa
7
(Vm )
−1
(5.2)
1
( x1 + x2 ) and
2
k2 = 1
( b + 2d + x0 ) x0
(5.3)
k2 is a nonlinearity factor in (5.1), the non-linearity term ε being
ε = k2 x 2
(5.4)
For a given accuracy and maximum displacement the over-all length of the transducer is
minimum for x0 = b ,assuming that at maximum displacement the armature does not
emerge from the secondary coils. Taking the length of armature La = 3b + 2d , neglecting 2d
compared with b and using (5.2), (5.1) can be simplified as
71
N O N L I N E A R
C O M P E N S A T I O N
O F
L V D T
U S I N G
16π 3 fI p n p ns 2b ⎛
x2 ⎞
e= 7
1
⋅
−
⎜
⎟
10 ln ( r0 ri ) 3m ⎝ 2b 2 ⎠
D I F F E R E N T A N N
T E C H N I Q U E S
(5.5)
For a given primary sinusoidal excitation, the secondary output voltage e is nonlinear with
respect to displacement x . This is shown in Fig. 5.2 in which the linear region of the plot is
indicated as xm . This limitation is inherent in all differential systems and methods of
extending the range have been proposed by various authors mainly by appropriate design
and arrangement of the coils. Some of these are:
(i)
Balanced linear tapered secondary coils. But the improvement in linearity is not
significant [5.3].
(ii)
Over-wound linear tapered secondary coil improves the linearity to a certain extent
[5.4].
(iii)
Balanced over-wound linear tapered secondary coils. Its performance is similar to
(ii) [5.5].
(iv)
Balanced profile secondary coils helps in extending linearity range by proper
profiling of the secondary coils [5.6].
(v)
Use of complementary tapered winding extends the linearity range but the winding
is quite complicated as it involves sectionalised winding [5.7].
All these methods improve the linearity performance to some extent and are based on
improvement on constructional design. The methods are nonelectronic in nature and hence are
nonadaptive. As a result they do not account for the nonlinearity contributed due to
environmental changes. Hence there is a need to devise intelligent methods for adaptively
compensating the nonlinearity contributed due to internal construction and temperature
variation of LVDT. Incorporating the sensing feature of malfunctioning and adaptively
compensating capability makes the LVDT an intelligent one.
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T E C H N I Q U E S
6
4
Voltage
2
Range of
linearity, xm
0
-2
-4
-6
-6
-4
-2
0
Displacement
2
4
6
Fig. 5.2 Range of linear region of LVDT
5.2 Nonlinearity Compensation of LVDT
The proposed nonlinearity compensation scheme is shown in Fig. 5.3. In this scheme the
LVDT can be controlled by a displacement actuator. The main controller gives an actuating
signal to the displacement actuator, which displaces the core of the LVDT. The differential
voltage of the LVDT after being demodulated does not keep linear relationship with the
displacement. The nonlinearity compensator can be developed by using different ANN
techniques like MLP, FLANN and cascaded FLANN. The output of the ANN based
nonlinearity compensator is compared with the desired signal (actuating signal of the
displacement actuator) to produce an error signal. With this error signal, the weight vectors
of the ANN model are updated. This process is repeated till the mean square error (MSE)
is minimized. Once the training is complete, the LVDT together with the ANN model acts
like a linear LVDT with enhanced dynamic range.
Displacement
Actuator
e
x
LVDT
ANN based
nonlinearity
compensator
x-
Σ
+
Controller
Fig. 5.3 Scheme of an intelligent nonlinearity compensation of LVDT
73
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O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
The practical set-up of the LVDT along with the ANN based nonlinearity compensator
after the training is shown in Fig. 5.4.
x
ANN
model
LVDT
x
Fig. 5.4 Practical set-up of LVDT after training
The percentage of linearity (PEL) of the LVDT can be computed as
PEL =
( desired output − estimated output ) *100
desired output
(5.6)
5.3 Simulation Studies
To demonstrate the effectiveness these ANN based nonlinear compensators, computer
simulation studies are carried out using experimental data obtained from a typical LVDT
having following specifications.
Number of turns in two secondary coils, S1 and S2, ns = 3300 turns each wound over it
uniformly with the two coils separated by a Teflon ring.
Core diameter is, 2ri = 4.4 mm,
Core length is, La = 4.5 mm.
The primary sinusoidal excitation voltage, Vp = 10V (peak-to-peak) of frequency, f = 5
kHz. Vp is 6.712 Vrms.
Primary winding resistance, rp = 260ohm.
Secondary winding resistance, sr1 = 426.8 ohm.
Secondary winding resistance, sr2 = 414 ohm.
Two secondaries are wound in opposite directions. The pitch is kept around 0.02-0.03 mm. The
experimentally measured data is presented in Table-5.1.
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N O N L I N E A R
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L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
TABLE-5.1: EXPERIMENTAL MEASURED DATA
Displacement
(x in mm)
-30
Differential Output voltage
(erms)
4.085
Demodulated voltage
output (e)
-5.185
-25
3.956
-5.017
-20
3.731
-4.717
-15
3.221
-4.039
-10
2.359
-2.896
-5
1.273
-1.494
Null position, 0
0.204
0.001
5
1.153
1.462
10
2.226
1.810
15
3.118
3.962
20
3.748
4.799
25
4.050
5.225
30
4.085
5.276
5.3.1 MLP based Nonlinearity Compensator
The differential or demodulated voltage e at the output of LVDT is normalized by dividing
each value with the maximum value. The normalized voltage output e is subjected to input to
the MLP based nonlinearity compensator. In case of the MLP, we used different neurons with
different layers. However, the 1-100-50-1 network is observed to perform better hence it is
chosen for simulation. Each hidden layer and the output layer contains tanh(.) type activation
function. The output of the MLP based nonlinearity compensator is compared with the
normalized input displacement of the LVDT. The BP algorithm, in which both the learning rate
and the momentum rate are chosen as 0.02 and 1 respectively, is used to adapt the weights of the
MLP. Applying various input patterns, the ANN weights are updated using the BP algorithm.
To enable complete learning, 10,000 iterations are made. Then, the weights of the various layers
of the MLP are frozen and stored in the memory. During the testing phase, the frozen weights
are used in the MLP model. Fig. 5.5 shows the nonlinearity compensation of LVDT by MLP. In
this model the PEL is obtained to be 3.1120 %.
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L V D T
U S I N G
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T E C H N I Q U E S
Compensation graph by MLP
Voltage output (Vdc), Output of ANN (y), Output of ANN (y)
1
Response of LVDT
Response of compensation network
Response after compensation
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Norm alised displacem ent (x), Voltage output (Vdc), Norm alised displacem ent (x)
Fig. 5.5 Response due to MLP (1-100-50-1) network
5.3.2 FLANN based Nonlinearity Compensator
In case of the FLANN model, the differential output of the LVDT is functionally
expanded. In this case we have used trigonometric functional expansion because it provides
better nonlinearity compensation compared to others. The expanded elements are then
multiplied with a set of weights, and then summed to yield the final output. The output of
the FLANN model is compared with the normalized input displacement of the LVDT to
give the error signal. Using the error and the input to the inverse model the weights are
updated by the simple Least Mean Square (LMS) algorithm. Two different experiments are
carried out using the same data set of Table-5.1. The first experiment is carried out with
less number of functional expansions i.e. P=10 with 10,000 iterations. In the second
experiment the number of functional expansions is chosen to be 50 and the number of
iterations is 5,000. Fig. 5.6 shows the response of the LVDT and the nonlinear
compensator for the first experiment. Fig. 5.7 shows the same results for the second
experiment. From these plots, it is observed that there is significant improvement in
linearity by FLANN based compensator with P=50. The model FLANN with 10 functional
expansions has achieved PEL about 0.3251 %; whereas with 50 functional expansions the
76
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O F
L V D T
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D I F F E R E N T A N N
T E C H N I Q U E S
PEL value has increased to 5.12×10-15 %. Therefore in the second experiment the
nonlinearity compensation is better than the first one [5.13].
Compensation graph by FLANN with P=10
Voltage output (Vdc), Output of ANN (y), Output of ANN (y)
1
Response of LVDT
Response of com pensation netw ork
Response after com pensation
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Norm alised displacem ent (x), Voltage output (Vdc), Norm alised displacem ent (x)
Fig. 5.6 Response due to trigonometric series FLANN with P = 10
Compensation graph by FLANN with P=50
Voltage output (Vdc), Output of ANN (y), Output of ANN (y)
1
Response of LVDT
0.8
Response of com pensation netw ork
Response after com pensation
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Norm alised displacem ent (x), Voltage output (Vdc), Norm alised displacem ent (x)
Fig. 5.7 Response due to trigonometric series FLANN with P = 50
77
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O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
5.3.3 CFLANN based Nonlinearity Compensator
The linearity characteristics of LVDT can further be improved by the proposed method by
employing Cascaded FLANN (CFLANN) structure, which essentially consists of two-stage
FLANN models. In this case each FLANN structure uses less number of nonlinear expansions
as compared to that needed for one-stage FLANN. Fig. 5.8 shows a 2-stage nonlinear expansion
scheme of the differential voltage e . The differential voltage e at the output of LVDT is
subjected to trigonometric expansion (M1 values) to achieve nonlinearly mapped values. These
are then multiplied with a set of weights and then added to produce an intermediate output
(stage-I). This output value undergoes further expansion (M2 values) which are then weighted
and summed to generate the final output (stage-II). It is compared with the desired signal
(actuating signal of the displacement actuator) to produce an error signal. With this error signal,
the weight vectors (2 sets) of the CFLANN model are updated. This process is repeated till the
mean square error (MSE) is minimized. Once the training is complete, the LVDT together with
the proposed model acts like a linear sensor with enhanced dynamic range.
SQ
WQ
1
en
y1 ( k )
Σ
Stage-I
WR
Functional Expansion
E
Functional Expansion
1
SR
Stage-II
d (k )
y2 ( k )
Σ
BP
Algorithm
-
+
Σ
ε (k )
Fig. 5.8 Structure of CFLANN model
Mathematical Derivation of CFLANN
Let E be the input vector of size N×1 which represents N number of elements; the nth
element is given by:
E ( n ) = en ,1 ≤ n ≤ N
(5.7)
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L V D T
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T E C H N I Q U E S
Each element undergoes nonlinear expansion to form M1 elements such that the resultant
matrix has the dimension of N×M1.
The trigonometric expansions of en are computed as
⎧
en
⎪
⎪
sq = ⎨ sin ( mπ en )
⎪
⎪⎩cos ( mπ en )
where m = 1, 2,
for q = 1
for q = 2, 4,… , M 1
for q = 3,5,… , M 1 +1
(5.8)
, M 1 2 . In the proposed expansion M1 is always an even number. For
example, if n = 1 and M 1 = 6 we have m = 1, 2, 3, and q = 1, 2,… , 7. Then the expanded
elements are
s1 = e1 ,
s2 = sin (π e1 ) ,
s3 = cos (π e1 ) ,
s4 = sin ( 2π e1 ) ,
s5 = cos ( 2π e1 ) ,
s6 = sin ( 3π e1 ) ,
s7 = cos ( 3π e1 ) .
In matrix notation the expanded elements of the input vector E, is denoted by SQ of size
N×(M1+1).
The bias input is unity. So an extra unity value is padded with the SQ matrix and the
dimension of the SQ matrix becomes N×Q, where Q = ( M 1 + 2 ) .
Let the weight vector of stage-I be represented by WQ having Q elements. The output of
stage-I, y1 is given as
Q
y1 = ∑ sq ⋅ wq
(5.9)
q =1
79
N O N L I N E A R
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O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
In matrix notation the output can be,
Y1 = SQ ⋅ WQT
(5.10)
In the ordinary FLANN model, the final output y1 is compared with the desired response.
The weights, wq are updated by using least mean square (LMS) algorithm [5.11].
However, in case of CFLANN, the output of stage-I, y1 is further expanded nonlinearly to
M2 elements each. Hence the functional expansion results
⎧
y1
⎪
⎪
sr = ⎨ sin ( lπ y1 )
⎪
⎪⎩cos ( lπ y1 )
where l = 1, 2,
for r = 1
for r = 2, 4,… , M 2
for r = 3,5,… , M 2 +1
(5.11)
, M 2 2. Here M2 is an even number. The matrix form of the nonlinearly
expanded elements is represented by SR having dimension N×(M2+1).
In the same way as stage-I, the bias input is included to form SR with dimension N×R,
where R = ( M 2 + 2 ) . Let WR be the weight vector of stage-II with R elements. The output
of IInd stage is given by
R
y2 = ∑ sr ⋅ wr
(5.12)
r =1
In matrix notation the final output is obtained as
Y2 = S R ⋅ WRT
(5.13)
At kth iteration the error signal ε ( k ) can be computed as
ε ( k ) = d ( k ) − y2 ( k )
(5.14)
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O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
where d ( k ) is the desired signal which is same as the control signal given to the
displacement actuator.
Let ξ ( k ) denote the cost function at iteration k and is given by
ξ (k ) =
1 J 2
∑ε j (k )
2 j =1
(5.15)
where J is the number of nodes at the final output layer.
The weight vectors (of both stage-I and stage-II) can be updated by back-propagation (BP)
algorithm, [5.12] as
wr ( k + 1) = wr ( k ) + ∆wr ( k )
(5.16)
wq ( k + 1) = wq ( k ) + ∆wq ( k )
(5.17)
where,
∆wr ( k ) = −η ⋅
∂ξ ( k )
(5.18)
∂wr ( k )
where η is the learning-rate parameter (0 ≤ η ≤ 1). The minus sign accounts for gradient
descent in weight space. (5.18) can be solved to yield
∆wr ( k ) = η ⋅ δ r ( k ) ⋅ sr
(5.19)
where δ r ( k ) is the local gradient and is defined as,
δr (k ) = −
∂ξ ( k )
∂y2 ( k )
(5.20)
= ε (k )
Similarly the change in weights of stage-I is given by
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N O N L I N E A R
C O M P E N S A T I O N
∆wq ( k ) = −η ⋅
O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
∂ξ ( k )
∂wq ( k )
(5.21)
∆wq ( k ) = η ⋅ δ q ( k ) ⋅ sq
(5.22)
where δ q ( k ) is the local gradient of stage-I and is defined as
δq (k ) = −
∂ξ ( k )
∂y1 ( k )
(5.23)
= ε ( k ) ⋅ w j ⋅ ϕ ′ ( y1 ( k ) )
where sr = ϕ ( y1 ( k ) ) and is obtained by functionally expanding y1 ( k ) . ϕ ′ ( y1 ( k ) ) is the
first order derivative of ϕ ( y1 ( k ) ) .
Simulation studies are carried out using the same experimental data of Table: 5.1, but
employing the CFLANN model developed in this Chapter. In this case 12 trigonometric
expansions are taken in stage-I and 4 expansions in stage-II respectively and the PEL is
found to be 3.41×10-15 % [5.15]. The plot of CFLANN model is shown in Fig. 5.9.
CFLANN model
Voltage output (Vdc), Output of ANN (y), Output of ANN (y)
1
Response of LVDT
Response of com pensation netw ork
0.8
Response after com pensation
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Norm alised displacem ent (x), Voltage output (Vdc), Norm alised displacem ent (x)
Fig. 5.9 Plots of response of the CFLANN model,
Nonlinearly mapped inputs = 12 (stage-I), 4 (stage-II)
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O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
5.4 Computational Complexity
In this section the computational complexity of MLP, FLANN and CFLANN models is
evaluated and compared. In case of MLP, the structure chosen is 1-H1-H2-1, i.e. number of
input, hidden layer-1, hidden layer-2, and output nodes are 1, H1, H2, 1, respectively. Three
basic computations, i.e. addition, multiplication and computation of tanh(.) are involved in
forward path of the MLP; whereas in FLANN and CFLANN, along with the additions and the
multiplications, some extra computations are required for calculating sin(.) and cos(.) functions.
In FLANN, the order chosen is P. The total number of elements after P numbers of
trigonometric expansion is M=2P+1. The computational complexity involved during the
training period is not considered as this operation is performed offline. Table 5.2 presents the
operation units during normal operation of various ANN models.
TABLE-5.2: COMPUTATIONAL COMPLEXITY
Network
Addition
Multiplication
tanh(.)
sin(.)/cos(.)
Percentage of Linearity
MLP
(1-100-50-1)
FLANN
(1-22-1)
FLANN
(1-52-1)
CFLANN
(1-14-6-1)
5150
(100+5000+50)
5301
(200+5050+51)
151
---
3.1120%
21
22
---
51
52
---
18
20
---
20
(10+10)
50
(25+25)
16
(12+4)
0.0033%
5.12×10-15%
3.41×10-15%
From Table-5.2, it is evident that the CFLANN model drastically reduces the computational
complexity. It also offers better linearity compared to MLP and FLANN models (1-22-1). The
1-52-1 FLANN model though offers almost similar linearity as CFLANN; its complexity is
much higher.
5.5 Summary and Discussion
Different efficient ANN based nonlinearity compensators for LVDT are developed in this
Chapter. The nonlinearity compensation capability of MLP is poor; the PEL is about 3.1120 %.
The high order FLANN based nonlinearity compensator improves the linearity quite
appreciably but involves more computations. To reduce the computational complexity of the
FLANN model, the CFLANN structure is proposed here. With reduced number of expansions
at each stage, the CFLANN model provides better performance compared to others. In Table-
83
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C O M P E N S A T I O N
O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
5.2, it is noted that the CFLANN (1-14-6-1) model employs 18 additions, 20 multiplications and
16 numbers of sin(.)/cos(.) expansions as compared to 51 additions, 52 multiplications and 50
numbers of sin(.)/cos(.) expansions needed for FLANN model. Thus, considering performance
and computational complexity, it is observed that the proposed 2-stage FLANN model is
preferred one compared to other models studied in this Chapter.
REFERENCES
[5.1]
Hermann K.P. Neubert, “Instrument Transducers: An Introduction to their performance &
design”, 2nd edition, Oxford University Press.
[5.2]
John P. Bentley, “Principles of Measurement Systems”, 3rd edition, Pearson Education.
[5.3]
Y. Kano, S. Hasebe, and H. Miyaji, “New linear variable differential transformer with
square coils,” IEEE Transactions on Magnetics, Vol. 26, Issue:5, pp.2020 – 2022, Sep 1990.
[5.4]
S.C. Saxena, and S.B.L. Sexena, “A self-compensated smart LVDT transducer,” IEEE
Transactions on Instrumentation and Measurement, Vol. 38, Issue: 3, pp.748 – 753, June 1989.
[5.5]
G. Y. Tian, Z, X. Zhao, R. W. Baines, N. Zhang, “Computational algorithms foe linear
variable differential transformers (LVDTs)” IEE Proc.-Sci. Measurement Technology, Vol.
144, No. 4,pp 189-192, July 1997.
[5.6]
D. Crescini, A. Flammini, D. Marioli, and A. Taroni, “Application of an FFT-based
algorithm to signal processing of LVDT position sensors” IEEE Transactions on
Instrumentation and Measurement, Vol. 47, No. 5, pp. 1119-1123 Oct. 1998.
[5.7]
R. M. Ford, R. S. Weissbach, and D. R. Loker, “A novel DSP-based LVDT signal
conditioner” IEEE Transactions on Instrumentation and Measurement, Vol. 50, No. 3, pp. 768774, June 2001.
[5.8]
J.C. Patra, A. C. Kot, and G. Panda, “An intelligent pressure sensor using neural
networks,” IEEE Transactions on Instrumentation and Measurement, vol.49, No.4, pp.829834, Aug.2000.
[5.9]
J. C. Patra, G. Panda, and R. Baliarsingh, “Artificial neural network-based nonlinearity
estimation of pressure sensors”, IEEE Transactions on Instrumentation and Measurement, vol.
43, No. 6, pp.874-881, Dec. 1994.
84
N O N L I N E A R
C O M P E N S A T I O N
O F
L V D T
U S I N G
D I F F E R E N T A N N
T E C H N I Q U E S
[5.10] J.C. Patra, and R.N. Pal, “A functional link artificial neural network for adaptive channel
equalization”, Signal Processing 43(1995) 181-195, vol.43, no.2, May 1995.
[5.11] B. Widrow and S.D. Sterns, “Adaptive Signal Processing” Prentice-Hall, Inc. Engle-wood
Cliffs, New Jersey, 1985.
[5.12] S. Haykin, “Neural Networks: A comprehensive foundation” 2nd Edition, Pearson Education
Asia, 2002.
[5.13] S.K. Mishra, G. Panda, D.P. Das, S.K. Pattanaik and M.R. Meher, “A Novel Method of
Designing LVDT using Artificial Neural Network,” Proceedings of IEEE, ICISIP-2005,
pp.223-227, Jan, 2005.
[5.14] G. Panda, S.K. Mishra, and D.P. Das, “A novel method of extending the linearity range
of LVDT using artificial neural network” (paper ID: SNA-D-05-00679) submitted
(under final revision) to Sensors & Actuators: A. Physical, Elsevier, on Nov, 2005.
[5.15] S.K. Mishra and G. Panda, “A Novel Method for Designing LVDT and its
Comparison with Conventional Design”,(paper ID: 6014) accepted in IEEE Sensors
Applications Symposium (SAS-2006) held on Houston Marriott West Loop by the Galleria,
Houston, Texas, February 7-9, 2006.
85
6
Chapter
Adaptive Nonlinear Compensation of
Control Valve Actuator using Soft
Computing Techniques
T
HE CONTROL VALVE ACTUATORS are the most common actuators in the
process industries. They are found in process plants, manipulating flows to maintain
controlled variable at their set points. A control valve acts as a variable restriction in a
process pipe. By changing its opening it changes the resistance to flow and, thus, the flow itself
[6.1]. The equal percentage control valve actuator devices have nonlinearities in their inherent
flow characteristics. But when it is put in a pipe line, it acts as nearly linear control valve, called
installed characteristics. The linearity depends on the pressure difference across the valve. As the
pressure difference across the value varies, it is always required to compensate to get very
accurate linear characteristics. Some of other fundamental actuator nonlinearities include friction,
deadzone, saturation, backlash and hysteresis. Amongst these the friction, the deadzone and the
saturation are static nonlinearities, whereas the backlash and the hysteresis are dynamic
86
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A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
nonlinearities. Various attempts have been made by many researchers to compensate the static
and dynamic nonlinearities present in the actuators [6.3-6.10]. The motivation of this Chapter is to
develop the efficient adaptive inverse models with less computational complexity to compensate
the nonlinearity effect of the equal percentage control valve actuator characteristics.
The rest of the Chapter is organized as follows. In Section 6.1, the operation of different control
valve characteristics is presented. Section 6.2 develops the proposed adaptive inverse algorithms
required to develop the intelligent model and their simulation studies. Finally, the summary and
discussion are presented in Section 6.3.
6.1 Control Valve Actuator
The control valve is essentially a
Air Signal
variable resistance to the flow of a
fluid, in which the resistance and
Diaphragm
therefore the flow, can be changed by
ACTUATOR
a signal from a process controller. The
Packing
cross sectional view of a control valve
is shown in Fig. 6.1. The function of a
control valve is to vary the flow of
fluid through the valve by means of a
change of pressure to the valve top.
Supply
Seat
Stem
VALVE
Delivery
Plug
Fig. 6.1 Cross-sectional view of a control valve actuator
The relation between the flow through the valve and the valve stem position (or lift) is
called the valve characteristics. It may be differentiated as inherent flow characteristics and
installed flow characteristics. The inherent flow characteristics refers to the characteristics observed
with constant pressure drop across the valve. Where as the installed flow characteristics refers to
the characteristic observed when the valve is in service with varying pressure drop and
other changes in the system. The relation between the flow through the valve and the valve
stem position (or lift) is called the valve characteristics [6.1, 6.2].
In general, the flow through a control valve for a specific fluid at a given temperature can
be expressed as
87
A D A P T I V E
N O N L I N E A R C O M P E N S A T I O N O F C O N T R O L V A L V E
A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
q = f1 ( L, p0 , p1 )
(6.1)
where q = volumetric flow rate,
L = valve stem position (or lift)
p0 = upstream pressure
p1 = downstream pressure
The inherent valve characteristics is determined for fixed values of p0 and p1, for which (6.1)
becomes
q = f2 ( L )
(6.2)
Let, m = q qmax and x = L Lmax , whe re qmax is the flow when the valve stem is at its
maximum lift Lmax (valve is full open), x is the fraction of maximum lift, and m is the
fraction of maximum flow.
Hence (6.2) can be rewritten as
m = q qmax = f ( L Lmax )
or,
m = f ( x)
(6.3)
The sensitivity may be represented as
s=
dm
dx
(6.4)
In terms of valve characteristics, valves can be divided into three types: decreasing
sensitivity, linear and increasing sensitivity. For the decreasing sensitivity type, the
sensitivity (slope) decreases with m and it is known as “quick opening flow characteristics”.
For the linear flow characteristics, the sensitivity is constant and the characteristic is a
straight line. For the increasing type, the sensitivity increases with flow and it is also known
as “equal percentage valve”.
Mathematically the linear flow characteristics can be represented as
88
A D A P T I V E
N O N L I N E A R C O M P E N S A T I O N O F C O N T R O L V A L V E
A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
dm
=α
dx
(6.5)
where α is a constant. And assuming that the valve is shut tight when the lift is at lowest
position, we have m = 0 at x = 0 . For a single-seated valve that is not badly worn, the valve
can be shut off for x = 0 .
Integrating (6.5) and introducing the limits m = 0 at x = 0 and m = 1 at x = 1 gives
1
1
0
0
∫ dm = ∫ α dx
(6.6)
By solving this we get α = 1 .
For α = 1 , (6.5) can now be integrated to give
m=x
(linear valve)
(6.7)
Hence the linear type valves are commonly used in liquid level loops and in other processes
where the pressure drop across the valve is fairly constant.
For the equal percentage valve, the slope can be
dm
= βm
dx
(6.8)
where β is constant. Integration of (6.8) gives
m
or,
x
dm
∫m m = ∫0 β dx
0
(6.9)
m
= βx
m0
(6.10)
ln
where m0 is the flow at x = 0 .
89
A D A P T I V E
N O N L I N E A R C O M P E N S A T I O N O F C O N T R O L V A L V E
A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
The basis for calling the valve characteristics equal percentage can be by rearranging (6.8) in
the form
dm
= β dx
m
and
∆m
= β ∆x
m
(6.11)
in this form it can be seen that an equal fractional (or percentage) change in flow ( ∆m m )
occurs for a specified increment of change in stem position ( ∆x ) , regardless of where the
change in stem position occurs along the change the characteristic curve.
The term β can be expressed in terms of m0 by inserting m = 1 at x = 1 into (6.10). The
result is
⎛ 1 ⎞
⎟
⎝ m0 ⎠
β = ln ⎜
Then (6.9) for m gives
m = m0 e β x
(equal percentage valve)
(6.12)
m0 is the flow at x = 0 . m0 cannot be zero since there may be some leakage when the stem is
at its lowest position. For some valves, especially large ones, the valve manufacturer
intentionally allows some leakage at minimum lift ( x = 0 ) to prevent binding and wearing
of the plug and seat surfaces.
The Rangeability (R) is defined as the ratio of maximum flow to minimum controllable
flow over which the valve characteristic is followed,
R=
mmax
(6.13)
mmin,controllable
The main constraint in equal percentage valve is low rangeability. The rangeability mainly
depends on the term m0. For example, if m0 = 0.02 , the rangeability is 50. The application of
90
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A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
these valves requires in services where large variations in pressure drop are expected or in
services where a small percentage of the total system pressure drop is to be taken.
6.2 Adaptive Inverse Models
The practical setup of the interfacing of the inverse model with the control valve actuator is
shown in Fig. 6.2. This figure depicts that the adaptive inverse model is connected in series with
the actuator to compensate the associated varying nonlinearity of the actuator.
In this section we have employed three types of inverse models (LMS, MLP and FLANN) for
nonlinearity compensation of the control valve actuator and their performance in terms of
convergence speed, MSE level, accuracy and computational complexity has been compared.
Brief discussion of each network and its mathematical derivation is mentioned in Chapter 2. The
simulation studies of all types of adaptive inverse model are dealt in this section.
error
x
Control valve
Actuator
m
Inverse
Model
x
Σ
x
Fig. 6.2 Set-up for compensating nonlinearity of control valve actuator
6.2.1 LMS based Inverse Model
The detail of the LMS algorithm [6.11] is already reported in Chapter 2. The input flow x varies
from 1 to 15 psig with an increment of 1 psig. Accordingly the output of the control valve is
calculated by (12). The normalized output of the control valve serves as the input to the inverse
model and the output of that is compared with the normalized pressure. In this case, the order
of the tap is 20. The learning rate of the updation algorithm is taken as 0.1. To make the model
effective, 5000 iterations are used to train the inverse model. The weights of the network are
frozen after updation. The PEL is achieved as 4.48×10-15 %. The overall response of the model
is shown in Fig. 6.3.
91
A D A P T I V E
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A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
LMS based Inverse model
1
Response of control valve
Response of com pensation netw ork
Response after com pensation
0.9
0.8
0.7
output
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
input
0.6
0.7
0.8
0.9
1
Fig. 6.3 Plots of response of the LMS based inverse model
6.2.2 MLP based Inverse Model
In case of the MLP, (1-50-30-1) network has been taken for the development of the inverse
model of the actuator. Each hidden and the output layer consists of a nonlinear activation
function of tanh(.) type [6.12]. The weights of the network are updated by using the back
propagation algorithm. The learning rate and the momentum parameter of the algorithm are
chosen as 0.02 and 1 respectively. The number of iterations is chosen as 10000 to effectively
train the network. The response of the actuator along with the response of the compensation
network is plotted in Fig. 6.4.
MLP based Inverse model
1
Response of control valve
Response of compensation netw ork
0.9
Response after compensation
0.8
0.7
Output
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Input
0.6
0.7
0.8
0.9
Fig. 6.4 Plots of response of the MLP based inverse model
92
1
A D A P T I V E
N O N L I N E A R C O M P E N S A T I O N O F C O N T R O L V A L V E
A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
Once the training is complete, the weights of the network are frozen in the memory. In this case,
the PEL is obtained as 0.3106 %.
6.2.3 FLANN based Inverse Model
In this case, the output of the control valve actuator is functionally expanded into 50 elements.
The FLANN structure is formed as 1-52-1, including the original input and the bias term. There
are 52 weight elements in the network, and these are updated by employing the LMS algorithm.
The learning rate is taken as 0.01. For successfully training the network, 10000 iterations are
chosen. In this case the PEL is 7.70×10-5 %. Once the training is over, the coefficients are stored
and frozen in the memory. For testing phase, the inverse model is connected in series with the
control valve. The change of output of the control valve with respect to the input pressure is fed
to this inverse model. The FLANN model output is computed and compared with the actual
output to verify the effectiveness of the model [6.13]. The plot of the FLANN based inverse
model is shown in Fig. 6.5.
FLANN based Inverse model
1
Response of control valve
Response of com pensation netw ork
Response after com pensation
0.9
0.8
0.7
Output
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Input
0.6
0.7
0.8
0.9
1
Fig. 6.5 Plots of response of the FLANN based inverse model
6.3 Summary and Discussion
In this Chapter, three adaptive inverse models have been proposed. A comparative study
between of these three models has been carried out through simulation study. It is observed that
the LMS based inverse model offers better linearity performance compared to others. The LMS
93
A D A P T I V E
N O N L I N E A R C O M P E N S A T I O N O F C O N T R O L V A L V E
A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
based inverse model is simple to implement and offers improved performance compared to
others. This is because the actuator offers less nonlinearity which does not require MLP and
FLANN structures.
REFERENCES
[6.1]
D. R. Coughanowr, “Process Systems Analysis and Control”, 2nd edition, McGraw-Hill
international editions, 1991.
[6.2]
G. Tao and P. V. Kokotovic, “Adaptive Control of Systems with Actuator and Sensor
Nonlinearities”, John Wiley & Sons, New York, 1996.
[6.3]
B. Armstrong-Htlouvry, P. Dupont, and C. Canudas de Wit, “A survey of models,
analysis tools and compensation methods for the control of machines with fiction,”
Automatica, vol.30, no.7, pp.1083-1138, 1994.
[6.4]
C. Canudas de Wit, P. Noel, A. Aubin, and B. Brogliato, “Adaptive friction
compensation in robot manipulators: Low velocities,” Int. J. Robot. Res., vol.10, no.3,
Jun. 1991.
[6.5]
S. W. Lee and J. H. Kim, “Robust adaptive stick-slip fiction compensation,” IEEE
Trans. Ind. Electron., vol. 42, no.5, pp. 474-479, Oct. 1995.
[6.6]
Y. H. Kim and F. L. Lewis, “Reinforcement adaptive learning neural-net-based
friction compensation control for high speed and precision,” IEEE Trans. Control
Sysi. Technol., vol. 8, no. l, pp.118-126, Jan. 2000.
[6.7]
C. A. Desoer and S. M. Shahruz, “Stability of dithered nonlinear systems with
backlash or hysteresis,” Int. J. Contr., vol. 43, no. 4, pp. 1045-1060, 1986.
[6.8]
R. R. Selmic, F. L. Lewis, “Deadzone Compensation in Nonlinear Systems using
Neural Network”, Proc. of the 37th IEEE conference of Decision & control, Dec, 1998.
[6.9]
Y. D. Song, T. L. Mitchell, and H. Y. Lai, “Control of a class of nonlinear uncertain
systems via compensated inverse dynamics approach,” IEEE Trans. Automat.
Control, vol. 39, no. 9, pp. 1866-1871, Sep. 1994.
[6.10] R. R. Selmic, V. V. Phoha and F. L. Lewis, “Intelligent control of Actuator
Nonlinearities”, Proc. of the 42nd IEEE Conf. on Decis. & Contr., Maui, Hawaii USA,
Dec, 2003.
94
A D A P T I V E
N O N L I N E A R C O M P E N S A T I O N O F C O N T R O L V A L V E
A C T U A T O R U S I N G S O F T C O M P U T I N G T E C H N I Q U E S
[6.11] B. Widrow, S. D. Stearns, “Adaptive Signal Processing”, Pearson Education Asia, 2002.
[6.12] S. Haykin, “Neural Networks: A Comprehensive Foundation”, 2nd edition, Pearson
Education Asia, 2002.
[6.13] S.K.Mishra, G.Panda, T.K.Dan and S.K.Pattanaik, “Artificial Neural Network
based Nonlinearity Estimation of Control Valve Actuators”, Proceeding of National
Conference on Soft Computing Techniques for Engineering Applications (SCT2006), pp.223232, 24th-26th Mar, 2006, at NIT, Rourkela.
95
7
Chapter
Conclusions
T HE overall conclusions of the thesis are the following:
(1)
It investigates on the nonlinearity issues relating to three typical sensors:
(i)
The CPS
(ii) The LVDT
(iii) The control valve actuators.
(2)
The nonlinearity problem gives rise to the following difficulties:
(i)
Non accuracy in measurement
(ii) Limitation of dynamic range (linearity region)
(iii) Full potentiality of the sensor cannot be utilized.
(3)
The nonlinearity problem arises due to:
96
C O N C L U S I O N S
(i)
Environmental changes such as change in temperature, humidity and atmospheric
pressure
(ii) Aging
(iii) Constructional limitations.
(4)
In this thesis adaptive and intelligent methods for compensation of nonlinearities have
been proposed and have been applied to three typical sensors.
(5)
These methods are based on the following structures:
(i)
MLP
(ii) FLANN
(iii) CFLANN
(iv) RBFNN
The learning algorithms employed in the thesis are:
(i)
LMS algorithm in FLANN
(ii) BP algorithm in MLP and CFLANN
(iii) RBF learning algorithm
(6)
Basically nonlinear compensation has been achieved through
(i)
Direct modeling of the sensors
(ii) Inverse modeling of the sensors
(7)
Exhaustive simulation studies of various methods show that MLP and RBFNN
structures provide improved linearity compensation performance but involves more
computations and tedious to implement. However, the proposed FLANN and
CFLANN based solutions offer quite satisfactory performance and involve low
complexity.
(8)
The thesis also presents the fixed-point implementation of the FLANN based direct and
inverse models of the sensors.
97
C O N C L U S I O N S
7.1 Scope for Further Research Work
Many further research work may be carried out on the same and related topics.
(1)
For developing the inverse model, supervised learning has been used. It has employed
training data. In many situations the training data is not available. In absence of such
data, training can be carried out by employing blind techniques which are known as
unsupervised method. Investigation is needed to develop nonlinear compensator using
blind techniques such as higher order statistics (HOS).
(2)
The investigation made in this thesis can also be extended to other types of sensors and
instruments.
(3)
Practical implementation of the inverse model as a plug-in-module, cascading it with
physical sensors and using the combined ones for real-time applications is very
important. Further research work can be carried out in this direction.
(4)
FPGA implementation of the proposed nonlinearity compensators can be carried out as
further research work.
98
C O N C L U S I O N S
Publications out of the Research Work
[P1] G. Panda, D.P. Das, S.K. Mishra, S.K. Panda, S. Meher and K.K. Mahapatra,
“Development of an Efficient Intelligent Pressure Sensor using RBF based ANN”,
Proceedings of International Conference on Intelligent Signal Processing & Robotics (ISPR-2004),
(ITS2 in CD), 20th-23rd Feb, 2004, IIIT, Allahabad.
[P2] S.K. Mishra, G. Panda, D.P. Das, S.K. Pattanaik and M.R. Meher, “Performance
Evaluation of Fixed-point Inverse FLANN model of Intelligent Pressure Sensors,”
Proceedings of NCC-2005, pp.647-650, 28th-30th Jan. 2005, IIT, Kharagpur.
[P3] S.K. Mishra, G. Panda, D.P. Das, S.K. Pattanaik and M.R. Meher, “A Novel Method
of Designing LVDT using Artificial Neural Network,” Proceedings of IEEE, ICISIP2005, pp.223-227, Jan, 2005.
[P4] S.K.Mishra and G.Panda, “A New Methodology for Design of LVDT and
Comparison with Conventional Design”,(paper ID: 520-133) accepted in IASTED
International Conference on Signal Processing, Pattern Recognition and Applications (SPPRA
2006), held on 15th Feb, 2006, in Innsbruck, Austria.
[P5] S.K. Mishra and G. Panda, “A Novel Method for Designing LVDT and its
Comparison with Conventional Design”,(paper ID: 6014) accepted in IEEE Sensors
Applications Symposium (SAS-2006) held on Houston Marriott West Loop by the
Galleria, Houston, Texas, February 7-9, 2006.
[P6] S.K.Mishra, G.Panda, T.K. Dan and S.K. Pattanaik, “Artificial Neural Network
based Nonlinearity Estimation of Control Valve Actuators”, Proceeding of National
Conference on Soft Computing Techniques for Engineering Applications (SCT2006), pp.223-232,
24th-26th Mar, 2006, at NIT, Rourkela.
[P7] G. Panda, S.K. Mishra, and D.P. Das, “A novel method of extending the linearity
range of LVDT using artificial neural network” (paper ID: SNA-D-05-00679)
submitted (under final revision) to Sensors & Actuators: A. Physical, Elsevier, on Nov,
2005.
[P8] G.Panda and S.K.Mishra, “Development of an Intelligent LVDT using CFLANN
Structure”, (paper ID: IM-8409) communicated to IEEE Transaction on Instrumentation
& Measurement, on Jan, 2006.
[P9] G.Panda and S.K.Mishra, “Performance Evaluation of Fixed-point FLANN model
of Intelligent Pressure Sensors”, Communicated to IETE Journal of Research, on Oct,
2005.
99
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