ORTHOGONAL FUNCTIONS IN
SYSTEMS AND CONTROL
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ORTHOGONAL FUNCTIONS IN
SYSTEMS AND CONTROL
Kanti B Datta & 6 M Mohan Department
of Electrical
Indian Institute Kharagpur-
of
Engineering Technology
721302,
India
1 1 8 * World Scientific V M Singapore • New Jersey • London • Hong Kong
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P reface
I n this b o o k , Orthogonal
Functions
in Systems
and Control,
the results of investiga-
tions o f some aspects of the i d e n t i f i c a t i o n o f l u m p e d parameter, d i s t r i b u t e d p a r a m e t e r , t i m e - v a r y i n g a n d nonlinear d i s t r i b u t e d parameter systems are presented. T h e analysis of time-delay systems and also the o p t i m a l c o n t r o l , m i n i m i z i n g a q u a d r a t i c
perfor-
mance index a n d using an observer for the estimated feedback are o u t l i n e d . These problems are s t u d i e d w i t h the help of piecewise constant basis f u n c t i o n s , o r t h o g o nal p o l y n o m i a l s and sine-cosine functions and a c o m p a r a t i v e s t u d y is made thereof c o n c l u d i n g , w i t h the help of a variety of n u m e r i c a l examples, t h a t the o r t h o g o n a l p o l y n o m i a l s are superior t o other o r t h o g o n a l functions for these problems. T h e state-of-art o f the o r t h o g o n a l functions for the s t u d y of problems arising i n Systems a n d C o n t r o l is i n c l u d e d i n C h a p t e r 1. T h e h i s t o r y of o r t h o g o n a l p o l y n o m i als and sine-cosine functions is q u i t e o l d . B u t s u r p r i s i n g l y enough, t h e i r extensive a p p l i c a t i o n t o the area of c o n t r o l and systems was s t a r t e d one decade ago.
The
least-squares a p p r o x i m a t i o n o f an a r b i t r a r y f u n c t i o n gives rise to a l l the o r t h o g o n a l p o l y n o m i a l s and sine-cosine functions.
A n i n t e g r a l framework is p r o v i d e d i n Chap-
ter 2 to derive these p o l y n o m i a l s as a set of generalized o r t h o g o n a l p o l y n o m i a l s , the t h r e e - t e r m recurrence formulae and the ( o r d i n a r y ) differential recurrence relations of these o r t h o g o n a l p o l y n o m i a l s are also p r o v i d e d as t h e y are i m p o r t a n t for the signal analysis.
T o c a r r y o u t the analysis of signals available over an a r b i t r a r y i n t e r v a l ,
shifted o r t h o g o n a l p o l y n o m i a l s are i n t r o d u c e d i n C h a p t e r 3, w h i c h are capable of describing signals over any i n t e r v a l of our interest. However, for i n f i n i t e range p o l y nomials, t h e t e r m shift has been used i n a different sense. A c o m p a r a t i v e s t u d y of the signal representation v i a different classes of o r t h o g o n a l systems is i n c l u d e d considering the effect of noise, and the
filtering
properties o f these o r t h o g o n a l systems
are also s t u d i e d . T h e t w o - d i m e n s i o n a l o r t h o g o n a l functions and t h e i r a p p l i c a t i o n to the representation o f t w o - d i m e n s i o n a l signals are also o u t l i n e d . T h e i n t e g r a l and the derivative o p e r a t i o n a l matrices are i n t r o d u c e d as the t w o t i m e - d o m a i n operators to reduce the i n t e g r a l and the derivative operations to an algebraic o p e r a t i o n i n the sense o f least squares. For each o r t h o g o n a l system, the error i n t r o d u c e d by the integ r a l o p e r a t o r is analyzed. A n i n t e g r a l framework is p r o v i d e d for the d e r i v a t i o n o f the i n t e g r a t i o n o p e r a t i o n a l m a t r i x v i a generalized o r t h o g o n a l p o l y n o m i a l s . T h e analysis o f c o n t i n u o u s - t i m e d y n a m i c a l systems c o n t a i n i n g time-delays is discussed i n C h a p t e r 4 v i a o r t h o g o n a l functions such as piecewise constant basis func-
vi
Preface
tions, o r t h o g o n a l p o l y n o m i a l s and sine-cosine functions. A n i n t e g r a l f r a m e w o r k is provided t o construct the delay and the d e l a y - i n t e g r a t i o n o p e r a t i o n a l matrices for the o r t h o g o n a l polynomials and sine-cosine functions needed for this purpose. To i m prove the c o m p u t a t i o n a l accuracy, the time-partition technique is i n t r o d u c e d w h i c h can be used w i t h any system o f o r t h o g o n a l functions. T h e above m e t h o d of analysis is also extended t o systems having m u l t i p l e delays and piecewise constant delays. T h e various possible sources of errors arrising i n the analysis o f t i m e - d e l a y systems and the means of reducing t h e m are also considered. T h e methods t o estimate the p a r a m eters and wherever possible, the i n i t i a l conditions, of linear t i m e - i n v a r i a n t systems from the measurements of the i n p u t s and the o u t p u t s are o u t l i n e d i n C h a p t e r 5 using the o r t h o g o n a l functions. F i r s t the single-input single-output systems are considered and then the m u l t i - i n p u t m u l t i - o u t p u t systems whose m o d e l is assumed i n the form of a transfer f u n c t i o n m a t r i x . To i m p r o v e the accuracy, one shot o p e r a t i o n a l m a t r i x for repeated integrations of the o r t h o g o n a l functions is i n t r o d u c e d . I n C h a p t e r 6, a general and i n t e g r a l framework is presented for the e s t i m a t i o n of parameters, and i n i t i a l and b o u n d a r y conditions of the first- and second-order linear t i m e - i n v a r i a n t single-input single-output continuous-time d i s t r i b u t e d parameter systems f r o m the measurements of the i n p u t and o u t p u t signals. I n Chapter 7, an i n t e g r a l approach for the i d e n t i f i c a t i o n of parameters, i n i t i a l and b o u n d a r y conditions of linear t i m e varying, and nonlinear d i s t r i b u t e d parameter systems is presented w h e n the i n p u t and o u t p u t signals of the system are available f r o m measurements. I n C h a p t e r 8, the use of o r t h o g o n a l p o l y n o m i a l s , block-pulse and sine-cosine functions t o determine the t i m e - v a r y i n g gain for a state variable feedback i n a linear c o n t r o l system m i n i m i z i n g a quadratic performance index is o u t l i n e d . T h i s t i m e - v a r y i n g g a i n K ( t ) can, i n fact, be expressed i n terms o f the s o l u t i o n of a m a t r i x R i c c a t i differential e q u a t i o n . T h e r e are two methods to c o m p u t e this t i m e - v a r y i n g gain. T h e first is a direct m e t h o d w h i c h does not require the c o m p u t a t i o n of the state t r a n s i t i o n m a t r i x . O n the c o n t r a r y , the second m e t h o d is based on first evaluating the state t r a n s i t i o n m a t r i x and t h e n the t i m e - v a r y i n g gain is expressed i n terms of the elements o f t h i s m a t r i x . T h e o p t i m a l control law can be expressed by u*(<) = AT(2)x(<). I f , however, a l l the states are not available, the estimated state x ( t ) must be generated v i a an observer and the corresponding c o n t r o l law can be c o m p u t e d v i a the system of o r t h o g o n a l functions. T h e investigations i n the analysis and the parameter i d e n t i f i c a t i o n i n c o n t r o l systems are classified i n t o forty seven different problems, and each p r o b l e m i n t o as m a n y number of basis functions as have been used t o s t u d y t h e m .
A c r i t i c a l appraisal
showing the h i s t o r i c a l development of this subject based on m o r e t h a n three-hundred papers is one o f the m a i n a t t r a c t i o n s of this b o o k . O t h e r i m p o r t a n t features are the unified approach based on a l l o r t h o g o n a l functions t o s t u d y t h e m a j o r problems i n Systems and C o n t r o l from the n u m e r i c a l p o i n t of view and a c o m p a r a t i v e evaluation of the p o t e n t i a l i t i e s of a l l o r t h o g o n a l functions t o these problems. T h e t i m e - p a r t i t i o n m e t h o d is an i m p o r t a n t c o n t r i b u t i o n i n this b o o k as the most p o w e r f u l m e t h o d k n o w n t i l l t o d a y for the analysis of time-delay systems. New a l g o r i t h m s for t h e i d e n t i f i c a t i o n
vi
Preface
tions, o r t h o g o n a l p o l y n o m i a l s and sine-cosine functions. A n i n t e g r a l f r a m e w o r k is provided t o construct the delay and the d e l a y - i n t e g r a t i o n o p e r a t i o n a l matrices for the o r t h o g o n a l polynomials and sine-cosine functions needed for this purpose. To i m prove the c o m p u t a t i o n a l accuracy, the time-partition technique is i n t r o d u c e d w h i c h can be used w i t h any system o f o r t h o g o n a l functions. T h e above m e t h o d of analysis is also extended t o systems having m u l t i p l e delays and piecewise constant delays. T h e various possible sources of errors arrising i n the analysis o f t i m e - d e l a y systems and the means of reducing t h e m are also considered. T h e methods t o estimate the p a r a m eters and wherever possible, the i n i t i a l conditions, of linear t i m e - i n v a r i a n t systems from the measurements of the i n p u t s and the o u t p u t s are o u t l i n e d i n C h a p t e r 5 using the o r t h o g o n a l functions. F i r s t the single-input single-output systems are considered and then the m u l t i - i n p u t m u l t i - o u t p u t systems whose m o d e l is assumed i n the form of a transfer f u n c t i o n m a t r i x . To i m p r o v e the accuracy, one shot o p e r a t i o n a l m a t r i x for repeated integrations of the o r t h o g o n a l functions is i n t r o d u c e d . I n C h a p t e r 6, a general and i n t e g r a l framework is presented for the e s t i m a t i o n of parameters, and i n i t i a l and b o u n d a r y conditions of the first- and second-order linear t i m e - i n v a r i a n t single-input single-output continuous-time d i s t r i b u t e d parameter systems f r o m the measurements of the i n p u t and o u t p u t signals. I n Chapter 7, an i n t e g r a l approach for the i d e n t i f i c a t i o n of parameters, i n i t i a l and b o u n d a r y conditions of linear t i m e varying, and nonlinear d i s t r i b u t e d parameter systems is presented w h e n the i n p u t and o u t p u t signals of the system are available f r o m measurements. I n C h a p t e r 8, the use of o r t h o g o n a l p o l y n o m i a l s , block-pulse and sine-cosine functions t o determine the t i m e - v a r y i n g gain for a state variable feedback i n a linear c o n t r o l system m i n i m i z i n g a quadratic performance index is o u t l i n e d . T h i s t i m e - v a r y i n g g a i n K ( t ) can, i n fact, be expressed i n terms o f the s o l u t i o n of a m a t r i x R i c c a t i differential e q u a t i o n . T h e r e are two methods to c o m p u t e this t i m e - v a r y i n g gain. T h e first is a direct m e t h o d w h i c h does not require the c o m p u t a t i o n of the state t r a n s i t i o n m a t r i x . O n the c o n t r a r y , the second m e t h o d is based on first evaluating the state t r a n s i t i o n m a t r i x and t h e n the t i m e - v a r y i n g gain is expressed i n terms of the elements o f t h i s m a t r i x . T h e o p t i m a l control law can be expressed by u*(<) = AT(2)x(<). I f , however, a l l the states are not available, the estimated state x ( t ) must be generated v i a an observer and the corresponding c o n t r o l law can be c o m p u t e d v i a the system of o r t h o g o n a l functions. T h e investigations i n the analysis and the parameter i d e n t i f i c a t i o n i n c o n t r o l systems are classified i n t o forty seven different problems, and each p r o b l e m i n t o as m a n y number of basis functions as have been used t o s t u d y t h e m .
A c r i t i c a l appraisal
showing the h i s t o r i c a l development of this subject based on m o r e t h a n three-hundred papers is one o f the m a i n a t t r a c t i o n s of this b o o k . O t h e r i m p o r t a n t features are the unified approach based on a l l o r t h o g o n a l functions t o s t u d y t h e m a j o r problems i n Systems and C o n t r o l from the n u m e r i c a l p o i n t of view and a c o m p a r a t i v e evaluation of the p o t e n t i a l i t i e s of a l l o r t h o g o n a l functions t o these problems. T h e t i m e - p a r t i t i o n m e t h o d is an i m p o r t a n t c o n t r i b u t i o n i n this b o o k as the most p o w e r f u l m e t h o d k n o w n t i l l t o d a y for the analysis of time-delay systems. New a l g o r i t h m s for t h e i d e n t i f i c a t i o n
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C ontents
Preface
v
Chapter 1
Orthogonal Functions i n Systems and C o n t r o l :
A Historical Perspective
1
1.1
H i s t o r y of O r t h o g o n a l Functions
1
1.2
O r t h o g o n a l Functions i n Systems and C o n t r o l
3
Chapter 2
Least Squares A p p r o x i m a t i o n o f Signals
2.1
Least Squares A p p r o x i m a t i o n
2.2
O r t h o g o n a l P o l y n o m i a l A p p r o x i m a t i o n : A Generalized A p p r o a c h
25 25 . .
36
2.3
Legendre P o l y n o m i a l s
2.4
Laguerre Polynomials
40 43
2.5
Hermite Polynomials
47
2.6
Tchebycheff P o l y n o m i a l s of the F i r s t K i n d
51
2.7
Tchebycheff P o l y n o m i a l s of the Second K i n d
54
2.8
Jacobi P o l y n o m i a l s
56
2.9
G a m m a , B e t a and H y p e r g e o m e t r i c Functions
58
2.10 A l t e r n a t i v e Representation of Jacobi Polynomials 2.10.1
Jacobi Differential E q u a t i o n
2.11 Gegenbauer P o l y n o m i a l s
61 66 67
2.11.1
Properties of G a m m a F u n c t i o n
70
2.11.2
N o r m a l i z a t i o n Factor of Gegenbauer Polynomials
72
2.12 R o d r i g u e s ' Formulae
73
2.13 D i f f e r e n t i a l Recurrence Relations
75
2.14 O r d i n a r y D i f f e r e n t i a l Recurrence Relations
78
2.15 I n t e g r a t i o n O p e r a t i o n a l M a t r i x
81
2.16 P r o b l e m s
84
Chapter 3 3.1
Signal Processing in Continuous T i m e D o m a i n
87
Shifted O r t h o g o n a l P o l y n o m i a l s
87
3.1.1
Shifted Laguerre Polynomials
91
3.1.2
Shifted H e r m i t e P o l y n o m i a l s
92
3.1.3
Sine-Cosine Functions
93
Contents
x
3.1.4
Block-Pulse Functions
3.1.5
Examples
95
3.2
Analysis of Noisy Signals
99
3.3
Analysis of Signals Characterized by T w o - D i m e n s i o n a l
3.4
I n t e g r a t i o n O p e r a t i o n a l M a t r i x of Shifted Functions
3.5
Derivative Operational M a t r i x
118
3.6
Conclusion
124
Square-Integrable
Chapter 4
94
Functions
103 114
Analysis of Time-Delay Systems
127
4.1
Introduction
4.2
Delay O p e r a t i o n a l M a t r i x
129
4.2.1
129
4.2.2
127
Delay O p e r a t i o n a l M a t r i x for Block-Pulse Functions Delay O p e r a t i o n a l M a t r i x for O r t h o g o n a l P o l y n o m i a l s and Sine-Cosine Functions
130
4.3
Delay-Integration Operational M a t r i x
131
4.4
Integration Operational M a t r i x E
131
4.5
Analysis of T i m e - D e l a y Systems
T
for Sine-Cosine Functions
132
4.5.1
Block-Pulse Function A p p r o a c h
4.5.2
Recursive A l g o r i t h m
4.5.3
Orthogonal Polynomial Approach
135
4.5.4
Sine-Cosine F u n c t i o n A p p r o a c h
136
4.6
Time-Partition Method
4.7
Problems
Chapter 5
133 . . .
134
141 .'
Identification of L u m p e d Parameter Systems
153 157
5.1
Introduction
5.2
One Shot O p e r a t i o n a l M a t r i x for Repeated I n t e g r a t i o n s ( O S O M R I ) .
159
5.2.1
160
5.2.2
O S O M R I of Block-Pulse Functions O S O M R I of Shifted Tchebycheff Polynomials o f the F i r s t K i n d :
5.2.3
5.3
157
161
O S O M R I of Shifted Tchebycheff Polynomials o f the Second K i n d :
163
5.2.4
O S O M R I of Shifted Legendre Polynomials
164
5.2.5
O S O M R I of Laguerre and H e r m i t e P o l y n o m i a l s
166
5.2.6
O S O M R I of Sine-Cosine Functions:
166
I d e n t i f i c a t i o n of L u m p e d Parameter Systems
167
5.3.1
169
Examples of L u m p e d Parameter System I d e n t i f i c a t i o n
5.4
Transfer F u n c t i o n M a t r i x I d e n t i f i c a t i o n
173
5.5
Conclusion
179
5.6
Problems
180
x l
Contents Chapter 6
Identification of D i s t r i b u t e d Parameter Systems
183
6.1
Introduction
183
6.2
U n i f i e d A p p r o a c h for I d e n t i f i c a t i o n
185
6.2.1
I d e n t i f i c a t i o n v i a Block-Pulse Functions
190
6.2.2
I d e n t i f i c a t i o n v i a Sine-Cosine Functions . . . . ,
190
6.3
P r a c t i c a l L i m i t a t i o n s of I d e n t i f i c a t i o n
6.4
I l l u s t r a t i v e Examples
192
6.5
Problems
201
Chapter 7
Identification of Linear T i m e - V a r y i n g and
191
Nonlinear
D i s t r i b u t e d Parameter Systems
203
7.1
Introduction
203
7.2
M a t h e m a t i c a l Preliminaries
203
7.3
I d e n t i f i c a t i o n o f T i m e - V a r y i n g Systems
206
7.4
I d e n t i f i c a t i o n of Nonlinear. Systems
207
7.4.1
209
I d e n t i f i c a t i o n v i a Block-Pulse Functions
7.5
P r a c t i c a l L i m i t a t i o n s of the I d e n t i f i c a t i o n A l g o r i t h m
209
7.6
Conclusion
210
7.7
Problems
211
Chapter 8
O p t i m a l C o n t r o l of Linear Systems
213
8.1
Introduction
213
8.2
S o l u t i o n of R i c c a t i E q u a t i o n W i t h o u t S T M
215
8.2.1
Laguerre P o l y n o m i a l s and Recursive C o m p u t a t i o n
218
8.2.2
B l o c k - P u l s e Functions and Recursive C o m p u t a t i o n
218
8.3
Riccati Equation Solution w i t h S T M
219
8.4
Examples
221
8.5
E s t i m a t e d State Feedback Using A n Observer
227
8.6
Orthogonal Function Approach
229
8.6.1
Laguerre Polynomials
230
8.6.2
B l o c k - P u l s e Functions
231
8.7
Problems
232
List of Figures
235
List o f Tables
237
List of Abbreviations
239
Bibliography
241
Author Index
268
Subject
272
Index
Chapter
1
Orthogonal and
Functions
Control:
A
in
Systems
Historical
P ersp ective
A b r i e f review of i m p o r t a n t developments i n the field of ' o r t h o g o n a l functions and t h e i r applications i n systems and c o n t r o l ' is i n c l u d e d i n this chapter. A classification of u p - t o - d a t e l i t e r a t u r e on the basis of problems studied and the basis functions used is p r o v i d e d i n a chronological order. F i n a l l y , an o u t l i n e of some i m p o r t a n t features of the research a c t i v i t y i n the area of systems and c o n t r o l w i t h the help o f o r t h o g o n a l functions is given.
1.1
History of Orthogonal
Functions
T h e h i s t o r y of o r t h o g o n a l p o l y n o m i a l s is very o l d . T h e Legendre p o l y n o m i a l s o r i g i n a t e d f r o m d e t e r m i n i n g the force of a t t r a c t i o n exerted by solids o f r e v o l u t i o n [162] and t h e i r o r t h o g o n a l properties were established by A d r i e n M a r i e Legendre d u r i n g (1784-90). T h e p r o b l e m of solving o r d i n a r y differential equations over i n f i n i t e or semii n f i n i t e intervals and of o b t a i n i n g expansion of a r b i t r a r y functions over such intervals a t t r a c t e d the a t t e n t i o n of famous mathematicians
in the nineteenth c e n t u r y and to
resolve i t , functions k n o w n t o d a y as H e r m i t e p o l y n o m i a l s were i n t r o d u c e d [162] i n 1864 by Charles H e r m i t e (1822-1905). T h e theory of continued f r a c t i o n gives rise to all the o r t h o g o n a l p o l y n o m i a l s [304] and, i n fact, E d m o n d Laguerre (1834-1866) i n his effort to convert a divergent power series i n t o a convergent c o n t i n u e d f r a c t i o n i n 1879 [5] discovered p o l y n o m i a l s k n o w n t o d a y as Laguerre p o l y n o m i a l s .
I n 1807, Joseph Fourier (1768-1830) while solving the p a r t i a l differential e q u a t i o n encountered
i n connection w i t h c o n d u c t i o n of heat i n a r o d discovered t h a t the so-
l u t i o n can be expressed as a series of e x p o n e n t i a l l y weighted sine f u n c t i o n s .
Later,
2
Chapter
1: A Historical
Perspective
he extended this idea to represent any a r b i t r a r y f u n c t i o n as an i n f i n i t e sum of sine and cosine functions. Pafnuti L . Tchebycheff (1821-1894) observed t h a t o f a l l p o l y nomials a p p r o x i m a t i o n of an a r b i t r a r y f u n c t i o n i n the i n t e r v a l — 1 < x < 1, the one t h a t minimizes the m a x i m u m error is a linear c o m b i n a t i o n of p o l y n o m i a l s k n o w n today as the Tchebycheff p o l y n o m i a l s [162]. T h e genesis of o r t h o g o n a l p o l y n o m i a l s and sine-cosine functions is a l t h o u g h different arising as t h e y are i n a b i d to solve a diversity of problems, they are employed here for the sole purpose of i d e n t i f i c a t i o n of a continuous t i m e d y n a m i c a l m o d e l . These o r t h o g o n a l functions dated more t h a n one century back b u t surprisingly enough their p o t e n t i a l a p p l i c a t i o n to the problems of i d e n t i f i c a t i o n is barely one decade o l d . T h e r e is another family of o r t h o g o n a l functions w h i c h has assumed considerable i m p o r t a n c e i n the i d e n t i f i c a t i o n of parameters and i n i t i a l conditions i n a large number of systems. These functions are k n o w n as piecewise constant basis functions whose f u n c t i o n a l values are constant w i t h i n any subinterval o f a t i m e p e r i o d . I n t h i s family, i n 1910, Haar first proposed a set of periodic, o r t h o n o r m a l and complete system of functions, called t o d a y as Haar functions [83]. T h e second set of o r t h o g o n a l functions i n this class is k n o w n as Rademacher functions i n t r o d u c e d by Rademacher i n 1922 [251], w h i c h is, u n f o r t u n a t e l y , not a complete system o f o r t h o g o n a l functions. I n his effort to make this set complete, Walsh subsequently i n 1923 i n t r o d u c e d another set of complete o r t h o g o n a l functions called Walsh functions w h i c h triggered i n t o action a massive effort to a p p l y these functions in continuous-time m o d e l i d e n t i f i c a t i o n . There is another class of complete set of o r t h o g o n a l functions k n o w n as block-pulse functions w h i c h is much more popular and efficient i n the area o f p a r a m e t e r i d e n t i f i c a t i o n . These functions were being used for a long t i m e in c o m m u n i c a t i o n engineering and were found extremely useful to solve numerous problems i n this discipline. T h e class of o r t h o g o n a l functions have found wide a p p l i c a t i o n i n science and engineering such as i n the c o m m u n i c a t i o n engineering [86], nuclear science and engineering [2],[278],[316],[314], d i g i t a l p r o t e c t i o n of transmission lines [163], d i g i t a l signal processing [1],[171], d i g i t a l filters [161], picture processing [169], c o m p u t e r aided design of electronic circuits [301] etc.. Recently, considerable interest has been shown i n app l y i n g these functions to a large n u m b e r of diverse problems i n the field of 'Systems and C o n t r o l ' . T h i s has been possible only due to the pioneering w o r k of C o r r i n g t o n , 1973 [75], w h i c h has d r a w n a large number of workers to c o n t r i b u t e generously i n this field. D e p e n d i n g u p o n the s t r u c t u r e , the o r t h o g o n a l functions m a y be b r o a d l y classified i n t o three families: (a) (b) (c)
Piecewise Constant O r t h o g o n a l Functions, O r t h o g o n a l Polynomials, and Sine-Cosine (Fourier) Functions.
T h e block-pulse functions, Haar functions and W a l s h functions belong to the f a m i l y of piecewise constant o r t h o g o n a l functions w h i l e Legendre p o l y n o m i a l s , Laguerre p o l y n o m i a l s , H e r m i t e p o l y n o m i a l s , Tchebycheff p o l y n o m i a l s o f the first k i n d ,
1.2 Orthogonal
Functions
in Systems
and
3
Control
Tchebycheff p o l y n o m i a l s of the second k i n d , Jacobi p o l y n o m i a l s , a n d Gegenbauer p o l y n o m i a l s belong t o the f a m i l y of o r t h o g o n a l p o l y n o m i a l s . Since each class o f these functions forms a basis for the series expansion o f a square-integrable f u n c t i o n , ort h o g o n a l functions are c o m m o n l y referred t o as basis functions.
I n a d d i t i o n to the
functions m e n t i o n e d above, there are other classes o f basis functions such as pulse o r t h o g o n a l f u n c t i o n s , delayed u n i t step functions, H a h n p o l y n o m i a l s , piecewise linear p o l y n o m i a l f u n c t i o n s , and T a y l o r p o l y n o m i a l s w h i c h have been used t o s t u d y numerous problems i n systems and c o n t r o l . T h e piecewise linear p o l y n o m i a l functions a n d T a y l o r p o l y n o m i a l s do n o t however belong to the f a m i l y of o r t h o g o n a l functions. T h e extensive research i n t h e field of 'systems and c o n t r o l v i a o r t h o g o n a l f u n c t i o n s ' has resulted i n a large n u m b e r of research publications i n the last t w o decades. I t is therefore e x t r e m e l y difficult to discuss i n d e t a i l a l l the developments made i n t h i s broad a n d specialised
field.
However, i n order to u n d e r s t a n d and appreciate the
problems w h i c h can be h a n d l e d and solved by a p p l y i n g o r t h o g o n a l f u n c t i o n s ,
an
o u t l i n e of research activities i n this field is p r o v i d e d i n t h e f o l l o w i n g section.
1.2
Orthogonal Functions in Systems and
Control
For the first t i m e i n t h e h i s t o r y o f o r t h o g o n a l functions, C o r r i n g t o n , 1973 [75] showed t h a t i n t e g r a l equations c o u l d be reduced t o linear algebraic equations w i t h an app r o x i m a t i o n i n the sense o f least-squares. T o d e m o n s t r a t e this fact, he t o o k a set of W a l s h f u n c t i o n s , i n t e g r a t e d t h e m a n a l y t i c a l l y a n d expressed the result a p p r o x i m a t e l y i n terms of t h e o r i g i n a l set of W a l s h functions. T h e n he arranged the coefficients i n the W a l s h series expansion i n a t w o - d i m e n s i o n a l array a n d called i t ' l o o k - u p table'. I n the process o f i n t e g r a t i o n , a l l the discontinuities of W a l s h functions were removed. Moreover, successive i n t e g r a t i o n of W a l s h functions smoothened the W a l s h functions even m o r e .
O n the other h a n d , when Walsh functions were differentiated, t h e re-
sult was a series of D i r a c delta-functions, one for each d i s c o n t i n u i t y , w i t h a l t e r n a t i n g signs. H i g h e r order derivatives w o u l d lead to higher order s i n g u l a r i t y functions whose W a l s h series expansions were usually divergent. Therefore, C o r r i n g t o n r u l e d o u t diff e r e n t i a t i o n o f W a l s h functions for the s o l u t i o n of differential equations. U s i n g t h e above concept o f repeated integrations of W a l s h functions, he [75] solved linear a n d n o n l i n e a r differential equations and i n t e g r a l equations.
I t appears, t h a t
t h i s paper i n s p i r e d remarkable research activities w i t h W a l s h functions i n systems and c o n t r o l . I n the f o l l o w i n g year Cheng and L i u , 1974 [55] analysed discrete dyadici n v a r i a n t systems by using W a l s h functions. I n C o r r i n g t o n ' s m e t h o d of solving differential equations, a c e r t a i n s o l u t i o n was i n i t i a l l y assumed a n d an i t e r a t i v e technique was used t o i m p r o v e this s o l u t i o n . Since i t e r a t i v e techniques consume m u c h t i m e a n d must be shown t o converge, V a n et al, 1975 [317] proposed a simple algebraic f o r m u l a as a s o l u t i o n to differential equations
4
Chapter
1: A Historical
Perspective
w i t h constant and variable coefficients. For this purpose, t w o n u m e r i c a l i n t e g r a l and differential operators were developed. I n the same year, Chen and Hsiao, 1975 [34] applied Walsh functions to estimate the parameters of linear t i m e - i n v a r i a n t systems w i t h zero i n i t i a l conditions to determine a suitable i n t e r n a l s t r u c t u r e for a system from its i n p u t - o u t p u t response. T h i s m e t h o d was based on repeated integrations of Walsh functions. Except for the difference in t e r m i n o l o g y , the o p e r a t i o n a l m a t r i x for the i n t e g r a t i o n of the Walsh basis vector used i n this paper is s i m i l a r t o the ' i n t e g r a l operator' used by Van et al. However, Chen and Hsiao gave a general expression for the i n t e g r a t i o n o p e r a t i o n a l m a t r i x of order m . T h e c o n t r o l of a linear system by m i n i m i z i n g a quadratic performance index gives rise t o a t i m e - v a r y i n g gain for the linear state variable feedback, and this gain is obtained by solving a m a t r i x R i c c a t i differential e q u a t i o n . Chen and Hsiao, 1975 [35], by a p p l y i n g the i n t e g r a t i o n o p e r a t i o n a l m a t r i x of Walsh functions, obtained a numerical s o l u t i o n of the m a t r i x Riccati differential equation and o b t a i n e d the solution of the t i m e - v a r y i n g gain in the form of a piecewise constant f u n c t i o n . I n this paper, the authors also presented a Kronecker p r o d u c t f o r m u l a for the solution of state equations. A l m o s t at the same t i m e , b u t independently Rao and Sivakumar, 1975 [264] also used the o p e r a t i o n a l m a t r i x of i n t e g r a t i o n v i a Walsh functions to identify the parameters as well as i n i t i a l conditions of a linear t i m e - i n v a r i a n t system. These papers [34],[264] on system i d e n t i f i c a t i o n v i a Walsh functions formed a significant step towards the use of Walsh funtctions for the i d e n t i f i c a t i o n of p a r a m e t r i c models. Power-law devices or systems, such as square law diodes, occur q u i t e frequently i n engineering problems. T h e analysis of such systems was studied v i a W a l s h functions by M a q u s i , 1977 [197]. Chen et a l , 1977 [37] established a r e l a t i o n between Walsh functions and block-pulse functions, the latter are o r t h o g o n a l and very much useful in c o m m u n i c a t i o n engineering. I n this paper, the o p e r a t i o n a l m a t r i x for i n t e g r a t i o n of block-pulse functions was also derived. Moreover, i t was shown t h a t block-pulse functions are more f u n d a m e n t a l t h a n Walsh functions, and the strucure of the i n t e g r a t i o n o p e r a t i o n a l m a t r i x of the former is simpler t h a n t h a t of Walsh f u n c t i o n s . I t is i n this paper, t h a t the derivative o p e r a t i o n a l m a t r i x was first i n t r o d u c e d for block-pulse functions and t h e n for Walsh functions t h r o u g h the linear r e l a t i o n e x i s t i n g between t h e m . T h i s derivative o p e r a t i o n a l m a t r i x appears t o be i d e n t i c a l w i t h the one f o u n d in Van et al [317] for Walsh functions. In the same year, Stavroulakis and Tzafestas, 1977 [299] extended Walsh series approach to the design of observers and filters in o p t i m a l c o n t r o l systems. Sannuti's papaer, 1977 [276] on the analysis and synthesis of d y n a m i c a l systems is a significant step towards the a p p l i c a t i o n of block-pulse functions. S a n n u t i showed t h a t the o p t i m a l c o n t r o l p r o b l e m studied by Chen and Hsiao [32] could be solved using block-pulse functions w i t h m i n i m a l c o m p u t a t i o n a l effort. He, at the same t i m e , i n d i c a t e d the use of block-pulse functions i n solving general nonlinear systems, and p o i n t e d out the connection between this m e t h o d and the well k n o w n one using trape-
1.2 Orthogonal
Functions
in Systems
and
5
Control
z o i d a l a p p r o x i m a t i o n . Moreover, he d e m o n s t r a t e d t h a t the final results
obtainable
v i a W a l s h a n d block-pulse functions were the same a l t h o u g h the c o m p u t a t i o n a l techniques used i n these m e t h o d s were different. W a l s h function-based a l g o r i t h m s for the i d e n t i f i c a t i o n of linear t i m e - i n v a r i a n t dist r i b u t e d p a r a m e t e r systems were for the first t i m e given by Paraskevopoulos
and
Bounas, 1978 [239]. L i k e finite dimensional systems, the i n c o m p l e t e state feedback is v e r y m u c h i m p o r t a n t i n time-delay systems also. T h e design o f observers and
fil-
ters, based on Walsh functions [299] was extended t o these time-delay systems by S t a v r o u l a k i s and Tzafestas, 1978 [300].
Tzafestas, 1978 [313], carried o u t an ex-
h a u s t i v e i n v e s t i g a t i o n i n t o the a p p l i c a t i o n of W a l s h functions to the i d e n t i f i c a t i o n of s i n g l e - i n p u t and single-output ( S I S O ) , and m u l t i - i n p u t and m u l t i - o u t p u t ( M I M O ) systems w i t h zero and nonzero i n i t i a l conditions; stochastic, t i m e - v a r y i n g and dist r i b u t e d p a r a m e t e r systems, a n d also state-space and i n t e g r a l e q u a t i o n models. T h i s paper gave a complete p i c t u r e of the state-of-art of Walsh functions at t h a t t i m e . B y i n t r o d u c i n g a p r o d u c t m a t r i x of the Walsh basis vector and its transpose, and its p r o p e r t i e s , Chen and S h i h , 1978 [48] applied Walsh series a p p r o x i m a t i o n to estimate the parameters of bilinear systems. I n consonance w i t h Paraskevopoulos and Bounas [239], Shih and H a n , 1978 [288] also i n t r o d u c e d double W a l s h series for t h e s o l u t i o n o f a single and also a set of simultaneous first-order p a r t i a l differential equations. Chen and S h i h , 1978 [47] i n t r o d u c e d an o p e r a t i o n a l m a t r i x for b a c k w a r d i n t e g r a t i o n i n a b i d t o integrate the adjoint equation b a c k w a r d i n t i m e and solved the s t a t e t r a n s i t i o n m a t r i x of o p t i m a l c o n t r o l of linear t i m e - v a r y i n g systems w i t h a q u a d r a t i c performance index. Moreover, m a k i n g use of the o p e r a t i o n a l m a t r i x of W a l s h p r o d u c t m a t r i x [48], they analysed linear t i m e - v a r y i n g systems. A l t h o u g h Chen et a l .
[37] t o o k the help of block-pulse functions i n d e r i v i n g
the o p e r a t i o n a l matrices for f r a c t i o n a l calculus v i a Walsh functions, t h e y expressed t h e i r reservations t o G o p a l s a m i and Deekshatulu [82], [81] on the use of the former o r t h o g o n a l functions w h i c h must not be preferred since they were t h o u g h t n o t t o f o r m a complete set. T h i s n o t i o n of block-pulse functions, t h a t they d i d n o t f o r m a complete set, prevailed for a long t i m e u n t i l Rao and Srinivasan, 1978 [269] proved the p r o p e r t y o f t h e i r completeness. T h e scope of a p p l i c a t i o n s of piecewise constant o r t h o g o n a l functions was enlarged, w i t h an i n t r o d u c t i o n to a new o p e r a t i o n a l m a t r i x for delay, by Rao and Srinivasan, 1978 [266].
T h i s concept o f delay o p e r a t i o n a l m a t r i x for block-pulse functions was
f r u i t f u l l y a p p l i e d to solve o p t i m a l l y the nonlinear f u n c t i o n a l equations of ' L i e n a r d ' t y p e c o n t a i n i n g constant t i m e delays. B y i n t r o d u c i n g the shift Walsh m a t r i x , Chen and S h i h , 1978 [49] analysed linear t i m e - i n v a r i a n t and t i m e - v a r y i n g delay-differential systems. Tzafestas, 1978 [312] proposed a W a l s h - G a l e r k i n expansion m e t h o d for the design of d i s t r i b u t e d p a r a m e t e r o p t i m a l controllers and
filters.
Using the o r t h o g o n a l
p o l y n o m i a l s as a set of basis functions, Maroulas and B a r n e t t , 1978 [198] showed t h a t the characteristic p o l y n o m i a l of any m a t r i x was the characteristic p o l y n o m i a l of a comrade m a t r i x w i t h the help of w h i c h a n u m b e r of system theoretic properties
Chapter
6
1: A Historical
Perspective
were s t u d i e d . Block-pulse functions were also used to i n v e r t n u m e r i c a l l y Laplace [279].
T h e technique of m o d e l r e d u c t i o n is o f p a r a m o u n t
transforms
i m p o r t a n c e i n the area
of process dynamics and c o n t r o l . Considering its i m p o r t a n c e , B r i s t r i z and Langholz, 1979 [10] gave a technique based on Tchebycheff p o l y n o m i a l s of the first k i n d for m o d e l r e d u c t i o n . I n the same year Rao and Sivakumar, 1979 [261] p r o v i d e d an a l g o r i t h m for the i d e n t i f i c a t i o n of time-lag systems v i a Walsh functions. A single-term
method
to o b t a i n the s o l u t i o n of a d y n a m i c a l system recursively
over any l e n g t h of t i m e i n t e r v a l was i n t r o d u c e d by Rao et a l , 1980 [260]. first
As the
t e r m , w h i c h is used in the single-term approach, is c o m m o n t o any system of
o r t h o g o n a l functions, the i d e n t i t y of o r t h o g o n a l functions is lost i n this approach. T h e dynamics of the current collection mechanism of an electric l o c o m o t i v e gives rise to differential equations w i t h stretched arguments. B y i n t r o d u c i n g an o p e r a t i o n a l m a t r i x for stretch, Rao and Srinivasan, 1980 [268] o b t a i n e d a s o l u t i o n of stretched systems i n a simple recursive f o r m v i a block-pulse functions.
T o solve p a r t i a l dif-
ferential equations, m u l t i - d i m e n s i o n a l block-pulse functions were proposed by Rao and Srinivasan, 1980 [267]. T h e use of o r t h o g o n a l functions may produce solution
for otherwise
stable partial
differential
equations.
an
unstable
Walsh functions were also
used i n the p r e d i c t i o n of l i m i t cycles i n nonlinear systems. T h e describing f u n c t i o n of a n o n l i n e a r i t y was determined using the Walsh series i n analogy w i t h the Fourier series and an a t t e m p t was made to predict the l i m i t cycles i n nonlinear systems by Cameron et al, 1980 [80]. Based on block-pulse functions, Shih et a l , 1980 [290] o b t a i n e d piecewise constant a p p r o x i m a t e solutions of linear delay-differential systems.
Moreover, for the given
i n p u t and the k n o w n state variables, the u n k n o w n parameters of these delay systems were d e t e r m i n e d .
For a class of nonlinear d i s t r i b u t e d parameter systems, t h a t the
i d e n t i f i c a t i o n was possible v i a Walsh functions was shown by Sinha et a l , 1980 [293]. Chen and Jeng, 1981 [44] analysed
a linear piecewise constant
delay
dynamic
system by i n t r o d u c i n g a delay o p e r a t i o n a l m a t r i x w h i c h takes care of time-delays, not a m u l t i p l e of 1 / m , where m is the number of terms i n the block-pulse series expansion. By using block-pulse functions K w o n g and Chen, 1981 [170] identified the impulse response of an u n k n o w n plant from the k n o w n measurement f u n c t i o n i n the feedback p a t h and the given i n p u t and the o u t p u t response.
T h e earliest
attempt
to replace Walsh functions w i t h block-pulse functions i n the system i d e n t i f i c a t i o n of p a r a m e t r i c models was made by Palanisamy and B h a t t a c h a r y a , 1981 [233]. Hwang and Shih, 1981 [131] found o p e r a t i o n a l matrices for i n t e g r a t i o n and differentiation for Laguerre p o l y n o m i a l s . Jan and W o n g , 1981 [144] proposed a m e t h o d based on block-pulse functions for the i d e n t i f i c a t i o n of bilinear systems w h i c h , for the same accuracy, required much less c o m p u t a t i o n a l t i m e compared w i t h t h a t based on W a l s h functions [153]. Nurges and Yaaksoo, 1981 [219] suggested an approach e m p l o y i n g Laguerre p o l y n o m i a l s for the analysis of m u l t i v a r i a b l e discrete systems. T h e m e t h o d of Cheng and Hsu, 1982 [53] for o b t a i n i n g the s o l u t i o n and e s t i m a t i n g
1.2 Orthogonal
Functions
in Systems
and
7
Control
the parameters of bilinear systems was based on block-pulse o p e r a t i o n a l matrices.
The
recursive a l g o r i t h m for the s o l u t i o n of bilinear equations requires the c o m p u t a t i o n of a m a t r i x inverse i n each s u b i n t e r v a l . Due to the b i d i a g o n a l s t r u c t u r e of the i n t e g r a t i o n o p e r a t i o n a l m a t r i x of Laguerre p o l y n o m i a l s , H w a n g and S h i h , 1982 [136] a r r i v e d at a recursive i d e n t i f i c a t i o n a l g o r i t h m to estimate the u n k n o w n parameters of linear t i m e - i n v a r i a n t systems.
Chen and Lee, 1982 [46] presented a general W a l s h series
approach r e q u i r i n g no t i m e - p a r t i t i o n to analyse linear systems w i t h t i m e delays. I n order to s t u d y the scaled systems v i a Walsh functions, Rao and Palanisamy, 1982 [259] i n t r o d u c e d a W a l s h stretch o p e r a t i o n a l m a t r i x and employed the same for the s o l u t i o n o f f u n c t i o n a l differential equations. Hsu and Cheng, 1982 [112] studied
the
i d e n t i f i c a t i o n of nonlinear d i s t r i b u t e d systems previously studied by Sinha et a l , 1980 [293], by using block-pulse functions. Rao and S i v a k u m a r , 1982 [263], i n t r o d u c e d a technique of i d e n t i f i c a t i o n for piecewise linear systems.
T h e same a u t h o r s , 1982 [262], developed
a m e t h o d for the
i d e n t i f i c a t i o n of order and parameters of continuous linear systems. I n order to solve f u n c t i o n a l differential equations v i a Laguerre p o l y n o m i a l s , H w a n g and S h i h , 1982 [133] i n t r o d u c e d a Laguerre stretch o p e r a t i o n a l m a t r i x . I n the same year C h a n g and W a n g , 1982 [19] derived an o p e r a t i o n a l m a t r i x for the i n t e g r a t i o n of Legendre p o l y nomials and applied the same to the p r o b l e m of parameter i d e n t i f i c a t i o n investigated by H w a n g and Shih [136]. K u n g and Lee, 1982 [167] applied Laguerre p o l y n o m i a l s to solve t i m e - i n v a r i a n t and t i m e - v a r y i n g state equations and to estimate the parameters of u n i t y feedback systems.
T h i s approach provided solutions of state equations i n
the f o r m o f s m o o t h continuous t i m e - f u n c t i o n s .
H w a n g and S h i h , 1982 [137] applied
block-pulse functions to solve p o p u l a t i o n balance equations w h i c h were described by stretched differential and integro-differential equations. T h e i n t e g r a l power of an o p e r a t i o n a l m a t r i x for i n t e g r a t i o n accomplishes the task of repeated i n t e g r a t i o n s . Despite the simple algebraic r e l a t i o n , the use of higher powers leads to a c c u m u l a t i o n of errors at each stage of i n t e g r a t i o n . Consequently, the use of i n t e g r a t i o n o p e r a t i o n a l m a t r i x to higher order integro-differential equation models, gives rise to considerable errors. T h e accuracy of repeated i n t e g r a t i o n was i m p r o v e d by Rao and Palanisamy, 1983 [258] by i n t r o d u c i n g the so-called one shot o p e r a t i o n a l m a t r i x for repeated i n t e g r a t i o n ( O S O M R I ) v i a block-pulse functions and W a l s h functions. T h e O S O M R I [258] improves the accuracy of i d e n t i f i c a t i o n i n the case o f higher order models. I n fact, this concept of O S O M R I is not new as, for example, t w o times and three times repeated i n t e g r a t i o n of Walsh basis vector gives a result w h i c h is exactly
the same
as the Corrington's
look-up
tables I I I and I V , respectively i n [75].
O h k i t a et a l , 1983 [222] applied Haar functions for the solution of differential equations. H w a n g and S h i h , 1983 [132] presented a direct m e t h o d for solving v a r i a t i o n a l problems v i a Laguerre series. S i m i l a r l y , for the same p r o b l e m Chang and W a n g , 1983 [20] used Legendre p o l y n o m i a l s . T h e same a u t h o r s , 1983 [18] developed a m e t h o d for m o d e l r e d u c t i o n based on Legendre p o l y n o m i a l a p p r o x i m a t i o n . I n this papaer, based on the m o d e l r e d u c t i o n technique a new algebraic m e t h o d was also proposed for the
Chapter
8
1: A Historical
Perspective
design of feedback c o n t r o l systems. K a w a j i , 1983 [154] suggested a block-pulse series approach to the analysis of linear systems i n c o r p o r a t i n g observers.
Using Tchebycheff p o l y n o m i a l s o f the
first
kind,
Paraskevopoulos, 1983 [237] studied the system i d e n t i f i c a t i o n , analysis, and o p t i m a l c o n t r o l . T h e a l g o r i t h m s derived were similar to those already developed w i t h W a l s h and block-pulse functions and Laguerre p o l y n o m i a l s .
B y using delayed u n i t step
functions, H w a n g , 1983 [113] solved f u n c t i o n a l differential equations. L i u and Shih, 1983 [189] derived o p e r a t i o n a l matrices o f f o r w a r d i n t e g r a t i o n and b a c k w a r d i n t e g r a t i o n and the p r o d u c t m a t r i x for Tchebycheff p o l y n o m i a l s of the first k i n d and applied these matrices to the analysis and o p t i m a l c o n t r o l of linear t i m e v a r y i n g systems. W a n g and C h a n g , 1983 [325] employed an a p p r o x i m a t e m e t h o d based on Legendre p o l y n o m i a l s to determine the o p t i m a l c o n t r o l law of linear dist r i b u t e d parameter systems. K u n g and Lee, 1983 [166] extended the a p p l i c a t i o n of Laguerre p o l y n o m i a l s [113],[167] to linear systems w i t h a t i m e delay. B y i n t r o d u c i n g the time-delay operator and using the t i m e - i n t e g r a t i o n operator, they estimated the parameters and o b t a i n e d the solution of systems w i t h t i m e delays. H w a n g and Shih's w o r k , 1983 [134] on the m o d e l reduction using Laguerre p o l y n o m i a l s was i n a s i m i l a r line w i t h [18]. T h e concept of the shift Walsh m a t r i x , i n t r o d u c e d by Chen and S h i h , 1978 [49], is related to t h a t of Walsh delay m a t r i x , first proposed by Rao and Srinivasan, 1978 [266] b u t its a p p l i c a t i o n is l i m i t e d to the case of only zero i n i t i a l functions as shown by Rao and Palanisamy, 1984 [257]. I n the same year Paraskevopoulos, 1984 [236] extended the k n o w n results on the state-space analysis using o r t h o g o n a l functions for regular systems to the case of singular systems. Chang and W a n g , 1984 [21] solved stretched f u n c t i o n a l o r d i n a r y differential equations of p o p u l a t i o n balance, w h i c h can also be used to describe the c r y s t a l l i z a t i o n process of breakage models, using Legendre p o l y n o m i a l s . H w a n g and Shih, 1984 [142] discussed the a p p l i c a t i o n of discrete Tchebycheff p o l y n o m i a l s to reduce the order of a linear t i m e - i n v a r i a n t discrete system described by z-tranfer f u n c t i o n . Sinha and Jie, 1984 [294] presented an a l g o r i t h m for e s t i m a t i n g the states of an observable linear continuous-time system f r o m the samples of i n p u t - o u t p u t d a t a using the theory of block-pulse functions and the p r i n c i p l e of a Luenberger observer. H w a n g and Guo, 1984 [128] enlarged the a p p l i c a b i l i t y o f Legendre polynomials i n system identification for single-input single-output systems [19] to m u l t i - i n p u t m u l t i - o u t p u t systems. Using the o p e r a t i o n a l properties of i n t e g r a t i o n and the p r o d u c t of Tchebycheff p o l y n o m i a l s of the first k i n d [189], L i u and S h i h , 1984 [190] studied the analysis and parameter e s t i m a t i o n of bilinear systems. W a n g et a l , 1984 [330] modified the Laguerre p o l y n o m i a l s by m u l t i p l y i n g the independent variable by a factor, obtained an i n t e g r a t i o n o p e r a t i o n a l m a t r i x for m o d i f i e d Laguerre p o l y n o m i a l s and applied the same to solve i n t e g r a l equations. H w a n g and G u o , 1984 [127] derived a new o p e r a t i o n a l m a t r i x for the t i m e m u l t i p l i c a t i o n of a Legendre vector.
Using this m a t r i x and the i n t e g r a t i o n o p e r a t i o n a l m a t r i x [19], an
identification procedure for a class of t i m e - v a r y i n g systems characterized by a linear
1.2 Orthogonal
Functions
in Systems
and
9
Control
differential e q u a t i o n w i t h coefficients w h i c h are p o l y n o m i a l s of a finite order i n the variable t i m e was presented. Jaw and K u n g , 1984 [145] applied block-pulse functions to e s t i m a t e the parameters o f a linear t i m e - v a r y i n g system described by a m o d i f i e d observable canonical s t r u c t u r e . H w a n g , 1984 [114] presented a new t i m e - d o m a i n approach for the d e r i v a t i o n of Tchebycheff first k i n d scaled m a t r i x and developed
a
recursive f o r m u l a for c o n s t r u c t i n g the entries of the Tchebycheff scaled m a t r i x . T h e n e w l y developed Tchebycheff scaled m a t r i x together w i t h the Tchebycheff i n t e g r a t i o n o p e r a t i o n a l m a t r i x [237] was next applied to analyse f u n c t i o n a l differential equations c o n t a i n i n g t e r m s w i t h a scaled a r g u m e n t . T s a i et a l , 1984 [307] analysed linear t i m e v a r y i n g systems w i t h Legendre p o l y n o m i a l s . Paraskevopoulos and Kekkeris, 1984 [242] proposed a new m e t h o d for the d e t e r m i n a t i o n of o u t p u t s e n s i t i v i t y f u n c t i o n , based on o r t h o g o n a l functions. Except for the basis f u n c t i o n s , the linear d i s t r i b u t e d parameter system i d e n t i f i c a t i o n procedure of R a n g a n a t h a n et a l , 1984 [252] was exactly the same as given i n [239]. M e r t z i o s , 1984 [207] proposed a pair o f discrete W a l s h o p e r a t i o n a l matrices w h i c h corresponded to the w e l l k n o w n o p e r a t i o n a l m a t r i x of continuous W a l s h system [34],[264]. U s i n g these o p e r a t i o n a l matrices, a procedure was presented for the s o l u t i o n and i d e n t i f i c a t i o n of discrete linear t i m e - i n v a r i a n t m u l t i v a r i a b l e systems described by the state-space model. T h e shift W a l s h m a t r i x approach [49] or the shift block-pulse m a t r i x approach [290] was o n l y applicable to the delay systems w i t h zero i n i t i a l c o n d i t i o n s . T h e coefficient
shift m a t r i x approach, proposed by H w a n g and S h i h , 1985 [135] for the c o n t r o l
of delay systems v i a block-pulse functions, could be used w i t h nonzero i n i t i a l cond i t i o n s . C h o u and H o r n g , 1985 [67] used shifted Tchebycheff series for the analysis of o p t i m a l c o n t r o l systems i n c o r p o r a t i n g observers. Chang and W a n g , 1985 [22] employed Legendre p o l y n o m i a l s to find the s o l u t i o n of i n t e g r a l equations.
T h e same
a u t h o r s , 1985 [326] studied the a p p l i c a t i o n of Legendre p o l y n o m i a l s to o p t i m a l cont r o l of l u m p e d p a r a m e t e r systems. H w a n g and Chen, 1985 [119] presented recursive a l g o r i t h m s to construct
the delay o p e r a t i o n a l m a t r i x r e l a t i n g the Legendre coeffi-
cient vector of a delay f u n c t i o n to t h a t of the nondelay one.
T h i s n e w l y developed
delay o p e r a t i o n a l m a t r i x together w i t h the i n t e g r a t i o n o p e r a t i o n a l m a t r i x was used to find the a p p r o x i m a t e solutions and to estimate the u n k n o w n parameters o f delay systems. Palanisamy and A r u n a c h a l a m , 1985 [225] d e m o n s t r a t e d the advantages of using single-term Walsh series m e t h o d over block-pulse f u n c t i o n and W a l s h f u n c t i o n m e t h o d s i n the analysis of bilinear systems. H o r n g and C h o u , 1985 [91] developed a design technique for the o p t i m a l observer w i t h specified d i s t i n c t and m u l t i p l e eigenvalues by a p p r o x i m a t i n g a q u a d r a t i c performance measure w i t h the help o f Tchebycheff p o l y n o m i a l s of the first k i n d . M o u r o u t s o s and Sparis, 1985 [218] proposed a T a y l o r series approach to the p r o b lems o f system i d e n t i f i c a t i o n , analysis and o p t i m a l c o n t r o l considered i n [237].
The
a l g o r i t h m s proposed here were exactly s i m i l a r to those already available for o r t h o g o n a l f u n c t i o n approaches. E x t e n d i n g the T a y l o r p o l y n o m i a l a p p r o a c h , the analysis
10
Chapter
1: A Historical
Perspective
and o p t i m a l c o n t r o l of t i m e - v a r y i n g systems were studied by the same a u t h o r s , 1985 [297].
T h e s i m p l i c i t y of Taylor p o l y n o m i a l s led to very simple forms for the oper-
a t i o n a l matrices of forward and backward integrations and the p r o d u c t of a T a y l o r vector w i t h its transpose. Zaman and Jha, 1985 [336] enlarged the scope of Laguerre p o l y n o m i a l s to parameter identification of nonlinear l u m p e d p a r a m e t e r systems. H o r n g and C h o u , 1985 [89] are the first authors to i n t r o d u c e the delay o p e r a t i o n a l m a t r i x for Tchebycheff p o l y n o m i a l s of the first k i n d . Using this m a t r i x , t i m e - i n v a r i a n t and t i m e - v a r y i n g linear delay-differential equations w i t h an a r b i t r a r y t i m e delay were analysed a p p r o x i m a t e l y . Based on the a l g o r i t h m for analysis, an i d e n t i f i c a t i o n technique was presented to estimate the u n k o w n parameters o f t i m e - i n v a r i a n t systems w i t h a fixed delay. I n a d d i t i o n , the Tchebycheff series approach was also applied to o b t a i n the o p t i m a l control of time-delay systems w i t h a q u a d r a t i c cost f u n c t i o n a l . Chen and W u , 1985 [52] applied block-pulse functions for the first t i m e t o the analysis of t i m e - i n v a r i a n t and t i m e - v a r y i n g sampled data c o n t r o l systems. L i u and Shih, 1985 [191] used Tchebycheff p o l y n o m i a l s of the second k i n d for the analysis and parameter identification of t i m e - i n v a r i a n t systems. I n this study, an o p e r a t i o n a l m a t r i x for i n t e g r a t i o n of the Tchebycheff vector of the second k i n d was derived. Discrete Laguerre polynomials were applied to a p p r o x i m a t e the o p t i m a l c o n t r o l laws of d i g i t a l systems by H o r n g and Ho, 1985 [107]. Since all classes of finite range o r t h o g o n a l p o l y n o m i a l s can be obtained from J a c o b i p o l y n o m i a l s , L i u and Shih, 1985 [193] applied Jacobi polynomials to analysis, parameter e s t i m a t i o n and o p t i m a l c o n t r o l of linear systems to make the approach more general. I t is i n this paper t h a t the o p e r a t i o n a l m a t r i x of i n t e g r a t i o n for Jacobi p o l y n o m i a l s was first i n t r o d u c e d . C h o u and H o r n g , 1985 [60] developed a linear, n o n i t e r a t i v e , nondiffere n t i a l , nonintegral and well-adapted m e t h o d v i a Tchebycheff p o l y n o m i a l s of the first k i n d for the solution of c o n v o l u t i o n integral, and V o l t e r r a and F r e d h o l m integral equations. Based on the Tchebycheff series of the first k i n d , a c o m p u t a t i o n a l m e t h o d for evaluating the o p t i m a l c o n t r o l and t r a j e c t o r y of a linear d i s t r i b u t e d parameter system was established by the same authors, 1985 [90]. Since the o p t i m a l discount factor was n o t needed for the a p p l i c a t i o n of discrete Tchebycheff series, the technique of H o r n g and Ho, 1985 [101] for the o p t i m a l c o n t r o l of d i g i t a l systems was more convenient t h a n the discrete Laguerre approach [107]. C h o u and H o r n g , 1985 [68] presented a Legendre polynomial-based approach for the analysis of linear systems i n c o r p o r a t i n g an observer studied by K a w a j i [154] v i a blockpulse series. T h e same authors, 1985 [59] investigated the i d e n t i f i c a t i o n and also the analysis of scaled systems v i a Tchebycheff p o l y n o m i a l s of the first k i n d . K a w a j i and S h i o t s u k i , 1985 [155] proposed a new m e t h o d v i a W a l s h functions for o b t a i n i n g a reduced order model for higher order systems. Legendre p o l y n o m i a l s were applied to analyse and identify nonlinear systems described by H a m m e r s t e i n models by Shih and K u n g , 1985 [284], B y i n t r o d u c i n g a shift t r a n s f o r m a t i o n m a t r i x , a direct p r o d u c t m a t r i x and a summ a t i o n m a t r i x of discrete Walsh p o l y n o m i a l s , the analysis of t i m e - v a r y i n g d i g i t a l con-
1.2 Orthogonal
Functions
in Systems
and
Control
11
t r o l systems and the s o l u t i o n of t i m e - i n v a r i a n t d i g i t a l o p t i m a l c o n t r o l systems were accomplished by H o r n g and H o , 1985 [104], H w a n g and Chen, 1985 [124] presented a c o m p u t a t i o n a l l y efficient m e t h o d based on Legendre p o l y n o m i a l s for o b t a i n i n g the s u b o p t i m a l c o n t r o l of linear t i m e - v a r y i n g systems w i t h m u l t i p l e delays in the state and c o n t r o l vectors. H w a n g and C h e n , 1986 [123] approached the p r o b l e m o f solving a f u n c t i o n a l differe n t i a l e q u a t i o n as i n [133] v i a Legendre p o l y n o m i a l s . A n elegant o p e r a t i o n a l m a t r i x w h i c h relates t h e Legendre p o l y n o m i a l s to t h e i r stretched forms was derived i n this c o n t e x t . Shih et a l , 1986 [286] studied the analysis of linear systems i n c o r p o r a t i n g an observer as done i n [154] b u t v i a Laguerre p o l y n o m i a l s . C h o u and H o r n g , 1986 [66] employed Tchebycheff p o l y n o m i a l s of the first k i n d for the anaysis and i d e n t i f i c a t i o n of t i m e - v a r y i n g bilinear systems.
K u n g and S h i h , 1986 [168] presented a l g o r i t h m s
for the analysis and i d e n t i f i c a t i o n of nonlinear delay systems described
by a H a m -
m e r s t e i n m o d e l , w h i c h consisted of a single-valued n o n l i n e a r i t y followed by a linear time-delay p l a n t . H o r n g and Ho, 1986 [106] i n t r o d u c e d a shift t r a n s f o r m a t i o n m a t r i x for discrete Legendre p o l y n o m i a l s and applied i t to the p r o b l e m of o p t i m a l c o n t r o l s t u d i e d i n [107]. S h i h a n d K u n g , 1986 [282] applied Tchebycheff p o l y n o m i a l s of the first k i n d to deal w i t h the analysis and p a r a m e t e r e s t i m a t i o n problems t r e a t e d i n [284] w h i l e R a n g a n a t h a n et a l , 1986 [253] used Laguerre p o l y n o m i a l s for the i d e n t i f i c a t i o n o f n o n l i n ear d i s t r i b u t e d p a r a m e t e r systems studied in [293] v i a Walsh functions. H o r n g and H o , 1986 [105] applied discrete Walsh p o l y n o m i a l s to solve d i g i t a l o p t i m a l c o n t r o l of t i m e - v a r y i n g systems. T h e design a l g o r i t h m s of d i g i t a l o p t i m a l c o n t r o l were based on the discrete v a r i a t i o n a l p r i n c i p l e combined w i t h the idea of p e n a l t y f u n c t i o n for e v a l u a t i n g o p t i m a l c o n t r o l and state t r a j e c t o r y . Shih and K u n g , 1986 [281] derived an o p e r a t i o n a l m a t r i x for ' s t r e t c h ' for Legendre p o l y n o m i a l s and discussed the problems of analysis and p a r a m e t e r e s t i m a t i o n o f l i n ear scaled systems. C h o u and H o r n g , 1986 [64] presented a t r e a t m e n t for d e t e r m i n i n g the parameters of linear t i m e - v a r y i n g systems v i a Tchebycheff p o l y n o m i a l s of the
first
k i n d . U s i n g a w e l l c o n s t r u c t e d t r a n s f o r m a t i o n m a t r i x of discrete Legendre p o l y n o m i als, t i m e - i n v a r i a n t discrete systems were identified by H w a n g and S h i h , 1986 [143]. T a k i n g the advantage of elegant o p e r a t i o n a l properties of Tchebycheff p o l y n o m i a l s of the first k i n d , H o r n g et a l , 1986 [98] developed a direct c o m p u t a t i o n a l a l g o r i t h m for e v a l u a t i n g the o p t i m a l c o n t r o l and the state t r a j e c t o r y of scaled systems. Since a l l the classes of o r t h o g o n a l p o l y n o m i a l s can be derived from the generalized o r t h o g o n a l p o l y n o m i a l s , i n recent years researchers s t a r t e d s t u d y i n g a variety of problems i n systems and c o n t r o l i n a very general f r a m e w o r k . T h i s k i n d of w o r k is essentially m o t i v a t e d w i t h a view to m a k i n g the o r t h o g o n a l p o l y n o m i a l s approach m o r e general and u n i f i e d .
I n this d i r e c t i o n the first a t t e m p t
was made by
and Lee, 1986 [29] for the o p t i m a l c o n t r o l of t i m e - v a r y i n g linear systems.
Chang I n this
paper the o p e r a t i o n a l matrices for the f o r w a r d as also for the b a c k w a r d i n t e g r a t i o n of general o r t h o g o n a l p o l y n o m i a l s were i n t r o d u c e d . Moreover, i t was shown t h a t the
Chapter
12
1: A Historical
Perspective
p r o d u c t of t w o a r b i t r a r y t i m e functions could be a p p r o x i m a t e d by general o r t h o g o n a l p o l y n o m i a l s . Shih and K u n g , 1986 [285] extended the Legendre expansion
method
[284] to attack the Hammerstein m o d e l of nonlinear systems w i t h a t i m e delay.
Lee
et a l , 1986 [180] made use of Jacobi p o l y n o m i a l s to analyse linear o p t i m a l c o n t r o l systems i n c o r p o r a t i n g an observer [156], [67]. C h o u and H o r n g , 1986 [63] presented a m e t h o d v i a Tchebycheff p o l y n o m i a l s of the first k i n d for i n v e r t i n g r a t i o n a l and i r r a t i o n a l Laplace transforms [37], [279], [321], [202]. Later Lee a n d Tsay, 1986 [182] derived a shift t r a n s f o r m a t i o n m a t r i x for general discrete o r t h o g o n a l p o l y n o m i a l s and applied i t to simplify the discrete Euler-Lagrange equations for the a p p r o x i m a t i o n of state and c o n t r o l variables of d i g i t a l systems. Shih and L i u , 1986 [284] used Tchebycheff p o l y n o m i a l s of the second k i n d to est i m a t e the parameters of p o l y n o m i a l coefficients of t i m e - v a r y i n g systems as i n i t i a t e d by Tzafestas, 1978 [313]. T h e key to this m e t h o d was the o p e r a t i o n a l m a t r i x for m u l t i p l i c a t i o n of Tchebycheff polynomials of the second k i n d w i t h a t i m e variable. Chang and Y a n g , 1986 [23] solved the state equation of a t w o - p o i n t b o u n d a r y - v a l u e p r o b l e m and the o p t i m a l c o n t r o l p r o b l e m of l u m p e d and d i s t r i b u t e d parameter systems by using generalized o r t h o g o n a l p o l y n o m i a l s . Perng, 1986 [247] proposed a direct approach using Legendre p o l y n o m i a l s to the o p t i m a l c o n t r o l p r o b l e m of linear time-delay systems. M o u r o u t s o s and Sparis, 1986 [216] used o p e r a t i o n a l m a t r i x of d i f f e r e n t i a t i o n for the identification of a class of t i m e - v a r y i n g systems using T a y l o r series. I n a d d i t i o n to the i d e n t i f i c a t i o n of a t i m e - i n v a r i a n t d i s t r i b u t e d parameter system, the p r o b l e m of analysis was also studied by H o r n g et al, 1986 [97]. H w a n g and Chen, 1986 [118] extended the a p p l i c a t i o n of Legendre polynomials to the analysis a n d i d e n t i f i c a t i o n of bilinear systems [190]. T h e extension was achieved t h r o u g h d e r i v i n g a Legendre series representation of the p r o d u c t of t w o t i m e functions. Lee and Chang, 1986 [175] used generalized o r t h o g o n a l p o l y n o m i a l s for the analysis and identification of linear d i s t r i b u t e d systems studied in [97] w h i l e J i a n g , 1986 [150] used block-pulse functions for the o u t p u t sensitivity investigated i n [242]. Rang a n a t h a n et a l , 1986 [254] presented recursive a l g o r i t h m s for the p a r a m e t e r e s t i m a t i o n i n bilinear and nonlinear systems v i a Laguerre p o l y n o m i a l s . Tsay and Lee, 1986 [309] i n t r o d u c e d Taylor o p e r a t i o n a l m a t r i x for i n t e g r a t i o n and T a y l o r p r o d u c t
operational
m a t r i x and applied t h e m to a p p r o x i m a t e solutions of i n t e g r a l equations [165],[22],[60]. C h o u and H o r n g , 1986 [69] presented a new approach using continuous
orthogonal
functions to f a c i l i t a t e research in state e s t i m a t i o n of linear systems [294].
Mouroutsos
and Sparis, 1986 [217] i n t r o d u c e d shift Fourier m a t r i x and p r o d u c t Fourier m a t r i x . Using these matrices, linear t i m e - i n v a r i a n t and t i m e - v a r y i n g systems w i t h a t i m e delay were analysed a p p r o x i m a t e l y .
H o r n g and C h o u , 1986 [92] applied Legendre
series to a p p r o x i m a t e the quadratic performance index and to design an o p t i m a l observer w i t h specified eigenvalues. W a n g et a l , 1986 [328] applied generalized o r t h o g onal p o l y n o m i a l s to the analysis and o p t i m a l c o n t r o l of t i m e - v a r y i n g linear systems studied i n [47]. T h i s paper presented o p e r a t i o n a l matrices for f o r w a r d and b a c k w a r d i n t e g r a t i o n of generalized o r t h o g o n a l p o l y n o m i a l s and o p e r a t i o n a l m a t r i x of p r o d u c t
1.2 Orthogonal
Functions
in Systems
and
13
Control
of t , where k is an integer, and generalized o r t h o g o n a l p o l y n o m i a l vector.
Jacobi
delay o p e r a t i o n a l m a t r i x was i n t r o d u c e d and applied to the anaysis and i d e n t i f i c a t i o n of t i m e - d e l a y systems by H o r n g and C h o u , 1986 [88]. H o r n g et a l , 1986 [100] discussed the a p p l i c a t i o n of discrete W a l s h series expansion to reduce the order of a linear t i m e - i n v a r i a n t d i g i t a l system described by
z-transfer
f u n c t i o n . Lee and C h a n g , 1986 [177] i n t r o d u c e d a nonlinear o p e r a t i o n a l m a t r i x for the generalized o r t h o g o n a l p o l y n o m i a l s and applied i t along w i t h the other o p e r a t i o n a l matrices [29] to the analysis, parameter i d e n t i f i c a t i o n and o p t i m a l c o n t r o l of nonlinear c o n t r o l systems. H o r n g et a l , 1986 [99] a p p l i e d T a y l o r series m e t h o d to the analysis of linear o p t i m a l c o n t r o l systems i n c o r p o r a t i n g observers [156],[67],[180]. Lee and Tsay, 1986 [181] presented a generalized discrete o r t h o g o n a l p o l y n o m i a l approach for the analysis a n d o p t i m a l c o n t r o l of discrete linear t i m e - v a r y i n g systems by d e r i v i n g a shift t r a n s f o r m a t i o n m a t r i x . T h e analysis o f linear t i m e - i n v a r i a n t o p t i m a l c o n t r o l systems i n c o r p o r a t i n g observers [156],[67],[180],[99] was treated using generalized o r t h o g o n a l p o l y n o m i a l s by C h a n g and Lee, 1986 [30] by developing an o p e r a t i o n a l m a t r i x for f o r w a r d i n t e g r a t i o n of generalized o r t h o g o n a l p o l y n o m i a l s . Perng, 1986 [246] a p p l i e d a single-step m e t h o d o f generalized block-pulse functions t o the analysis of linear d i s t r i b u t e d systems governed by simultaneous
first-order
linear p a r t i a l differential equations w i t h variable coefficients. I n t h i s w o r k , generalized block-pulse f u n c t i o n expansion of the p r o d u c t of t w o functions was i n t r o d u c e d . C h o u a n d H o r n g , 1986 [61] extended
Legendre series approach
[122] to t h e i d e n t i f i c a t i o n
of t i m e - v a r y i n g b i l i n e a r systems. Tsay and Lee, 1986 [311] derived a discrete scaled m a t r i x for the generalized discrete o r t h o g o n a l p o l y n o m i a l s and used i t for t h e analysis of d i g i t a l scaled systems. C h a n g et a l , 1986 [25] presented a very effective m e t h o d based on the d e r i v a t i v e o p e r a t i o n a l m a t r i x of generalized o r t h o g o n a l p o l y n o m i a l s for i d e n t i f y i n g the parameters of a process whose behavior could be modeled by a linear differential e q u a t i o n w i t h t i m e - v a r y i n g coefficients i n the f o r m of finite order polynomials.
Tsay and Lee, 1986 [311] used H e r m i t e p o l y n o m i a l s for the o p t i m a l
c o n t r o l p r o b l e m i n c o r p o r a t i n g an observer, i n i t i a t e d earlier i n [156],[67],[180],[99],[30] and Tchebycheff p o l y n o m i a l s of the second k i n d [178] for the analysis of linear t i m e v a r y i n g a n d b i l i n e a r systems. T h e scaled o p e r a t i o n a l m a t r i x for Jacobi p o l y n o m i a l s was
first
i n t r o d u c e d by
C h o u , 1987 [56] and a systematic m e t h o d was presented as a p o w e r f u l t o o l to analyse and i d e n t i f y a class o f scaled systems.
A n o p e r a t i o n a l m a t r i x for the linear trans-
f o r m a t i o n of generalized o r t h o g o n a l p o l y n o m i a l s was i n t r o d u c e d by Lee a n d C h a n g , 1987 [176], and an efficient m e t h o d was o u t l i n e d to analyse a class of t i m e - v a r y i n g delay systems. H o r n g and C h o u , 1987 [87] established an a l g o r i t h m for the analysis and i d e n t i f i c a t i o n of nonlinear systems using Jacobi series. A t this stage another class of basis functions namely discrete pulse o r t h o g o n a l f u n c t i o n s came i n t o existence for s t u d y i n g a variety o f problems related to discrete t i m e systems.
H o r n g and H o , 1987 [103] for the first t i m e i n t r o d u c e d a new set of
discrete pulse o r t h o g o n a l functions and t h e i r shift t r a n s f o r m a t i o n m a t r i x to study
Chapter
14
1: A Historical
Perspective
the analysis, parameter e s t i m a t i o n and o p t i m a l c o n t r o l of d i g i t a l c o n t r o l systems. W a n g et a l , 1987 [334] derived derivative o p e r a t i o n a l m a t r i x of generalized o r t h o g o n a l p o l y n o m i a l s and demonstrated its m e r i t i n parameter e s t i m a t i o n of t i m e - i n v a r i a n t and bilinear systems. A n approach based on Tchebycheff p o l y n o m i a l s of the first k i n d for the p r o b l e m of sensitivity analysis of linear t i m e - i n v a r i a n t systems [242] was proposed by K e k k e r i s , 1987 [157]. I n this paper, the s e n s i t i v i t y analysis of linear systems w i t h t i m e - v a r y i n g elements using o r t h o g o n a l functions was i n t r o d u c e d for the first t i m e . Chang et a l , 1987 [27] employed generalized o r t h o g o n a l p o l y n o m i a l s for solving integral equations [22] by developing an i n t e g r a t i o n o p e r a t i o n a l m a t r i x of the generalized o r t h o g o n a l p o l y n o m i a l s . C h u n g and Sun, 1987 [73] extended the delay o p e r a t i o n a l m a t r i x of T a y l o r series to the analysis of m u l t i - d e l a y d y n a m i c systems. A g a i n another class of basis functions, namely piecewise linear p o l y n o m i a l functions, not o r t h o g o n a l in nature, was added to the g l o b a l f a m i l y of basis functions. C l a i m i n g t h a t piecewise linear p o l y n o m i a l functions had the advantages o f piecewise constant o r t h o g o n a l functions and also o f o r t h o g o n a l p o l y n o m i a l s , L i o u a n d C h o u , 1987 [187] applied these new basis functions to the o p t i m a l c o n t r o l of linear t i m e v a r y i n g systems. I n this paper, the authors i n t r o d u c e d an o p e r a t i o n a l m a t r i x of integ r a t i o n and of the p r o d u c t of two t i m e functions for this new set of basis functions. H w a n g and Shyu, 1987 [141] used discrete Legendre p o l y n o m i a l s for the analysis of t i m e - v a r y i n g discrete systems. Chen and W u , 1987 [50] considered block-pulse functions for the analysis and o p t i m a l c o n t r o l of a class o f pulse w i d t h m o d u l a t e d systems. Lee and Tsay, 1987 [179] solved the analysis, parameter e s t i m a t i o n , and o p t i m a l c o n t r o l problems of time-delay systems by a p p l y i n g Pade a p p r o x i m a t i o n of the
first
order to t r a n s f o r m the delay differential equation i n t o a differential e q u a t i o n , and t h e n using generalized o r t h o g o n a l p o l y n o m i a l s to further reduce the differential equation i n t o an algebraic f o r m .
H o r n g and Ho, 1987 [102] employed discrete pulse o r t h o g -
onal functions for the analysis, parameter e s t i m a t i o n and o p t i m a l c o n t r o l o f linear t i m e - v a r y i n g d i g i t a l systems. I n the paper by Chen and Y a n g , 1987 [38] a delay ope r a t i o n a l m a t r i x of T a y l o r p o l y n o m i a l s was derived and a p p l i e d for the analysis and i d e n t i f i c a t i o n of time-delay systems.
Using discrete Legendre p o l y n o m i a l s , H w a n g
and S h y u , 1987 [140] proposed a new m e t h o d for the analysis and i d e n t i f i c a t i o n of linear t i m e - i n v a r i a n t discrete t i m e systems. T h i s m e t h o d was based on representing the spectral vector of the delayed f u n c t i o n y(k — 1) i n terms of t h a t of the o r i g i n a l function
y{k).
Jiang, 1987 [148] presented several a l g o r i t h m s for i d e n t i f y i n g m u l t i - i n p u t m u l t i o u t p u t systems w i t h and w i t h o u t time-delays from t h e i r i n p u t - o u t p u t d a t a . T h e basis of these a l g o r i t h m s was to set up a difference e q u a t i o n of the continuous systems using block-pulse functions. L i o u and C h o u , 1987 [188] first derived the i n t e g r a t i o n operat i o n a l m a t r i x of piecewise linear p o l y n o m i a l functions and then studied t h e problems of analysis and i d e n t i f i c a t i o n . Later the same authors, 1987 [186] extended the above study to linear t i m e - v a r y i n g systems by representing the i n t e g r a t i o n o f the p r o d u c t
1.2 Orthogonal
Functions
in Systems
and
Control
15
of t w o piecewise linear p o l y n o m i a l series i n a new piecewise linear p o l y n o m i a l series. A g a i n the same a u t h o r s , 1987 [185] f u r t h e r extended the above approach t o m i n i m u m energy c o n t r o l of linear systems w i t h time-delays. I n this c o n t e x t , they extended the concept of delay o p e r a t i o n a l m a t r i x t o piecewise linear p o l y n o m i a l f u n c t i o n s . T h e w o r k of H w a n g et a l , 1987 [116] w i t h H a h n p o l y n o m i a l s gave recently a m o r a l boost t o the a p p l i c a t i o n s of basis f u n c t i o n s . T h e y presented a procedure of a p p l y i n g H a h n p o l y n o m i a l s for the analysis and parameter e s t i m a t i o n o f linear discrete-time s i n g l e - i n p u t s i n g l e - o u t p u t systems described by difference equations.
I n t h i s study,
t h e y first derived an o p e r a t i o n a l m a t r i x for p e r f o r m i n g the t i m e shift on the H a h n vector and t h e n used this m a t r i x t o establish an expression w h i c h related the H a h n coefficient vector of a discrete variable t o t h a t of the t i m e - s h i f t e d f u n c t i o n and the i n i t i a l c o n d i t i o n s . C h u n g and Sun, 1987 [71] extended the T a y l o r series approach t o the anaysis and p a r a m e t e r e s t i m a t i o n of general linear scaled systems by i n t r o d u c i n g a scaled o p e r a t i o n a l m a t r i x of T a y l o r series. O h k i t a , 1987 [220] applied r a t i o n a l i z e d Haar functions t o solve differential equations c o n t a i n i n g t i m e delays. For this purpose, delay vectors a n d m a t r i x based on the rationalized Haar functions were constructed. Palanisamy and B a l a c h a n d r a n , 1987 [230] presented a new m e t h o d for the analysis of linear singular systems v i a the single-term Walsh series approach.
By deriving a
stretch o p e r a t i o n a l m a t r i x of generalized o r t h o g o n a l p o l y n o m i a l s , Chang et a l , 1987 [26] solved f u n c t i o n a l differential equations. H o r n g and C h o u , 1988 [93] employed generalized discrete o r t h o g o n a l p o l y n o m i a l s t o design a d i g i t a l P I D controller w h i l e Vlassenbroeck and Van D o o r e n , 1988 [319] suggested a technique based on Tchebycheff p o l y n o m i a l s for nonlinear o p t i m a l c o n t r o l p r o b l e m s . T h e use o f discrete Legendre p o l y n o m i a l s for the f i n i t e - t i m e o p t i m a l c o n t r o l of t i m e - v a r y i n g discrete systems w i t h a q u a d r a t i c performance i n d e x was emphasized by Shyu and H w a n g , 1988 [292].
Z h u and L i u , 1988 [338] a n d Palanisamy
et al,
1988 [231] f u r t h e r broadened the scope of single-term Walsh series [260] t o develop hierarchical strategy for nonlinear o p t i m a l c o n t r o l systems [338] and o p t i m a l c o n t r o l of linear t i m e - v a r y i n g delay systems. Based o n the d e r i v a t i v e o p e r a t i o n a l properties of the e x p o n e n t i a l Fourier series A r d e k a n i and K e y h a n i , 1989 [4] established an i d e n t i f i c a t i o n a l g o r i t h m for n o n l i n ear t i m e - i n v a r i a n t single-input single-output systems. M o h a n and D a t t a , 1989 [213] i n t r o d u c e d O S O M R I for Fourier series i n order to i m p r o v e the accuracy o f an opera t i o n a l m a t r i x for the repeated i n t e g r a t i o n of a Fourier basis vector. Based on this O S O M R I , t h e y presented an a l g o r i t h m for the i d e n t i f i c a t i o n of linear t i m e - i n v a r i a n t s i n g l e - i n p u t s i n g l e - o u t p u t d i s t r i b u t e d parameter systems.
Razzaghi and
Razzaghi,
1989 [271] showed t h a t the a p p l i c a t i o n of T a y l o r series for the heat c o n d u c t i o n p r o b lem produces i n s t a b i l i t i e s i n the s o l u t i o n due t o the occurrence o f an H i l b e r t m a t r i x w h i c h is k n o w n t o be i l l - c o n d i t i o n e d . T o overcome this difficulty, later t h e y used shifted Legendre p o l y n o m i a l s and shifted Tchebycheff p o l y n o m i a l s of the first k i n d by e x p a n d i n g t h e m i n T a y l o r series, and o b t a i n e d satisfactory results. P a t r a a n d Rao, 1989 [245] proposed general h y b r i d o r t h o g o n a l functions w h i c h
16
Chapter
1: A Historical
Perspective
combine the inherent characteristics o f piecewise constant o r t h o g o n a l functions and o r t h o g o n a l p o l y n o m i a l s and demonstrated t h e i r power i n the analysis o f linear t i m e invariant systems and the p r e d i c t i o n of a l i m i t cycle i n the h i g h l y nonlinear V a n der Pol's oscillator. T h e same authors, 1989 [244] later presented a completely continuoust i m e explicit self-tuning a l g o r i t h m based on block-pulse c h a r a c t e r i z a t i o n o f process signals. D i n g and Frank, 1989 [77] demonstrated t h a t a system a p p r o x i m a t e d by o r t h o g o n a l functions preserved the p r o p e r t y of the c o n t r o l l a b i l i t y and observability of a state-space system provided t h a t some prescribed c r i t e r i a were satisfied. Lewis et a l , 1990 [184] showed how to use Walsh functions t o a p p r o x i m a t e the s o l u t i o n of m u l t i v a r i a b l e bilinear systems. Paraskevopoulos et a l , 1990 [243] derived a general expression for the o p e r a t i o n a l m a t r i x of i n t e g r a t i o n for the Bessel functions for the first t i m e . T h e a p p l i c a b i l i t y of this m a t r i x was d e m o n s t r a t e d i n the problems of i d e n t i f i c a t i o n and analysis of linear t i m e - i n v a r i a n t systems. So far the applications of piecewise constant o r t h o g o n a l functions, o r t h o g o n a l p o l y n o m i a l s , sine-cosine functions and other basis functions n o t necessarily o r t h o g o n a l to a variety of problems i n systems and c o n t r o l are h i g h l i g h t e d . T h e m e t h o d i n v o l v e d i n any p r o b l e m veers r o u n d the development of proper o p e r a t i o n a l matrices such as i n t e g r a t i o n , d i f f e r e n t i a t i o n , delay, stretch and so on and the conversion of the o r i g i n a l p r o b l e m i n t o an algebraic p r o b l e m . T h e classification of a l l the problems w i t h the system of o r t h o g o n a l functions used to solve t h e m is given below t o get q u i c k l y a complete p i c t u r e of numerous c o n t r o l problems w h i c h already been solved and w h i c h need further investigations. Pi.
Analysis of linear time-invariant continuous
systems
via WF:
[75],[35],[33],[32],[260], [223],[295],[226],[228],[215]
BPF:
[276],[280],[289],[45], [232],[41]
LaP:
[167]
HF:
[222],[221]
TP1:
[237]
LeP:
[327],[325],[15],[17], [238]
SCF:
[275]
TP:
[218],[273]
TP2:
[191]
JP:
[193]
GOP:
[23],[28],[24]
PLP:
[188]
BF:
[243]
P2. Analysis of singular systems via OF:
[236],[12],[204],[158],[183], [208]
BPF:
[306],[205]
WF:
[230],[205]
1.2 Orthogonal P3.
P4.
Functions
P6.
and
Control
Analysis of linear time-invariant discrete systems v i a LaP:
[219],[139]
WF:
[207]
POF:
[103],[84]
LeP:
[140]
HaP:
[116]
Analysis of linear time-varying continuous systems via WF:
P5.
in Systems
[317],[47],[223],[227],[228]
BPF:
[111],[289],[138],[43], [152]
LaP:
[167]
TP1:
[189],[117],[272]
LeP:
[307]
TP:
[297],[298]
GOP:
[328] ,[308]
TP2:
[178]
PLP:
[186]
Analysis o f linear t i m e - v a r y i n g discrete systems v i a WF:
[104]
GOP:
[181]
LeP:
[141]
POF:
[102]
Analysis of linear t i m e - v a r y i n g sampled-data systems via BPF:
P7.
WF:
P8.
P9.
[52],[51]
Solution of integral equations via [75],[110],[109]
BPF:
[165], [324], [126]
LaP:
[330]
TP1:
[60]
TP:
[309]
GOP:
[27]
Analysis o f time-invariant bilinear systems via BPF:
[53],[199]
WF:
[184]
Analysis o f t i m e - v a r y i n g bilinear systems v i a TP1:
[190],[66]
WF:
[225]
LeP:
[118]
TP2:
[178]
PlO. Analysis of nonlinear
systems via
WF:
[75],[196],[224],[227]
BPF:
[276],[270],[289],[203]
Chapter
18 LeP: TP1: GOP: JP: Pll.
BPF: LaP: LeP:
P12.
[284] [282] [177] [87]
Analysis of linear time-invariant time-delay via WF:
1: A Historical
systems
[49],[224],[235],[257] [290] [166],[13]
TP1:
[326], [119], [16], [174] [89]
SCF:
[217]
JP: GOP: TP:
[88] [179],[331]
[38],[188] HF: [220] Analysis of linear time-invariant piecewise constant or multi-delay systems via
P13.
P14.
BPF:
[44],[332]
WF:
[46],[40],[235],[257]
TP:
[73]
Analysis of linear time-varying time-delay systems via WF:
[49]
TP1:
[89]
BPF:
[203]
SCF:
[217]
GOP:
[176],[179]
TP:
[72]
Analysis of linear time-varying multi-delay systems via LeP:
P15.
P16.
[124],[120]
Analysis of nonlinear systems w i t h delay BPF:
[266],[168]
LeP:
[285]
A n a l y s i s o f scaled systems v i a BPF:
[268],[39],[333]
WF:
[259]
LaP:
[133],[115]
DUSF:
[113]
TP1:
[114],[59]
LeP:
[123],[281]
JP:
[56]
via
Perspective
1.2 Orthogonal
P17:
P18.
Functions
TP:
[71]
GOP:
[26]
Solution of population BPF:
[137], [14]
LeP:
[21]
and
19
Control
balance equations via
Solution of partial differential equations via WF:
P19.
in Systems
[288]
BPF:
[267],[289],[250],[42]
TP1:
[97]
GOP:
[175]
TP:
[271]
Identification of linear time-invariant
continuous
systems via
P20.
WF:
[34],[264],[82],[313],[7], [8],[262],[258],[78],[9]
BPF:
[233],[54],[323],[151]
LaP: TP1:
[19],[238]
SCF:
[70],[213]
TP:
[218]
TP2:
[191]
JP:
[193]
GOP:
[334],[335]
PLP:
[188]
[237]
BF: [243] Identification of linear time-invariant
discrete systems
via
P21.
LaP:
[195],[139]
WF:
[207],[108]
Lep:
[143],[140]
POF:
[103],[84]
HaP:
[116]
Identification of linear time-varying
continuous
systems via
P22.
WF:
[313]
LeP:
[127]
BPF:
[145],[125]
TP1:
[64]
TP2:
[291]
TP:
[216],[296]
GOP:
[25],[329]
Identification of integral equation models via WF:
[48]
Chapter
20
P23.
BPF:
[53]
TP1:
[190]
LeP: LaP: GOP: Identification WF: BPF:
[53] [190]
LaP:
P26.
[153] [144] [66]
LeP: [61] Identification of nonlinear systems via WF: LaP: LeP: TP1:
[263] [336],[254] [284],[94] [282],[65]
GOP: JP:
[177] [87]
SCF: [4] Identification of linear systems containing via WF: BPF: LaP:
P27.
[122],[118] [254]
GOP: [334] Identification of time-varying bilinear systems v i a WF: BPF: TP1:
P25.
bilinear systems v i a
[48]
TP1: LeP:
P24.
[122],[118] [254] [334] of time-invariant
time-delays
[261],[258]
LeP:
[290],[148] [166],[13] [119],[16],[174]
TP1:
[89]
JP:
[88]
GOP:
[179],[331]
TP: [38] Identification of nonlinear systems time-delays via
P28.
1: A Historical
BPF:
[168]
LeP:
[285]
I d e n t i f i c a t i o n o f scaled systems v i a TP1:
[59]
LeP:
[281]
containing
Perspective
1.2 Orthogonal
P29.
P30.
Functions
JP:
[56]
TP:
[71]
in Systems
and
Control
I d e n t i f i c a t i o n of m u l t i - i n p u t m u l t i - o u t p u t s y s t e m s v i a WF:
[313],[265]
LeP:
[128]
BPF:
[148]
I d e n t i f i c a t i o n of l i n e a r t i m e - i n v a r i a n t d i s t r i b u t e d parameter systems via
P31.
WF:
[239],[313]
TPl:
[97]
LaP:
[252],[146]
GOP:
[175],[172]
SCF:
[213]
I d e n t i f i c a t i o n of n o n l i n e a r d i s t r i b u t e d p a r a m e t e r systems via
P32.
WF:
[293]
BPF:
[112]
LaP:
[253],[147]
O p t i m a l c o n t r o l of l i n e a r t i m e - i n v a r i a n t c o n t i n u o u s systems via WF:
P33.
[35],[36],[32],[81],[315], [76],[269],[223],[302]
BPF:
[276],[229],[130]
TPl:
[237],[96],[287],[318]
LaP:
[132]
LeP:
[20],[327],[326],[283],[57]
HeP:
[159]
SCF:
[79],[275]
TP:
[218],[248],[74],[274]
JP:
[193]
GOP:
[23]
O p t i m a l c o n t r o l of l i n e a r t i m e - i n v a r i a n t d i s c r e t e systems via LaP:
P34.
[107]
TPl:
[101]
WF:
[104]
LeP:
[106],[41]
GOP:
[182]
POF:
[103],[84]
O p t i m a l c o n r t o l of l i n e a r t i m e - v a r y i n g c o n t i n u o u s systems via BPF:
[111]
WF:
[47],[223]
21
22
P35.
Chapter TPl:
[189],[58],[117],[95],[272]
TP: GOP: PLP:
[297] [29],[31],[328],[308] [187]
O p t i m a l control of linear time-varying discrete systems via WF: GOP: POF:
P36.
P38.
P39.
[105] [181] [102]
LeP: [292] O p t i m a l control of nonlinear systems v i a GOP: TPl:
P37.
1: A Historical
[177] [319]
WF: [338] Optimal control incorporating WF: BPF:
[299],[156] [154]
TPl:
[67]
LeP: LaP:
[68] [286],[249]
JP: TP:
[180] [99]
observers via
GOP: [30] HeP: [310] O p t i m a l control of linear time-invariant systems via WF:
[234],[235],[206],[255]
TPl:
[89],[62]
LeP: PLP:
[247] [185]
O p t i m a l control of linear time-varying
time-delay
time-delay
systems via
P40.
BPF: LeP:
[135] [124],[121]
GOP:
[179]
WF: [231] O p t i m a l control of linear distributed systems via WF:
[312],[194]
LeP:
[325]
TPl:
[90]
GOP:
[23]
parameter
Perspective
1.2 Orthogonal
Functions
in Systems
and
23
Control
P 4 1 . N u m e r i c a l inversion of Laplace transforms
via
BPF:
[279],[129],[201],[202],[200], [6],[149]
TPl:
[63]
P42. M o d e l reduction of continuous systems via TPl:
P43.
[10],[192]
BPF:
[322]
LeP:
[18]
LaP:
[134]
WF:
[155]
M o d e l reduction of discrete systems via TPl:
[142]
WF:
[100]
P44.
P r e d i c t i o n o f l i m i t cycles v i a
P45.
C o n t r o l system design
WF:
P46.
[80],[164],[11]
LeP:
[18]
LaP:
[337]
via
State estimation via BPF:
[294]
TPl:
[91]
OF:
[69]
LeP:
[92]
P47. Sensitivity analysis OF:
via
[242],[160],[303]
BPF:
[150]
TPl:
[157]
T h e use of o r t h o g o n a l functions to diverse problems i n systems and c o n t r o l is m a i n l y m o t i v a t e d by their c o m p u t a t i o n a l s i m p l i c i t y .
A m i d s t the b o o m i n g n u m b e r
of p u b l i c a t i o n s , there has been an inadvertent d u p l i c a t i o n of research efforts.
The
piecewise constant basis functions d o m i n a t e d the scene t i l l mid-eighties after w h i c h the o r t h o g o n a l p o l y n o m i a l s and sine-cosine functions t o o k over.
Chapter Least
2
Squares
Approximation
of
S ignals
T h e least squares a p p r o x i m a t i o n of an a r b i t r a r y f u n c t i o n v i a a set o f generalized ort h o g o n a l p o l y n o m i a l s is presented i n t h i s chapter. T h e t h r e e - t e r m recurrence formulae a n d t h e ( o r d i n a r y ) differential recurrence relations of these o r t h o g o n a l p o l y n o m i a l s are also d e r i v e d . B y selecting the w e i g h t i n g f u n c t i o n w(t) as well as the i n t e r v a l [
2.1
Least Squares Approximation
E v e r y system receives i n f o r m a t i o n , signals
or s t i m u l i at its b o u n d a r y from the ex-
t e r n a l w o r l d and after i n t e r a c t i o n w i t h t h e m produces i n t e r n a l l y a different k i n d of i n f o r m a t i o n or signals i n general, some of w h i c h are available again at the b o u n d a r y . T h e e x t e r n a l s t i m u l i a p p l i e d to the system are called i n p u t s and those i n t e r n a l signals or responses w h i c h are thereby generated and are available at the t e r m i n a l s are called the o u t p u t s of t h e system.
Since a system deals exclusively w i t h various kinds of
signals, i t is i m p e r a t i v e to know how to describe an a r b i t r a r y f u n c t i o n
representing
t h e signal a n a l y t i c a l l y . T h i s representation is generally done w i t h the help o f o r t h o g onal functions.
As o n l y continuous signals w i l l take p a r t in our discussion, we shall
describe below h o w an a r b i t r a r y f u n c t i o n can be represented as a series of o r t h o g o n a l f u n c t i o n s such as block-pulse first
and second
guerre
polynomials
Gegenbauer
( B P F ) , Walsh functions
( T P l and T P 2 ) , Legendre
( L a P ) , Hermite
polynomials
D e f i n i t i o n 2.1 t i o n w(t).A
functions
kind polynomials
polynomials
( H e P ) , Jacobi
polynomials
Tchebycheff ( L e P ) , La(JP), and
GP.
Orthogonal Functions with Respect
set of
(WF),
polynomials
functions {4> (t)}, m
m = 0 , 1 , • • • ,n
to a W e i g h t i n g
Func-
Chapter
26 are said to be orthogonal the interval
m
Squares
with respect to a nonnegative
Approximation
weighting
of
function
Signals
w(t)
over
[ t , t / ] if 0
f
where j
2:Least
w(t)^(t)<j> (t)dt
is a nonzero
positive
constant.
in [ t o , * / ] - If 7m = 1. the set {(j> {t)}
r
^
It is assumed
is known
m
I f the set {4> }
m
= { ° '
T
'
(2.1)
that w(t)
is not zero
as the orthonormal
represents a set of o r t h o g o n a l p o l y n o m i a l s then the
m
identically
set. following
theorem is true. Theorem
2 . 1 Any polynomial
set of orthogonal polynomial
polynomials
p (t)
of degree n can be represented
n
{(b (t),
0 i ( t ) , . . . , 0 „ ( t ) } over [t ,t ]
0
of degree m in t,m
0
f
uniquely
by the
where 4> {t) m
is a
= 0 , 1 , . . . ,n.
Proof. Let p „ ( t ) = a < M * ) + « i 0 i ( t ) + • • • + a„<j> (t) 0
where i n view of Eq.2.1 the scalars o ' s
are
m
a
=
m
(2.2)
n
1 f'j — / 7m <0
w(t)p (t)4> (t)dt. n
m
T h i s representation is unique, because i f p (t) n
= M {t)
+ ( 9 , ^ , ( 0 + • • - + M {t)
0
(2.3)
n
is another representation, s u b t r a c t i n g Eq.2.3 from Eq.2.2 we have
(a
0
- 0 )Mi)
+ ( " i - Pi)4>dt) + ••• + (a
a
- P„)d>„(t)
n
Using the p r o p e r t y of E q . 2 . 1 , i t is easy to see t h a t a
m
= (3
m
=
0
for a l l m .
Hence the
representation of Eq.2.2 is unique. O n the other h a n d , any continuous f u n c t i o n f(t) can be a p p r o x i m a t e d by a p o l y n o m i a l p(t)
over a finite closed i n t e r v a l [ t o , t / ]
w i t h a prescribed degree o f accuracy i n
view of the following theorem established by Weirstrass in 1885. Theorem
2.2 If f(t)
{ P n ( t ) } which exists
is continuous
is uniformly
a polynomial
p ( t ) such c
in [ Z , t / ] , then there is a sequence
convergent
0
to f{t)
in [ t , i / ] , i.e., 0
that
I / ( 0 - J » « ( 0 l< M o
given
of
polynomials
any e > 0
there
2.1 Least Squares
Approximation
However, i f we select Fourier
27
series to a p p r o x i m a t e a f u n c t i o n f(t),
t h e o r e m gives sufficient conditions for its
the f o l l o w i n g
representation.
T h e o r e m 2.3 Let f(t) be (a) a single-valued bounded function in the interval t < t < tj where it is (b) piecewise continuous having a finite number of finite discontinuities and has (c) a finite number of maxima and minima. Then f{t) can be described over the interval [ t , t / ] by 0
0
(2.4)
where (2.5) (2.6) V and at the point
of
=
if — t
0
discontinuity ,
The conditions
(a)-(c)
are known
/(«+) + /(«-) 2
as D i r i c h l e t ' s conditions .
T h e o r e m s 2.2 and 2.3 a l t h o u g h give sufficient conditions under w h i c h a f u n c t i o n can be a p p r o x i m a t e d by a set of o r t h o g o n a l functions, t h e y do not say a n y t h i n g regarding the error consequent on the a p p r o x i m a t i o n by a finite n u m b e r of t e r m s . T o have some idea on this aspect, let the f u n c t i o n f(t) functions
be represented as a series of the
{
{t), i(t),. . . , <^„(t)}. T h e n we can w r i t e Q
f{t)
M
S {t) n
where n *«(*) = £ / m < M * ) m=0
(2.7)
W e call s „ ( t ) an a p p r o x i m a t i o n of order n o f the f u n c t i o n f(t)
.The residual or error
i n t h e a p p r o x i m a t i o n is therefore equal to
R(t)
T h e coefficients / ' s m
= f{t)
- s (t) n
= f{t)
-
£ / <Mt). m=0 m
(2.8)
are so d e t e r m i n e d t h a t the i n t e g r a l square error w i t h respect to
the nonnegative w e i g h t i n g f u n c t i o n u ; ( i ) over [ t , t / ] defined by 0
28
Chapter
el
2:Least
Squares
Approximation
of
Signals
2
=
w(t)R (t)dt
=
f
»(*)[/(*) '°
E /„*.(*)]*«« m=0
is a m i n i m u m . D i f f e r e n t i a t i n g (2.9) w i t h respect t o f
(2.9)
and s e t t i n g t h e result t o zero
T
we have 2
de —P-
=
f'J -2 / «(«) /(«) -
=
-2 / '
=
0
* £
w{t)R{t)4> (t)dt T
(2.10)
for r = 0 , 1 , . . . , n as t h e c o n d i t i o n for a m i n i m u m i n t e g r a l square ( M I S ) error because
— f
=
2/
tof.*)*,^)*
is always positive over [ t o . * / ] - T h e c o n d i t i o n of Eq.2.10 for M I S error w h i c h states t h a t the residual R(t) should be o r t h o g o n a l to (t)'s, can be r e w r i t t e n as T
/
for a l l r = 0 , 1 ,
' «"(*) £ f {t)4>r{t)dt m=0 m
'«
=
m
J ' w{t)f{t) {t)dt
(2.11)
T
, n . These equations are o f the form Au = v
where A = [ a
m r
(2.12)
] is a s y m m e t r i c m a t r i x of order (n + 1), a
=
m r
y ' u>{i)4> {t)<j> (t)dt m
=
/ '
=
«™
v
=
[ d d, • • • d „ ]
d
=
J ' w(t)f(t)<j> (t)dt
r
w(t) (t)dt r
m
T
0
m
(2.14)
T
Once the functions < £ ( t ) ' s and the w e i g h t i n g f u n c t i o n w(t) Eq.2.12 can be p r e d e t e r m i n e d .
(2.13)
r
fixed,
then A in
T h e m a t r i x A can be used repeatedly
are
t o get the
2.1 Least
Squares
Approximation
29
coefficients / ' s i n the a p p r o x i m a t i n g f u n c t i o n s„(t) given by Eq.2.7 for different values of n , w h i c h decides on the order of a p p r o x i m a t i o n . m
A n o u t s t a n d i n g advantage results in selecting the functions 's to be o r t h o g o n a l because i n t h i s case m
a
= a
m r
r m
= 0 , m ^ r;
o-rr = t r . as follows f r o m Eq.2.1 and Eq.2.13. Consequently from Eq.2.11 we have fr = ~
/ ' w(t)f(t)
(2.15)
r
I T
<°
where
w h i c h ensures the m i n i m u m value of Eq.2.9. We shall call 7 the normalization r
factor
of the o r t h o g o n a l p o l y n o m i a l s . I n view of Eq.2.1 and Eq.2.8 - Eq.2.11 the M I S error can be w r i t t e n as
min[el\
=
f'w(t)\f(t)
£
-
'»
=
f '
=
f ' w(t)[f(t)
=
f
m
f ' ' w{t)[f\t) 0
£ m=0
- f(t)
w{t)f{t)dt
,
m
w(t)f(t)R(t)dt
£
-
'«
=
f 4> {t)\R{t)dt
m-0
f l
f 4> (t)]dt m
l
m
m
m = 0
-
jZ
fMfflt
2 16
(- )
m = 0
T h e accuracy i n a p p r o x i m a t i o n should i m p r o v e w i t h increasing n. I f n tends to i n f i n i t y , the finite sum given by Eq.2.7 goes over to an infinite series, w h i c h w i l l t r u l y represent the parent series reducing the M I S error to zero according to the expected converging p r o p e r t y of the series i f f
w{t)f\t)dt '°
= £
l
m
fl
(2.17)
m = 0
A set of o r t h o g o n a l functions
is said to be complete i f for any continuous f u n c t i o n f(t), the M I S error represented by Eq.2.16 tends t o zero as n tends to infinity. T h e n Eq.2.17 represents the r e l a t i o n
Chapter
30
2:Least
Squares
Approximation
of
Signals
of completeness or Parseval's r e l a t i o n . However, i f n is finite, we have the Bessel's inequality
f
w(t)f(t)dt
>
E
7
m
/
(2.18)
m
T h e class of functions satisfying the c o n d i t i o n 2
/ ' w(t)f (t)dt •'to
< oo
is k n o w n as square integrable over the i n t e r v a l [ i , * / ] 0
E x a m p l e 2.1
S i n e - C o s i n e F u n c t i o n s The set of
functions
{ 1 , cos 6, sinff, cos 20, sin 2 0 , . . . } ore orthogonal
over the interval
[ 0 , 2 t t ] with the weighting
function
w(6) = 1.
Indeed,
we have J
/ a
cos mddff = 7r; / o
2
f' /
sin m8d9 = n\
J
ft* cos m 0 cos r6d6 = 0; /
sin m0 sinrSdO
= 0;
cos mO sin r9d6 = 0; u where r ^ m.But
the set {
1
sin0
cos 26
cos 6 sin 26
is orthonormal. The above sets are called Fourier each of them is a complete set of functions . E x a m p l e 2.2 H a a r F u n c t i o n s
har(r,n,t)
sine-cosine
The set of functions
har(0,0,t)
=1,
|
2f,
==i < t <
-2*,
< ( <
r
defined
functions
(SCF)
and
by [83]
16[0,1)
= \
I. 0,
\
elsewhere
^ p for t G [ 0 , 1 )
where 0 < r < l o g m , 1 < n < 2 , m = 2*, and A; is u positive integer, are known as Haar functions. This set is periodic, orthogonal and complete with w(t) ' 1 and was 2
2.1 Least
Squares
poroposed where
in 1910 by Haar.
31
Approximation A set of first eight Haar functions
are given
h (t)
=
har{0,0,t),
h^t)
= har(0,1,t),
h {t)
=
har[\,\,t),
h (t)
=
har{\,2,t),
h (t)
= har(2,l,t),
/i (i) =
har(2,2,t),
h (t)
=
har(2,Z,t),
h (t)
=
a
3
e
4
7
2
5
pulses
with 2
m _ 1
cycles
in the unit interval
or —1 in each half cycle is orthonormal pulse
of amplitude
A set of four
functions
were developed
are shown
\,p
is an
functions
values
r (t)
+1
is a unit
0
in 1922
as described
u> (t) generated n
[251].
in Example
by
T
2.3, n = 0 , 1 , 2 , . . . , 2 —
integer, integer
of [.]} ,
of n in the binary
system:
2
6, 's are the coefficients
in the representation k 1
k
n = b2' system
In order
to make complete,
0
w,(t)
= =
n(t),
=
r (t),
pulse functions.
A set of four
a
complete A set of
2
r (t) (t) 2
ri
described
by
otherwise
orthonormal
orthonormal
Paley
o(').
=
1
order-
where p = 2 and r
I "0,
or
functions,
as Walsh functions.
E x a m p l e 2.5 B l o c k - P u l s e F u n c t i o n s The set of functions
for all k = 1 , 2 , . . . m, is a complete
to dyadic
set of Rademacher
and is known
in Fig.2.3
w (<)
three types of
is according
by Walsh [320] in 1923 to form
functions,
are included
b.2
0
There are generally
the incomplete
was introduced
set of rectangular
Walsh functions
functions.
and the above representation
the above set of functions orthonormal
+ ••• +
k
of orthogonal
ing of Walsh functions
1
+ b _i2 ~'
k
is a complete
four
The function by Rademacher
m = { | [ l o g n] \ + 1 , where | [.] |:= greatest
ordering.
which is a train
m
in Fig. 2.2.
E x a m p l e 2.4 A V a l s h F u n c t i o n s The set of functions
where r ; ( t ) 's are Rademacher
r (t)
[ 1 , 0 ) and takes
but not complete.
+ 1 . These functions
Rademacher
Fig.2.1,
har(2,4,t).
E x a m p l e 2.3 R a d e m a c h e r F u n c t i o n s The set of functions of rectangular
in
system
block-pulse
of functions
functions
known
are given in
as blockFig.2.4-
32
Chapter
2:Least
Squares
Approximation
Figure 2 . 1 : A set of eight Haar functions
of
Signals
2.1 Least
Squares
Approximation
r (t) n
r (t) 2
r, ( t ) 1
Figure 2.2: A set of four Rademacher functions
Chapter
-t
2:Least
Squares
Approximation
1-
Figure 2.3: A set of four Walsh functions
of
Signals
2.1 Least
Squares
Approximation
-J-^ t b
2
( t ) n
2
b , ( t ) i
b (t), t
2
F i g u r e 2.4: A set o f four o r t h o n o r m a l block-pulse functions.
Chapter
36
2.2
2:Least
Squares
Approximation
of
Signals
Orthogonal Polynomial Approximation : A
Generalized Approach
A p a r t from the above o r t h o g o n a l functions, there are different classes of polynomials each of w h i c h forms an o r t h o g o n a l set. T h e i n t e r v a l [ i , t / ] over w h i c h each class is defined depends on the n a t u r e of the b o u n d a r y of a physical p r o b l e m and the mean integral square error is a g u i d i n g factor i n selecting the w e i g h t i n g f u n c t i o n w(t). We shall now t r y to get a solution t o the following p r o b l e m : P r o b l e m Given \to,tj) and w(t), determine the set of orthogonal polynomials 4> (t)'s of degree r,r = 0 , 1 , . . . , n . w h i c h w i l l m i n i m i z e the i n t e g r a l square error i n representing an a r b i t r a r y square integrable f u n c t i o n . Let (t) belong to the set of o r t h o g o n a l p o l y n o m i a l s { (t)} a n d q -i(t) be an a r b i t r a r y p o l y n o m i a l of degree r — 1 or less. B y T h e o r e m 2.1 there are scalars a,'s such t h a t 0
r
r
m
? _!(*) r
= a 4> {t) 0
0
+ a ^ t )
r
+ ••• + a _ ! * _ i ( t ) r
(2.19)
r
Since the f u n c t i o n (t) is o r t h o g o n a l t o each of 4>k(t), k = 0 , 1 , . . . , r — 1 contained in Eq.2.19, (j> {t) is also o r t h o g o n a l to q -i(t) w i t h respect to w(t) as defined by Eq.2.1,i.e., r
T
r
/ Let a new f u n c t i o n u {t) T
w{t)ct> (t)q _
r
= a.
1
(2.20)
be defined by w(t) (t) =
(
— dV
T
= u ;\t).
(2.21)
Hence i n view o f Eq.2.20 we get r )
f ' u ' ( i ) g _ ( i ) d t = 0. io r
(2.22)
1
-/
Now g _ i ( < ) being a p o l y n o m i a l of degree ( r — 1), its r t h derivative is zero. i n t e g r a t i n g left h a n d side of Eq.2.22 by parts we get r
[«l
r _ 1 )
a )
(<)9,-i(o - u£" (*)ft-i(o + • • • + ( - i r v w ^ ' w i L '
= o
Hence
(2.23)
F r o m Eq.2.21 T
1
d u {t) T
w(t)
dt
T
and since 4> {i) is a p o l y n o m i a l of degree r differentiating Eq.2.24, ( r + 1) times we have r
d
T+1
\
1
r
d u {t) — T
= 0
(2.25)
2.2 Orthogonal
Polynomial
Approx.:
A Generalized
Approach
w h i c h is a d i f f e r e n t i a l e q u a t i o n for the new f u n c t i o n u .
37
As g _ i ( t ) is a r b i t r a r y , for
T
r
Eq.2.23 t o be satisfied we must have u (t )
=
u (t )
= i i ( l ) = • • • = u<
u (t )
=
u (tj)
= u (tj)
r
T
0
}
T
0
r
r
r _ I )
0
r
(
1)r
= • • - . = ui ~ t )
T
(2.26)
= 0
f
(2.27)
I f t h e 2 r n u m b e r of c o n d i t i o n s of Eq.2.26 and Eq.2.27 are i m p o s e d , t h e s o l u t i o n of ( 2 r + l ) t h order differential e q u a t i o n Eq.2.25 w i l l give u (t)
c o n t a i n i n g one a r b i t r a r y
r
c o n s t a n t . T h i s s o l u t i o n w h e n inserted in Eq.2.24 w i l l generate the o r t h o g o n a l p o l y n o mials 0 ( i ) ' s . T h e r e l a t i o n o f Eq.2.24 for g e n e r a t i n g t h e set of o r t h o g o n a l p o l y n o m i a l s r
is k n o w n as Rodrigues' s o l u t i o n even w h e n i
formula.
I t can be shown t h a t t h e above p r o b l e m has also a
a n d / or i ; is i n f i n i t e p r o v i d e d in [ r , t / ]
0
0
w(t) and
> 0
. /
fw{t)dt
where r is a p o s i t i v e integer, exists. T h u s we have shown t h a t given the w e i g h t i n g f u n c t i o n w{t) and the i n t e r v a l [ t o , * / ] how we can generate t h e o r t h o g o n a l p o l y n o m i a l s <^ (t)'s b u t this m e t h o d relies on the r
u n k o w n p o l y n o m i a l u (t) r
equation.
t o be o b t a i n e d
by solving a (2r + l ) s t order differential
F o r t u n a t e l y , these o r t h o g o n a l p o l y n o m i a l s can be generated recursively by
the f o l l o w i n g three-term T h e o r e m 2.4
formula.
T h r e e T e r m R e c u r r e n c e F o r m u l a Let {
onal polynomials as defined
recurrence
in Eq.2.1.
to the weighting
function
T
r
T
and c
w(t)
orthog[t ,t/] 0
Then <*V+i(t) = (<M + b )4> (t)
where a , b
be a set of
over the interval
T
with respect
are constants,
r
+ c 0 _!(t)
r
r
(2.28)
r
r = 0 , 1 , . . . , and _[(<) : = 0.
Proof. L e t t h e r t h degree p o l y n o m i a l 4> (t) be represented by r
'
4> (t) = /3 T
r0
+ 0 it r
2
+ • • • + T f~ r
+ a i r
r _ 1
+ ji f.
(2.29)
r
I f we choose (2.30) then
is a p o l y n o m i a l of degree at most r and cosequently can be expressed as a linear c o m b i n a t i o n of d> (i), d>i(t), • • • ,&•(<)• 0
have
Therefore for some constants b ,c r
T
and e we
Chapter
38
d> (t)
- a td> (t)
r+l
T
=
r
2:Least
Squares
Approximation
of
Signals
b {t) + c _!(t) + e _ _ (t) + • • • r
r
r
r
r
2
r
2
+ £ ^ . ( t ) + £o<^o(i) I
M u l t i p l y i n g b o t h sides by w(t)(p (t)
(2-31)
1
and i n t e g r a t i n g over [ i o , * / ]
m
w
e
have i n view of
Eq.2.1 / ' w(t)4> {t)[ {t) Io m
for m = 0, l , . . . , r — 2.
- a t (t)]dt = e
r+l
T
T
m 7 m
On the left side, the first t e r m vanishes because of or-
t h o g o n a l i t y of < £ ( i ) ' s , w h i l e the second t e r m vanishes because of Eq.2.20, since m
tcj> {t) m
e
m
is a p o l y n o m i a l of degree less t h a n r.
As 7 ' s are positive, we must have m
= 0 , m = 0 , 1 , . . . , r - 2. Hence the relation of Eq.2.28 follows f r o m E q . 2 . 3 1 . B y v i r t u e of Eq.2.29 we can evidently w r i t e -Z~Mi)
*<£r-l(*) =
+
(2.32)
?r-l(<)
where g , - - ^ ) is a p o l y n o m i a l of degree at most r — I . We now m u l t i p l y b o t h sides of Eq.2.28 by w(t)(j> _ (t) r
and t a k i n g E q . 2 . 1 , Eq.2.20 and Eq.2.32 into cosideration, we
1
integrate this p r o d u c t over [*o,*/] to get 0
=
a f'tw{t) _ (t) {t)dt
=
a
T
T
T
I Io
l
0r-
w{t)
flr7rA-
+
r
c
T
T
i
- {t) + g _ , ( t ) 4> (t)dt + c T
+
c ', _
r
r
r 7 r
_!
r7r
Pr
Hence s u b s t i t u t i n g from Eq.2.30 /3,
c
r
+
1
/?r-l
PI T
E q u a t i n g coefficients of t
7
r
(2.33)
= 7r- ,
from b o t h sides of Eq.2.28 we get Q
R + 1
= aa T
+
T
bP T
T
Hence s u b s t i t u t i n g from Eq.2.30, b is expressed by T
Pr
Q +
l
r+1
Pr
+
(2.34)
l
T h e r e l a t i o n of Eq.2.28 w i t h o , 6 and c determined by Eq.2.30, Eq.2.34 and Eq.2.33 respecively is k n o w n as the three-term recurrence relation for a set of o r t h o g o n a l polynomials. r
r
r
T h e following theorem contains another i m p o r t a n t p r o p e r t y of o r t h o g o n a l p o l y nomials.
2.2 Orthogonal
Polynomial
Approx.:
T h e o r e m 2.5 in the interval
The orthogonal [t ,tj].
A Generalized
polynomials
Approach
39
<j> (t) 's, r > 1 possess
r distinct
r
real
zeros
0
Proof. Since (t) is a constant and w(t) 0
is nonnegative, for the o r t h o g o n a l r e l a t i o n
to be t r u e , 4> (t) must change sign at least once in [r-o>*/] and consequently 4> {t) must T
r
have at least one real zero i n [ i , i / ] . Let 0
4> (t) =
f} {t-t )---{t-t )
T
where 0
T
i
r
is the coefficient of the leading t e r m of ^> (z), see Eq.2.29. Factors of 4> (t)
T
r
T
associated w i t h complex conjugate zeros o c c u r r i n g in pairs and factors associated w i t h real zeros o f even m u l t i p l i c i t y do not change for t v a r y i n g i n [ t , i / ] . T h e r e m a i n i n g 0
factors c o n t a i n real zeros o f o d d m u l t i p l c i t y in [ t , * / ] w h i c h we denote by ( i — t i ) , ( i — 0
t ), 2
. . . , ( < — < ) where t\,t ,. m
2
.. , t
(t>r(t)
where g(t),
=
m
are d i s t i n c t and m < r. Let »(*)(<
• • • • ( * - * « )
c o n s t i t u t e d by the factors associated w i t h real zeros of even m u l t i p l i c i t y ,
complex conjugate zeros and factors (t — t , ) , . . . , ( i — < ) w i t h even powers, does not m
change sign w i t h t i n [ t , i / ] . 0
Suppose t h a t m < r . T h e n i n view of the r e l a t i o n i n
Eq.2.20 we have t )4> (t)dt m
r
= 0
or
Since the i n t e g r a n d is nonnegative, i t does n o t change sign w i t h t in [ i , i / ] and the 0
left h a n d side of the last r e l a t i o n is positive . T h i s c o n t r a d i c t i o n ensures t h a t d> (t) T
has r d i s t i n c t zeros i n [ < , t / ] . 0
Eq.2.1 does n o t give a convenient expression
for c o m p u t i n g the n o r m a l i z a t i o n
factor 7 . A suitable expression is o b t a i n e d using Eq.2.29, Eq.2.20 and E q . 2 . 2 1 . T h u s r
we get
(2.35)
40
Chapter
2:Least
Squares
Approximation
of
Signals
where the last step follows f r o m i n t e g r a t i o n by p a r t s . F r o m the foregoing discussion t i o n w(t)
we observe t h a t f r o m t h e given w e i g h t i n g func-
and the i n t e r v a l [ i , * / ] we can d e t e r m i n e the associated set of o r t h o g o n a l 0
p o l y n o m i a l s d> (t)'s w i t h the help of Eq.2.24-Eq.2.27.
A n y f u n c t i o n f(t)
r
p r o x i m a t e d by Eq.2.7 where / ' s
is t h e n ap-
are d e t e r m i n e d by Eq.2.15 and Eq.2.35.
r
above b a c k g r o u n d , we shall now see how different selections of w(t)
W i t h the
a n d the i n t e r v a l
[*o,*/] give rise t o the different classes of o r t h o g o n a l p o l y n o m i a l s .
2.3
Legendre
Polynomials
I n t h e case of Legendre p o l y n o m i a l s we assume t h a t w(t)
= 1 and [t , t/\ = [—1,1]. By 0
the linear t r a n s f o r m a t i o n (a — 6 ) r = — 2t + (a + 6), we can always change t belonging t o any a r b i t r a r y i n t e r v a l [a, 6] to r € [—1,1]. B y v i r t u e of Eq.2.25-Eq.2.27 we have to solve 2r+l
d u {t) r
— m)
ui (-l)
= u<
m )
= 0
(l)
=
(2.36) 0
(2.37)
for m = 0 , 1 , . . . , r — 1. T h e s o l u t i o n of Eq.2.36 is a p o l y n o m i a l of degree 2r having the form u (i)
= Q
r
2 r
2 r
r
+ • • • + Q, i + a
To satisfy t h e conditions i n Eq.2.37, u (t)
must have ( t
T
(2.38)
0
2
— 1) as a factor.
Also the
conditions r
= ul -"(-l) = 0 are satisfied i f u {t) T
2
r
= ( t — l ) . As a result, the s o l u t i o n of Eq.2.36 can be expressed
as u (t)
= d (i
r
where d
r
2
r
- l)
r
(2.39)
is an a r b i t r a r y constant. T a k i n g 1
d
T
= rW
the set of o r t h o g o n a l p o l y n o m i a l s , k n o w n as the r t h Legendre p o l y n o m i a l s , in view of Eq.2.24 can be w r i t t e n as r
d u (t) — dr
r
=
r
1
T
P (i
H2
7
d (t "
= r
-
iy (2.40)
df
V
;
w h i c h is the Rodrigues' f o r m u l a for Legendre p o l y n o m i a l s . T h e o r t h o g o n a l properties of these p o l y n o m i a l s follow f r o m E q . 2 . 1 . T h u s /
P {t)P (t)dt m
r
=
0,m^r
(2.41)
2.3 Legendre
41
Polynomials r
for n o n n e g a t i v e i n t e g r a l values of m and r . To d e t e r m i n e 0 ,
the coefficient of t
r
P {t), r
in
see Eq.2.29, we note from Eq.2.40 t h a t
Pr(t)
!2
r
dr
(2r)! r
2 (r!)
i
2
r ( r - 1) r
2
-f
2(2r-l)
+
Hence, c o m p a r i n g t h e above p o l y n o m i a l w i t h Eq.2.29 i t follows t h a t (2r)!
r
(1 - r )
r
(2.42)
and the relations Eq.2.35, Eq.2.39 and Eq.2.42 are used to give us
=
f
p?(t)dt r
J-i
-t)l-L f 2 r! 2 r! r
(2r)! 2!
{
=
w
2 "(r!)
2 2
2 r + 1
(H)
(i _
2
t ydt
r
2
_
2
(2.43)
( 2 r + 1 ) ! ~ 2r + 1
where t h e last b u t one step follows f r o m i n t e g r a t i o n by parts.
P u t t i n g Eq.2.42 and
Eq.2.43 i n Eq.2.30,Eq.2.34 and Eq.2.33 we get respectively, Pr+i ^
2r+ 1
=
Pr
K
=
c
=
r
r + 1 '
0, r 4- 1
w h i c h w h e n s u b s t i t u t e d i n Eq.2.28 gives Pr M +
t h e three-term
recurrence
= ^^-tPr(t) r + 1
formula
- -^—Pr-At), r + 1
(2.44)
for Legendre p o l y n o m i a l s . W i t h the help of Eq.2.40
and Eq.2.44, these p o l y n o m i a l s can be generated recursively, the first eight of w h i c h are s h o w n i n Table 2.1 T h e nth degree Legendre p o l y n o m i a l a p p r o x i m a t i o n of an a r b i t r a r y f u n c t i o n
f[t)
over [—1,1] is represented by n « J „ ( i ) = £ frPr(t) r=0
fit)
where i n v i e w of Eq.2.15 and Eq.2.43 f
r
is given by
t
fr = ~~Z~~ f V
(2-45)
J
f(t)PAt)dt _i
(2.46)
Chapter
42
2:Least
Squares
Approximation
of
Signals
Table 2 . 1 : Legendre P o l y n o m i a l s
1
Pot*)
t PAt)
(3t
p (t) P (r)
(5i
3
2
-
3
l)/2
- 3i)/2
4
(35« - 3 0 i
4
5
2
+ 3)/8
3
(63« - 70 + 15i)/8
Ps(t) PM
6
2
( 2 3 1 t - 315«* + 105r -
J°T(0
7
5
5)/16
3
( 4 2 9 t - 693* + 3 1 5 i -
35r)/16
T h e M I S error i n this a p p r o x i m a t i o n as follows f r o m Eq.2.16 is
2
mm[e ]
=
/ ' -t
f{t)dt
-
E
J
r
=
0
2 r + l
T h e Legendre p o l y n o m i a l s satisfy a second order differential e q u a t i o n w h i c h w i l l now be derived. D i f f e r e n t i a t i n g Eq.2.39 w i t h respect to t we get 2
u (t)
= 2rtd (t
T
-
r
\y->
w h i c h , using Eq.2.39 again, can be w r i t t e n as 2
(1 - t )u {t)
+ 2rtu (t)
T
= 0
r
D i f f e r e n t i a t i n g again the above e q u a t i o n w i t h respect t o t we get 2
(1 - t )il (t)
+ 2(r -
T
\.)tii (t) r
+ 2ru (l) = 0 r
w h i c h w h e n differentiated r times results in the differential equation 2
l
- (2i)rz4
+2)
(1 - t )u ; (t) +
2 ( r - l)tu \t)
r + 1 )
r)
(i) - r(r -
l)u (t) r
+ 2 r ( r - l ) u < ( i ) + 2ru \t)
T+1
r )
r
r
T
= 0
where we have used L e i b n i t z f o r m u l a for the r t h d e r i v a t i v e of a p r o d u c t , v i z . , T
d (xi v} ——dt
T
=
uD v
=
E
r
l
+ (I) DuD ~ v
+ Q
2
r
2
D uD ~ v
+ --•+
r
D uv
r
m
r
m
C )D uD - v m
(2.47)
2.4 Laguerre
Polynomials
43
where D
d — dt ( a - m + l)...« _
= =
(-l)"(- ) a
1 - 2 - m (a)
=
t
w
1 -2--m
a ( a + 1) • - . ( a + fc - l ) , ( a )
0
= 1
and a is a real n u m b e r . B y v i r t u e of Rodrigues' f o r m u l a contained i n Eq.2.40, above differential e q u a t i o n simplifies to 2
- d P (t) (l-t )—£r -M—£dt' 2
1
r
dP {t) T
L
+ r(r+l)P (t)
= 0
r
dt
the
(2.48)
w h i c h is k n o w n as Legendre differential equation. To express P (t) as a p o l y n o m i a l , we expand (Z — l ) in a b i n o m i a l series to get 2
r
r
2
(t -i)
r
= =
2 r t
- Qt
£ ( - i )
2
r
2
k
2r
2k
- + ••• + (-\) c )t - +
•••
k
k
G)*
2 r
-
+(-iy
2 t
k=0
Hence P {t) T
given by Eq.2.40 takes the f o r m 1 W _ £ ( - ! ) * Q(2r-2A0(2r-2fc-l)---(r-2fc + l)i 2 r ! k=0 W
P (t) T
r
=
2
r
M £(-!)* ' ~ * 2 ' * ! ( r - fc)!(r (
=
2
r
2 f c )
(2.49) 2k)\
where
r is o d d . T h u s P ( i ) = P ( - i ) i f r is even and P ( i ) = - P (-t) r
2.4
r
r
r
i f r is o d d .
Laguerre Polynomials
To derive the Laguerre p o l y n o m i a l s we assume t h a t w(t)
= e
_0,<
;
[to,*/] = [0,oo]
where a is a positive constant. We note t h a t the change of variable defined by
44
Chapter
2:Least
Squares
Approximation
of
Signals
magnifies the i n t e r v a l [a, 6] to [0,oo], Therefore, the differential equation Eq.2.25 to be satisfied by the u n k n o w n p o l y n o m i a l s u (t) becomes T
T
d ->
al
e
= 0
dr
(2.51)
w h i c h is to be solved w i t h the i n i t i a l and t e r m i n a l conditions given by T
l
u ( 0 ) = u ( 0 ) = • • • = u ~ \u) r
r
T
T l
« ( o o ) = i i ( o o ) = • • • = u ' \oo) r
r
T
= 0,
(2.52)
= 0.
(2.53)
I n t e g r a t i n g Eq.2.51 r + 1 times we get
_ , = e
[a + a,( + • • • + at ] 0
r
dr where a , a , 0
. . . , ct are constants. I n t e g r a t i n g r times again, the s o l u t i o n of Eq.2.51
t
T
for u ( r ) takes the form r
u (t) r
a i
T
=
e'
[d + i i + • • • +
+
[b + 6,Z + • • • + 6 . , r ' ]
a
dt ]
x
0
r
(2.54)
r
where b , b ,.. . , b _ are integration constants and the constants d ,d . . . ,d depend on a, a , ct . . ., ce . T h e t e r m i n a l conditions i n Eq.2.53 at ( = oo give us 6 = b = • • • = b = 0 and the i n i t i a l conditions in Eq.2.52 at t = 0 provide us do = d = •• • — d _ = 0. Hence 0
:
r
1
0
T
a
0
%t
l
x
lt
r
r - 1
r
t
r
u (t)
at
= d t e~
r
(2.55)
T
where d is an a r b i t r a r y constant. T h e o r t h o g o n a l p o l y n o m i a l (t) given by Eq.2.25 i n c o r p o r a t i n g Eq.2.55 in i t w i t h d = — becomes r
r
T
, L (at)
r
T
1 d u (t) = — ^ w{t) dr
s
T
r
e
=
a i
T
d
r ! df
r L
* e"
.i
(2.56)
1
w h i c h is the Rodrigues' formula for the r t h degree Laguerre p o l y n o m i a l when a = 1. I n view of E q . 2 . 1 , i t is obvious t h a t
y
r oo at
e~ L (at)L (at) m
r
dt = 0,m
^
r.
where m and r are nonnegative integers. I n the expanded f o r m , the p o l y n o m i a l has the expression
r\L {at) T
T
= {-a) f
2
r
T y
+ r (-a) -'t -
+ * '~ ~ ^ (
( - a ) - i - + • • • + r!. r
2
r
2
L (at) r
2.4 Laguerre
Polynomials
45
T a b l e 2.2: Laguerre P o l y n o m i a l s
L (t)
1
Ldt) L {t) L {t)
1 - t
0
2
(2 - 4 i + t ) / 2
2
2
3
(6 - 18* + 9* -
3
L (i)
t )/6
2
3
4
4
(24 - 96* + 72* - 16* + * / 2 4
L (t)
(120 - 600* + 600* - 200* + 25* - i ) / 1 2 0
Lett) L {t)
(720 - 43204 + 5400* - 2400r + 4 5 0 i " - 3 6 i + < ) / 7 2 0
2
5
3
2
4
5
2
(5040 - 35280* + 52920* -
7
5
3
4
5
29400*
6
6
3
7
+ 7350i - 8 8 2 i + 49r -
f )/5040
C o m p a r i n g t h e above p o l y n o m i a l w i t h Eq.2.29 we get (-a) Pr= —^,a =
r
r
2
( - Q )
K
r
R
'
" '
2
2
r (r-l) (-a)- , -!2
,T =
2
T
(2.57)
and f r o m Eq.2.35 a n d Eq.2.55
ai
/
2
e-
L (at)dt r
T
(-l) r\' 1 K
(r!)
2
/ o
at
fe- dt
J
(2.58)
a where t h e last step follows f r o m i n t e g r a t i o n by p a r t s .
We now use Eq.2.30, Eq.2.34
and Eq.2.33 t o get respectively
a
T
A-+1 = —— =
a 1 + 2r r, b = —, c r - : r + 1 T
r r
r + 1
w h i c h w h e n p u t i n Eq.2.28 give us ( r + l)L (at) r+l
= ( 1 + I T - at)L (at) r
T h e above e q u a t i o n Eq.2.59 represents t h e three-term
- rL _i{at)
(2.59)
T
recurrence
formula
of Laguerre
polynomials. T h e Laguerre p o l y n o m i a l s can be generated recursively w i t h t h e help of Eq.2.56 and Eq.2.59, the first eight o f these are i n c l u d e d i n Table2.2
46
Chapter
2:Least
Squares
Approximation
of
Signals
T h e n t h order a p p r o x i m a t i o n of an a r b i t r a r y f u n c t i o n / ( < ) over [0,oo] i n a series of the Laguerre p o l y n o m i a l s { Z ( a t ) } is represented by r
/ ( < ) w s (t)
= £
n
f L (at) T
(2.60)
T
r=0
where f
r
is given by, see Eq.2.15 and Eq.2.57, oo
e""f(t)L (at)dt.
/u T h e M I S error i n this representation
(2.61)
r
as a result of Eq.2.16 is 71
oo
/
2
f
- J2 —
at
e- f\t)dt
r=0 a
w h i c h exists i f | f(t)
| grows less r a p i d l y t h a n e '^
2
(2.62)
a
as t tends t o i n f i n i t y .
T h e second order differential equation satisfied by the Laguerre p o l y n o m i a l s can be derived as follows. D i f f e r e n t i a t i n g Eq.2.55 once w i t h respect t o t we get, u (t)
=
r
l
a
d rf- e- ' r — u {t) —
=
T
-
T
d t ae-
at
T
au (t).
r
r
After rearrangement, the above equation becomes tu (t)
+ (at - r)u (t)
T
= 0,
r
w h i c h w h e n differentiated again gives us, tu (t)
+ ( 1 + at - r)u (t)
r
+ au (t)
T
= 0.
r
T h i s equation is differentiated r times using L e i b n i t z f o r m u l a o f Eq.2.47 t o give us + (1 + a i ) t 4
r+2
tu \t) T
r + 1 )
( i ) + a(l
W e now make use o f Eq.2.56 to replace u (t) r
ai
t^[L (at)e- ] r
a(l
+ 7-)L (Qi)e~
0 , <
r
r
+ r ) « < ' ( t ) = 0.
by L (ctt)
t o get
T
at
+
(l+at)j [L (ot)e- }
=
0
t
+
r
w h i c h on s i m p l i f i c a t i o n becomes, 2
d t — L (at) dt T
+ (l-at)
d — L (ctt) at T
+ arL {at) r
=
0
(2.63)
2.5 Hermite
47
Polynomials
T h e e q u a t i o n Eq.2.63 at a = 1 is called the Laguerre differential e q u a t i o n . Eq.2.56 is now used to w r i t e the p o l y n o m i a l expression for L (t). T h u s using L e i b n i t z f o r m u l a of Eq.2.47 we get, T
L {t)
e* d
=
T
r\ dt
r
fe~<
T
-\e-'r\
+ (:)(-\)e~'r\t
+
r'
+
r
-i
G ) ( - i ) V
+ --
r
+(-l) e-'t'
(2.64)
0'
2.5
Hermite
Polynomials
To derive the H e r m i t e p o l y n o m i a l s , we assume t h a t w(t)
= e"°
2 < 2
,
[t ,t/] =
[-00,00]
0
where a is a real constant. We note t h a t the change of variable defined by
(f - a ) ( t - b) magnifies the i n t e r v a l [a, 6] to
[—00,00]
. W i t h the above a s s u m p t i o n , the differential
e q u a t i o n Eq.2.25 takes the f o r m , i2
a t
r
(2.65)
0
dr
dr w i t h t e r m i n a l conditions
u ( — 00) T
u (oo) T
=
« ( R
— 00)
= • • • = u, r
l
= u ( o o ) = • • • = u ~ \oo) r
r
•(— 0 0 )
=
0,
= 0.
(2.66) (2.67)
We observe t h a t i f 2
— a *
u (t) r
= e
2
,
t h e n b o t h the given conditions of Eq.2.66 and Eq.2.67 are satisfied and
moreover
Chapter
48
2:Least
Squares
Approximation
of
Signals
e
df is a p o l y n o m i a l of degree r. I t follows, therefore, t h a t « (t) =
(2.68)
de
r
r
is a solution of Eq.2.65, and the associated Eq.2.24 w i t h d = ( - l ) are
o r t h o g o n a l p o l y n o m i a l s expressed by
r
T
H {at)
=
r
T
1
d uJt) —
r
= (-l) e
dr
w{t)
Q
2,2 d '
T
(2.69)
dr 2
w h i c h is the Rodrigues' f o r m u l a for the r t h degree H e r m i t e p o l y n o m i a l i f a — 1, (sometimes a = 1/2 is also used) and is designated by H (t). I t follows f r o m Eq.2.1 that 2
T
H {at)H (at)dt m
m ± r,
= 0,
T
where m and r are nonnegative integers. We note t h a t the Taylor series expansion of a function / ( x ) is T
d f(x) dx
T
and t h a t d - a - f ( a t - x ) ox (-a)
—/{at ax T
=
- x)
=
d dt
—f(at-x)
—- •f(at dr
-
x)
Hence i t follows from Eq.2.69 t h a t
H (at) T
=
(-l)'e- ' — e at c*x
r
( 2 Q I ( - I
3
)
ox =
(a) r\ times the coefficient of x a
x
r
in the a
Taylor series expansion o f ( ° ' - * ) e
2.5 Hermite
49
Polynomials
Now 2
=
xt
2
e ° e-* y , (2axt)"
2
~
(-x )*
fc'
D'
_
=o
P-
P
,To*T„
For a fixed value of k, we o b t a i n x
T
K
k=o
-
? w
by s e t t i n g p + 2k = r when
r
k
coefficient of x
=
( — 1) r
for t h i s value o f k. T h e t o t a l coefficient of x
(2ai) -" ( r - 2fc)!fc!
is o b t a i n e d by s u m m i n g over a l l allowed
values o f k. Since r — 2k = power of 2o;i should be p o s i t i v e , k can vary f r o m 0 t o [ r ] , see Eq.2.50. Hence
coefficient of x
T
r
M T\{-\)
(2at) ~
k
=
2k
and
*'<-°
=
g
n
(
-
1
( r - a W .
}
( 2
-
7 0 )
C o m p a r i n g t h i s p o l y n o m i a l w i t h Eq.2.29, we identify, 2
/J = ( 2 a r , a r
=
r
0, r
r
1
=
)
2
(2a r'
(2.71)
W i t h /? as o b t a i n e d above, we use Eq.2.35 and Eq.2.68 to c o m p u t e y r
r
oo
/ =
w h i c h is
2 r
_
( - l ) e ° ' di
2
(2a )V!— a
(2.72)
W i t h t h e help of Eq.2.30, Eq.2.34 and Eq.2.33 we c o m p u t e respectively a
r
=
2
2
£ ± i = 2 a , 6, = 0, c = ft-
-2a r
r
T h e t h r e e - t e r m recurrence f o r m u l a for t h e H e r m i t e p o l y n o m i a l s H (at) T
now follows
f r o m Eq.2.28 and is given b y H {at) r+1
=
2
2a tH (at) T
- 2o?rH _ {at) r
i
(2.73)
50
Chapter
2:Least
Squares
Approximation
of
Signals
Table 2.3: H e r m i t e Polynomials
ffo(0
1
Bi(t) Hi{t)
2t At
2
3
H (t)
8t
3
- 2 -
12t 2
16<" - 4 8 i + 12
H,{t)
5
3
32« - 160r + 120< 6
2
6 4 t - 4 8 0 f + 720E -
He(t) H (t)
7
5
120
3
1 2 8 i - 1344« + 3360r -
7
1680*
T h e first eight H e r m i t e p o l y n o m i a l s generated w i t h the help of Eq.2.69 and Eq.2.73 are i n c l u d e d i n Table 2.3 . T h e n t h order a p p r o x i m a t i o n of an a r b i t r a r y f u n c t i o n / ( ( ) over ( — 0 0 , 0 0 ) in a series of the H e r m i t e p o l y n o m i a l s H (at)
is
T
n
/
=
n
£/ ff,(ai)
(2.74)
r
r= 0
where i n view of Eq.2.15 and Eq.2.72
e
=
a
2
or 2r \ , r- r ~ *' 2Tec" r!"V7T -oo
t{t)H {at)dt
(2.75)
r
J
T h e M I S error is given by Eq.2.16 w h i c h in the present case is
/ where i t is assumed t h a t | f(t)
°°
e'
a
22 ' f (t)dt 2
71 2
-
r
2
2
\ grows less r a p i d l y t h a n e " ' '
I t w i l l now be shown t h a t the H e r m i t e p o l y n o m i a l s H (ctt) T
differential equation.
2
v ^ E / (2 a -'r!)
2
(2.76)
as t tends t o infinity. satisfy a second order
To prove this, we differentiate Eq.2.68 w i t h respect t o t and
rearrange to get
2
i i ( r ) + 2ct tu (t) r
=
r
0
T h i s equation is differentiated again to give us 2
« ( i ) + 2a tu {t) r
T
2
+ 2a u (t) r
=
0
2.6 Tchebycheff
Polynomials
of the First
Kind
51
D i f f e r e n t i a t i n g t h e above e q u a t i o n r times by using the L e i b n i t z f o r m u l a Eq.2.47 we get
u
r + 2 T
2
r+1
2
\ t ) + 2a tu \t)
r )
+ 2a (r
T
+ l)u< (i)
=
0
w h i c h , after s u b s t i t u t i o n f r o m Eq.2.69, takes the f o r m
^[ -^HAat)}
+
e
2 0 ^ - ^ ( 0 , ) ] 2 a ( r + \)e-"
+
2
2>2
H {at)
= 0
T
A f t e r s i m p l i f i c a t i o n , t h e differential e q u a t i o n satisfied by the H e r m i t e p o l y n o m i a l s assumes t h e f o r m , 2
d —H (at)-2a t dt
, d — H (at) dt
2
2
2.6
r
2
+ 2a rH (at)
T
r
=
0
Tchebycheff Polynomials of the F i r s t
To a p p r o x i m a t e any f u n c t i o n f(t) k i n d we t a k e w(t)
2
= 1/\f{\
—t )-
(2.77)
Kind
over [—1, 1] by Tchebycheff p o l y n o m i a l s of the first Since w(t) assumes very h i g h values as t tends t o ± 1 ,
i n such an a p p r o x i m a t i o n t h e errors at the ends of the i n t e r v a l are made very s m a l l . To generate t h e o r t h o g o n a l p o l y n o m i a l s w i t h the w e i g h t i n g f u n c t i o n and i n t e r v a l as defined above, w h i c h w o u l d a p p r o x i m a t e any a r b i t r a r y f u n c t i o n i n t h e least squares sense we s h a l l , t o avoid c o m p l i c a t i o n , employ Eq.2.20 and n o t Eq.2.25 as done i n the foregoing cases. T h u s we get 2
i/2
(1 - t )- 4> (t)qr-i(t)dt
= 0
r
(2.78)
1
P u t t i n g t = cos 8 we have f r o m above 4> {cos9)q _ {cos9)d9
i: T h e p o l y n o m i a l q -i(t) r
q _ (cos8) T
l
r
r
= 0
i
(2.79)
is of degree at most r — 1. Therefore we have =
ct + cti cos 9 + a
=
ct
Q
0
+ a
y
2
2
cos 9 +
cos 8 + 0 . 5 o ; 2 ( l + cos 29) + • • •
• cos(r - 1)8] =
a
0
+ a.i cos 9 + a cos 29 + • • • 2
+ a _ ! cos(r - 1)9 r
(2.80)
Chapter
52
2:Least
Squares
Approximation
of
Signals
I n view of the o r t h o g o n a l properties of sine-cosine functions, see E x a m p l e 2 . 1 , to satisfy Eq.2.79 i t is obvious t h a t we must have (cos 8) = d cos rO r
r
and we can choose {4> (t) = d cos(r c o s r
d
r
- 1
<)} as the set of o r t h o g o n a l p o l y n o m i a l s . I f
T
= 1, these p o l y n o m i a l s are k n o w n as Tchebycheff p o l y n o m i a l s of the first k i n d of
degree r and are designated by T (t)
= cos(rcos
T
_ 1
t)
(2.81)
I t is evident t h a t 2
f J
1/2
(1 - t )~ T {t)T (t)dt m
= f cosm8 o
r
cos r8d9 = 0 , m y£ r
J
-i
where we have set t = cos 9. I f m = r , we get
2
= f
7 r
i/2
2
( 1 - t )~ T (t)dt
=
f"
cos
2
r8d0,
TT/2,
(2.82)
r # 0
To generate the three-term recurrence f o r m u l a , for r > 1 we note t h a t cos(r + 1)8 + cos(r — 1)9 = 2 cos r 0 cos 8 We now p u t t = cos 9 and substitute from Eq.2.80 to get T (t)
= 2tT (t)
T+1
- T _,(t)
r
(2.83)
r
w h i c h is the three-term recurrence f o r m u l a for the Tchebycheff p o l y n o m i a l s of the first k i n d . P u t t i n g r = 0 , 1 , . . . i n Eq.2.81 and Eq.2.83 we can generate recursively all the Tchebycheff polynomials of the first k i n d , the first eight o f w h i c h are included in Table 2.4 T h e Rodrigues' f o r m u l a for X ( t ) ' s is given in Section 2.9. Tchebycheff polynomials of the first k i n d satisfy a second-order differential equat i o n . To derive i t , we note t h a t i f T = cosr9,t = cos 8 t h e n r
T
2
2
2
r
=
, „ dt dt - r c o s r S = - r T ; — = -sintf, = - cos 8. d8 d8 2
r
2
V
(2.84) '
Also we have the i d e n t i t y d d8
dT (t) r
dt
dt d8
2
d T {t) r
(dt\"
~d?~\de)
x
2
dT (t) r
+
2
^r~de ~
2
dt (
2
8
5
)
2.6 Tchebycheff
Polynomials
of the First
53
Kind
Table 2.4: Tchebycheff Polynomials of F i r s t K i n d
1 t
T (t) 0
2
T (t)
2t
Ti(t)
At - 3<
2
- 1
3
T (t)
2
8t* - 8t
4
5
+ 1 3
16z - 2 0 t + St
m ) T (t)
6
2
3 2 i - 48t* + 1 8 i - 1
6
7
5
3
6 4 t - 112< + 5 6 i -
It
S u b s t i t u t i n g f r o m Eq.2.84 i n t o Eq.2.85 we get 2
2
-r T (t) T
V
d T {t) , i-t-sine dt
dT {t) — cos 6 dt
r
=
;
T
2
Hence we o b t a i n 2
, d TJt) (1 - t )—±± dt' 2
- t
dT (t) — + dt r
, r T (t) 2
= 0
r
(2.86)
w h i c h is the differential equation satisfied by the Tchebycheff p o l y n o m i a l s of the first kind.
T h e n t h order a p p r o x i m a t i o n of an a r b i t r a r y f u n c t i o n f(t)
over [—1,1] as a
series of Tchebycheff first k i n d p o l y n o m i a l s is described by /(*) » » ( 0 m
=
£
/ T (i) r
(2.87)
r
where the p r i m e signifies t h a t the first coefficient is / / 2 and the others are as usual. 0
T h e coefficient f
r
is expressed by fr
=
2 - J (1 TT
J
2
=
2
l/2
t y f(t)T (t)dt r
- l
-
r
f(cosO)(cosr9)dB
(2.88)
7T *B
w h i c h follows f r o m Eq.2.15 and Eq.2.82. T h e error i n the a p p r o x i m a t i o n is oo
e„(*) = / ( * ) - * « ( * ) =
£
/rT (0
and since T ( i ) = cos(r0) we have r
oo
I «.(*)!< E l/r|.
r
54
Chapter
2:Least
Squares
Approximation
of
Signals
T h e relations of Eq.2.87 and Eq.2.88 show t h a t the representation of an even f u n c t i o n i n a series of Tchebycheff first k i n d p o l y n o m i a l s T (t) T
function
2.7
is the same as the Fourier cosine
expansion.
Tchebycheff Polynomials of the Second K i n d
T h e Tchebycheff p o l y n o m i a l s of the second k i n d arise in a p p r o x i m a t i n g an a r b i t r a r y f u n c t i o n over the i n t e r v a l [—1,1] w i t h a w e i g h t i n g f u n c t i o n w(t)
2
1
2
= ( 1 — t ) ' . There-
fore Eq.2.20 can be w r i t t e n as
l
/
(1 -
t ^ ^ r W g r - ^ d t
-1
=
2
f
<£ .(cos0)g _ (cos0)sin 0s'0 = 0 r
r
(2.89)
1
w i t h t = cost?. Eq.2.89 is satisfied i f 4> (cos 0) : = s i n ( r + 1)0/ sin 0, r
since sin
0<£ (cos 0) = - [cos rO - cos(r + 2)0] r
w h i c h is o r t h o g o n a l t o <2 _i(cos0) represented by Eq.2.80 over [0,TT]. T h e p o l y n o m i a l r
is defined t o be the Tchebycheff p o l y n o m i a l of the second k i n d of degree r
and is designated by U (t).
Thus
r
s i n ( r + 1)0 <7 (t)
=
r
,cos0 = *
sin 0
(2.90)
We recall t h a t T (t) T
= cosr0,cos0 = t
Hence, differentiating w i t h respect to t we get •. , , T (t)
=
r
-rsinr0—=
dO
r sin rO
dt
sin 0
= rt/ _,(t)
(2.91)
r
On the other h a n d , differentiating Eq.2.90 w i t h respect to t we o b t a i n •
( r + l ) s i n 9 c o s ( r + 1)0 - c o s g s i n ( r + 1)0 sin
0
-1 sin 0
Hence i t follows t h a t 2
(1 - t )U (t) r
= tU (t) r
- (r + l ) T
r + 1
(0
(2.92)
D i f f e r e n t i a t i n g Eq.2.92 w i t h respect to t and s u b s t i t u t i n g from E q . 2 . 9 1 , we get 2
(1 - i ) ( V ( i ) - 3tU (t) r
T
+ r ( r + 2)U {t) r
= 0,
(2.93)
2.7 Tchebycheff
Polynomials
of the Second
55
Kind
a second-order differential e q u a t i o n for Tchebycheff p o l y n o m i a l s of the second k i n d . I n the i d e n t i t y s i n ( r + 1)8 + s i n ( r - 1)8 = 2 sin rd cos 8 i f we s u b s t i t u t e f r o m Eq.2.90 t h e n there follows the t h r e e - t e r m recurrence f o r m u l a for U (t) w h i c h is given b y T
U (t)
- 2W (t)
r+1
-+• t ^ - i ( t )
r
=
(2.94)
0.
M a k i n g use of Eq.2.90 and Eq.2.94, these o r t h o g o n a l p o l y n o m i a l s can be c o m p u t e d , the first eight of w h i c h are i n c l u d e d i n Table 2.5 . These p o l y n o m i a l s can also be
Table 2.5: Tchebycheff Polynomials of Second K i n d
1 2t
U (t) Ux(t) U (t) 0
2
At - 1
2
3
u (t)
8 « - At
U (i) u (t) Ue(t) U (t)
16< - 1 2 t + 1
3
4
t
2
5
3
32i - 32i + 6i
t
6
4
6 4 i - 80< + 2At 7
2
5
- 1 3
1 2 8 I - 192r + 80E - 8t
7
generated by R o d r i g u e s ' f o r m u l a given i n Section 2.9. T h e o r t h o g o n a l relations for U (t) T
can be easily verified. T h u s f J
2 1/2
_i
( 1 - t ) U {t)U (t)dt m
7
r
=
= f s i n ( m + l ) 0 s i n ( r + \)8d8 •'o
T
2
J
1/2
2
/ ( 1 - t ) U (t)dt -i
A n a r b i t r a r y f u n c t i o n f(t)
2
= f" s\n {r o J
+ \)8d8
= 0 , m ^ r;
= TT/2
(2.95)
over [—1,1] can be a p p r o x i m a t e d in a series of Tchebycheff
p o l y n o m i a l s o f the second k i n d f/ 's given by r
(2.96) where f
r
=
- f
=
-
(i -
1/2
r
2 -7T
2
t ) f(t)UAt)dt
/ ( c o s 8) s i n ( r + 1)8 sin n
8d8
(2.97)
Chapter
56
2.8
2:Least
Squares
Approximation
of
Signals
Jacobi Polynomials
For Jacobi p o l y n o m i a l s we choose [*o,«/l =
[ - M l .
and w
(t)
=
(1 - <)°(1 + tf-a
> -1,0
> - 1 .
(2.98)
To compute the o r t h o g o n a l polynomials 0 ( t ) ' s given by Eq.2.24, we have to solve r
r
1 dt^
1
d u {t) r
w(t)
=
0
(2.99)
dV
subject to the conditions u (-l) T
r
=
u ( - l ) = ••• = u < - ° ( - l ) = 0
=
u ( l ) = • • • = u[ - (\)
r
(2.100) u (l) r
r l)
= 0
r
2
Based on our discussion in Section 2.3, we conclude t h a t u (t) w o u l d contain (1 — i ) as a factor. Moreover, to satisfy the Rodrigues' f o r m u l a , v i z . ,
r
T
4>Jt) v
(1 - « ) ° ( 1 + tY
w h i c h is a p o l y n o m i a l of degree r, u (t) r
=
-
d u (t) — T
dr
(2.101)
should be of the form
T
u (i)
T
1
=
;
2
d (l -
+ 0"(1 - * ) '
r
(2.102)
since in this case, the r i g h t hand side of Eq.2.101 w o u l d be a p o l y n o m i a l of degree r after r times of differentiation as can be verified by L e i b n i t z f o r m u l a Eq.2.47. Let us write v(t)
= u (t)
r +
=
r
d (l - <) °(l +
(2.103)
r
T h e o r t h o g o n a l polynomials 4> (t) in Eq.2.101 are defined as the Jacobi polynoT
mials of degree r and are represented by
if d = r
(-l) z r!
r K
—r~
(2.104)
2.8 Jacobi
57
Polynomials
T h e Rodrigv.es'
formula
for Jacobi p o l y n o m i a l s is therefore given by
2"r!
(1 -
i)P
+
^
dt
T
J
I n v i e w of E q . 2 . 1 0 1 , Eq.2.103 a n d the L e i b n i t z f o r m u l a of Eq.2.47, we have the relation r
r
2 r!(l - i)°(l + t f P ^ \ t ) =
( - l )
r
£
m
C) ^
[(1 + 0
= (-l) D 1
r +
r
r +
[(1 + i )
[(1 -
t)
r +
r+a
"(l -
t) ]
"]
m = 0
=
r
("l) £
( L ) ( r + 0){r
+ /? - 1) • • • ( r + /? - m + 1)(1 + « )
r +
"-
m
m = 0 r
m
( - l ) ~ ( r + o;)(r + a - 1) • • • ( r + a - r + m + 1)(1 a
Hence t h e J a c o b i P o l y n o m i a l s P^ ^(t) r
t + l \
« -
are represented by
t
(fi
a + m
( a + 1) • • • (a + m)(a + m + 1 ) • • • (a + r )
m (
t)
(a + 1 ) • • • (a + m)r!
+ r - m + ! ) • • •(/? + ;•)
1
f E - 1
(—y V
m!
'
(
r
^ -
r
(r - m ) ! E + 1 '
_ i ) . . . (
( +l). Q
r
_
m
+
i)
-(a + m)
/ i U
n -
' W+ l '
t - ) ( ! ± l ) ' F ( - , ^ - f t .
+
l
!
(2.105)
£ ^ )
where F(a,6;c;r)
:=
(
£ m
=
0
~
m
a
1-2
(a)
m
(a)o
m
(2.106)
••••m
1) 6 ( 6 + ! ) • • - ( 6 + m - 1)
m
c(c + 1) • - • (c + m - 1)
1 + ~
(-J»)(l)
)
( a + 1 ) • • - (a + m -
a ( q + ! ) • • - (a + m - 1) 6 ( 6 + 1) • • - ( 6 + m - 1)
' ~ F
6
m
~~ =
(
°\ "« (c) m!
=
1-2
m
m
c(c + 1) • • • (c + m - 1)
(;+»)
=
a(a + 1) • • • (a + m - 1)
=
1
T h e f u n c t i o n F ( a , 6 ; c ; t ) is defined as the h y p e r g e o m e t r i c f u n c t i o n . r e p r e s e n t a t i o n of J a c o b i p o l y n o m i a l s is derived in Section 2.10.
(2.107)
A n alternative
58
Chapter
2.9
2:Least
Squares
Approximation
G a m m a , Beta andHypergeometric
of
Signals
Functions
To derive an a l t e r n a t i v e representation o f Jacobi p o l y n o m i a l s we must k n o w w h a t are G a m m a and Beta functions. A brief description of these functions w i l l now be given. T h e G a m m a f u n c t i o n T ( z ) is defined by the Euler's i n t e g r a l oo 1
e"'t*" dc,Re z > 0
/
(2.108)
u
I n t e g r a t i n g by parts we get r(z + l )
z
=
/
e'*t dt,
=
zT(z).
(2.109)
O b v i o u s l y for an integer m we can easily derive t h a t T ( m + 1)
=
m!
(2.110)
T(a + m )
=
(a + m - l ) T ( a + m - 1)
=
(o + m - 1) • • • o r ( a )
=
(a) T(a)
(2.111)
m
where a is any number, real or complex and we have used the r e l a t i o n Eq.2.107. For two numbers a and b p r o p e r l y restricted as i n the definition of a G a m m a function, the p r o d u c t of t w o G a m m a functions T ( a ) and T(6) is foo
oo
(i
/
/
u
e-
+
T)
1
t"- T'-'dtdT
•'O
Setting t = x ;r = y , 2
2
where x and y are two a u x i l i a r y variables, the above p r o d u c t becomes
foo
oo
/ u
(
/
e" * +» V ' - y - ' d x d y
•'o
I n t r o d u c i n g another variable 0 defined by x = r cos 9,
y = r sin 6
and replacing the elemental area dx dy w i t h r dO dr, we get oo
/
u
/-ir/2
/ •'O
e-
r
r
2 a + 2 4
2
- cos
2 a
-' 9sin
2 4
- 0rd0dr
2.9 Gamma,
Beta and Hypeigeometric
Functions
59
F i n a l l y using a last change of variable by s e t t i n g u = r the f o r m
r(o)r(6)
2 r
=
u
a+b
r •* n
l
e- u - du ft
J
the above p r o d u c t assumes
2
2
cos -
1
2
6 s i n ' - ' 0 d6 (2.112)
T h e second i n t e g r a l o f the above expression is, in fact, equal to
2
2/ where t =
- 1
2 6
1
c o s " 9 s i n - Bd8
=
/
(1 - t )
a _ 1
i
t _ 1
(2.113)
2
sin 8.
T h e B e t a f u n c t i o n is defined by
B{a,b)
=
l
f
k
l
( 1 - t y t - d t , Re a > 0, Re 6 > 0.
(2.114)
Therefore the relations of Eq.2.108 and Eq.2.11 l-Eq.2.113 give us l
k
i
f (i-ty- t - dt
=
[
t—\\
-
tf-'di
0 =
5 ( 6 , a)
r(q)r(6)
(2.115)
r(a + 6) T h e h y p e r g e o m e t r i c f u n c t i o n F(a,b;c;z)
rpf
i
\
f-
can be w r i t t e n as, see Eq.2.106
(«)»(*) m!(c)
m = 0
m
r(c) S Z^r E r(a)
m = 0
/ \ T(a + m ) „ ("') - l ) ^ f(c + m) k
m
m
(2.116)
w h i c h follows f r o m E q . 2 . I l l and the fact t h a t
(->) ( - i r = -
6
M
-
1
}
- ! -
6
-
r
o
+
1
)
( - i r
=
I n view o f Eq.2.115, we have T(a + m ) T(c + m )
1 — / r r ( c - a) •'o
+
m
-'(i -
c
t) -°-'dt,
^
Chapter
60
2:Least
Squares
Approximation
of
Signals
w h i c h when inserted i n Eq.2.116 gives us
F(a,6;c; ) 2
I
=
f
1
°° / \ - t ) — £ ("') e
- /
r(c)
/
r ( o ) r ( c - a) Q
1
l
c
( - z ^ d t
_ 1
t — ( l - < ) - ° ( l - zt)-'
J
(2.117) P u t t i n g z = l , and then using Eq.2.115, we get
.,, , F(a,o;c;l)
=
T(c)
r(a)r(c-6-a)
r ( a ) T ( c - a)
r(c - 6)
r(c)r(c-
b-q)
r(c - a)r(c - 6) I t w i l l now be proved t h a t the hypergeometric f u n c t i o n F(a,b;c; pressed by F(o,6;c;z)
=
(1 - z)~° F ( a , c - b; c; -
z) can be ex-
)
(2.118)
For this purpose, the following identities w i l l be useful: (-o)(-o-l)---(-o-n+l)
n
& - * ) - '
=
£
( a ) i ( a + k)
=
a(a + 1) • • • (a + k - l ) ( a + A;) • • • (a + k + n - 1)
=
(a)*+„
n
( r ) (-*)" = £
o ( a + 1) • ' • (a + n - 1)
^
(2.120)
D e n o t i n g the r i g h t hand side of Eq.2.118 by X, Eq.2.106 and E q . 2 . I l l we have
~
^
(a) (c-b)„
„
k
— 7 T T < — (c-6) (a) t
(2-119)
t +
z
and by using the above relations,
k
z
»(-l)*g*
— I — + n
z
2.10 Alternative
Representation oo
=
of Jacobi
Polynomials
61
m
i , ..,
1^1^ m = 0
J
0
=
(
C
) i J
!
777— z
( " l
-
= J > 0,n = m - j > 0
i)!
ff(-*)j(«-»M^' (
= 0;=0
=
£
C
TO!
W
F(-m,(c-6);c;l) m
m = 0
y\ m = o
m
=
2.10
r(c)r(6 + m ) ( a )
r(c+ )r(6)
(
c
2
!
M
m!
m
o
=
R O
!
) m ™
F(a,6;c;z)
(2.121)
Alternative Representation of Jacobi Polynomials
W e now make use o f Eq.2.105 and Eq.2.108 to w r i t e P™«)
=
( r
=
(;
a
+
a
) ( ^ ) ^ ( - » , - ^ n
;
l
a ~ )
+
r
) i ( - n , l + a + ^ + n;l +
(2.122)
a ; ^ )
T h e Jacobi p o l y n o m i a l s Eq.2.105 can be w r i t t e n as f
= r i
( l + « ) , ( ! + /?),
^ T ' m ! ( r - m ) ! ( l 4 - o ) ( l + /3) _ 0
;
m
r
m
/ t - n - z t
+ i y
V
2
2
'
v
' (2.123)
Hence we o b t a i n ~ r
P <°^(r>
r
r
t ; ( l + a ) ( l + /3) r
oo
r
to to
f i_
m
y j y j i j
t
r
~
-(
a
r
.
+ « U 1 +
) * b J l a l f
n!(l + a ) .
l \ r — m - r
,j = r - m > 0 , n =
m > 0
(2.124) L - . i ' ( i + /»)iJ
62
Chapter
2:Least
Squares
Approximation
of
Signals
w h i c h is k n o w n as Bateman's generating function for the Jacobi p o l y n o m i a l s . I n the above i d e n t i t y , i f t is replaced by —t and x by — x we have Pl"^\-t)x
T
(1 +a) {\
+ /?)„
T
r!(l +
T = 0
a)
(2.125) r
Hence c o m p a r i n g w i t h Eq.2.124, we find t h a t the right hand side of Eq.2.125 is the same as the r i g h t hand side of Eq.2.124 b u t w i t h a and /3 interchanged. Therefore it follows t h a t r
(-l) P
( a
'f'(-Z)
r
=
p ^ " \ )
(2.126)
t
and from Eq.2.122 we o b t a i n pO*J>\ ) t
r
(
"'
o )
=
(-l) F
=
(-l) (: ^)F(-r,l +
r
r
(-0
+
+ /3 +
Q
r
r ; l + / ? ; ^ )
m
J2 ( - l ) ( / ? + l ) ( - l ) ( r - m + l ) r!m!(l + /J) r
m
m
(1 + y
(^p)
+ /3 + r)
a
m
(-ir"
m
( l + / ? ) „ ( ! + a + (3)
ft + I V
r+m
^ m ! ( r - m ) ! ( l +/3) (l + a +/?) 0
m
^
r
2
' (2.127)
(1 + a + / J ) ( l + a + 0 + r ) r
m
= (1 + a + /?)
r
Similarly pM(t)
=
t+a)
F ( - r , l + a + / i+
r ; l + o : ; ^ )
(1 + o ) ( - i r ( r - m + l ) r
m
(-1)" (i + « ) »
^ " i
1 + a) (l + a + /3) r
„=o
m
!
(
r
-
' ft - 1 \ *
r + m
- m ) ! ( l + o ) ( l + a + /J) V m
r
2
* (2.128)
I d e n t i f y i n g the coefficient o f f i n t h e above p o l y n o m i a l , we differentiate i t r times to get
2.10 Alternative
Representation
of Jacobi
63
Polynomials
(1 + a ) ( l + a + / ? ) r
T
D P^\t)
2r
2 ' r l ( l + a ) ( l + a + /J) r
( l + a + /?)
r
2r
(2.129)
r
2 ( l + a + /?), C o n s i d e r i n g the value of m = r on the r i g h t h a n d side of Eq.2.128 we get 0
T
=
T
coefficient of t (1 + a +
0)
tT
r
2 r!(l + a +
(2.130)
0)
r
A g a i n considering the value of m = r and m = r— 1 on the r i g h t h a n d side of Eq.2.128 we get a
r
=
T
coefficient of t (1 + a + 0)
r
2r
r ! ( l + a + 0)
+
r
T
2
(1+ «),(!
+ a +
1
(r - 1)!(1 + a ) _ , ( l + a + /?), 2 ' -
1
r
(a - / ? ) ( ! + a +
0) _, 3r
(2.131)
r
2 ( r - 1)1(1 + a + /?)
r
S i m i l a r l y considering m = r, ( r — 1), and ( r — 2) on the r i g h t h a n d side o f Eq.2.128 we o b t a i n TV
=
coefficient of t ( l + c + /3)
r _ 2
r(r-l)
2 r
r
2 r ! ( l + o + /3)
2
r
(1 + a ) ( l + q + / ? ) _ r
2 r
1
(r-1)
I
2 ' - ( r - 1)!(1 + a ) _ , ( l + a + 0% r
(1 + q ) ( l + q + / 3 ) _ r
r
2 r
2
2
2 - ( r - 2)!2!(1 + a ) _ ( l + a + 0) r
(1 + a + / 3 ) 2 r
2
T
2
r
2 - » ( r - - 2)!(1 + a + /?) (Q
+
r
0 + 2 r ) ( a + /? + 2r - 1)
(a
+
r){0 + r) (2.132)
Chapter
2:Least
Squares
Approximation
of
Signals
2.6: Coefficients of T h r e e - t e r m Recurrence F o r m u l a of O r t h o g o n a l Polynomials: + b )4> {t) +
t * + i ( * ) = (a t r
(a)
T
T
Crfc^'t)
r
Legendre
Polynomials:
to = " I
tj = 1
Pr{t)
6 = 0 P
00
Laguerre
Polynomials:
L (Ql)
io = 0
tj = oo 6 _ ('+"•)
i u ( t ) = exp( - a t )
Polynomials:
r = -(7+7) tf (at)
t
i / = oo b = 0
= exp( - a t ) c = —2a r
Polynomials Kind: t, = 1
w(t)
= -
6
c =
-1
= — oo
0
a (d)
r
= 2a
2
r
Tchebycheff of F i r s t to = - 1 l , a = 2(r > 1) a Tchebycheff r
0
(e)
of Second
r
0
Jacobi to
= -1 =
(g)
r
tf (t) w(0 = v/(i - < )
b = 0
c =
Polynomials:
rwi(tj
t, = 1
i»(<) = (1 - t ) ° ( l + i ) " _ (o.+r)OH-r)(
M»-0 cr = A + 2r,
Gegenbauer
Polynomials:
to = - 1
= 1 6 = 0
'
2
Kind:
A= 1+ a +
M
2
2
r
tj = 1
olo+il 2
r
Polynomials
T
(f)
C
Hermite
(r+1)
(c)
r
r
2
r
-1
/i = (r + l)(A + r ) 2
r
u,(t) = ( i - 1 ) — ? _ 2./+r-l ~ r+1 r
2.10 Alternative
Representation
of Jacobi
65
Polynomials
N o w , we use Eq.2.35 along w i t h Eq.2.103, Eq.2.104 and Eq.2.130 to get
r
J
2 r\
-\
U s i n g a new variable r defined by 2 r = t + 1, we have 22r + a + / 3 + l
Q
f t / '+/»(i _ •'o
IT
r
r
+
a
+
2
r
i
+
f
i
0
r
)'+°dr
r ( l + q + r ) r ( l + / 3 + r) r ( 2 + a + 3 + 2r)
+l3+l
2" T(l
+ g + r)T(l
+ 0 + r ) ( l + a + /?)
2r
r ! ( l + a + /9) T(2 + a + 0 + 2 r ) r
£ , +
+ 1
2 " r ( l + a + r ) T ( l + 0 + r) r ! ( l + a + 0 + 2 r ) T ( l + a + 0 + r)
(2.133)
S u b s t i t u t i n g f r o m Eq.2.131-Eq.2.133 i n t o Eq.2.30, Eq.2.34 and Eq.2.33, we get (1 + a + 0 + 2 r ) ( 2 + a + 0 + 2 r )
(2.134)
2 ( r + 1)(1 + a + 0 + r) K
=
+
«
+
+ 2 0
(
2
.
1
3
5
)
2 ( r 4- 1)(1 + a + /S + r){a + 0 + 2 r ) C
( a + r ) ( / J + r){2 + a + 0 + 2r) =
T
( r + 1)(1 + a + 0 + r)(a
(2.13D)
+ 0 + 2r)
respectively, w h i c h w h e n p u t in Eq.2.28 gives 2 ( r + 1)(1 + a + 0 + r ) ( « + 0 + 2 r ) P =
(
+
f'(t)
( 1 + a + 0 + 2 r ) [ ( a + /? + 2 r ) ( 2 + a + 0 + 2r)t + ( a P^^it)
the three-term
r
- 2 ( a + r)(/3 + r ) ( 2 + a + 0 + 2r)P^f
recurrence
formula
'(<).
2
2
- /J )] (2.137)
for Jacobi p o l y n o m i a l s .
T h e n t h degree Jacobi p o l y n o m i a l a p p r o x i m a t i o n of an a r b i t r a r y f u n c t i o n
f(t)
over [—1,1] is represented by
/tfl
=
n £
f r P ^ K t )
(2.138)
r= 0
where i n view of Eq.2.15 a n d ST
=
Eq.2.133 r ! ( l + a + 0 + 2 r ) f ( l + a + 0 + r) 2 + ° T ( l + a + r ) r ( l + j9 + r ) 1
/
+
a
0
(1 - i ) ( l + t) f(t)P^\t)dt
(2.139)
Chapter
66
2.10.1
Jacobi Differential
2:Least
Squares
Approximation
of
Signals
Equation
Differentiating Eq.2.103 once w i t h respect to t and rearranging we get 2
(1 - t )v(t)
+ [(a - 0) + (a +0
+ 2r)t]v(t)
= 0
Differentiating again, we get 2
(1 - t )v(t)
+ [(a - 0) + [a + 0 + 2r - 2)t\v{t)
+ (a + 0 + 2r)v{t)
= 0
Using the L e i b n i t z f o r m u l a i n Eq.2.47, we differentiate the above equation r times to get (1 - « V
+
2
)
(0
+ [(a + 0 -
1)
2)t + (a -
0)}v^ (t) (r
+
[(a + 0 + 2r) + r(a + 0 + 2r - 2) - r(r - l)]v \t)
I n this equation, we replace v(t)
(1 -
•e_
t * ) £ -
= 0
by 4>r(t) using Eq.2.101 and Eq.2.103 to get
[(1 - ! ) - ( ! +
t )
0
M t ) }
2
'dt + +
d [(1 - * ) " ( 1 + tfMt)} dt [(a + 0 + 2 r ) + r{ct + 0 + 2r - 2 ) - r(r - 1)]
{( -0) a
+ {a + 0 -
(i -
2)t\-
+ t?d>
After s i m p l i f i c a t i o n , the above equation gives rise to the Jacobi differential equation described by 2
0 +
, d 4>Jt) ^ — l j t ~ ^ -{{a-0) dt'
da>Jt) + {2 + a + 0)t] dt
r ( l + a + 0 + r)4> (t) = 0
(2.140)
r
Assigning p a r t i c u l a r values to a and 0 we get (a) Legendre differential equation: a = 0 = 0 (b) Tchebycheff differential equation of the First K i n d : a = 0 = — i (c) Tchebycheff differential equation of the Second K i n d : ct = 0 =
|
Moreover, as a result of Eq.2.101 and Eq.2.102, the Rodrigues' formulae for generating the Tchebycheff p o l y n o m i a l s of the first k i n d T ( i ) ' s and the Tchebycheff polynomials of the second k i n d f / ( t ) ' s take the f o r m r
r
2
1
2
T,(*)
=
d,(l - t ) ' ^ -
tf (<)
=
«*r(l - t ) ~
r
2
1 / 2
2
[(1 - i y 2
^ - [(1 - r )
1
/
2
]
T + I / 3
(2.141) ]
(2.142)
2.11 Gegenbauer
Polynomials
67
where <
A for the p o l y n o m i a l s T (t)
_
i
r
and
T
r
( - l ) ( r + 1) 2'(f), for the p o l y n o m i a l s
U (t). T
I n view of the expression i n Eq.2.104 for d
of Jacobi p o l y n o m i a l s , i t is easy to
r
verify t h a t r! TJt)
=
(--
-i
2
-r—Pr
1
( r + 1)!
(t)
(1,1)
t 2 Jr
I n t r o d u c i n g proper scaling factors, the coefficients / 3 , a r
and r
r
r
of Tchebycheff p o l y -
n o m i a l s can be c o m p u t e d by using Eq.2.130, E q . 2 . 1 3 1 , and Eq.2.132
respectively.
I n d e e d , for Tchebycheff p o l y n o m i a l s of the first k i n d we have a + (3 = —1 and hence, „
_
(l + a + Phr
r!
=
2
r
_
1
2V!(l + a + /3) (i) r
Q r
=
T
0,
=
T
r
-r2 -\
S i m i l a r l y , for Tchebycheff p o l y n o m i a l s o f the second k i n d , since a + 0 = 1, we have
ft-
=
( g j g , ( r + 1)! S . ' l '„Y 2'H(2) (f), V
r
a
2.11
r
=
0,
r
=
-(r
T
-
T
1)2 ~\
Gegenbauer Polynomials
T h e Gegenbauer p o l y n o m i a l s are a generalization of the Legendre p o l y n o m i a l s a n d are denoted by C"(t)
where r is an integer and u is a real n u m b e r . These p o l y n o m i a l s
are defined by the r e l a t i o n
oo 7
(1 - 2tx + x ) ' "
= £ r= 0
c;(t)x
r
(2.143)
Chapter
Table 2.7: Coefficients ce ,/3 ,f r
T
and r
T
2:Least
Squares
Approximation
of
Signals
of O r t h o g o n a l P o l y n o m i a l s , see Eq.2.29 and
r
Eq.2.35
(t)
a
r
PAt)
0
L (at)
-
r
T
r
2 (r!)
2
2r+l
2 r
{2c?y
[2a ) r
H (at) T
0
T (t)
0
t/ (t)
0
2
P < '"'(i)
Eq.2.131
Eq.2.130
c;(0
o
T
r
a
r
2
r - l
r
7r,
r = 0
?,
r 5^ 0
T
-r2 -*
f
-r(r - l)2
Eq.2.133
Eq.2.132
^
r
! ( „ + r ) r V )
r -
(r-2)!
B y b i n o m i a l expansion of the left hand side, we have n
[1 - ( 2 i - * ) * ] " "
=
£
{u) x (2t n
-
»)"
(-l)"» (2«)»—s"
E E
n
m\(n
— m)!
Set A; = m > 0 and r — k = n, then n — m = r — 2fc and k can v a r y f r o m 0 t o [ r ] , where I f ,
if r S is even i f r iis o d d
W, Hence we get, oo
[1 - (2t -
x)x}'
M
E E r=0
fc=0
fc!(r-2fc)!
(2.144)
2.11 Gegenbauer
Polynomials
69
and c o m p a r i n g w i t h Eq.2.143 we get
w h i c h can be identified w i t h the Legendre p o l y n o m i a l of Eq.2.49 i f i / =
1/2.
For Gegenbauer p o l y n o m i a l s we choose [
=
2
(\-t y-
1 / 2
(2.146)
where i> > —1/2, w h i c h is a special case of Jacobi p o l y n o m i a l s o b t a i n e d by s e t t i n g ct = P — v — 1/2. T o c o m p u t e c6 (t)'s given by Eq.2.24, we have to solve r
T
1 r
dt
d u {t) r
w(t)
dt
=
T
0
(2.147)
subject t o the c o n d i t i o n s u«(-l)
k
=
u< \\)
= 0
r
(2.148)
for k = 0 , 1 , . . . , ( r — 1). As shown i n Section 2.3, u (t) Since by R o d r i g u e s ' f o r m u l a
2
4>Jt) K
'
is a p o l y n o m i a l of degree r, u (t) r
u {t) r
as a factor.
r
1
=
T
contains ( 1 — t )
T
2
1
(1 - t ) - /
d u (t) — dt' T
;
2
y
(2.149) '
should be of the f o r m =
2
d (l - « ) r
r +
"-
1 / 2
(2.150)
As i n the case of Jacobi p o l y n o m i a l s we can show t h a t Gegenbauer p o l y n o m i a l s satisfy the differential e q u a t i o n 2
(1 - t )C" (t)
- (1v + l ) t C ; ( t ) + r ( 2 : / + r ) C " ( i )
r
=
r
0
(2.151)
w h i c h is the same as Eq.2.140 at a = /? = v — 1/2. For Gegenbauer p o l y n o m i a l s of degree r the constant d is t a k e n as T
dr
=
(-l) \ 2 r! T
r
r
(2i/) \ \ (u + - ) ,
r
(2.152)
Since this constant differs f r o m t h a t of Jacobi by the last r a t i o n a l factor, i n view of Eq.2.128 we can w r i t e C "W r
=
Pr
"
(2i/)
2
r + m
\t)
(2.153) f t - l \
m
Chapter
70 T h e p o l y n o m i a l C"(t)
2:Least
Squares
Approximation
of
Signals
has the expansion Eq.2.145. To see t h i s , we consider the series
£c w r
= E E
(2t/)
,.
:p ,
(2.155)
4
„ m\(r
- mY.lu
+
We now set k = m, n = r — m so t h a t r + m = n + 2k. T h e n the last series takes the form oo
oo
fc!ra!(i/ + - )
71=0 ifc== 0 oo
y , y , (2w + 2fc)
"
n
2*
k
oo
n
( 2 i / ) ^ ( r - 1)*
(2i/) (« -
k
t
l
l) x
a t
^2**1(1/+ |)*(l-x)**+» k
(1-x)
k
k
fc!(l -
1 2
(I-*)
k
(u) 2 (t-l) x
E
2
2(t (1-x)
x) l)x 2
2
=
2k
[1 - 2xt + x ] - "
(2.156)
where we have used the i d e n t i t y (2v)
2k
2k
=
2 {u)
(u + i )
k
(2.157)
Hence the series i n Eq.2.145 and Eq.2.154 are i d e n t i c a l . For the convenience of comp u t a t i o n , the series Eq.2.145 w i l l be used i n the sequel. To derive the three-term recurrence f o r m u l a for Gegenbauer p o l y n o m i a l s , comp a r i n g Eq.2.145 w i t h Eq.2.29 we get 2 »
r
(2.158) (2.159) '(•Or-l
(2.160)
0 - 2 ) 1
2.11.1
Properties
of G a m m a
Function
To c o m p u t e the n o r m a l i z a t i o n factor -y for the Gegenbauer p o l y n o m i a l s we need some i m p o r t a n t properties of G a m m a functions w h i c h w i l l now be discussed. r
Let us now define the G a m m a f u n c t i o n T(z, n) by the f o l l o w i n g a l t e r n a t i v e relation, see Section 2.9. 7(*,n)
=
n ' j \ \ - r T n J
(2.161)
2.11 Gegenbauer
71
Polynomials
Using the general f o r m u l a o f repeated integrations by parts of t w o functions U(T) a n d V(T)
viz., = UV) — u'v
J uvdr
u
v
+ "3
2
— •• •
where a p r i m e represents d i f f e r e n t i a t i o n once and the subscript v
k
denotes k times
i n t e g r a t i o n , we have
r(z,n)
TT
=
n
+
n ( n - 1) • • • 1 z(z + 1) • • • (z + n)
n
(1 - r ) — + n ( l -
z(z + 1) -i I
(2.162) S e t t i n g t = " T , and t a k i n g the l i m i t as rt tends to i n f i n i t y , the r e l a t i o n of Eq.2.161 can also be w e i t t e n as lim T ( z , n )
=
lim /
(1
)"i'
n—KX) J
n — oo
- 1
di,
JI
N
/ oo (2.163)
u
= r(z), where T ( z ) is defined i n Eq.2.108. Hence i t follows t h a t n\n*
T(z) = l i m
( « ) n + l
Therefore, w i t h t h e help of Eq.2.111 we can derive t h a t , 1
=
lim
(z)„
+ 1
r(z)
n =
T(n)n*
lim i - *
0
0
n
+
z
T(rt
+
z)
z
=
r(n)n lim — » T ( n + z)
N o w , m a k i n g use of E q . 2 . I l l , we get 2rc!
=
2rt(2n - 2) • • -2 • ( 2 n - l ) ( 2 n - 3) • • • 1 2
2 "n!(n-i)(r .-f)...ir(i) i
r(|) m
2
2 "n!
r(n + j)
(2.164)
Chapter
72
2:Least
Squares
Approximation
of
Signals
Also, we observe t h a t 2
2n!(2n)
r(2z,2n)
2z(2z + 1) • • • (2z + 2rt) 2n
2 n\T(n
2z
+
TC-)2^[z{z
\){2n)
+ \)---{z
+ n)} 1
[
2
2 2 _ 1
(
+
z
I ) . . .
n!n 2 z +
iilii'
{
z
+
„ _ ! ) ]
T(n + j )
z +
I X f ) (*)«+, ( z + f ^ v ^ n - l ) !
n + j rT~
T a k i n g the l i m i t of b o t h sides as n tends to i n f i n i t y we get r(2z)
=
^r(z)r(z + i)
where we have used Eq.2.164 at z = i 1 ! > + !/+-)
(2.165)
Moreover, we have
=
1 3 1 1 (r + i / - - ) ( r + i / - - ) • • • ( ! / + - ) r ( i / + - )
=
2 " ( 2 r + 2u-
r
l ) ( 2 r + 2i/ - 3) • • • (2v + \)V(v
+
1
(2r + 2t/ - l ) ( 2 r + 2 i / - 2) • • • (2t/ + l ) 2 i / r ( » / + f ) r
2 ( 2 r + 2 f - 2) • • • ( 2 i / + 2 ) 2 i /
^ ^ I > 2r
+ i)
V
2 (v)
2
r
r(2r + 2 i / + l )
(2.166)
;
=
(2i/)
2 r + 1
r(2i/)
(2.167)
Hence
r(r
+
i
>+i)
r(i)
T ( 2 r + 2i/ + l ) 2.11.2
2
2 r + 2
"(K) (// + r)r(^)
Normalization Factor of Gegenbauer
To derive the n o r m a l i z a t i o n factor 7
r
(2.168)
r
Polynomials
of the Gegenbauer p o l y n o m i a l s we use the
r e l a t i o n Eq.2.35 and t a k i n g i n t o consideration of Eq.2.150, Eq.2.156 a n d Eq.2.166Eq.2.168, we get T
(-l) r!A. /
u (t)dt T
2
+
(-iy \/3 c f\i-t y "r
T
r
1 / 2
dt
2.12 Rodrigues'
Formula
2 r + 2
73 r
=
2
=
a ^ » .
r +
"(-l) r!/? c f r
_
(
V
i
r
r
W
r
e
r
T
r t r
+
l l 2
"'
(l
- r)
-i/a)r(r
y
'
r +
+
"y
1 / 2
d Y , i f 2r = 1 + t
- i / 2 )
T ( 2 r + 2 i / + 1)
( 2 i / ) r ( i / 2 ) r ( » + i/2) r
r!(i/ + r ) I V ) w h i c h gives the n o r m a U z a t i o n factor o f the Gegenbauer p o l y n o m i a l s . Using Eq.2.30, Eq.2.34 a n d Eq.2.33 we now get 2
r
=
^r+i ft 0,
c,
=
ft ft._i —-
6
r
0 + ) r + 1 '
=
7r
2i/ + r - 1
7r-l
r + 1
+1
;
respectively. I n view o f Eq.2.28, the t h r e e - t e r m recurrence f o r m u l a for Gegenbauer p o l y n o m i a l s becomes
(r + l)C " (f) = 2( / + r)iC "(<)-(2i/ + r-l)C;_ (0 r
+1
I
r
1
(2.169) T h e expressions for the coefficients a ,b and c i n the t h r e e - t e r m recurrence form u l a Eq.2.28 as also a , / 3 , 7 and r of a l l o r t h o g o n a l p o l y n o m i a l s are listed i n Table 2.6 and Table 2.7 respectively. T
r
2.12
r
r
T
r
r
Rodrigues' Formulae
A l l t h e o r t h o g o n a l p o l y n o m i a l s w h i c h have been i n t r o d u c e d i n the foregoing pages can be derived f r o m the Rodrigues'
formula,
viz., T
1
Mi)
d
= ^ — — MOs'W] w(t)
dt
(2.170)
T
where g(t) has the f o r m
g(t)
2
= at + bt + c
(2.171)
Indeed, we have for (a) Legendre P o l y n o m i a l s : a = 1,6
= 0,c = — 1 ;
(b) Laguerre Polynomials:!! = 0,6 = l , c = 0; (c) H e r m i t e Polynomials:*! = 0,6 = 0,c = 1; (d) Tchebycheff P o l y n o m i a l s : F i r s t and Second K i n d s : a = —1,6 = 0, c = l ; (e) Jacobi P o l y n o m i a l s :a = —1,6 = 0, c = l ; (f) Gegenbauer Polynomials:*! = —1,6 = 0,c = 1; T h e p a r a m e t e r s a,b, c and d for the o r t h o g o n a l p o l y n o m i a l s are listed i n Table r
2.8.
Chapter
74
Table 2.8:
Parameters a,b,c
W e i g h t i n g F u n c t i o n w(t)
2:Least
in Rodrigues'
of O r t h o g o n a l
Squares
Approximation
T
b
c
Pr(t)
1
0
-1
L (at)
0
1
0
1_
H {oct)
0
0
1
c-ir
T (t)
-1
0
1
(1 -
UM
-1
0
1
(1 "
-1
0
1
T
r
d
«"(«)
r
l
1
r
2 r!
e"«
2
a
r
2 r!
a > -1
(1 - 0 ( i + * ) " 0 > -1
r
c;(t)
1
0
1
Signals
F o r m u l a , N o r m a l i z a t i o n Factor d
Polynomials
a
T
of
(-i) W »*>! ( » + | ) '
2
(1 - i ) " - J
and
2.13 Differential
Recurrence
Relations
75
Table 2.9: Parameters a,6 ,ri T
and Function
T
g(t)
in the D i f f e r e n t i a l Recurrence R e l a t i o n g(t) (t) = ( r a t + 6 ) (t) + T) d> _,(t) of T
Orthogonal
Pr(t)
i)
u
K
»7r
1
0
—r
L {at)
t
0
r
—r
B (at)
i
0
0
2a r
T (t)
( i - t)
2
-1
0
r
U (t)
(i
2
-1
0
1 + r
pK/*)(t)
( i - t)
2
-1
a+B + 2r
c:(t)
(i
-1
0
r
T
r
r
2.13
T
Differential
- t)
2
- t)
T
T
T
Polynomials
2
r(o-/9) a-r3+2r
2v + r - 1
Recurrence
Relations
T h e class of a l l o r t h o g o n a l p o l y n o m i a l s satisfy a differential recurrence r e l a t i o n of the form g(t)d\ (t)
=
T
(rat + 6 ) {t) + Vrr-i{t) r
(2.172)
r
where o,<5 , r/ are constants, g(t) is a f u n c t i o n of t i m e w h i c h are listed i n Table 2.9. To prove this r e l a t i o n we note t h a t r
r
h(t)
:=
g(t)4> (t) T
l
-
-rg{t)t4> (t) T
is a p o l y n o m i a l o f degree r , as this follows f r o m Eq.2.29 and Eq.2.171 a n d the fact t h a t the coefficient o f t
r+1
i n h(t)
is i d e n t i c a l l y zero. I n fact, this coefficient is
1 ar0
r
r2a(3
r
=
0
Chapter
76
2:Least
Squares
Approximation
of
Signals
Therefore m a k i n g use of T h e o r e m 1 we can w r i t e g{t)4> {t)
- art {t)
T
r
=
(2.173)
£«*<£*(*)
over [ i , i ^ ] . N o w for k < r — 2, 0
I'"' =
-
g(t)4> (t)w(t)<j> (t)dt r
k
[ ' 4> {t)—
[g{t)ct> {t)w{t)\dt
r
J
f
k
Hi
4.
4>r{t)
w{t)—{g{t)4> {t)) dt
g{t) (t)w(t) dt
+
k
k
«"W— dr (ff(0<^(<)) + O(«)Wi(«) - s(0)»W
dt
t
where i n the first step the first t e r m arising f r o m i n t e g r a t i o n b y parts is zero because of Eq.2.24, Eq.2.26 and
Eq.2.27, and i n the t h i r d step we have used Eq.2.170 for
r = 1. I f k < ( r — 2 ) , all the terms on the r i g h t h a n d side are zero i n view of Eq.2.20. Also J
td> {t)4> (t)dt r
=
k
0, k < (r -
2)
as a result again of Eq.2.20. Hence m u l t i p l y i n g b o t h sides of Eq.2.173 by we have i n t e g r a t i n g f r o m t
0
to
bklk
= 0,
T
k < (r — 2)
and the r e l a t i o n of Eq.2.172 follows f r o m Eq.2.173 where r/ = 6 _ . r
r
T
r/ and S can be c o m p u t e d as follows. E q u a t i n g coefficients of t r
w(t)
tj,
T
T h e unknowns
l
T
and t ~
l
from both
sides of Eq.2.172 we have i n view of Eq.2.29 and Eq.2.171 f r
t~
L
:
act (r
:
a r ( r — 2) + ba (r
— 1) + br/3
T
r
r
r
= S /3 T
T
+
raa
T
— 1) + cr/3
T
= arr
r
+ Sa T
r
+ r/ 3 _i r /
r
Hence <5
r
=
br — a
(2.174) Jr
cr(3
r
— 2 a r — ba
cr/3
T
— 2 a r — ba
r
brc
T
Jr-i r
r
(2.175)
2.13 Differentia.!
Recurrence
Relations
77
T h e differential recurrence relations of a l l the o r t h o g o n a l p o l y n o m i a l s o f our interest can be derived w i t h the help of the relations given by Eq.2.171 and Eq.2.172. T h e values o f S and n using the f o r m u l a given i n Eq.2.174 and Eq.2.175 are c o m p u t e d as shown below. r
r
(a) Legendre:
S
= br = 0 crB — 2aT — ba . = r + Q -lT —r Pr-l = 1 •r — 0 = r
r
r
nr
r
T
( b ) Laguerre:
S
(c) H e r m i t e :
T
6 6
=
r
n
(e) Tchebycheff
<5 r/
= 0
= j ^ - + f±-
T
= r
=0
r
Second K i n d :
2
= 2a r
= 0 •r - ^
T
First K i n d :
( f ) Jacobi:
r
=0
r
T) ( d ) Tchebycheff
2
"~
r
= ! ^ ± ^ > = (1 + r )
r
S
=
T
1 rB _
^
3r
_
1
r
.
J
,
=
M
_2T£
-r
?
r
-
o
2
PrBr-l
1
1 (a+ff + 2r)(o+/3 + 2 r - 1 )
a+S-t-r + (^+7+7) [K*»
+ ^ +
2
r
a
^
*- /J + 3 r -
1)
- 2(a + r)(/3 + r ) ] 1 ( a + f l + 2t-)(>+fl + 2 , r - l ) "
l _
=
2
(a+0+r) 21> + r W + r ) ( a + ^ + 2r)
(g) Gegenbauer:
<5
=
^
=^
r
= 0 2
i /
r
1
r
1
+ ^zr = ( + - ) - ( - )
= ( 2 v + r - 1) T h e differential recurrence relations of the o r t h o g o n a l p o l y n o m i a l s are therefore: 2
Legendre
( i - l)P (/)
=
rtP (t)
Laguerre
i£ (o;t)
=
r£ (af) — rZ^.^at)
Hermite
H {ott)
=
r
r
T
2
Tchebycheff I
(1 - t )t {t)
Tchebycheff I I
Jacobi
rft.^t)
r
2
2a rH _ (at) r
1
=
-rtT (t)
+ rT _i(t)
(1 - t )U {t)
=
-rtV {t)
+ (1 +
( l - O ^ W
=
[~ri +
r
2
T
+
Gegenbauer
-
r
7
(1 - t )C?(t)
= +
r
T
l>+,J + 2r)
r
r)U ^(t) T
/*"«(*)
* > - !
W
-rtC?(t) (2i/ + r - l ) C " _ ( t ) r
l
78
2.14
Chapter
2:Least
Squares
Approximation
of
Signals
O r d i n a r y Differential R e c u r r e n c e Relations
T h e class of o r t h o g o n a l p o l y n o m i a l s also satisfies ordinary differential recurrence relations w h i c h w i l l now be established. These relations are of p r i m e i m p o r t a n c e to us as the i n t e g r a t i o n o p r a t i o n a l matrices are derived from t h e m . (a) Legendre
polynomials
For Legendre p o l y n o m i a l s , the three-term recurrence f o r m u l a of Eq.2.44, the differential recurrence r e l a t i o n and the differential equation Eq.2.48 are respectively Pr+dt)
=
r
^~rtPr(t)
—P - (t)
2
(i -l)P (r) r
2
(\-t )P {t) r
(2.176)
r l
r + 1
r + 1
=
r t P ^ - r P ^ t )
-
2tP (t)
(2.177)
+ r(r + l)P (i) = 0
r
r
(2.178)
We differentiate b o t h sides of Eq.2.177 w i t h respect to t and add the r e s u l t i n g equation to Eq.2.178 and simplify to get rP (t) r
+ 4_t(<)
=
tPr(t)
(2.179)
Now differentiating Eq.2.176 and s u b s t i t u t i n g from Eq.2.179 we get after simplification P (t)
- Pr-M
T+1
=
(2r+\)P {t),
(2.180)
T
w h i c h is the o r d i n a r y differential recurrence r e l a t i o n for Legendre p o l y n o m i a l s . (b) Laguerre
Polynomials
For Laguerre p o l y n o m i a l s , the differential recurrence r e l a t i o n and the differential equation Eq.2.63 are respectively, tL (at)
=
rL (at)
t
+
(1 - at)L (ctt)
T
'L (at) r
r
- rL ^(at)
(2.181)
T
+ arL {at)
r
= 0
T
(2.182)
Differentiating Eq.2.181 and s u b t r a c t i n g the resultant equation f r o m Eq.2.182 we get atL (at) r
Now s u b s t i t u t i n g atL (at) r
=
arL {at)
+ rt (at)
T
r
- rL _ {at) T
1
(2.183)
from Eq.2.181 i n t o Eq.2.183 and s i m p l i f y i n g we o b t a i n aL^^at)
=
i _ ( o ; « ) - L (at) r
a
r
(2.184)
w h i c h is the o r d i n a r y differential recurrence r e l a t i o n for Laguerre p o l y n o m i a l s .
2.14 Ordinary
Differential
Recurrence
(c) H e r m i t e
Polynomials
79
Relations
For H e r m i t e p o l y n o m i a l s , the o r d i n a r y differential recurrence r e l a t i o n is the same as the d i f f e r e n t i a l recurrence r e l a t i o n , v i z . , H (at)
2
=
r
2a rH _ (at) r
(2.185)
1
( d ) T c h e b y c h e f f P o l y n o m i a l s of F i r s t K i n d For Tchebycheff p o l y n o m i a l s of the first k i n d , the t h r e e - t e r m recurrence f o r m u l a of Eq.2.83, the differential recurrence r e l a t i o n and the differential e q u a t i o n
Eq.2.86
are respectively, T {t)
- 2tT {t)
T+l
T
2
(1 - t )f (t)
+ rtT (t)
r
2
(1 - t )f {t)
- tf {t)
T
+ T _M T
- rT^it)
r
T
2
+ r T {t) T
=
0,
(2.186)
=
0,
(2.187)
=
0.
(2.188)
D i f f e r e n t i a t i n g Eq.2.187 a n d s u b t r a c t i n g the resultant e q u a t i o n f r o m Eq.2.188, a n d after s i m p l i f i c a t i o n , we get tf (t)
=
r
r T ( 0 + -^—f ^{t) r — 1 r
(2.189)
r
W e n o w differentiate Eq.2.186 and s u b s t i t u t e f r o m Eq.2.189 t o get
T
j
t
r
k
)
2W^_7V_ (£)_
=
i
J
2(r + l )
V
2(r-l)
1
as the o r d i n a r y differential recurrence r e l a t i o n of Tchebycheff p o l y n o m i a l s . ( e ) T c h e b y c h e f f P o l y n o m i a l s of S e c o n d
Kind
W e a p p l y the above technique to the f o l l o w i n g relations of Tchebycheff second k i n d p o l y n o m i a l s o b t a i n e d from Eq.2.94, differential recurrence r e l a t i o n and Eq.2.93 U (t)
=
r+l
2
{l-t )U {t) T
2
(1 - t )ij {t) r
2tU {t)-U _ {t) T
=
-rtU {t)
=
3tU (t)
r
r
T
l
+ ( r + l)
t o get t h e o r d i n a r y differential recurrence r e l a t i o n as
UJt) K
'
=
+
U
\ 2(r + l )
, 2(r + l )
(2.191) '
Chapter
80
(f) Jacobi
2:Least
Squares
Approximation
of
Signals
Polynomials
For Jacobi p o l y n o m i a l s , the t h r e e - t e r m recurrence f o r m u l a of Eq.2.137, the differe n t i a l recurrence r e l a t i o n and the differential e q u a t i o n Eq.2.140 are respectively (
2(r + l ) ( a + 0 + 2 r ) ( l + a + 0 + =
r)P lf(t) r
(1 + a + 0 + 2r) [ ( a + 0 + 2 r ) ( 2 + a + 0 + 2r)t + ( a
2
-
(
P ^ \ t )
- 2 ( a + r)(0 + r ) ( 2 + a + 0 + 2r)P l'?\t),
' (2.192)
r
{a + 0 + 2r){\
2
-
7
0 )]
a 0
t )Pl ' \t)
=
-r[0
+
2(a + r)(0 + r)P Zf\t),
0)
- Q + (a + 0 +
2r)t\P^ {t)
(
(2.193)
r
2
(1 -
a 0
t )Pl - \t)
+
[0-a-{2
+
r ( l + a + 0 + r)P^' (t)
+ a + 0)t\
pW\t)
0}
= 0
(2.194)
Differentiating Eq.2.193 w i t h respect to t and then s u b t r a c t i n g the resultant equation from Eq.2.194 and after s i m p l i f i c a t i o n , we get
t
^ \ t )
P
=
—Llel—pt'
p
)
m
T
+
r P ^ \ t ) V
+ ;
(a + 0 + 2r) 2(a r)(/? r ) P^f(t) ( a + 0 + 2 r ) ( a + 0 + r) +
(2.195)
+
r _ 1
V
N o w , differentiating Eq.2.192 w i t h respect to t and s u b s t i t u t i n g f r o m Eq.2.195 we get # * i
M
2(l+
=
a
+ 0 + r)
^
(i -t- a + (3 zr]( + a + 0 + 2r) (\-ra 0 +•+-2r){2
r + 1
2(a-/3) ( a + 3 + 2 r ) ( 2 + a + 0 + 2r) -
2(et + r)(0 + r) . — P (a + 0 + r ) ( l + a + 0 + 2 r ) ( a + 0 + 2 r ) r
! a B l -
1
s '(t)
(2 196) '
W
V
as the o r d i n a r y differential recurrence r e l a t i o n for Jacobi p o l y n o m i a l s . (g) Gegenbauer Polynomials T h e equations Eq.2.180, Eq.2.190 and Eq.2.191 are special cases of Eq.2.196. For Gegenbauer p o l y n o m i a l s , from Eq.2.169, differential recurrence r e l a t i o n and Eq.2.151 we have (r+l)C;
(«)
=
2(i- + r ) i C ; ( ( ) - ( 2 / + r - l ) C ; . ( 0
(1 - z ) C ; ( t )
2
=
-rtC" {t)
(l-t )C;(t)
-
(21/ + l ) i C ; ( 4 ) + r ( 2 i / + r ) C ; ( t ) = 0
2
+ 1
1
T
+ ( 2 i / + r - 1.
1
(*3
2.15 Integration
Operational
81
Matrix
F o l l o w i n g the same procedure as used i n the case of Jacobi p o l y n o m i a l s , the o r d i n a r y differential recurrence r e l a t i o n for Gegenbauer p o l y n o m i a l s can be o b t a i n e d as,
c
2.15
"
( i )
=
^ r r ^ ^ - ^ r r - \ 2(is + v) 2(u + r )
d
w
^
( 2
-
1 9 7 )
Integration Operational Matrix
T h e o r d i n a r y differential recurrence relations of the o r t h o g o n a l p o l y n o m i a l s are der i v e d i n the last section and are reproduced (2r +
=
P
l
=
L _ (at)
H (at)
=
aL _ (at) T
r
r + 1
T
(t) -
below.
Pr-M, -
l
L (at) r
2
2a rH _ (at) T
1
t (t)
f ^(t)
r+1
r
2 ( r + 1)
2 ( r - 1)
U (t)
l/ _ (<)
T+l
U (t) r
l*J»
P
(t)
r
r
2{r + l )
t
2(r + l )
2{l 0 + r) (1 + a + 0 + 2 r ) ( 2 + a + 0 + 2 r )
=
+
W
2
+
{
a
a
^
+
0
-
)
r + 1
P ^ X t ) r
{a + 0 + 2 r ) ( 2 + a + 0 + 2 r )
P !f (
2(a + r)09 + r ) (a + 0 + r ) ( l + a + 0 + 2r)(a + 2(i/ + r )
0+
>(t)
2r)
2(i/ + r )
A careful s c r u t i n y of these relations reveals t h a t they have the general f o r m given by <M0
=
where t h e coefficients A , B r
r
A,<> and C
r
r + 1
(t) + B « ( t ) + C 0 _i(t) r
r
r
(2.198)
r
are given i n Tables 2.10 and 2 . 1 1 .
I n t e g r a t i n g b o t h sides o f the general differential recurrene r e l a t i o n of Eq.2.198 f r o m to t o t we have / ' c6 (t)dt r
=
A d> (t) T
r+l
+ B {t) + C rS) T
T
T
+ D (t) T
0
(2.199) where o(t) = 1 and D
r
= -\A <j> ,{t ) T
T+
0
+ B 4> (t ) r
r
0
+
C 4r-i(to)\ T
Chapter
2:Least
Squares
Approximation
of
Table 2.10: Coefficients of Differential Recurrence R e l a t i o n : r = 0
Polynomials
A
0
So
Co
D
m )
1
1
0
0
L (t)
-1 r
1
0
0
0
0
0
1
0
0
1
0
0
0
H (t) To(t) 0
2
1 I
U (t) 0
2
0
Table 2.11: Coefficients of Differential Recurrence R e l a t i o n s > 1
Ml}
A
Pr(t)
(2r+l)
0
L (ctt)
-1/a
1/a
0
0
0
r
H (oct) r
Br
T
1
1 2(r+l) 2
C
r
1 (2r +
0
l)
0 0, r = e v e n
Q
2
Tr(t)
1 2(r+l)
I 4
u {t) r
2(r-l)
0 1
2(r+l)
2(r +
2{X+r)
2( -B) «(l+
1
a
0
(4i)!'
r * - l
r
_
o
+1
d
. ,
r
>
d
r
>
1
0
2(„+r)
)
2
1
0,r = 1
4
» = A + 2r,
c:(t)
2
(-ir
1
0
(
a
l)
2(o+r)(^+r)
(a+e+r)
1 2(„+ r)
r+1
Signals
2.15 Integration
Operational
Matrix
83
for r = 1, 2 , . . . a n d
J
d> (t)dt = t - t 0
0
=
Ao4>i(t) +
=
6 - ( — +
(2.200)
B (t) 0
0
io where 1 = —, B
A
0
0
0
t) 0
For Tchebycheff first k i n d p o l y n o m i a l s , i f r = 1
Jt
T,{t)dt
=
l
-\To
+ T] 2
-U T 2
a
0
C o m b i n i n g Eq.2.199 and Eq.2.200 we can w r i t e f
4>{t)dt =
E4{t)
(2.201)
ifhere 4>(t)
=
\4> {t)
•••
0
0m-l(*)]
(2.202)
T
and E is the i n t e g r a t i o n o p e r a t i o n a l m a t r i x h a v i n g the f o r m : Bo
Ao
D
Si c
2
2
0 A B
l
2
0
0
0
0
0
0
0
0
0
0
0
A
2
•
(2.203) D.
m 2
0
0
0
0
0
0
Bm0
2
Cm-l
A-
m 2
B
M
- L
T h u s i t is shown t h a t any a r b i t r a r y signal can be a p p r o x i m a t e d as a series of o r t h o g o n a l f u n c t i o n s . I f the n u m b e r of terms i n the series is f i n i t e , t h e n the i n t e g r a l of the square o f the error is a m i n i m u m i n t h i s a p p r o x i m a t i o n . T h e p o l y n o m i a l d> (t) r
of degree r i n any system of o r t h o g o n a l p o l y n o m i a l s can be generated by solving a differential e q u a t i o n of order 2r + 1. F o r t u n a t e l y , a t h r e e - t e r m recurrence f o r m u l a is available by means of w h i c h the o r t h o g o n a l p o l y n o m i a l s i n any system can be generated recursively, and consequently, one should not go for the s o l u t i o n of the differential e q u a t i o n . B y choosing the w e i g h t i n g f u n c t i o n w(t) and the i n t e r v a l [ t , tA, any system 0
of o r t h o g o n a l p o l y n o m i a l s e.g., Legendre, Laguerre, H e r m i t e , Tchebycheff first a n d second k i n d s , Jacobi, and Gegenbauer p o l y n o m i a l s can be generated. Each o f these o r t h o g o n a l p o l y n o m i a l s is shown to satisfy a differential e q u a t i o n and an ( o r d i n a r y ) differential recurrence r e l a t i o n , the l a t t e r is found to be useful i n the d e r i v a t i o n of the integration operational matrix.
Chapter
84
2.16
2:Least
Squares
Approximation
Problems
l.Show t h a t r
M
2
m
=
f _ T
_ l)f
m
m!(r — 2m)!
0
m
W
^(o=E(-i) -^—-!—(2ty-*" m
=o
m!(r — 2m)!
2.Verify the following generating functions (a)
, I V I - 2«x + x l
W
2
= E r
=
S (t)x
r
r
0
X
~ r = T (t) + 2 E 2 t x + x^
T (r)x
0
1
—
r
1 ( C )
r
r
=
1
°°
l - ^ x +
2
x
^
^
1
(d)
2
2
V T - 2 i x + x { l - x + V l - 2rx + x } ° 2
a +
" = E
2
{ 1 + x + V I - 2tx + x } '
expjtxjx - 1) ~ j = 2^ L (t)x 1-x OO j exp(2rx - x ) = E -H (t)x
(e)
^ ( t ) .
r
2
(/)
r
T
3. Prove the following special values: T ( l ) = 1;
(7 (l) = r + 1
T (-1) = (-1)'; T (0) = ( - ! ) ' ; T (0) = 0;
(7 (_l) = ( - i r ( r + t) <7 (0) = ( - l ) U (0) = 0
r
r
r
r
r
S r
2r
2r+1
2r+l
4. Show t h a t (a)
T ( i ) = U (t) -
(6)
T ( < ) = t.7,._ (*) - < 7 _ ( r )
(c)
7 , ( 0 = j [ t f ( t ) - t/,-a(01
r
r
r
t
«/,._,(<) r
2
r
(d)
(l-
(e)
E
2 I
)(7 _,(/)= [tT (()-T r
r
2 m
r
( t ) = d l + (7 (t)] 2r
r + 1
(t)]
T
of
Signals
2.16
Problems
5. Prove t h e f o l l o w i n g relations: (a)
2T (t)T (t)
(6)
2T?(t)
T
= T (t)
k
+ r _*(t),(r > k)
r+k
r
= 1 + T (<) 2 r
2
2
(c) T ( )-r (0^_ (<)= 1 - t r
(d)
<
r+1
I
T {T (r)} = r {T (t)} = m
r
r
T (t)
m
mr
6. Prove t h e f o l l o w i n g recurrence relations for t h e Jacobi p o l y n o m i a l s : (a)
(2r + a + / ? ) P
( o r
a
1
i a p
- ^ ' ( * ) = ( r + a + p)P ' \t)
(6)
( 2 r + a + 0)P< -''- 't)
(c)
P^" '^)
1
- (r +
r
1)
fi
- p(*-W(t)
/S)P^f\t) (
= ( r + a + /3)P^' \t)
+ (r + a)P !' r
=
(d) p{"-")(f) = 1(1 + + + /})j£f<**%3 r
(e) (/) { 9
)
a
(
( t + l ) P ^ ' « ( t ) = r P ^ « ( ( ) 4- 0 » + r ) P : t r
I , W
r
(
(< - l ) P ° " ^ ( < ) = rP^Xt) r
(t) +1
- (a +
r)p£f \t)
W
P < - « ( t ) = ! { ( / , + r ) P < : r ' ( * ) + (a +
(
r)P !f
+ 1 )
r
(t)}
7. Prove t h e f o l l o w i n g relations for the Gegenbauer p o l y n o m i a l s : (a)
r C " ( i ) = ( r - 1 + 2 i y ) i C " _ ( r ) - 2v(l r
r
+
1
(6)
r C ; ( t ) = 2„{rC;_
(c)
(r + 2 * ) C ; ( t ) = 2 ^ { C ;
(d)
c;(i)
=
1
(t) -
2
-
1
t )C Z\{t) T
C^ \t)} 2
+ 1
(t) -
tC^{t)}
2»c;X(t)
8. B y m a t h e m a t i c a l i n d u c t i o n prove t h a t (r + 2 „ ) C ; ( t ) - ( r + l ) C ; 2(1
+ 1
(i)
-t)
9. Show t h a t
(a)
L {t) r
m -
£ i™(«) m = 0
(6)
/ (t - * ) - . & , ( * ) « • * =
. - f ( m + r + 1)!
+
X *
+
1
( t )
/ 3 ) 1
(i)
86
Chapter
2:Least
Squares
Approximation
10. Prove t h a t
(6)
f = — 2 ' ~ fc!(r - 2 k ) ! (
r-(r.)'t -
w
(a)
t
r
=
l L ± ^ 2' ~
1 ) > £
+ ' -
( t
* > 2
^ - u ( * ) fc!M i_* r+
of
Signals
C hapter
3
Signal
Processing
Time
D o m a i n
in
Continuous
T h e o r t h o g o n a l p o l y n o m i a l s developed i n C h a p t e r 2 were seen t o be o r t h o g o n a l over a specified range. To carry o u t the analysis of signals available over an a r b i t r a r y i n t e r v a l , shifted o r t h o g o n a l p o l y n o m i a l s are i n t r o d u c e d w h i c h are capable of describing signals over any i n t e r v a l of o u r interest. However, for i n f i n i t e range p o l y n o m i a l s , the t e r m shift has been used i n a different sense. A c o m p a r a t i v e s t u d y of the signal representations v i a different classes of o r t h o g o n a l systems is i n c l u d e d considering the effect of noise, a n d the
filtering
properties of these o r t h o g o n a l systems are also s t u d i e d . T h e
t w o - d i m e n s i o n a l o r t h o g o n a l functions and t h e i r a p p l i c a t i o n t o the representation of two d i m e n s i o n a l signals are also o u t l i n e d . T h e i n t e g r a l a n d the derivative o p e r a t i o n a l m a t r i x are i n t r o d u c e d as the t w o t i m e - d o m a i n operators t o reduce the i n t e g r a l a n d the d e r i v a t i v e operations t o algebraic operations i n the sense of least squares.
For
each o r t h o g o n a l system, the error i n t r o d u c e d by the i n t e g r a l o p e r a t o r is analyzed. A n i n t e g r a l f r a m e w o r k is p r o v i d e d for the d e r i v a t i o n o f the i n t e g r a t i o n o p e r a t i o n a l m a t r i x v i a generalized o r t h o g o n a l p o l y n o m i a l s .
3.1
Shifted Orthogonal
In Chapter
Polynomials
2, we have studied i n details how t o represent an a r b i t r a r y square-
i n t e g r a b l e f u n c t i o n i n a series of an o r t h o g o n a l system of functions { < / > ( t ) } , r r
0 , 1 , 2 . . . over the i n t e r v a l [ t o , * / ] thogonality
T h i s set of o r t h o g o n a l functions satisfies t h e
= or-
relation
(3.1) where j
T
is k n o w n as the normalization
factor.
For a complete system of o r t h o g o n a l
f u n c t i o n s , t h e m i n i m u m i n t e g r a l square error is zero. T h i s result can be s u m m a r i z e d i n the f o l l o w i n g t h e o r e m .
Chapter
88
3: Signal
T h e o r e m 3 . 1 Let { (t)} be a complete r
to the weighting function
function
w(t)
in this interval.
Processing
system
in the interval
Then f(t)
in Continuous
of orthogonal
[ i o , * / ] and f(t)
can be represented
Time
functions be any
as an infinite
Domain
with
respect
square-integrable series
represented
by OO
/(*)=
E/rM*), r=0
(3.2)
f
(3.3)
where f
r
7 r
In other words,
if s (t)
=
-
=
f
represents
m
w(t)f(t)4>At)dt, w{t)l{t)dt.
(3.4)
an approximation
of f(t)
described
by
m - l
then a- (*) converges m
in the mean to f(t) lim
/
»(*)[/(*) <0
as m tends £
to infinity,
MrWfdt
i.e.,
= 0.
r=0
T h e set of coefficients / o , / i , • • • , f -i, • • • i n the series representation of f(t) i n Eq.3.2 is called the spectrum of / ( < ) w i t h respect t o the system of o r t h o g o n a l functions {,.(*)}• T h e necessary and sufficient c o n d i t i o n t h a t the finite s u m s (t) converges i n the mean to f(t) as m tends to i n f i n i t y is t h a t the Parseval's identity ft, °° J w{t)f{t)dt=Y. fl '° r=0 m
m
lT
be satisfied. A n o r t h o g o n a l system of functions satisfying the Parseval's i d e n t i t y is defined to be a complete system. We have observed i n Chapter 2, t h a t a l l the finite range o r t h o g o n a l p o l y n o m i a b , viz., L e g e n d r e ( L e P ) , Tchebycheff first(TPl) and second k i n d ( T P 2 ) , J a c o b i ( J P ) and G e g e n b a u e r ( G P ) are defined over the i n t e r v a l [ 1 , — 1], w h i l e the values of a l l the signals in w h i c h we are interested are o b t a i n e d from p r a c t i c a l measurements over an arbitrary i n t e r v a l [ T , T J ] , where r and ry may not necessarily be —1 and 1 respectively. To suit our needs, the above o r t h o g o n a l p o l y n o m i a l s m a y be redefined over an arbitrary i n t e r v a l [Tr,,Tj] when they are k n o w n as shifted orthogonal polynomials. 0
0
T h e shifted o r t h o g o n a l p o l y n o m i a l s {rp (r)} are defined over an a r b i t r a r y interval [ r , T y ] and may be o b t a i n e d f r o m the unshifted ones by setting i = r * defined by T
0
2(r T
T) 0
=
1 Tf - T
0
=
AT + B,
(3.5)
3.1 Shifted
Orthogonal
Polynomials
89
where
A
- -J_,B-J2±4 Tj
-
T
(3.6)
(r, - r ) 0
0
i f the i n t e r v a l over w h i c h t h e functions are o r t h o g o n a l is
finite.
The orthogonality
r e l a t i o n for t h e shifted p o l y n o m i a l s therefore changes t o
T
[ ' w(r)^„(r)^ (r)iir = { ° T t A r
0
,
'
m
TO
r
*
' r,
—
(3.7)
where w(r) and 7
r
=
w ( r * ) , (r) = W ( 0
(3.8)
r
is listed i n T a b l e 2.7. 7
Let / ( r ) be an a r b i t r a r y square-integrable f u n c t i o n defined for r € [ T ' O I " / ] - T h e n i t can be a p p r o x i m a t e d by a system of complete shifted o r t h o g o n a l p o l y n o m i a l s
/
N
r
E
,
{V'I-(T)}
/ V (T) = f ^(T), r
r
where m n u m b e r of p o l y n o m i a l s V o ( ' ) ,
(3.9)
r
r
V'I(T), . . . , V'm-i( )
are used for the a p p r o x i -
m a t i o n , t h e spectral vector f is given by f = [fo fx V>(r) = [ f , ( r )
• • /m-lf i
Mr)
(3.10) 4> -i(r)] , T
•••
m
and /
=
r
-
f
(T)f(r)Mr)dT.
(3.11)
W
7 , ' ' .
T h e w e i g h t i n g functions W ( T ) of the shifted o r t h o g o n a l p o l y n o m i a l s are shown i n T a b l e 3.1 a n d the n o r m a l i z a t i o n factors -y i n T a b l e 2.7. T
T h e series expansion
a l l t h e recursive relations can be o b t a i n e d by s e t t i n g t = r * i n the
and
corresponding
expressions for t h e unshifted o r t h o g o n a l p o l y n o m i a l s given i n C h a p t e r 2. T o evaluate f
r
n u m e r i c a l l y , the closed loop q u a d r a t u r e f o r m u l a may be used, i f
W ( T ) has finite values at T
0
and T j . However, i n the case o f Tchebycheff p o l y n o m i a l s
of t h e first k i n d , for r = r
0
and r = T y , w ( T ) becomes i n f i n i t y . Hence, t h e closed loop
quadrature formula to compute f t o t h e Gauss-Tchebycheff
first
r
i n Eq.3.11 is not valid and one has t o take recourse
k i n d open q u a d r a t u r e f o r m u l a , given by
f i
I
flt)dt
, °
T
\J{t
-
T )(T, 0
-
t)
=E»>/(<>), k
=
1
(3.12)
Chapter
90
3: Signal
Processing
in Continuous
Time
Domain
Table 3 . 1 : W e i g h t i n g Functions for Shifted O r t h o g o n a l P o l y n o m i a l s
polynomials
w(r)
V {r) r
C (r)
e
-(r-r„)
r
H {r) T
%{r) V((*-—TO)(T/-T)]
U {T) T
(r/-r ) 0
p(«J>}( ) r
[(T/-r)(r-ro))-
C<">(r)
( T / - T O )
2
where
are the w e i g h t i n g factors of an £-th degree Tchebycheff p o l y n o m i a l o f the first kind and t
k
(r, + T ) = —= 2 0
(T) - Tp) 2
J ( 2 f c - 1)7T cos \ I 2£
are the zeros of the ^ - t h degree Tchebycheff p o l y n o m i a l of the first k i n d . T h e value of I i n Eq.3.12 can be a r b i t r a r i l y chosen according to the accuracy r e q u i r e d . Eq.3.12 can be applied only w h e n j(t) any desired value oft.
is e x p l i c i t l y k n o w n , so t h a t f(t)
can be evaluated for
As the i n p u t signal to a d y n a m i c a l system is e x p l i c i t l y k n o w n ,
the evaluation of the s p e c t r u m of f(t)
is not difficult.
For Tchebycheff polynomials
of the first k i n d , either of the two q u a d r a t u r e formulae can be used b u t c o m p u t a t i o n w i t h the closed q u a d r a t u r e f o r m u l a t u r n s o u t to be more l a b o r i u s . T h e Laguerre p o l y n o m i a l s are o r t h o n o r m a l over 0 < t < oo w h i l e the Hermite p o l y n o m i a l s are o r t h o g o n a l over —oo < t < oo.
T h e w e i g h t i n g functions w(t) of 2
Laguerre and H e r m i t e p o l y n o m i a l s are respectively exp( — t) and exp( — t ) . N o w , it m a y be observed t h a t i n b o t h the cases w(t)
= 1 at i = 0 and w(t)
—• 0 as t —* oo.This
3.1 Shifted
Orthogonal
91
Polynomials
means t h a t the weightage offered by these w e i g h t i n g functions at large values o f * is not as large as t h a t offered at t = 0.Therefore, t o represent a square-integrable f u n c t i o n / ( * ) , * € [ T , T ; ] w i t h b e t t e r accuracy, the o r i g i n (t = 0) is shifted t o r„ by s e t t i n g 0
T — T
t =
0
so t h a t the new w e i g h t i n g f u n c t i o n i n b o t h the cases becomes u n i t y for r = T . 0
The
r e s u l t i n g p o l y n o m i a l s are called shifted p o l y n o m i a l s . T h e s h i f t i n g defined above for Laguerre a n d H e r m i t e p o l y n o m i a l s , is completely different f r o m t h a t of finite range o r t h o g o n a l systems, seeEq.3.5.
3.1.1
Shifted
Laguerre
Polynomials
A system o f shifted Laguerre p o l y n o m i a l s ( L a P ) { £ ( r ) } is called o r t h o n o r m a l on r
T
0
< T < oo w i t h respect t o the w e i g h t i n g f u n c t i o n w ( r ) : w(r) = -
( T
e
-
T o )
(3.13)
if / "
w(r)£ (t)£ (*)<W = m
r
TO
f 0,
if m ^ r,
I
i f m = r.
1,
(3.14)
Shifted Laguerre p o l y n o m i a l s are e x p l i c i t l y defined as „
.
,
,y>
(-1)*(T -
T.)*
C T ' T ) = r ! 2 r^ - ;( r - * ), !, (, ,*,!, ), », k
. . . . .
•
(3.15)
0
These p o l y n o m i a l s have the i m p o r t a n t relations: oo
/
e-^C (t)dt
( r
T
=
e- - °>[£ (r) - £ _,(r)l,
(3.16)
=
C (T)
(3.17)
r
r
r
T
C {t)dt
f
r
T
- £
r + 1
(r),
TO
dC (r)
(3.18)
r
A square-integrable f u n c t i o n / ( r ) i n r
0
< r < ry can be a p p r o x i m a t e d i n fintie series
of shifted Laguerre p o l y n o m i a l s as m - l
f(r)
*
E
fkC (r) k
T
= f £(r),
(3.19)
92
Chapter
3: Signal
Processing
in Continuous
Time
Domain
where fk
C(t)
=
f
*>{t)f(t)C„{t)dt,
=
[£„(*) £,(*) •••
(3.20)
C - (t)f, rn
1
and f is given by Eq.3.10. I n Eq.3.20, i f 77 = 00 a n d / ( t ) is e x p l i c i t l y k n o w n i n this i n f i n i t e range, t h e n Gauss-Laguerre open q u a d r a t u r e f o r m u l a can be used t o compute f
k
efficiently.
T h e m a i n drawback of these i n f i n i t e range p o l y n o m i a l s is t h a t
the
representation of a f u n c t i o n defined over a finite range is, i n general, n o t exact.
3.1.2
Shifted
Hermite
Polynomials
A system of shifted H e r m i t e p o l y n o m i a l s ( H e P ) { W ( r ) } is called o r t h o g o n a l on —00 < r
T < 00 w i t h respect t o the w e i g h t i n g f u n c t i o n w ( r ) : [
w(r) =
-
e
( T
T
- °'
2 1
(3.21)
if
/
°°
f n w(0««(t)« (i)* =
jf m J
f -
r
v
.
i
T
r '
m
(3-22)
Shifted H e r m i t e p o l y n o m i a l s are e x p l i c i t l y denned as 1
W - n E ' to
^ - ^
( 3
.
2 3 )
k>.(r-2k)\
These p o l y n o m i a l s have the special value , , J (—l) ^ K,(r,)= 1 „ tO, r
2
r t
i f r is zero or even, ., . it r is o d d ,
( r / 2 ) !
(3.24)
and the i m p o r t a n t relations: f
n (t)dt T
= [n (T)
- W
R+L
r + 1
( r ) ] / [ 2 ( r + 1)]
(3.25)
= 2r?i _ (r)
(3.26)
0
TO
dri {T) T
dr
r
A square-integrable f u n c t i o n / ( r ) i n T < T < r 0
f
1
can be a p p r o x i m a t e d i n a
finite
series of shifted H e r m i t e p o l y n o m i a l s as m - l
fir)
«
£
r
/*W*(r) = f W ( r ) ,
(3.27)
/J' "W/(0W*(0*.
(3-28)
k=0
where /* =
3.1 Shifted
Orthogonal
93
Polynomials = [H (T)
W(r)
T
W , ( r ) ••• « _ , ( r ) ] ,
0
and f is given by Eq.3.10.
m
I n the above e q u a t i o n , i f To = —oo,T
F
= oo a n d f(r)
is
e x p l i c i t l y k n o w n i n t h i s i n f i n i t e range, t h e n G a u s s - H e r m i t e open q u a d r a t u r e f o r m u l a m a y be used t o c o m p u t e f .
These p o l y n o m i a l s also have t h e same d r a w b a c k of
k
Laguerre p o l y n o m i a l s i n representing t h e functions over a finite range. I n general,
i n f i n i t e range p o l y n o m i a l s should be avoided t o represent square-
integrable functions i n the
finite
interval, r
0
< r
< r,
as i t necessitates a large
f
n u m b e r of basis functions t o represent an a r b i t r a r y f u n c t i o n w i t h a resonably g o o d degree of accuracy.
3.1.3
Sine-Cosine
Functions
T h e Fourier series representation i.e., t h e representation i n terms of sine-cosine funct i o n s ( S C F ) of a square-integrable f u n c t i o n / ( r ) defined over t h e i n t e r v a l t
0
< t < tj
is given by /(*)
~
/ o F o ( i ) + H [fkhW
=
f F(i),
+
f F {t)] k
k
T
(3.29)
where for k = 0 , 1 , 2, • • • , m — 1, F (t)
= cos{A;7r[2i - {t
k
f
+ i )]c},
(3.30)
0
w h i l e for fc = 1, 2, • • • , m — 1 F (t)
= sin{fc7r[2t - {t, + t )]c},
k
(3.31)
0
and t h e Fourier coefficients fo,f
and f
k
are
k
h = cf ' w h i c h is t h e mean of / ( < ) over t
0
f(t)dt,
(3.32)
< t < t
f
A
=
2c f ' f(t)F (t)dt,
(3.33)
A
=
2 c / ' f(t)F 't)dt
(3.34)
k
k
with 1 (3.35)
(t,-t )' 0
f
=
r
[/o,A,---,/m-i./i,---,/ -i] , m
(3.36)
Chapter
94
3: Signal
Processing
in Continuous
Time
Domain
w h i c h is a ( 2 m — l ) - d i m e n s i o n a l spectral vector of / ( t ) , and T
F(0 = [F„(0, A ( < ) , • • ' . £ » - i ( * ) , A ( 0 , • • • , F _ ( < ) ] , m
1
(3.37)
w h i c h is a ( 2 m — 1)— d i m e n s i o n a l Fourier basis vector . T h e series i n Eq.3.29 w i t h coefficients given i n Eq.3.32-Eq.3.34 converges t o (a) / ( « ) i f t is a p o i n t o f continuity, and t o ( b ) [f(t
+ 0) + f(t
— 0 ) ] / 2 i f t is a p o i n t of d i s c o n t i n u i t y .
c o m p u t a t i o n of Fourier s p e c t r u m of f(t)
T h e numerical
is n o t a difficult p r o b l e m . T h e sine-cosine
functions have the special values: F (< ) = (-l)*;Ft(«o) = 0 t
(3.38)
0
and the recurrence relations for k = 1,2, • • • , m — 1: dF {t) k
= -2kircF {t),
(3.39)
k
dt dF {t) k
2kivcF (t),
(3.40)
= - J — F (t), 2kirc
(3.41)
k
dt
F {r)dT k
and
/ i
t
F (r)dr = [(-1)* - F (*)]—— 2K7TC t
0
(3.42)
t
T h e results of the signal analysis may be compared w i t h those o b t a i n e d by piecewise constant basis functions. T h e o r t h o g o n a l system of block-pulse f u n c t i o n s ( B P F ) i n t r o d u c e d i n E x a m p l e 2.5 w i l l be taken for this purpose. T h i s is because, the representation of any system of functions i n this f a m i l y is piecewise constant and the set of block-pulse functions is a good representative of t h i s f a m i l y .
Moreover, the
c o m p u t a t i o n w i t h i t is simpler t h a n the other systems i n this f a m i l y .
3.1.4
Block-Pulse
Functions
A set of m block-pulse f u n c t i o n s ( B P F ) , o r t h o g o n a l over i € [t ,t/), 0
b it) 1
I
J
o + i A t < t < i + (j + l ) A t ; otherwise; 0
I 0,
for j = 0 , 1 , 2, • • • , m — 1, where At
=
(t -t )/m f
0
These functions are disjoint and o r t h o g o n a l i.e.,
I bj(t),
if i = j ;
is defined as
3.1 Shifted
Orthogonal
Polynomials
95
and <6,.(o,6
j
j (
A square-integrable f u n c t i o n f(t) block-pulse series as
o > = { °> *\* :< I At if i = j . i n <„ < t < t m a y be a p p r o x i m a t e d i n a t
finite
m - l fT
/(*)
* £ fM*) = ° W
where f
=
:/
fx
0
•••
T
/ -ii m
is an m - d i m e n s i o n a l block-pulse spectral vector of / ( < ) and b(r)
=
[6 (t)
6,(0
0
•••
6
m
_ (0]
T
1
is an m - d i m e n s i o n a l block-pulse basis vector. T h e block-pulse s p e c t r u m components fj are d e t e r m i n e d such t h a t the f o l l o w i n g integral-squared-error is m i n i m i z e d T
£ =
2
f'[f(t)-f b(t)] dt.
Then /j
=
w h i c h is t h e average value of f(t)
1 — At
r'o+G'+ilAi /
f(t)dt, J
t +j£\t 0
over t + jAt
< t < t
0
0
+ (j + l)At.
T h e residual
e r r o r i n v o l v e d i n this a p p r o x i m a t i o n is given by e(0
=
/ ( 0 - f
T
b ( 0
where e ( 0 —• 0 as m —> oo. T h e n u m e r i c a l c o m p u t a t i o n o f f is q u i t e simple. 3.1.5
Examples
T h e f o l l o w i n g examples i l l u s t r a t e the representation of some square-integrable
func-
tions w i t h the help of the basis functions i n t r o d u c e d above and the accuracy w h i c h can be achieved w i t h t h e m . E x a m p l e 3 . 1 It is required 2 with the help of the first above.
to approximate
the function
eight basis functions
f(t)
= exp( — 0 ouerO < t <
of each orthogonal
system
introduced
Chapter
96
3: Signal
Processing
in Continuous
Time
Domain
T h e spectra of / ( « ) w i t h respect t o B P F , T P l , T P 2 , L e P , L a P and HeP are given i n Table 3.2, w h i l e the Fourier s p e c t r u m is [0.43233,
- 7 . 9 5 4 8 9 . E - 2,
- 9 . 6 2 5 9 5 F - 3,
5.4411E-3,
2.13611E - 2, - 3 . 4 9 0 2 1 £ - 3,
2.42675B - 3,
- 1 . 7 8 4 2 5 £ - 3,
-0.24991,
0.13422,
- 9 . 0 7 2 2 4 £ - 2,
6 . 8 3 7 4 8 £ - 2,
-5.48241F-2,
4.57431 E - 2,
-3.92376£ -
T
2]
Table 3.2: Spectra of e x p ( - r )
h
BPF
TPl
TP2
fo
0.88479
0.41582
h h h
0.68908
0.46576 -0.41582
ft
0.32549
h h
0.25349 0.19742
fi
0.15375
0.53665 0.41794
9.98772E-2
-0.19975 4.89316E-2
-1.63103E-2 2.01360E-2
-8.05529E-3 9.98639E-4
-1.99559E-4
-9.92575E-5 8.17106E-6
1.63520E-5 -6.87924E-7
-5.35648E-7
fk
LeP
LaP
HeP
fo
0.43233
0.49448
0.30758
fl
-0.40600
0.26050
0.12760
h h
0.13163 -2.59176E-2
0.12796 5.799631E-2
-3.26001E-2
U ft fo
3.66404E-3 -4.13495E-4
2.45402E-2 1.09501E-2
2.16464E-3
1.38170E-5 6.81642E-5
6.90146E-3 6.47924E-3
fl
-1.84824E-2 1.51895E-3 -1.26094E-4 -9.41530E-5
For each o r t h o g o n a l system, the representation of f(t)
w i t h t h e help o f t h e first
eight basis functions and t h e residual error e ( t ) i n t h e a p p r o x i m a t i o n are shown in F i g . 3 . 1 .
I t is clear from Fig.3.1 t h a t the B P F a p p r o x i m a t i o n , see p l o t no.2, of
exp( — t) is a staircase function w i t h the residual error shown by p l o t no.6.
The
a p p r o x i m a t i n g functions o b t a i n e d by using t h e o r t h o g o n a l p o l y n o m i a l s T P 1 , T P 2 , and LeP are exactly i n agreement w i t h the a c t u a l signal / ( < ) , see p l o t n o . l . T h e Laguerre p o l y n o m i a l representation of f(t),
p l o t no.3, is a very g o o d m a t c h w i t h
3.1 Shifted
Orthogonal
Polynomials
97
the a c t u a l f u n c t i o n w i t h a s m a l l residual error described by p l o t no.7. However, the H e r m i t e p o l y n o m i a l representation o f f{t),
plot no.4, is very poor w i t h a residual
error given by p l o t no.8. E x c e p t at the end p o i n t s w h i c h are, i n fact, t h e p o i n t s of d i s c o n t i n u i t y a n d i n t h e i r n e i g h b o r h o o d , the representation by sine-cosine f u n c t i o n s , p l o t no.5, almost matches w i t h E x a m p l e 3.2 It is required
f(t).
to approximate
of the first eight basis functions
f(t)
= exp t for t € [0,2] with the help
in each orthogonal
system.
T h e spectra o f exp i w i t h respcet t o B P F , T P l , T P 2 , LeP, L a P and HeP are given i n T a b l e 3.3 w h i l e t h e Fourier s p e c t r u m is [3.1945, - 0 . 5 8 7 7 9 , 0 . 1 5 7 8 4 , - 7 . 1 1 2 6 7 £ - 2 , 4 . 0 2 0 4 6 E - 2, - 2 . 5 7 8 9 4 F - 2,1.79314F - 2 . - 1 . 3 1 8 3 9 F -
2,1.8466,-0.99173,
0.67035, - 0 . 5 0 5 2 3 , 0 . 4 0 5 1 , - 0 . 3 3 8 , 0 . 2 8 9 9 3 ]
1-
Table 3.3: Spectra o f exp t
fk
BPF
TPl
TP2
fo
1.13610
3.44150
3.07250
fl
1.45878
h h
1.87311 2.40512
3.07250 0.73800 0.12052
1.47600 0.36156 5.95229E-2
ft
3.08824
1.48825E-2
7.38029E-3
h
3.96538
1.47863E-3
7.34534E-4
fo
5.09165
1.24867E-4
6.15481E-5
fl
6.53781
1.28639E-5
4.58723E-6
fk
LeP
LaP
HeP
fo
3.19450
0.97644
fl
3.00000 0.97264
2.00000 3.59853E-7
h h h h h fl
-0.66667
0.75911 0.16862
0.19153
-0.66667
-9.19877E-3
2.70581E-2
-0.40000 -8.88891E-2
-6.31874E-3 -3.57502E-4
0.15873
-3.42355 E-4
0.30476
-1.30250E-4
2.81374E-3 -4.36790E-4 1.29050 E-4
T h e a p p r o x i m a t i o n of exp t o b t a i n e d w i t h t h e first eight basis functions i n each o r t h o g o n a l system is shown in Fig.3.2. T h e o r t h o g o n a l p o l y n o m i a l s T P l , T P 2 , and
98
Chapter
3: Signal
Processing
in Continuous
Time
Domain
- 0 . 50 Figure 3.1: A p p r o x i m a t i o n of exp( — t) by o r t h o g o n a l functions and the residual error.
3.2 Analysis
of Noisy
99
Signals
LeP offer a g o o d a p p r o x i m a t i o n represented by p l o t n o . l , w h i l e the a p p r o x i m a t i o n by the p o l y n o m i a l s L a P , see p l o t no.3, is far away f r o m the a c t u a l f u n c t i o n w i t h the residual error shown by p l o t no.7. I t m a y be recalled t h a t i n E x a m p l e 3 . 1 , the representation of exp(—t) was f o u n d t o be satisfactory by p o l y n o m i a l s L a P w h i l e t h a t by HeP was worse t h a n others. So, the representation of square i n t r g r a b l e functions using i n f i n i t e range p o l y n o m i a l s is, i n general, not acceptable. T h e f o l l o w i n g example demonstrates the s u p e r i o r i t y of shifted i n f i n i t e range p o l y nomials t o unshifted ones for the representation of square-integrable E x a m p l e 3.3 Approximate
f(t)
= exp( — t ) , l < t < 3 in terms
with shift equal to one and (ii) without
functions.
of LaP and HeP
(i)
shift.
W i t h t h e first eight basis f u n c t i o n s , the a p p r o x i m a t i o n s of exp(—t) v i a u n s h i f t e d L a P and HeP a n d shifted L a P and HeP are shown together w i t h exp( — t) i n Fig.3.3. W i t h s h i f t , L a P a p p r o x i m a t i o n of exp( — i ) , p l o t no.3, is almost coincident w i t h
f(t)
shown by p l o t n o . l , w i t h the residual error denoted by p l o t no.7, zero. T h e first eight unshifted HeP have c o m p l e t e l y failed i n representing f(t),
see p l o t no.4. W i t h shift,
this representation is very m u c h i m p r o v e d , a l t h o u g h i t is not as m u c h accurate as the representation by shifted L a P as may be observed f r o m plots no.5 and no.9.
3.2
Analysis of Noisy Signals
I t is k n o w n t h a t W a l s h functions have inherent f i l t e r i n g properties [256]. I f the Walsh s p e c t r u m of a signal s-(r) is {s^}, r a n d o m noise r / ( l ) h a v i n g
and i f the signal is c o r r u p t e d w i t h a zero mean
as the W a l s h s p e c t r u m of s(t)
= s(t)
+ r/(i), then
•St —* s/c for a l l nonnegative i n t e g r a l values of k, i f t is sufficiently large.
I n other
words, i f the W a l s h s p e c t r u m of i ( r ) is m u l t i p l i e d by the W a l s h basis vector, the o r i g i n a l signal s(t)
m a y be recovered from the noisy signal. T h i s technique is very
i n t e r e s t i n g and is of considerable i m p o r t a n c e as i t employs p h y s i c a l l y no filters t o e l i m i n a t e the noise f r o m the c o r r u p t e d signal. T h e representation of signals characterized by piecewise s m o o t h
square-integrable
functions i n t e r m s of piecewise constant basis functions such as W F , B P F , and H F is never s m o o t h . I t is already established t h a t the accuracy i n the representation o f a signal is i m p r o v e d by increasing the n u m b e r of basis functions i f the a p p r o x i m a t i o n is made by a complete system of o r t h o g o n a l functions. However, i f the signal is corr u p t e d w i t h noise and block-pulse functions are used, the accuracy does not i m p r o v e by increasing the n u m b e r of terms i n the basis functions. Consequently, block-pulse functions do not possess g o o d f i l t e r i n g properties. T h e
filtering
properties of o r t h o g -
o n a l p o l y n o m i a l s as also sine-cosine functions w i l l now be investigated. Fig.3.4 is the block d i a g r a m o f a signal reconstructor w h i c h is s i m p l y a software loaded i n a d i g i t a l c o m p u t e r a n d i t computes the s p e c t r u m of the c o r r u p t e d signal 5(f) w i t h respect t o the chosen o r t h o g o n a l system and m u l t i p l i e s the s p e c t r u m by the corresponding basis vector to produce s(t).
T h i s s(t)
is expected t o follow
s(t)
100
Chapter
3: Signal
Processing
in Continuous
Time
Domain
3.2 Analysis
of Noisy
Signals
101
F i g u r e 3.3: A p p r o x i m a t i o n of exp(—t) by shifted Laguerre and H e r m i t e p o l y n o m i a l s and t h e residual error.
Chapter
102
3: Signal
if the chosen o r t h o g o n a l system has
filtering
Processing
in Continuous
Time
features.The f u n c t i o n e(t)
Domain
i n F i g . 3.4
represents the residual error.
Signal Rec o n s t r u c t o r
Figure 3.4: Schematic diagram to study the inherent f i l t e r i n g properties of orthogonal functions.
T h e p o l y n o m i a l T P l is not included i n this study as the s p e c t r u m evaluation i n noisy environment is not possible. For the noise-free signals, this spectrum is evaluated by using the Gauss-Tchebycheff first k i n d open q u a d r a t u r e f o r m u l a , which requires the value of the signal at the the zeros of the fcth degree T P l . I n practice, the values of the signal at these zeros ( not u n i f o r m l y d i s t r i b u t e d over the interval of o r t h o g o n a l i t y ) are not available as the signal is n o r m a l l y sampled w i t h a constant sampling rate. Moreover, i n t e r p o l a t i n g a noisy signal at the zeros of the A; t h degree p o l y n o m i a l of T P l using sampled data has no significance. As the p o l y n o m i a l s LaP and HeP cannot ensure a reasonably good a p p r o x i m a t i o n even i n the case of noise-free signals, these polynomials are kept outside our consideration for representing a noisy signal. E x a m p l e 3.4 Let us now investigate cheff polynomials the sine-cosine
of the second functions
s(t)
the inherent
kind
filtering
TPS and the Legendre
SCF for the -0
= 1 - 2.2941573e -
properties polynomials
of the
Tcheby-
LeP as also
signal 9<
sin(0.4358898* + 0.4510268),
(3.43)
3.3 Two-Dimensional
Square-integrable
t € [ 0 , 7 . 5 ] , if 7)(t) is a zero mean equal to 0, 0.05,
0.10,
eight basis functions
0.15,
0.20,
Gaussian or 0.25.
of each orthogonal
Fig.3.5 shows the signal s(t),
103
Functions noise
with noise-to-signal
For the purpose
system
are
ratio
of investigation,
(NSR) the
first
considered.
given by Eq.3.43, the c o r r u p t e d signal s(t) w i t h N S R
= 0.05, the a p p r o x i m a t e d signal s(t) and the residual error e(t) for the a p p r o x i m a t i o n of s(t)
via T P 2 .
Fig.3.6 shows e ( i ) at different noise levels. I t is observed t h a t the
m a x i m u m value o f e ( f ) , for N S R = 0.25, is a p p r o x i m a t e l y equal t o j u s t 3 per cent o f the m a x i m u m value o f s(t)
a n d the m a x i m u m value of s(t)
LeP as t h e basis, s(t),s(t),s(t)
is u n i t y . S i m i l a r l y w i t h
and e(t) are shown i n Fig.3.7 w h i l e e ( i ) at different
levels o f noise is shown i n Fig.3.8.
I n t h i s case, the m a x i m u m value o f e(t),
N S R = 0.25 is s l i g h t l y less t h a n 3 per centof the m a x i m u m value o f s(t). w h e n S C F is considered, s(t),s(t)
for
Similarly,
w i t h N S R = 0.25, s(t) a n d e(t) are shown i n Fig.3.9.
For different levels of noise, the behavior o f e ( i ) is shown i n Fig.3.10. N o w e(t)
for
N S R = 0.0 is n o t as g o o d as t h a t of T P 2 a n d LeP. T h i s implies poorer convergence rate of Fourier series i n t h e present s i t u a t i o n . M o r e n u m b e r o f basis functions i n the series expansion are r e q u i r e d t o i m p r o v e the accuracy. However, as seen i n the case of noise-free signals, the representation at the end points ( p o i n t s o f d i s c o n t i n u i t y ) and i n t h e i r n e i g h b o r h o o d w i l l never be exact for any n u m b e r o f basis functions used i n the series expansion. T h i s is the m a i n drawback w i t h SCF i n the analysis of signals. T h i s example has clearly demonstrated
t h a t the finite range o r t h o g o n a l p o l y n o -
mials a n d sine-cosine functions have excellent
filtering
properties a n d therefore, these
basis functions can safely be used for the analysis of signals i n a noisy e n v i r o n m e n t .
3.3
Analysis
of Signals
Characterized
Dimensional Square-integrable
by T w o -
Functions
I f the complete o r t h o g o n a l system of functions i n one variable is k n o w n , i t is possible t o c o n s t r u c t a complete o r t h o g o n a l system o f functions i n t w o or m o r e variables by the f o l l o w i n g t h e o r e m : T h e o r e m 3.2
Let {tp (x),
in the interval
x
ip (x),..
0
0
{4>o(t)> i(t), • • • , } be a similar weighting
function
w (t) 2
. Then
ifii(x)4>j(t), form
a complete
Xf,t
< t < t
0
square-integrable
f
., } be a complete
l
< x < Xj with respect
orthogonal with
the
in the interval
t
a
systems
function
< t < tj
of
functions
w^x),
and
with respect
let
to the
functions
i = 0,1,2,
system
respect
function
system
orthogonal
to the weighting
••; j =
of functions
to the weighting
in this region,
0,1,2,---,
in x and t in the rectangle function
the relation
w (x)w (J) 1
2
of completeness
.
x
0
If f(x,t) is
< x < is a
represented
104
Chapter
3: Signal
Processing
in Continuous
Time
Figure 3.5: F i l t e r i n g properties of p o l y n o m i a l s T P 2 . a ( t ) = a c t u a l signal, s(t)= signal, s(t)
= reconstructed signal, e(t) = residual error.
Domain
noisy
3.3 Two-Dimensional
Square-integrable
Functions
Figure 3.6: Residual error i n signal r e c o n s t r u c t i o n by T P 2 .
105
Chapter
106
3: Signal
Processing
in Continuous
Time
Figure 3.7: F i l t e r i n g properties of p o l y n o m i a l s LeP. ,s(i) = a c t u a l signal, s(t)= signal, s(t)
reconstructed
signal, e(t) = residual error.
Domain
noisy
3.3 Two-Dimensional
Square-integrable
Functions
F i g u r e 3.8: Residual error i n signal r e c o n s t r u c t i o n by LeP.
107
Chapter 3: Signal Processing in Continuous
108
N SR - 0.2S
\
IllTf
Time
Domain
,
3.Tit)
Hue to SCF
Figure 3.9: Filtering properties of S C F . j ( i ) = actual signal, i ( ( ) = noisy signal, = reconstructed signal, e(t) = residual error.
/
3.3 Two-Dimensional
Square-integrable
Functions
F i g u r e 3.10: Residual error i n signal r e c o n s t r u c t i o n by S C F .
109
Chapter
110
3: Signal
Processing
in Continuous
Time
Domain
by J
' j io
=
2
'
w (x)w (t)f (x,t)dxdt l
2
xo
£ £ [ /
2
j
» (x)«; (l)/(i,i)V (x)^(i)
2
i
I n view of this t h e o r e m , a square-integrable f u n c t i o n / ( x , i ) i n t h e region x Xf,t
0
< x <
< t < tf can be a p p r o x i m a t e d i n a finite series of o r t h o g o n a l functions as
0
/(x,t) « £
= V- (x)F0(r)
ft(i)E/i,*i(0
T
(3.44)
j=0
1=0
where V(X) =
lVo(*),V'l(x),...,Vm-l(x)]
(3.45)
r
is an m—dimensional basis vector in x , 0(0 =
[
(3.46)
T
an n—dimensional basis vector in I, and foo F =
/ l . n - l
(3.47) -
in mxn
/o,„-i
fn
/io
/m-1,0
/m-1,1
• ' •
/m-l.n
coefficient m a t r i x of / ( x , t ) w i t h respect t o t h e o r t h o g o n a l system employed.
B y m i n i m i z i n g the i n t e g r a l squared error given by e = f
' f io
'\f{x,t)
T
-
0
the block-pulse coefficients / , j may be o b t a i n e d
fi
-i
=
1 1 i n r J
0
+ J'AX < x
0
t
A
x
=
(3.48)
+ j A i < i < *„ 4- ( j + l ) A i ,
where x a n d x are the i n i t i a l and final values of x , i values o f t respectively a n d 0
f(x,t)dxdt
over the subregion
+ (i + l)Ax,<
0
as
r'o+O + l l A i z-io+tH /-io+(i+l)Ai /-
w h i c h is the average value of f(x,t) x
2
^ (x)F4>(t)] dxdt,
x
( x ^ - x o )
A
<
=
0
and t
(«/-«.)_
f
are t h e i n i t i a l and final
(3.49)
3.3 Two-Dimensional
Square-integrable
Functions
111
S i m i l a r l y , by m i n i m i z i n g the i n t e g r a l weighted square error given by tf
fx, J
/ 10
w (x)w (t)[f(x,t) 1
T
T
-
2
7
^ {x)F<j,{t)] dxdt,
o
the elements / , , o f the m a t r i x F can be o b t a i n e d for a l l the classes of o r t h o g o n a l p o l y n o m i a l s . I n the above expression, u>i(x) and w (t)
are the w e i g h t i n g functions of
2
the o r t h o g o n a l system chosen. For Tchebycheff p o l y n o m i a l s of the first k i n d , we have fi, = Cic J ' j
' t U ! ( x ) i ( ; ( i ) / ( x , t)V\(x)<^(t)dx
2
2
(3.50)
/here _ | 2 / [ * • { * / - x ) ] , for i = 0, I 4 / [ T T ( X - i ) ] , for i = l , 2 , - - - , ( m - 1); 0
1
/
=
c
2
(3 51)
0
J 2 / [ i r ( i - t ) ] , for j = 0, I 4 / [ j r ( t - t „ ) ] , for j = 1 , 2 , • • • , ( » » - 1); /
0
/
%
M
=
======= 2\/(x - x ) ( x / -
(3.53) x)
0
and W2
(t)
=
( < /
t o )
= 2\/(t - t ) ( t / - t)
(3.54)
0
For Tchebycheff p o l y n o m i a l s of the second k i n d , we have fn = -T, v 7T7 / Wi(0»!W/(i,«)^,^)^(<)W T T ^ X ^ - X ) ( t / - t ) "'io ''io 0
(3-55)
0
where 5
« j , ( x ) = 2[(x - z ) ( x / - x ) ] ° - / ( x / - x ) 0
0
(3.56)
and w (t)
= 2[(t - f ) ( t
2
0
/
- t)]
0
5
/ ( t / - to)
(3.57)
I f t h e o r t h o g o n a l system is the Legendre p o l y n o m i a l s t h e n we have f*i = , ?f/ A / ' F' f{x,t)4>M,(t)dxdt (Xf - X ) ( t - t ) ''io ' . i l functions are the Laguerre p o l y n o m i a l s , t h e n we have +
+
0
fi, = J ' J
/
0
' w {x)w (t)f{x,t)ip (x) it)dxdt l
(3.58)
0
2
i
J
(3.59)
112
Chapter
3: Signal Processing
in Continuous
Time
Domain
where
( <
w (t)
0
= e- -' '
2
For t h e o r t h o g o n a l system of H e r m i t e p o l y n o m i a l s , we have fij =
„ • • ! • . , . , f ' I ' w (x)w (t)f(x,t)ib (x)d. it)dxdt l
2
i
(3.60)
J
where w {x)
= -
s
e
(t)
( l
-
I o ) 2
= e-"-'"'
W2
2
I n t h e same way, a square integrable f u n c t i o n f(x,t) xj,t
0
i n t h e region x
0
< x <
< t < tj can be a p p r o x i m a t e d i n a finite Fourier series as
f(x,t)
«
V>o(x){Ao<M<) + Y,[Lj4>j{t)
£
^(x){/ ^o(0 +
+
+ /«*,(*)]} +
i o
i=i
=
+ /„>,(')]}
j=i
-0 (i)F0(i) T
(3.61)
where
= [lM*).tfl(*),- • •,^m- (l),^l(l), • • ..^m-lC*)!*" 1
(3.62)
is a ( 2 m — 1)— dimensional Fourier basis vector i n i , *(<) = [<M*).0t(<), • ' ' ,
tf—j(0.
tfi(0.
•' •.
T
tfn-i(*)] ,
a ( 2 n — 1) —dimensional Fourier basis vector i n t a n d /o,n-l
/oi
/o,n-i •f 1 , n - l
F =
/ m - 1 , 0 /i*0
/m-1,1 / l l
/ m - l , n - l
/m-1,1
/ l , n - l
A ,
l,n- 1 / l , n - l
(3.63)
3.3 Two-Dimensional
Square-integrable
Functions
113
a ( 2 m — 1) x (2ra — 1) constant m a t r i x whose elements are given by
foo
=
cJ
J to
foj
=
f(x,t)dxdt, xo
2c J ' J
f*j
=
2c j
'j' io
/,„
=
fij
=
/*
= =
hi
'
f(x,t)4>j(t)dxdt,
'
f(x,t)i>i(x)dxdt,
to
x
r'i
rJ
0
x
J
to
X
io
*o '
=
f{x t)i>i{t)dxdt, t
XO
A c J ' J ' io
*0
io
*o l
/ ( x . i ^ x ^ t ^ x d t ,
= c
/(x,t)V\(a:)<^-(t)
0
2c / ' ' J iQ
Si,
f(x,t)4>j(t)dxdt,
XQ
2c / ' ' J 4c J
' *0
to
4,cJ'f'f{z,t)ii>i{x)4>j(t)dxdt,
(x
f
- X )(tf
-
0
t)
.
0
T h e n u m e r i c a l c o m p u t a t i o n o f / y for each system of o r t h o g o n a l p o l y n o m i a l s is exactly the same as t h a t of fj i n the single variable case w i t h the exception t h a t now an a p p r o p r i a t e double i n t e g r a t i o n f o r m u l a is to be used. For example, i f the o r t h o g o n a l systems are block-pulse functions, Legendre p o l y n o m i a l s , Laguerre p o l y n o m i a l s , H e r m i t e p o l y n o m i a l s , Tchebycheff p o l y n o m i a l s of the second k i n d , and sine-cosine f u n c t i o n s , Simpson's | double i n t e g r a t i o n f o r m u l a m a y be used.
I f the o r t h o g o n a l
system is the Tchebycheff p o l y n o m i a l s of the first k i n d , then the open q u a d r a t u r e f o r m u l a of Gauss-Tchebycheff
first k i n d is to be used. T h i s f o r m u l a is given by
f(x,t)dxdt J
<0
J
I Xo
- X )(Xf 0
- X)(t - t )(t 0
where
* = l^Wi l
- t)
f
7T
i =
' 1^ j =
V>jf(Xi,tj)
l
7T
Wi = —; W: = —,
k
(Xf+Xo) x, = — (tf + tp) h =
1
'
(x -x ) f
0
{tf - tg)
t
cos
/(2«-l)ir\
I
2k
J(2j -
i ' l)tr
(3.64)
Chapter
114
3: Signal
Processing
in Continuous
Time
Domain
A n integro-differential equation can be reduced w i t h some a p p r o x i m a t i o n i n the least squares sense t o a simple linear algebraic equation w i t h t h e help o f t w o operators k n o w n as (a) t h e I n t e g r a t i o n O p e r a t i o n a l M a t r i x and ( b ) t h e D e r i v a t i v e O p e r a t i o n a l M a t r i x . These operators w i l l now be described.
3.4
Integration Operational Matrix of Shifted Functions
T h e i n t e g r a t i o n o p e r a t i o n a l m a t r i x is a square m a t r i x o b t a i n e d by i n t e g r a t i n g each element i n the basis vector ip(r)
and expressing t h e result i n t e r m s of t h e o r i g i n a l set
of basis functions. T h i s is expressed m a t h e m a t i c a l l y as J
*(t)dt
E.^{T)
a
TO
T h i s implies t h a t an a p p r o x i m a t e i n t e g r a t i o n of t h e basis vector is achieved t h r o u g h p r e m u l t i p l y i n g the basis vector by the i n t e g r a t i o n o p e r a t i o n a l m a t r i x . I t can be shown t h a t the i n t e g r a t i o n o p e r a t i o n a l m a t r i x for the block-pulse functions is given by [256], 0.5
1
1
0
0.5
1
0
0
0.5
0
0
0
0
• •
1
1 1
1
• •
1
1
0
• •
0.5
0
•
0
1 0.5
S u b s t i t u t i n g f r o m Eq.3.5 i n t o Eq.2.28 the t h r e e - t e r m recurrence r e l a t i o n of shifted o r t h o g o n a l p o l y n o m i a l s becomes
0r+i(r)
=
(^r +
WWrJ + ^
W
(3.65)
where 4>T(T*) = ib {r),
a
ib { )
= (a r
r
a
= I,Mr)
T
T
= Aa , r
0
(3
T
=
Ba
r
+ b ,j r
r
= c
r
+ 0) O
A g a i n i n view of Eq.2.198 and Eq.3.5, the differential recurrence r e l a t i o n for shifted o r t h o g o n a l p o l y n o m i a l s becomes,
A4> (T) t
=
A xP (r) r
r+1
w h i c h on i n t e g r a t i n g f r o m T - t o r yields 0
+ B,i(r) +
C Vv_,(r) r
3.4 Integration
T
Operational
115
Matrix
A
0
(3.66) where D
f
T
R
ib (r)dT 0
=
-A XI> (T )
=
r - r
T
t+1
8
''TO
-
B
S,Wr„) - C,V>,-i(r )
(3.67)
lA(r)]
(3.68)
0
= -[4„i(r) + A
where B„ = - ( B + — ) - A T .
(3.69)
0
C o m b i n i n g equations Eq.3.66-Eq.3.68 we can w r i t e f\(T)dr
=
£.V(r)
(3.70)
''TO
where T
V>(r) = [Mr)
Mr)---ip -i(r)} m
and B„ is t h e i n t e g r a t i o n o p e r a t i o n a l m a t r i x for shifted o r t h o g o n a l p o l y n o m i a l s given by
A„
Bo D, D
+
2
C,
B, c 2
0 B
2
0
0
0
0
0
0
0
0
0
0
0
A
2
•
0
0
0
•
0
0
0
•
C _2 m
B
m
A
_2
0
B
-
M
m
2
_i
(3.71) B y selecting t h e a p p r o p r i a t e
values of the parameters A , A , B , C , K
K
k
and D
K
from
Table 2.10 and Table 2 . 1 1 , t h e i n t e g r a t i o n o p e r a t i o n a l m a t r i x E , for each system of t h e shifted o r t h o g o n a l p o l y n o m i a l s can be o b t a i n e d . T h i s m a t r i x for some o r t h o g o n a l p o l y n o m i a l s a n d sine-cosine functions is shown below. For i n f i n i t e range o r t h o g o n a l p o l y n o m i a l s , t h e i n t e g r a t i o n o p e r a t i o n a l matrices for t h e shifted and t h e
unshifted
case are t h e same. However, for the finite range p o l y n o m i a l s E , = (1/A)E,
where A
is defined by Eq.3.6.
Chapter
116
3: Signal
Processing
in Continuous
Shifted Legendre Polynomials
E,
=
1
i
0
0
o _ i 5
h
o
0
0
0
0
0
0
A 0
2 m —3
0
Shifted L a g u e r r e Polynomials
1
-1 1
0 -1
• • • •
0 0
0 0
0
1
• •
0
0
0
0
0
• •
1
0
0
0
•
0
• • • • •
0
0 '
0 0
0 0 0
0 0 E.
Shifted H e r m i t e
= -1 1
Polynomials
i_
' 0 2^
2
0 0
2
0 _
E,
3
=
2
a
m
0 4
0
0 0 I 6
0
0
0
• •
0
0
0
0
• •
0
0
0
0
• •
0
1 2 ( m - l )
where
1
—^,
^ 0
3 2 m
i f m j t 0 or even, i f m is o d d .
0 .
Time
Domain
3.4 Integration
Operational
Matrix
117
S h i f t e d T c h e b y c h e f f P o l y n o m i a l s of t h e F i r s t
0
0
7
0
0
0
0
0
Kind
E. ( m - l ) ( m - 3 ) m ( m - 2 )
2(m-3)
-1)
0
0
2(m-2)
S h i f t e d T c h e b y c h e f f P o l y n o m i a l s of the S e c o n d K i n d
E.
0
0
1
°
o 5
= m - 1
0
0
0
0
0
0
2 ( m - l )
a ( m - i )
0
0
J
Fourier Sine-Cosine Functions T h e Fourier series a p p r o x i m a t i o n o f the square integrable f u n c t i o n / ( i ) defined i n the interval t
0
< t < tf is given by
/(*)
+ E[/rF (t) +
*
f.f.(t)
=
f F(t),
r
where F (f)
=
c o i { r 7 r [ 2 i - (t
r
=
0,1,2, • • • , m
=
sin{rir[2t
r
F {t) r
r
=
c
= =
Sr
=
0
- (t, +
l,2,--,m 1 ,
*/ £
+ t )]c},
f
—
'o cf'f(t)dt,
2 c / ' J
4-
f(t)F (t)dt, T
t )]c}, 0
f F (t)] r
r
Chapter
118
f
r
f
3: Signal
=
2c / '
=
[A./l, •••,/», L
Processing
in Continuous
Time
Domain
f(t)F (t)dt, r
•••,/mf,
F(t) = [F (t),F (t),---,F (t) F (t) ---,F (f)] 0
1
m
>
1
)
T
m
F r o m these equations the i n t e g r a t i o n o p e r a t i o n a l m a t r i x associated w i t h the Fourier sine-cosine functions can be easily derived [213], a n d is represented by
where
U
=
diag [ 1
|
•- •
i
] ,
0 is an m-vector w i t h zero elements, a n d O an m x m n u l l m a t r i x .
3.5
Derivative Operational
Matrix
T h e d e r i v a t i v e o p e r a t i o n a l m a t r i x is o b t a i n e d by d i f f e r e n t i a t i n g every element of the basis vector (t) w i t h respect t o t and expressing the result i n t e r m s of the original set of basis functions. Expressed m a t h e m a t i c a l l y , this can be w r i t t e n as d(t) at T h i s implies t h a t the differentiation of the basis vector (t) w i t h respect t o t is obt a i n e d t h r o u g h p r e m u l t i p l y i n g 4>{t) by a square m a t r i x D. For t h e piecewise constant basis functions, this d e f i n i t i o n t o c o m p u t e D cannot be d i r e c t l y a p p l i e d . for the block-pulse functions we have the f o l l o w i n g i m p o r t a n t T h e o r e m 3.3 derivative the inverse
/ / the initial
operational
matrix
of the integration
value
of a square
D associated operational
with
matrix
integrable the
However,
theorem.
function
block-pulse
is zero
functions
then is
the
simply
E.
Proof. F r o m the t h e o r y of i n t e g r a t i o n we know t h a t /
f(T)dT
= f{t)-
/(«„)
(3.72)
119
3.5 D e r i v a t i v e OperationaJ M a t r i x Let r
f(t)
«f 0(t),
T
/(to) = / ( t ) e 0 ( t ) , o
and /(t)«fj*(t), where
(3.73)
is t h e m—dimensional s p e c t r u m vector of / ( t ) and e = [ l l
... I f
is an m - v e c t o r . Therefore, the r e l a t i o n takes the f o r m T
tjE*[t)
= f (t) - / ( t ) e
T
0
0(f)
(3.74)
N o w , / ( t ) can be w r i t t e n as r
T
/(t)«f *(0 =
f D(t),
w h i c h w h e n compared w i t h Eq.3.73 gives us r
f j = f £>. Hence, i n view of Eq.3.74, we have T
f DE w h i c h gives us D = J 5
_ 1
T
= f
-
r
/(t )e , 0
i f / ( t ) = 0. 0
W e shall derive below the derivative o p e r a t i o n a l matrices for some o r t h o g o n a l p o l y n o m i a l s a n d sine-cosine functions. For this d e r i v a t i o n we s h a l l assume t h a t the u n d e r l y i n g f u n c t i o n is defined over the i n t e r v a l [T ,Ty] and the d e r i v a t i v e o p e r a t i o n a l 0
m a t r i x for a l l f i n i t e range p o l y n o m i a l s and sine-cosine functions like the i n t e g r a t i o n o p e r a t i o n a l m a t r i x is a f u n c t i o n of this i n t e r v a l . Shifted Tchebycheff Polynomials of the First
Kind
To derive the d e r i v a t i v e o p e r a t i o n a l m a t r i x for Tchebycheff p o l y n o m i a l s of the first
k i n d , we take the help of the identities , sin2r0
=
2 sin 0[cos(2r - 1)8 + cos(2r - 3)0 + cos30 + cosfl]
s i n ( 2 r + 1)8
=
2 s i n 0 [ c o s 2 r 0 + cos(2r - 2)8 +
1(3.75)
h
cos20 + O.5]
(3.76)
Chapter
120
3: Signal
Processing
in Continuous
Time
Domain
Differentiating the r t h degree Tchebycheff p o l y n o m i a l of the first k i n d defined by T {t)
= cos r0,t
r
=
cos0,
w i t h respect to t we get T (t) r
= r
sin T0 sin 0
, t == cos 0.
(3.77)
S u b s t i t u t i n g Eq.3.77 i n t o Eq.3.75 and Eq.3.76 we get respectively, T
T (t) st
Ti (t)
= ^EVu-iC), =
r+l
2(2r + l ) £ T
2 r
T_,(i):=0,
_ , ( i ) - ( 2 r + l)T 2
0
for a l l r = 0 , 1 , . . . . Setting t = r' defined by Eq.3.5, we have the f o l l o w i n g derivative o p e r a t i o n a l m a t r i x for the Tchebycheff p o l y n o m i a l s of the first k i n d i f m is an even number representing the number of functions i n the basis vector.
D
=
A
0
0
0
0
1 0
0 4
0 0
0
3 0
0 8
6 0
0 m — 1
0 0 0
0 0
0 0
8
2 ( m - 2)
0
2 ( m - 2)
0
2 ( m - 1)
0
• ••
0 2 ( m - 1)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2 ( m - 2)
0
0
0
2 ( m - 1)
0 .
A similar m a t r i x when m is an odd number can be w r i t t e n . S h i f t e d T c h e b y c h e f f P o l y n o m i a l s of t h e S e c o n d
Kind
T h e f o l l o w i n g t r i g o n o m e t r i c i d e n t i t y can be easily proved sin(2r + 2)0 cos 0 = sin 0[cos(2r + 2)0 + 2 cos 2r8 +
2 cos(2r - 2)0 + • • • + 2 cos 49 + 2 cos 20 + 1]
3.5 Derivative OperatiojiaV
Matrix
121
Dividing both sides by sin 9 and using the definition of the Tchebycheff polynomials of the second kind given by
U (t)
=
r
sin(r + 1)9 —, sin 9
t = cos 9
We have U i{t) 2r+
cos 9 - 2(r + 1) cos(2r + 2)9
=
- ( 2 r + l ) c o s ( 2 r + 2)9 + 2 cos 2r9 +
=
(2r + l)[cos 2r9 - cos(2r + 2)9]
1- 2 cos 29 + 1
+
(2r - l)[cos(2r - 2)8 - cos 2r8]
+
3(cos 28 - cos 48) + 1 - cos 28
=
2 s i n 9 [ ( 2 r + l ) s i n ( 2 r + 1)9 + (2r - l ) s i n ( 2 r - 1)9 + ••• + 3 sin 39 + sin 9]
Hence
U dt) 2r+
cos 9 dt)^ --(2r sin 9
;
U
=
2[(2r + l ) l / ( t ) + ( 2 r - l ) l / _ ( t )
+
• • • + 3U (t)
=
2j^(2k
2
T
+
T
+ 2)
cos(2r + 2)9 \ sin 9
=
2 r
2
2 r
+
2
U (t)] 0
+ \)U (t)
(3.78)
2k
For polynomials of even degree we use the identity sin(2r + 1)9 cos 9 =
sin 9[cos(2r + 1)9 + 2 cos(2r - 1)9 +
1- 2 cos 39 + 2 cos 9],
from which it follows that, U (t) 2r
=
cos 9 - (2r + l ) c o s ( 2 r + 1)9
- 2 r cos(2r + 1)9 + 2 cos(2r - 1)9 + • • • + 2 cos 39 + 2 cos 9
=
2r[cos(2r - 1)9 - cos(2r + 1)9]
+
(2r - 2)[cos(2r - 3)9 - cos(2r - 1)9]
+
2[cos9 - cos 39]
=
2 sin 9[2r sin 2r9 + (2r - 2) sin(2r - 2)9 +
h 2 sin 29]
Chapter
122
3: Signal
Processing
in Continuous
Time
Domain
Hence, cos 6 Virtt)
=
U (t) 2r
2
cos(2r + 1)0 (2r + 1) " sin 6 + (2r - 2 ) l / _ ( 0 + • • • + 2
=
sin 6 2 \2rU Mt) r- 1
=
2E(2H2)f
2r
2 r
! l t
2U (t)\
3
1
i(l)
k=0 From Eq.3.78 and Eq.3.79 the derivative o p e r a t i o n a l m a t r i x for the Tchebycheff polynomials of the second k i n d can be w r i t t e n as 0 1 0
0 0 2
0 0 0
0 L 1
2 0
0 3
0 0 0
0 0 0
0 m - 1
0 0
2A
D
m - 2 0
where m is an even number. A similar expression for m o d d can be w r i t t e n . Shifted Legendre Polynomials For shifted Legendre p o l y n o m i a l s , the following recurrence relation can be easily proved by m a t h e m a t i c a l i n d u c t i o n t a k i n g i n t o consideration of Eq.2.180 and the scaling factor for shifted polynomials:
[r-2J Pr(t)
= A E ( 2 r - 4k -
l)P - Mt) r 2k
F r o m this relation, the derivative o p e r a t i o n a l m a t r i x for Legendre p o l y n o m i a l s takes the f o r m :
D
=
A
0 0 1 0
0 0
0 0
0 3 1 0
0 5
0 0
0
3
0
7
2m - 5
0
0
L 1
0
5
0
0
2m - 3
0
0 1 0 0 0
i n w h i c h m is assumed to be an even number.
3.5 Derivative Shifted
Operational
Matrix
123
Laguerre Polynomials
I n v i e w o f Eq.2.178, the R o d r i g u e s ' f o r m u l a for Laguerre p o l y n o m i a l s
can
be
w r i t t e n as L (t) T
= -e'D' r!
[e-V] ,D = at
B y d i f f e r e n t i a t i n g b o t h sides w i t h respect to t, the above e q u a t i o n gives
L (t)
= 0,
0
L {t)
r + 1
= -e'D
T
[ e " V ] + L „ ( t ) , r > 1.
D e n o t i n g the first t e r m on the r i g h t h a n d side by F we have on s i m p l i f i c a t i o n
F
=
W
=
1 : e'D (r-1)!
r
-LJt)+
-L (i) - i,-i(i) + r
-L («) - L _!(r) r
r
e-'i"" 1
J
(r-2)!
-
1
V l T
r
e-'i'
- L (t)
+ e'D [e~
-
-
y
-Z (i) - i r - l { * ) ~
1
lilt)
- 2
L (t) 0
F r o m the above development, the derivative o p e r a t i o n a l m a t r i x for Laguerre p o l y n o mials for b o t h the shifted and the unshifted cases assumes the f o r m
0
0
0
0
0
0
0
0
0
0
0
0
0
1 1 1 0
0
0
1 1 1 1
0
0
1 1 1 1
1
0
1 0 1 1 0 D
Shifted
Hermite
=
Polynomials
The Hermite polynomials
satisfies the f o l l o w i n g differential recurrence r e l a t i o n
represented by Eq.2.185 and is given by
H (t) T
=
2rH _ (t) r
1
as
Chapter
124
3: Signal
Processing
in Continuous
Time
Domain
As a consequence of this r e l a t i o n , we have the d e r i v a t i v e o p e r a t i o n a l m a t r i x for the shifted and unshifted H e r m i t e p o l y n o m i a l s represented b y :
Fourier Sine-Cosine
" 0
0
0
0
•
0
0
0 "
1
0
0
0
•
0
0
0
0
2
0
0
0
0
3
• •
0
0
0 0
0
0
0
0
0
0
0
• •
. 0
0
0
0
•
0 0 m — 1 0 .
m - 2 0
Functions
T h e first element i n the Fourier basis vector is a constant f u n c t i o n , and so its derivative is zero. Considering t h i s , the derivative o p e r a t i o n a l m a t r i x for sine-cosine functions follows d i r e c t l y from Eq.3.39 and Eq.3.40 and is given by :
[ 0 D
=
TT
o
r
A \ 0
O
0
M
I
0
T
-M O
]
j J
where
M
=
diag [ 1
2
••• m - 1 ] ,
0 is an ( m — l ) - v e c t o r w i t h zero elements, and O a n u l l m a t r i x of order m — 1.
3.6
Conclusion
A n i n t e g r a l framework is p r o v i d e d for the signal analysis v i a the i n t e g r a t i o n opera t i o n a l m a t r i x for any system of o r t h o g o n a l functions.
I t has been shown t h a t the
Legendre p o l y n o m i a l s , the Tchebycheff p o l y n o m i a l s of the second k i n d and the sinecosine functions have excellent
filtering
properties i n contrast w i t h the
and other piecewise constant basis functions whose
filtering
block-pulse
properties are inferior.
T h e use of shifted o r t h o g o n a l p o l y n o m i a l s is found to be very convenient for the representation of signals over an a r b i t r a r y i n t e r v a l . T h e shifted Laguerre and Hermite p o l y n o m i a l s defined i n a manner different from t h a t of the finite range orthogonal p o l y n o m i a l s are shown to be superior to the unshifted ones for the representation of square-integrable signals.
For the evaluation of spectra v i a o r t h o g o n a l polynomials,
i f the w e i g h t i n g f u n c t i o n is finite at the t e r m i n a l values, the closed loop quadrature f o r m u l a can be used. However, for Tchebycheff p o l y n o m i a l s of t h e first k i n d , as the
3.6
Conclusion
w e i g h t i n g f u n c t i o n assumes i n f i n i t e values at the end p o i n t s , Gauss-Tchebycheff
125 open
q u a d r a t u r e f o r m u l a should be used. T h e i n f i n i t e range o r t h o g o n a l p o l y n o m i a l s such as Laguerre and H e r m i t e can represent any signal accurately i f t h e signal is k n o w n over the i n f i n i t e range of the d e f i n i t i o n of the p o l y n o m i a l s . O t h e r w i s e , a large n u m b e r of basis functions are required to a t t a i n a reasonably g o o d degree of accuracy.
Pro-
v i d e d t h a t the signal is k n o w n over the entire range of the i n f i n i t e i n t e r v a l , the shifted i n f i n i t e range o r t h o g o n a l p o l y n o m i a l s are found to be superior to the unshifted ones i n the d e s c r i p t i o n of t h i s signal. T h e derivative o p e r a t i o n a l m a t r i x is f o u n d useful i n the s y s t e m a t i c e v a l u a t i o n of a delay-operational m a t r i x i n c l u d e d i n C h a p t e r 4.
C hapter
4
Analysis
of Time-Delay
Systems
T h e analysis o f c o n t i n u o u s - t i m e d y n a m i c a l systems c o n t a i n i n g time-delays is discussed i n this chapter v i a o r t h o g o n a l functions such as piecewise constant basis funct i o n s , o r t h o g o n a l p o l y n o m i a l s a n d sine-cosine functions. A n i n t e g r a l f r a m e w o r k is p r o v i d e d t o construct the delay and t h e d e l a y - i n t e g r a t i o n o p e r a t i o n a l m a t r i x for the o r t h o g o n a l p o l y n o m i a l s and sine-cosine functions needed for t h i s purpose. T o i m p r o v e t h e c o m p u t a t i o n a l accuracy, the time-partition technique is i n t r o d u c e d w h i c h can be used w i t h any system of o r t h o g o n a l fucnctions. T h e above m e t h o d o f analysis is also extended t o systems h a v i n g m u l t i p l e delays and piecewise constant delays. T h e various possible sources of errors arising i n the analysis of time-delay systems and t h e means o f reducing t h e m are also considered.
4.1
Introduction
T i m e - d e l a y s occur frequently i n chemical processes, mechanical systems, transmission lines and i n d u s t r i a l processes [305].
T h e r e are m a n y systems such as p o p u l a t i o n
g r o w t h , epidemic g r o w t h , economic g r o w t h , n e u r a l networks etc. m o d e l l e d by delaydifferential equations
[85]. T h e delay is caused by mass or energy transfer i n t h e
physical systems or i t results from the signal transmission delays. T h e analysis o f time-delay systems is not as simple as t h a t of lag-free systems. Sometimes, the c o m p u t a t i o n of the response of time-delay systems becomes e x t r e m e l y difficult. However, i n spite of a l l t h e difficulties, t h e analysis of these systems becomes i n e v i t a b l e i f one wishes t o s t u d y t h e behaviour o f delay systems. T h e r e exist a n a l y t i c a l as also n u m e r i c a l methods t o analyse delay-systems [85]. A m o n g t h e n u m e r i c a l m e t h o d s , the approach based on the o r t h o g o n a l functions reduces t h e delay differential e q u a t i o n describing t h e system t o a set of linear algebraic equations whose s o l u t i o n gives, i n general, an a p p r o x i m a t e response of the system. Rao and Srinivasan, 1978 [266] i n t r o d u c e d the delay operational
matrix
t o solve
delay-differential equations v i a block-pulse functions. A l m o s t at the same t i m e , b u t i n d e p e n d e n t l y , t h e same k i n d of o p e r a t i o n a l m a t r i x for the W a l s h functions was i n -
128
Chapter
4: Analysis
of Time-Delay
Systems
t r o d u c e d by Chen and Shih, 1978 [49] for s i m i l a r p r o b l e m s . I n b o t h t h e cases, its a p p l i c a t i o n was l i m i t e d to solutions w i t h zero i n i t i a l functions. Subsequently, the delay systems were analysed v i a block-pulse functions by Shih et a l , 1980 [290] and piecewise constant delay systems by Chen and Jeng, 1981 [44] by i n t r o d u c i n g a new delay o p e r a t i o n a l m a t r i x for any delay ( n o t necessarily equal t o t h e block-pulse w i d t h or a m u l t i p l e of i t ) . L a t e r , Chen and Lee, 1982 [46] and C h e n , 1982 [40] used Walsh functions t o analyse m u l t i - d e l a y systems. K u n g and Lee, 1983 [166] f o u n d i t convenient t o a p p l y unshifted Laguerre p o l y n o m i a l s for the analysis o f single-delay causal systems i f the time-delay was s m a l l . Rao and Palanisamy, 1984 [257] presented a W a l s h f u n c t i o n based approach by i n c o r p o r a t i n g the nonzero i n i t i a l conditions. H w a n g and Chen, 1985 [119], Chang and W a n g , 1985 [16] a n d Lee and K u n g , 1985 [174] presented a Legendre p o l y n o m i a l approach for the analysis of systems c o n t a i n i n g delays i n b o t h the state and the c o n t r o l . A l t h o u g h t h e approach of Hwang and Chen [119] is c o m p u t a t i o n a l l y a t t r a c t i v e for the analysis of m u l t i - d e l a y systems and piecewise constant time-delay systems, t h e accuracy appears t o have suffered as an a p p r o x i m a t e delay o p e r a t i o n a l m a t r i x is used. I n c o r p o r a t i n g the effect of delay together with integration, an o p e r a t i o n a l m a t r i x was suggested by Lee and Kung [174], an i m p r o v e d version of w h i c h is o u t l i n e d i n Section 4.3. Delay systems were also investigated by H o r n g and C h o u , 1985 [89] v i a Tchebycheff p o l y n o m i a l s of the first k i n d and by C h a n g et a l , 1985 [13] v i a modified Laguerre p o l y n o m i a l s . A n i n t e g r a l approach to o b t a i n the delay o p e r a t i o n a l m a t r i x of any class of ort h o g o n a l p o l y n o m i a l s and sine-cosine functions f r o m its d e r i v a t i v e o p e r a t i o n a l m a t r i x was proposed i n [209] for the analysis of delay systems. H w a n g and C h e n , 1986 [120] used Legendre p o l y n o m i a l s together w i t h Galerkin's m e t h o d t o s t u d y delay systems. L a t e r , M o u r o u t s o s and Sparis, 1986 [217] also investigated this p r o b l e m v i a sinecosine functions. T h e i r delay o p e r a t i o n a l m a t r i x , a p p r o x i m a t e i n its representation, is q u i t e different from the one i n t r o d u c e d i n this chapter. Since each of the finite range o r t h o g o n a l p o l y n o m i a l systems is a special class of Jacobi p o l y n o m i a l system, H r o n g and C h o u , 1986 [88] used Jacobi p o l y n o m i a l s w i t h a view t o m a k i n g the approach more general c u l m i n a t i n g i n the use of generalized o r t h o g o n a l p o l y n o m i a l s by Lee and C h a n g , 1987 [176] and Lee and Tsay, 1987 [179] and W a n g et a l , 1987 [331]. W a n g et a l , 1987 [332] demonstrated the advantage of t h e generalized block-pulse functions over o r d i n a r y block-pulse functions i n t h e analysis o f systems w i t h multiple t i m e - v a r y i n g delays. T h e Taylor p o l y n o m i a l s , not o r t h o g o n a l i n n a t u r e , were also applied for the analysis of the delay systems by C h u n g and Sun, 1987 [73], Chen and Y a n g , 1987 [38] and C h u n g and Sun, 1987 [72]. O h k i t a , 1987 [220] used Haar functions for the analysis of delay systems. F r o m the above publications i t appears t h a t the c o m p u t e d response of the delay systems v i a the above o r t h o g o n a l f u n c t i o n approach is not i n good agreement
with
the exact response of the system. A l t h o u g h the accuracy can be i m p r o v e d by using a large n u m b e r of basis functions, this is n o t economical because o f t h e requirement of large c o m p u t e r memory. T h i s p r o b l e m was resolved by Rao and Palanisamy, 1984
4.2 Delay
Operational
129
Matrix
[257] by i n t r o d u c i n g a recursive m e t h o d called single-term function
approach.
piecewise
constant
basis
B u t this approach suffers f r o m the d r a w b a c k t h a t t h e c o m p u t e d
response o f a delay system is o b t a i n e d i n the f o r m of a piecewise constant f u n c t i o n . A new technique, called time-partition
method,
w h i c h seems t o be e x t r e m e l y p o w e r f u l
in a n a l y s i n g any k i n d of time-delay systems v i a any class of o r t h o g o n a l p o l y n o m i a l s is i n t r o d u c e d i n t h i s chapter. T h e c o m p u t e d response by this m e t h o d seems t o be i n a very g o o d agreement w i t h the exact response. A p a r t i c u l a r case of t h i s m e t h o d becomes e x a c t l y i d e n t i c a l w i t h the single-term P C B F approach i f the zeroth degree Legendre p o l y n o m i a l or sine-cosine f u n c t i o n alone is used i n the analysis.
An im-
p o r t a n t a n d i n t e r e s t i n g feature o f this chapter is the c o m p a r a t i v e s t u d y of various o r t h o g o n a l f u n c t i o n approaches t o the analysis of time-delay systems.
4.2
Delay Operational
Matrix
T h e delay o p e r a t i o n a l m a t r i x L o f any o r t h o g o n a l system is o b t a i n e d by d e l a y i n g each element of the basis vector */>(t) (see Section 2.15, equations Eq.2.201 and Eq.2.202) by a k n o w n constant r and e x p a n d i n g the result i n terms of the o r i g i n a l set of basis functions. T h e m e t h o d o f o b t a i n i n g the delay o p e r a t i o n a l m a t r i x L for block-pulse functions and any class o f o r t h o g o n a l p o l y n o m i a l s or sine-cosine functions is o u t l i n e d below. 4.2.1
Delay
Operational
Matrix
for B l o c k - P u l s e
Functions
To o b t a i n the delay o p e r a t i o n a l m a t r i x for block-pulse functions [256], \J> (t — T) for a constant r o b t a i n e d from the given m-vector r/>(t) for t € [ t o , * / ] , is expressed i n terms of the components of V (*) as rj,(t -
T)
= L^(t)
for t
< t < t
0
f
where L is given by N
L
=
(1 — a)A
T
=
(N + ct)(tf
N
=
1,2, • • • , ( m - 1),
A
=
A'
=
0 -l m
0 0
+ aA
N
+
l
,
— t ) / m , 0 < a < 1, 0
(4.1)
^ ( m - l ) x ( m - l )
oi
for i > m.
I t is evident f r o m Eq.4.1 t h a t , i f a = 0 or 1, the delay r is an i n t e g r a l m u l t i p l e of the b l o c k pulse w i d t h (tf — t )/m, 0
and no error is i n t r o d u c e d i n representing V (* — T)
due t o delay, any error whatsoever is on account o f the
finite
term approximation
Chapter
130
4: Analysis
of Time-Delay
Systems
via o r t h o g o n a l functions. I t m a y be n o t e d t h a t by a suitable t i m e scaling and segm e n t a t i o n for the given T, a should be adjusted t o zero or u n i t y for t h e error i n the delay o p e r a t i o n t o vanish. T h i s error free o p e r a t i o n a l m a t r i x can be employed for any i n t e r v a l t € [to,t ] f
4.2.2
Delay
of our interest.
Operational Matrix
Polynomials
for
a n d Sine-Cosine
Orthogonal Functions
I n Section 3.5, i t is already seen t h a t t h e derivative o p e r a t i o n a l m a t r i x D of any class of o r t h o g o n a l p o l y n o m i a l s or sine-cosine functions is always exact.
Hence, j times
successive differentiation of -0 ( i ) represented by Eq.2.202 gives us
W
- at £ P = *<0 W =
(4-2)
1
for j = 0, l , 2 , - - , ( m — 1) when i> ( i ) is an m - d i m e n s i o n a l o r t h o g o n a l p o l y n o m i a l vector. For j > m , t h e o p e r a t i o n a l m a t r i x D
1
reduces t o a n u l l m a t r i x . T h i s property,
however, is not shared by sine-cosine functions. Now the T a y l o r series expansion of V (* — 7") is CO
V(« - 7 > ( t - T) = E ( - l ) V ' # ( < ) / i ! tt)
where u(t) is a u n i t step f u n c t i o n . S u b s t i t u t i n g Eq.4.2 i n t h e above e q u a t i o n we get CO
*(.*-
r)u(t
- T ) =
[J2(-l) T D /j\}^(t) i=o J
j
j
= Ll,(t)
(4.3)
where t > t + r a n d 0
CO
L = £(-l)V'l>Vjl
(4.4)
i=o is the delay operational Eq.4.4 reduces t o
matrix of V (*)> the basis vector. For o r t h o g o n a l polynomials m - l
Tti-iW&m
(4.5)
i=o w h i c h is exact and is given i n terms of r a n d D. However, for sine-cosine functions, L is given by the i n f i n i t e series of Eq.4.4 w h i c h s h o u l d be t r u n c a t e d p r o p e r l y for the purpose of c o m p u t a t i o n . I t is interesting t o note t h a t the recursive formulae of Tchebycheff polynomials of the first k i n d t o generate L developed b y H o r n g a n d C h o u , 1985 [89] a n d that of Legendre p o l y n o m i a l s , developed by Lee a n d K u n g , 1985 [174] can be obtained d i r e c t l y f r o m Eq.4.5. C h a n g et a l , 1985 [13] suggested a new delay o p e r a t i o n a l m a t r i x for modified Laguerre p o l y n o m i a l s w h i c h c o u l d be used over t h e w h o l e i n t e r v a l of orthogonality.
U n l i k e the one derived above, i t is never exact i n its representation
and hence i t always introduces an error i n t h e delay o p e r a t i o n .
4.3 Dela.y-Integra.tion
4.3
Operational
131
Matrix
Delay-Integration Operational
Matrix
I n t e g r a t i n g Eq.4.3 w i t h respect to t from t + T t o t we get 0
I
it> (
— r)da
= L I
<0+T
where E
V ( T
(4.6)
(i)
<0 + r
is t h e i n t e g r a t i o n o p e r a t i o n a l m a t r i x o b t a i n e d by i n t e g r a t i n g V> (t)
T
to + T t o t. Since £
T
is different f r o m E, discussed i n Section 3.4 a n d as E
k n o w n , i t is convenient t o use for LE
from
is n o t yet
r
an a l t e r n a t i v e expression i n terms of k n o w n
T
matrices t o o b t a i n t h e equivalent of LE . T
For Legendre p o l y n o m i a l s Lee and K u n g ,
1985 [174] showed t h a t LE
r
=
E,L
and used t h e same i n the analysis of delay systems. A careful s c r u t i n y reveals t h a t t h e last r o w elements of E,L
are different f r o m those of LE .
elements of E,L,
Therefore LE
T
Since a l l t h e elements o f first ( m — 1) rows of LE
T
T
=fi E,L.
are equal t o t h e corresponding
a m o d i f i e d approach is t o be adopted so t h a t E,L
is equal t o
LE . T
T h i s can be accomplished by first o b t a i n i n g E, and L of (m + 1) - d i m e n s i o n a l basis vector V > + i ( t ) a n d t h e n disregarding t h e last row a n d last c o l u m n of t h e (m + 1) x m
(TO + 1 ) m a t r i x E,L operational
matrix
t o f o r m m x m m a t r i x P w h i c h w i l l be called
delay-integration
. T h i s o p e r a t i o n a l m a t r i x is exactly equal t o LE
for a l l classes of
T
o r t h o g o n a l p o l y n o m i a l s . Therefore using m a t r i x P, Eq.4.6 can be r e w r i t t e n as (4.7)
4.4
Integration Operational Matrix
E
T
for S i n e -
Cosine Functions U n f o r t u n a t e l y , for sine-cosine functions, m a t r i x P cannot be o b t a i n e d i n t h e m a n n e r described above for o r t h o g o n a l p o l y n o m i a l s . Therefore, E
r
is r e q u i r e d t o be derived
i n d e p e n d e n t l y for t h e analysis of delay systems v i a sine-cosine functions.
T o this
end, let t h e sine-cosine f u n c t i o n vector F ( ( ) , see Section 3.1.3, be o r t h o g o n a l over t
0
< t < t.
Then
}
where a = t + r such t h a t t 0
0
< a < tf.
N o w ( i — a) can be expressed i n terms of
sine-cosine functions as m - l
(i -
a)
« / Fo(t) + E 0
UjPjit)
+
f,Fi(t)}
Chapter
132
4: Analysis
of Time-Delay
Systems
where fo = [(*/ + < „ ) - 2 a ] / 2 , / /; = ( * / - « o ) ( - l )
i +
j
= 0,
70>)-
Therefore «
. . *=* , / F„(i) + £
/ ' F»d
=
( t , - t )[-Fj(a)F (t)
f
=
« -
J
m
F (o)da 0
F,{o)do
(4.8)
D
0
/
I o
+ F^t)) / (2j TT)
0
(4.9)
)[F ( )F ( )-F,(t)l/(2j7r) J
a
0
(4.10)
I
for j = 1,2, •• • , ( m - 1). T h e equations Eq.4.8-Eq.4.10 can be c o m p a c t l y w r i t t e n as J
«
F(a)dcr
E F(t) T
where
\{t
+ t-
}
0
2a)
0
£•('/-<»)* t}
2
z
^— F _,(a) m —1 m
l
-h{a)-
r
0
O
a
JF (O)
i )e
^(*/-*o)v
-j-( -t )U
0
F,(a)
±{tf -
°
Mt,-t )e
F,(a)
T
m —1
and e and t / are defined i n Section 3.4. I f r = 0, F
T
F _!(a) m
is exactly equal t o F i n Section
3.4.
4.5
A n a l y s i s of T i m e - D e l a y
Systems
T h e linear t i m e - i n v a r i a n t time-delay system w i t h a time-delay r i n the state and c o n t r o l considered for our analysis can be described by x(i)
=
i x ( t ) + Bu(t)
x(0)
=
c
x(i)
=
x (t),
- r < t < 0;
u(t)
=
u (/),
- r < t < 0;
t
t
+ Fx(i - r) + Gu(i - r)
(4.11)
4.5 Analysis
of Time-Delay
where x ( i ) is an
133
Systems
ra-dimensional
state vector, u ( i ) is an r - d i m e n s i o n a l c o n t r o l vec-
t o r , A, B, F a n d G are constant matrices of a p p r o p r i a t e dimensions, x ( 0 ) is an rad i m e n s i o n a l i n i t i a l state vector, and x ( t ) and u ( t ) are respectively an n - d i m e n s i o n a l 0
t
i n i t i a l state vector f u n c t i o n and an r - d i m e n s i o n a l i n i t i a l c o n t r o l vector f u n c t i o n .
4.5.1
Block-Pulse
Function
Approach
I n t e g r a t i n g Eq.4.11 w i t h respect t o t, we have x(f)
=
c + /
+
Fx(o
[Ax(cr) +
Bu(a)
- T) + Gu(o--
T)\ da
(4.12)
A p p r o x i m a t i n g a l l t h e vector functions i n Eq.4.12 by block-pulse f u n c t i o n s , we have, x(«)
ss
X4{t)
(4.13)
where X Xj(j
= 0 , 1 , . . . , m — 1) is an
—
[x
Xi ••• x _ i ] ,
0
m
ra-dimensional
c o l u m n vector,
c
=
O ( i ) ,
C
=
[c c • • • c ] ,
(4.14)
u(r)
«
U^,(t),
U
=
[u
(4.15)
U j •••u _ ],
0
m
1
a n d U j , ( i = 0 , 1 , . . . , m — 1) is an r - d i m e n s i o n a l c o l u m n vector. Since x(t v
— T)
'
x(t-r) -^6
/ x ( i - r ) for 0 < t <
T
I x ( i - r ) for r < r <
t
t
=
\
,
.
as
[ * -I- XL]^(t)
=
[ 60
.
s
(4.16)
t
x
X
0
• • •
l
X
o
m
_ i ]
Similarly u(i-r) l/
t
n
[t/ + i7L]V(*)
=
[UJO
(4.17)
6
Uii • • • u
6
m
_j]
S u b s t i t u t i o n of Eq.4.13-Eq.4.17 i n t o Eq.4.12 gives us
=
C^(t)
+{AX
+
G[U + UL}}
+ BU + F[X [\(o-)d(T
b
J
n
b
+XL]
Chapter
134
4: Analysis
of Time-Delay
Systems
Replacing the i n t e g r a t i o n w i t h an i n t e g r a t i o n o p e r a t i o n a l m a t r i x , see Section 3.4, the above r e l a t i o n takes the f o r m
X
=
C + {AX
+ BU + F[X
+ XL] + G[U + UL]}
t
b
T h e o n l y u n k n o w n i n the above m a t r i x equation is X. V
=
C +[BU
+ FX
+ G{U
b
Hence, by s e t t i n g
+
b
E.
UL)]E„
i t is possible t o w r i t e X - AXE,
- FXLE,
=
V
T a k i n g Kronecker p r o d u c t [173] on b o t h sides, we have vec(X)
T
=
[l -Ej®A-(LE.) ®F]~\ec(V) mn
(4.18)
where
vec(X)
; vec(V) =
=
T h u s i t is always possible t o compute x ( t ) for 0 < t < t
f r o m Eq.4.18 and Eq.4.13.
s
4.5.2
Recursive
Algorithm
I n Eq.4.18, a large m a t r i x of order mn is needed t o be i n v e r t e d . Since E, and L are upper t r i a n g u l a r matrices, they are helpful i n o b t a i n i n g recursive formulae and so a c o m p u t a t i o n a l l y laborious large m a t r i x inversion can be avoided. For 0 < t < r , the recursive relations are: l
x
0
=
[/ - Aw]~
x
t
=
[/ - 4UJ]" { [ / + Am] X i _ ! + [ B ( u . , + u )
+
F(x
where k = 1,2,
[c + (Bu
+ Fxj,o +
0
Gu )w] b0
1
t
t t
_
1
+ x
t
t
) + G(u
4 i
_
1
t
+ u )]uj} t t
(N — 1). I f T < t < tf, the recursive relations are:
x
N
x
t
1
=
[I - Aw]'
+
F(x
{ [ I + Aw] x _ ! + [ B ( u _ , + u j v ) +
=
[/ - Aw]'
+
F ( x _ _ ! + x _ ^ ) + G(u _ _
N
jv-i + x ) + G(u
t
0
t
N
1
t
N
jv-i +
{ [ I + Aw]x _i
0
+ [ B ( u _ + u,fc)
k
t
U )]«J} f c
k
N
l
+
1
IU-JV)]™}
4.5 Analysis
of Time-Delay
135
Systems
for k = N + 1, N + 2 , . . . , ( m — 1), where / is an n x n i d e n t i t y m a t r i x , St = tf/m t h e block-pulse w i d t h , w = it/2
=
and N is the n u m b e r of block-pulse functions over
0 < t < T. T h e above recursive relations are derived on t h e basis of t h e c o n d i t i o n , T
=
N
St.
For delay systems, t h e above c o n d i t i o n can be satisfied by a p r o p e r selection o f m over 0 < i < tf such t h a t N always becomes an integer. I n this a l g o r i t h m , a m a t r i x inversion of order n is r e q u i r e d o n l y once.
4.5.3
Orthogonal Polynomial
Approach
For 0 < t < r, the state vector x ( i ) can be c o m p u t e d f r o m vec(X)
- 1
=
[J - E? ® A ] v e c ( V O ,
V
=
C + iBU + FXt
C
=
[c
+
GUtjE,
0_^_0 ] . ( m — 1 ) cola.
For T < t < tf, Eq.4.11 can be integrated w i t h respect t o t t o give us x(r)
=
q + /
+
f
[Ax(o-) + Bu(o-)] da
[Fx(a
- r ) + Gu(a
[Fx (t
-T)
- r)]da
(4.19)
where q = c + /
b
+ G U , ( ( - r ) ] dt
A l l t h e vector functions i n Eq.4.19 are now a p p r o x i m a t e d as a finite series expansion of o r t h o g o n a l p o l y n o m i a l s as
where u(t)
q
=
Q
=
x(t - r )
«
Xi, (t - r)u(t
- T),
(4.21)
u(r - r )
ss
U^{t-
- r),
(4.22)
is a u n i t step f u n c t i o n .
Qi>{t),
(4.20)
[qO-O],
r)u(t
S u b s t i t u t i n g Eq.4.20, Eq.4.21 and Eq.4.22 i n t o
Eq.4.19 a n d m a k i n g use of t h e i n t e g r a t i o n o p e r a t i o n a l m a t r i x of Section 3.4 a n d Eq.4.7, we get X - AXE,
- FXP
= W,
where W = Q + BUE,
+
GUP.
136
Chapter
4: Analysis
of Time-Delay
Systems
Therefore vec(AT)
=
T
[/ - E? 0 A - P
_ 1
® F] vec(VK).
(4.23)
For the above analysis using o r t h o g o n a l p o l y n o m i a l s , no recursive relations can be derived.
4.5.4
Sine-Cosine
Function
Approach
T h i s approach is exactly similar to the o r t h o g o n a l p o l y n o m i a l approach w i t h P i n Eq.4.23 replaced by LE . T
I n this approach, a l l the vectors have (2m — 1) n u m b e r of
elements. For example, Q = [q
0_-_0
J,
2(m-l)col».
Therefore a m a t r i x of order ( 2 m — l ) r t is required to be i n v e r t e d w h i c h is comput a t i o n a l l y not as much a t t r a c t i v e as t h a t of o r t h o g o n a l p o l y n o m i a l approach. As a m a t t e r of fact, the o r t h o g o n a l p o l y n o m i a l approach itself is not c o m p u t a t i o n a l l y a t t r a c t i v e over the block-pulse f u n c t i o n approach as i t involves a m a t r i x inversion of order m r t . However, due to the s u p e r i o r i t y of o r t h o g o n a l p o l y n o m i a l s over blockpulse functions i n a p p r o x i m a t i n g piecewise s m o o t h square-integrable functions, for the analysis of continuous d y n a m i c a l systems, methods must be devised to use the orthogonal polynomials. T h e following two examples show the results of a c o m p a r a t i v e s t u d y using a l l the three families o f o r t h o g o n a l functions. E x a m p l e 4.1
It is required
to obtain the response
of the delay system
x(t)
=
4x(t - 0.25),
x(0)
=
1,
x(t)
=
0, for - 0 . 2 5 < i < 0.
described
by
T h e exact response of the above delay system can be easily c o m p u t e d a n d is given by
X (
( . . _ I ' ~ j I
1 4t 8 t - 4t + 2 ft - 1 6 i + 14f - 2.5 2
3
2
for for for for
0 < t < 0.25, 0.25 < t < 0.5, 0.5 < i < 0.75, 0.75 < t < 1.
B y t a k i n g m = 4, we now a p p l y the a l g o r i t h m s presented i n Sections 4.5.1 — 4.5.3. T h e c o m p u t e d results along w i t h the exact results are as shown i n Tables 4.1 and 4.2. T h e results o b t a i n e d v i a W a l s h functions [49] are exactly i d e n t i c a l w i t h those given i n Table 4 . 1 . T h i s is not surprising as block-pulse functions a n d W a l s h functions have
4.5 Analysis
of Time-Delay
T a b l e 4 . 1 : T h e response x(t) t BPF
x(t)
T a b l e 4.2: T h e response x(t) and sine-cosine f u n c t i o n s . t
137
Systems
i n E x a m p l e 4.1 v i a block-pulse functions
0-0.25
0.25-0.5
0.5-0.75
0.75-1
1
1.5
2.75
4.875
i n E x a m p l e 4.1 o b t a i n e d v i a o r t h o g o n a l p o l y n o m i a l s
Actual
TPl
TP2
LeP
x(t)
x(t)
x(t)
x(t)
0
1
1
1
1
0.25
1
1
1
1
0.5
2
1.76
1.7662338
1.7649269
0.75
3.5
3.1466667
3.1428571
3.146827
1
6.1666667
5.4933333
5.4545455
5.4750282
t
Actual x(t)
LaP x(t)
HeP x(t)
SCF x(t)
0
1
1
1
1
0.25
1
1
1
1
0.5
2
0.7057823
0.5128209
1.5472159
0.75
3.5
-3.8716578
2.0854705
2.3607629
1
6.1666667
-7.0374332
4.6837609
5.5362648
a one-to-one correspondence [256] and t h e y produce the same results i f the n u m b e r of basis functions is the same i n b o t h the cases [277]. T h e response x(t) o b t a i n e d v i a Tchebycheff p o l y n o m i a l s of the first k i n d by H o r n g and C h o u [89] is inferior t o t h a t shown i n Table 4.2. W i t h m = 8, the response
x(t)
o b t a i n e d v i a Tchebycheff a n d Legendre p o l y n o m i a l s by Lee and C h a n g [176] are m u c h inferior t o those given i n Table 4.2. T h e Laguerre approach even w i t h higher values for m c o u l d not produce acceptable results i n this example. T h i s fact was also confirmed by Lee and C h a n g , 1987 [176].
O n the other h a n d , the response x(t)
obtained via
H e r m i t e p o l y n o m i a l s shows m a r k e d i m p r o v e m e n t w i t h increasing m . T h i s is shown i n F i g . 4.1 w h i c h also shows the response x(t) D u e t o the p o o r convergence
o b t a i n e d v i a other o r t h o g o n a l f u n c t i o n s .
p r o p e r t y , the response x(t)
obtained via
sine-cosine
f u n c t i o n s is n o t as accurate as t h a t offered by Tchebycheff p o l y n o m i a l s of t h e
first
a n d the second k i n d and Legendre p o l y n o m i a l s . W i t h a larger value of m , the response x(t)
i n each case, except for Laguerre p o l y n o m i a l s can be very m u c h i m p r o v e d .
E x a m p l e 4.2
Obtain
the response
x(t)
of the linear time-invariant
delay system
de-
138
Chapter
Figure 4 . 1 : Response of x(t)
4: Analysis
of Time-Delay
= 4x(t - 0.25) v i a o r t h o g o n a l functions
Systems
4.5 Analysis scribed
of Time-Delay
Systems
139
by
x(t)
=
-x(t)
x(0)
=
0,
x(t)
=
u(t) = 0 for - 0 . 2 5 < t < 0,
to a unit step control
input
- 2x{t - 0.25) + 2u{t - 0.25),
over 0 < t < 1 tna Laguerre
and Hermite
polynomials.
T h e exact response o f t h e above system is given by 0
for 0 < t < 0.25, <
2 5
2 - 2e-' -°- > x(t)
for 0.25 < t < 0.5, 5
<
2 5
= { - 2 + (2 + 4 t ) e - ( ' - ° - ' - 2 e - < - ° - ' 2
for 0.5 < t < 0.75,
7 5
6 - (4.25 + 2 i + 4 t ) e - " - ° - > + <
5
2 5
(2 + 4 t ) e - < - ° - > - 2 e - ( ' - ° - )
for 0.75 < t < 1.
W i t h m = 6, t h e response x ( t ) c o m p u t e d v i a Laguerre a n d H e r m i t e p o l y n o m i a l s are shown i n T a b l e 4.3 a n d Fig.4.2.
Table 4.3: T h e response x(t) i n E x a m p l e 4.2 v i a Laguerre a n d H e r m i t e p o l y n o m i a l s t
Actual x(t)
Lap x ( t )
HeP x ( t )
0
0
0
0
0.125
0
0
0
0.25
0
0
0
0.375
0.23501
0.23486
0.59456
0.5
0.44293
0.40827
1.6414
0.625
0.59666
0.54855
2.2788
0.75
0.68094
0.65916
2.387
0.875
0.71194
0.74333
1.8849
1.0
0.71174
0.80414
0.74092
C o n t r a r y t o the last e x a m p l e , Laguerre p o l y n o m i a l s have offered satisfactory results w h i l e H e r m i t e p o l y n o m i a l s n o t . These examples suggest t h a t t h e i n f i n i t e range p o l y n o m i a l s do n o t always gurantee acceptable accuracy w h e n applied t o t h e analysis of delay systems. T h e response x(t) o b t a i n e d v i a Laguerre approach by Lee a n d Tsay [179] is f o u n d t o be inferior t o t h e one presented i n T a b l e 4.3. T h i s is perhaps due t o t h e Pade a p p r o x i m a t i o n made by t h e m i n t h e analysis of t h i s delay-differential equation.
140
Figure 4.2: Response of i ( < ) = Hermite polynomials.
Chapter
4: Analysis
of Time-Delay
Systems
- 2x(t - 0.25) + 2 u ( i - 0.25) v i a Laguerre and
4.6 Time-Partition
4.6
Method
141
Time-Partition
Method
To i m p r o v e the accuracy of the response c o m p u t e d for the state vector x ( t ) o f the delay systems i n t h e E x a m p l e s 4.1 and 4.2, p a r t i c u l a r l y over a large t i m e i n t e r v a l , t h e n u m b e r of f u n c t i o n s m i n the basis vector must be increased.
T h e r e is a l i m i t a t i o n
t o this a p p r o a c h , because w i t h an increase i n the value of m , the order of the mat r i x t o be i n v e r t e d increases, thereby i n v i t i n g a n u m b e r of c o n c o m i t a n t disadvantages. Therefore, using the above approach v i a o r t h o g o n a l p o l y n o m i a l s and sine-cosine funct i o n s , t h e c o m p u t a t i o n a l accuracy for the response of a time-delay system cannot be increased
indefinitely.
A careful s c r u t i n y o f the a n a l y t i c a l solutions i n Examples 4.1 and 4.2 reveals t h a t the expressions of the response for a time-delay system over different subintervals are different. T h e o r t h o g o n a l f u n c t i o n representation of such a set o f different functions over t h e whole i n t e r v a l is, i n general, poor. However, the f u n c t i o n over each subint e r v a l can be represented w i t h a reasonably good degree of accuracy by means of o r t h o g o n a l f u n c t i o n s . T h i s suggests t h a t , for a good a p p r o x i m a t i o n of the response x{t)
over the whole i n t e r v a l of our interest, the i n t e r v a l under consideration has to
be d i v i d e d i n t o some convenient n u m b e r of subintervals, over each of w h i c h x(t)
is
t o be c a l c u l a t e d recursively w i t h the i n i t i a l value taken to be the final value of the previous s u b i n t e r v a l . T h e single-term piecewise constant basis f u n c t i o n ( P C B F ) approach proposed i n [257] is based on this p r i n c i p l e . T h i s approach requires a t i m e scaling and n o r m a l i z a t i o n of each s u b i n t e r v a l . Since o n l y the first t e r m of P C B F is used i n t h i s approach, a higher n u m b e r o f divisions to be called time-partitions
of the
i n t e r v a l are required for b e t t e r accuracy. Moreover, the c o m p u t e d response is i n the f o r m of a staircase f u n c t i o n due to the n a t u r e o f the a p p r o x i m a t i o n made by these basis f u n c t i o n s . To get a r o u n d the above difficulties, the time partition the carry-over
spectrum
a < t < b is i n t r o d u c e d . I f {ip (t)}
defined over the i n t e r v a l
is a complete system o f o r t h o g o n a l functions, t h e n
T
f(t)
method w h i c h is based on
of a square-integrable f u n c t i o n f(t)
can be a p p r o x i m a t e d as
m - l
f(t)
n
T
£ / , 0 , ( O = f *(O, r= 0
where m n u m b e r o f functions t/> (t), Vi(<)> • • • > V \ n - i M 0
the set {f
0
f(t)
f,
• • • f -i} m
is called the spectrum
of f(t)
a
r
e
used for the a p p r o x i m a t i o n ,
and f, the spectral vector of
is given by f = [fo fl
•••
/—if.
and
^(t) = [tM<) V>i(0 ••• t i - i ( * ) f m
I t is evident f r o m the expression for / ( i ) t h a t the s p e c t r u m of f(t — r ) i n 6 < t < 2b —a is the same as t h a t of f(t)
in a < t < b if T = b — a. T h i s shows t h a t the s p e c t r u m
142
Chapter
4: Analysis
of Time-Delay
Systems
of / ( < ) over o < t < b is carried over to define the s p e c t r u m o f t h e delayed function f(t — r ) over the next i n t e r v a l b < t < 2b — a and is called the carry-over spectrum. T h e t i m e - p a r t i t i o n technique i n t r o d u c e d above w i l l now be applied t o t h e analysis of three different scalar systems, v i z . , ( i ) a delay system excited by a piecewise continuous signal, ( i i ) a m u l t i - d e l a y system a n d ( i i i ) a piecewise constant delay systyem. E x a m p l e 4.3 It is required scribed by
to compute
x(t)
=
x(t - T) +
x{t ) 0
=
c,
x(t)
=
x (t)
if r = 1,< — 0, c = x (t) 0
the response
u(t) K
'
system
de-
u(t), (4.24)
for t —
b
a
= 1, and the input
b
of the time-delay
T
< t < t. 0
is
J - 2 . 1 + 1.05r, = < I -1.05,
0 < t < 1 1 < t < 2
T h e exact response o f the system is given by J l - l . l i - t - 0.525i , ' ~ I - 0 . 2 5 + 1.575< - 1.075i + 0 . 1 7 5 t , 2
X (
2
3
0 < I < 1 l < t < 2
^ '
'
To c o m p u t e the response x ( i ) over t < t < tf for tf > r , we first check whether or not (tf — t ) is equal to r t i T , where is an integer. I f i t is n o t , tf is increased such t h a t (tf — r ) = r t j r as the final t i m e of our interest w i l l not be affected by such a choice. N o w the delay r is d i v i d e d i n t o n equal number of subintervals such that T = n h, where h is the l e n g t h of the subinterval. T h e n any s u b i n t e r v a l i n t 6 [*o,'/j can be represented as a
0
0
2
2
t + k,r + k h 0
2
+ k,r + (k
a
2
+
l)h
where hi = 0 , 1 , , . . . . , (n.i. — 1) and k = 0 , 1 , . . . , ( n - 1). T h e response x(t) of the syst e m Eq.4.24 is now c o m p u t e d recursively i n each s u b i n t e r v a l i n the following manner. Eq.4.24 is first integrated w i t h respect to r to give us 2
x(t)
= x(t
2
+ k^r + k h)
0
[x(a - T) + u(o)]da
+ J
2
(4.26)
T h e spectra of a l l t i m e functions in Eq.4.26 w i t h respect to the chosen orthogonal system are now o b t a i n e d as follows: x(t) x(t
0
+ k
l T
«
+ k h) 2
T
[x""'*»>] *(0, = [xi*
, , t 3 ,
] ^ (i)
x^"* x(*-r)«[x
( t ,
1 ,
(4.27)
T
2)
= c
(4.28) for k, = k
2
= 0
s
- *»>J V(0
(4.29) for k, = 0
4.6
Time-Partition
143
Method
T h e last expression represents the carry-over s p e c t r u m evaluated f r o m x ( i ) a n d 0
(
u(t)«
1 ,
,
I
lu * *» ] >(t)
(4.30)
We s u b s t i t u t e Eq.4.27-Eq.4.30 i n t o Eq.4.26 and make use of the i n t e g r a t i o n operat i o n a l m a t r i x of shifted functions i n Section 3.4 to o b t a i n [ (*»,*»)]T X
1
=
[
^ ) f
x
+
{ [ x
(*i-i,*>)|T
[ frMfy
+
n
Et
(
4.
3 1
)
2
T h u s once x ' * ' * ' is k n o w n , the response x(t) can be c o m p u t e d f r o m Eq.4.27. Since the s u b i n t e r v a l l e n g t h h is m a i n t a i n e d constant t h r o u g h o u t the c o m p u t a t i o n , the i n t e g r a t i o n o p e r a t i o n a l m a t r i x E, in Eq.4.31 remains u n a l t e r e d . T h e above recursive a l g o r i t h m is now applied v i a different o r t h o g o n a l systems t o time-delay e q u a t i o n Eq.4.24 for o b t a i n i n g the response
x(t).
(i) Block-pulse functions D u e t o the disjoint n a t u r e of block-pulse functions a n d the u p p e r t r i a n g u l a r struct u r e o f the m a t r i x E,, the f o l l o w i n g recursive relations can be easily o b t a i n e d from Eq.4.31: iHM)
=
JHM
=
VnM)
+
l£M
+
x
+
h[y
,toM
+
h[u
4*x.*a) „(*»•*»)
] / ( 2 m ) i
+
1
where k = 1,2, • • • , ( m — 1 ) , xj;* '**' is the fcth component of the vector jrt*»»W a n d 1
2
t h a t o f u ' * ' * ' . A t the end o f each s u b i n t e r v a l the response x(t) k X
f
M
'
=
X[t + fcjT + (fc + l)h]
=
x(t
0
is given by
2
m-l
+ k,r + k h) + 2 £ ( - l )
0
m
+
t
_
1
2
1
1
[ l ^ ' * ' - x(t
+ fcjT +
0
fcjft)]
for Jfc, = 0 , 1 , • • • , (ra, — 1) and fc = 0 , 1 , • • • , (ra — 1). For the n u m e r i c a l p r o b l e m under consideration we choose n, = 2 , n = 1 a n d m = 4 for tf = 2. T h e n /t = l,fej = 0 , 1 and fc = 0. T h e value k, = fc = 0 denotes the s u b i n t e r v a l 0 < t < 1 i n w h i c h the i n p u t u(<) = —2.1 + 1.05t. Hence, 2
2
2
2
u
Since x (t) b
(o,o)
2
[_i.96875,-1.70625,-1.44375,-1.18125]
=
= 1, I
r
x<- -°J = [ l l , l , l , ] ; a f ' )
!
0 )
= c = l
Therefore x °'
0 )
0
x
(
0 , 0 ) 2
= 0.8789062, x[°' = 0.5257812, x x<°'
0)
0)
0 , 0 ) 3
= 0.6695312 = 0.4476562
= x ( l ) = 0.425.
T
Chapter
144 N o w ki = 1 and k Hence
(1,0)
_
of Time-Delay
Systems
= 0 denote the s u b i n t e r v a l 1 < ( < 2 i n w h i c h u ( t ) = —1.05.
2
u
Since x
4: Analysis
(i.°)
_[i.05,1.05,1.05,1.05]
=
T
(0,0)
x
1 , 0 )
l e)
= 0.4036132, x[ '
0
„(i.o)
0.221582, xi
= 0.3346679,
1,0)
= 0.0807616.
Therefore, the response x ( i ) over 0 < t < 2 is as shown i n the Table 4.4 and Fig.4.3.
Table 4.4: Response x ( f ) i n E x a m p l e 4.3 v i a block-pulse functions. t
0-0.25
0.25-0.5
0.5-0.75
0.75-1.0
BPF x(i)
0.8789062
0.6695312
0.5257812
0.4476562
t
1.0-1.25
1.25-1.5
1.5 -1.75
1.75 - 2.0
BPF x(i)
0.4036132
0.3346679
0.221582
0.0807616
( i i ) S h i f t e d T c h e b y c h e f f p o l y n o m i a l s of t h e first k i n d For 0 < t < 1,
= k
= 0 and
2
(0
x, '
0)
1
0
T
= [1,0, O . O f i x ' - ' ' = [ 1 , 0 , 0 , 0 ]
(0,0)
_ j- 1 . 5 7 5 , 0 . 5 2 5 , 0 , 0 , ] 1
1
0
0
-1/4
0
1/4
0
-1/3
-1/2
0
0
-1/4
T
E. = ( 1 / 2 )
.
1/8
1/6 0
Therefore, f r o m Eq.4.31 ( 0
x '
0 )
=
[0.646875,-0.2875,0.065625,0]
Since i n this i n t e r v a l , t/>o(*) = l , 0 i ( t ) = 2 1 — 1 and xb (t) 2
x(t)
2
= 2(2< - l ) - 1 we have
2
,0)
= 1 - l . l t + 0 . 5 2 5 i w h i c h is the exact s o l u t i o n . Also x ° f
For 1 < t < 2,fc, = 1 and k
2
X
S
1,0)
= 0,x
(1.0)
(0,0)
'
T
1
= [0.425,0,0,0] ,u( '
0 )
T
= x ( l ) = 0.425.
Therefore
= [-1.05,0,0,0]'
4.6 Time-Partition
145
Method
F i g u r e 4.3: Response of x(t)
= x(t — 1) + u ( t ) v i a block-pulse
functions.
Chapter
146
4: Analysis
of Time-Delay
Systems
T h e n , f r o m Eq.4.31 ( 1
x '
0 )
3
= [0.2484375,-0.2179687,-0.0359375,5.46875 2
Since 0 ( < ) = l,ti(t) = 2 i - 3 , ^ ( t ) = 2 ( 2 i - 3 ) - 1 and jp (t) i n this i n t e r v a l , the response is O
2
x(t)
3
x 10~ ]
T
3
= 4 ( 2 i - 3 ) - 3(2r - 3)
2
= - 0 . 2 5 0 0 0 0 3 + 1.57500021 - 1.075i + 0 . 1 7 5 i
3
w h i l e the exact response x(t) is given by Eq.4.25. T h e very s m a l l difference between the a c t u a l x(t) and t h e c o m p u t e d x(t) v i a shifted Tchebycheff p o l y n o m i a l s of the first k i n d is due t o the t r u n c a t i o n and round-off errors i n the c o m p u t a t i o n . ( i i i ) S h i f t e d T c h e b y c h e f f p o l y n o m i a l s of t h e s e c o n d k i n d T h e response x(t) c o m p u t e d v i a shifted Tchebycheff p o l y n o m i a l s o f the second k i n d for 0 < t < 1, is s i m p l y the a c t u a l response. B u t , for 1 < t < 2, x(t)
= - 0 . 2 4 9 9 9 6 2 + 1.5749926r - 1.0749956*
(iv) Shifted Legendre
2
3
-I- 0.174999t .
polynomials
T h e response x(t) c o m p u t e d v i a shifted Legendre p o l y n o m i a l s for 0 < t < 1 is the same as the a c t u a l response, while for 1 < t < 2 x(t)
= -0.2499992 + 1.5749988* - 1.0749996*
2
3
-I- 0 . 1 7 5 i .
(v) Shifted L a g u e r r e polynomials Whenever i n f i n i t e range o r t h o g o n a l p o l y n o m i a l s are used t o analyse time-delay systems v i a the t i m e - p a r t i t i o n m e t h o d , the i n i t i a l f u n c t i o n x (t) and the control f u n c t i o n u(t) are assumed t o be k n o w n i n the whole i n t e r v a l of o r t h o g o n a l i t y for a very accurate evaluation of the spectra. T h e functions k n o w n over a finite interval may be s u i t a b l y e x t r a p o l a t e d for this purpose. t
For the p r o b l e m under consideration, ki = k = 0 over the i n t e r v a l 0 < t < 1. A l t h o u g h x (t) = 1 and u(t) = —2.1 + 1.051 are respectively k n o w n over the intervals 2
b
— 1 < t < 0 and 0 < t < 1, i t is assumed t h a t t h e y are given over — 1 < t < oo and 0 < t < oo respectively. Using the shifted Laguerre p o l y n o m i a l s defined over — 1 < t < oo, the carry-over s p e c t r u m is given by 1
0
x'- ' ' = [1,0,0,0]
T
and 0
T
u< '°) = - [ 1 . 0 5 , 1 . 0 5 , 0 , 0 ] ; x < ° ' 1 - 1 0 E.
0
o )
= [1,0,0,0] 0
1 - 1 0
0
0
L 0
0
1 - 1 0
1
T
4.6 Time-Partition
147
Method
Therefore f r o m Eq.4.31 [0.95,-1,1.05, Since ip (t)
= 1,
0
f u n c t i o n x(t)
T
0] . 2
= 1 - i and i> {t)
= 1 - 2 i + 0 . 5 i i n the i n t e r v a l t € [0, oo], the
2
t u r n s o u t to be the exact response given by Eq.4.25. Also x<°'
0)
= x(l)
= 0.425.
For the i n t e r v a l 1 < t < 2, we have fcj = 1 and k Here t o o , a l t h o u g h u(t)
1
2
0
= 0, and hence x j ' ^ = x ^ ° ' ° \
= —1.05 is k n o w n over t € [1,2] we assume i t to be
available over t 6 [ 1 , oo] and t h e n by using shifted Laguerre p o l y n o m i a l s denned over t € [ l , o o ] , we have ( 1
u '
0 )
T
U
= -[1.05,0,0,0] ;x, '
0 )
= [0.425,0,0, 0 ]
T
and f r o m Eq.4.31 ( 1
x ' Since if> (t) = l,ipi(t) 0
1 - 3(< -
0 )
= [0.325,-0.9,2.05,-1.05]
= 1 - ( i - l),ip (t)
= 1 - 2(< - 1) 4- 0.5(t - l )
2
2
T
2
and tp (t)
1) + 1.5(i - l ) - ( t - l ) / 6 for 1 < t < oo, the c o m p u t e d x(t)
the same as the a c t u a l response of the system.
=
3
3
is e x a c t l y
T h e power of the shifted
Laguerre
p o l y n o m i a l s defined i n Section 3.1.1 is w e l l demonstrated i n this example. (vi) Shifted H e r m i t e polynomials F o l l o w i n g a s i m i l a r k i n d of approach as for the shifted Laguerre p o l y n o m i a l s discussed above, the c o m p u t e d response x(t)
v i a shifted H e r m i t e p o l y n o m i a l s ( o r i g i n is
shifted to the desired p o i n t s , see Section 3.1.2) is completely i n agreement w i t h the a c t u a l response. (vii) Sine-cosine functions I n the t i m e - p a r t i t i o n m e t h o d , the s o l u t i o n f u n c t i o n over any s u b i n t e r v a l is cont i n u o u s at the end p o i n t s .
As the Fourier series cannot produce the t r u e value of
the f u n c t i o n at the end points of a d i s c o n t i n u i t y , the t i m e - p a r t i t i o n m e t h o d v i a sine1
cosine functions cannot be a p p l i e d .
2
T h i s is because, the c o m p u t a t i o n of x^* '* ' is
never precise and therefore recursion is not possible. E x a m p l e 4 . 4 It is required described
to compute
the response
x(t)
of a multi-delay
x(<) = x ( i - 0.35) + x(t - 0.7) + u(t), subject
to the conditions
0
x ( i ) = x (t) b
u(t)
orthogonal
is a unit step function functions
(4.32)
that x(t )
where
system
by
approach.
= x ( 0 ) = 0, = 0, for
t < 0,
over 0 < t < 1 using
time-partition
method
via
148
Chapter The
4: Analysis
m
for 0 < t < 0.35;
=I
2
+ 0.5(i - 0.35) ,
for 0.35 < t < 0.7;
, W . 7 6 1 2 5 + 1.35(i - 0.7)
0
2
3
I + (t - 0.7) + J ( I - 0.7) , Let
Systems
exact response o f this system is f tt,,
and
of Time-Delay
7"i = 0.35 a n d r
= 0.7. Since r
2
2
= 2r
L
for 0.7 < t < 1.05.
a n d tj = 1, we can e x t e n d tj t o 1.05
c o m p u t e x{t) over 0 < t < 1.05. Therefore rtj = 3. For n
considering m = 4, the response x(t)
2
=
= r
t
= 0.35. By
o f the system i n Eq.4.32 is c o m p u t e d i n steps
of h = 0.35 w i t h the help o f different o r t h o g o n a l functions as follows. (i)
Block-pulse functions:
I t is possible t o a r r i v e at the following recursive relations w h i c h are v a l i d over each s u b i n t e r v a l defined by feii'i + k h 2
< t < k,Ti + (fc +
\)h
2
where fci = 0 , 1 , 2 and fc = 0: 2
x
(HM
=
^ ^ )
x
+
f-*M
u
^ ^ )
+
(*>-'^)
I
+ ,(*_»,-».*•) +
X
for
M
+
^ -
2
^ )
]
/
(
2
m
)
-'•**>]/(2m)
it = 0 , 1 , 2 , 3 . The
above recursive a l g o r i t h m produces the response x(t) o f t h e s y s t e m i n Eq.4.32
as shown i n Table 4.5 a n d Fig.4.4.
Table 4.5: Response x(t)
in E x a m p l e 4.4 o b t a i n e d v i a block-pulse f u n c t i o n s .
t
0.0-0.0875
0.0875-0.175
0.175-0.2625
0.2625-0.35
x(t)
0.04375
0.13125
0.21875
0.30625
t
0.35-0.4375
0.4375-0.525
0.525-0.6125
0.6125-0.7
x(t)
0.395664
0.4908203
0.5936328
0.7041015
t
0.7-0.7875
0.7875-0.875
0.875-0.9625
0.9625-1.0
x{t)
0.8242243
0.9581643
1.1084216
1.2756662
Orthogonal polynomials The
resursive r e l a t i o n for o b t a i n i n g the response x(t)
o f t h e system given by
Eq.4.32 using t h e o r t h o g o n a l p o l y n o m i a l s can be shown t o be
4.6 Time-Partition
Method
149
Chapter
150
[x<^)]
T
il
=
4: Analysis
of Time-Delay
Systems
l)
[x< '* f + {
j
x
( t a - i . * * ) ] : r
+
[
x
( *
1
-
2
, *
2
)
]
T
[
+
u
( * t , * , ) ] r
}
£
j
T h i s recursive r e l a t i o n can be specialized t o different o r t h o g o n a l systems, the c o m p u t e d response x(t)
i n each case is as given below.
(ii)
S h i f t e d T c h e b y c h e f f p o l y n o m i a l s of t h e f i r s t k i n d :
For
0 < t < 0.7, the c o m p u t e d response x ( i ) is exactly t h e same as the actual
response x(t)
while
x{t)
=
2
0.7612499 + 1.3500009(2 - 0.7) + 0.9999973(i - 0 . 7 ) + 3
(/ - 0.7) /6.0000096 (iii)
S h i f t e d T c h e b y c h e f f p o l y n o m i a l s of t h e s e c o n d
x( ~
(iv)
for 0.7 < t < 1
(i I j I
t,
kind:
for 0 < t < 0.35; 2
1.00000031 + 0.4999993(t - 0 . 3 5 ) , 0.76125 + 1 . 3 5 0 0 0 0 3 ( i - 0 . 7 ) + 0.9999976(t - 0 . 7 ) + [t - 0 . 7 ) / 6 . 0 0 0 0 3 4 7 , 2
for 0.35 < t < 0.7;
3
for 0.7 < t < 1.05.
Shifted Legendre polynomials: ( t, I
for 0 < i < 0.35 2
1.0000011* + 0.4999967(i - 0 . 3 5 ) ,
for 0.35 < t < 0.7
j 0.7612503 + 1.3500004(t - 0 . 7 ) + 2
3
I 0.9999975(i - 0 . 7 ) + (r - 0.7) /6.000028, (v)
for 0.7 < t < 1.05.
Shifted L a g u e r r e polynomials:
T h e exact response is obtained using the Laguerre p o l y n o m i a l s . (vi)
Shifted H e r m i t e polynomials:
For 0 < t < 0.7, the c o m p u t e d response x(t)
is t h e same as the exact response of
t h e system. However for 0.7 < t < 1.05, x(t)
2
= 0.76125 + 1.3500004(t - 0.7) + {t - 0 . 7 ) + (t -
E x a m p l e 4.5 It is required
to analyse
the piecewise
constant
3
0.7) /6.0000096 delay system
described
by x(t)
=
x[t - a ( i ) ] + u ( t ) •]
x(0) =
0
[
x(t)
0 for t < 0
J
=
where f 0.1, a{t)
= \
0.3,
I 0.5,
0 < t < 0.35, 0.35 < t < 0.7, 0.7 < i < 1.
(4.33)
4.6 Time-Partition
Method
151
I n order t o analyse this delay system, a different t i m e - p a r t i t i o n i n g is r e q u i r e d . Since t
0
= 0,t
= l . T j = 0.1, r
f
2
= 0.3 and r
step size so t h a t the response x(t) k,h
< t < ( & ! + l)h,
3
= 0.5 we m a y select h = 0.05 as the
of the system i n Eq.4.33 can be c o m p u t e d over
where fe = 0 , 1 , . . . , 19. T h e recursive formulae i n this case can x
be d i r e c t l y o b t a i n e d f r o m E x a m p l e 4.3. F r o m Eq.4.31 we w r i t e
r
T
T
[x<*'>] = [ x ^ ' f + {[x<*>-'>] + [ u ^ ] } £ . , where
f 2 I = { 6 [ 10
for k, = 0 , 1 6, for A:, = 7 , 8 , . . . , 13, for fc, = 1 4 , 1 5 , . . . , 19.
For the block-pulse functions, the simplified recursive relations are: 4'°
where k = 1,2,
=
^
)
+ M4*
l )
+ 4
k l
-
1 )
]y(2m)
, ( m — 1). A t the end of each s u b i n t e r v a l , the response x(t) is given
by,
4*°
=
*K*j + i)A]
=
X(k,h)
+ 2 £
(-l)
m +
*
_ 1
[xi*
, )
-
*(*!&)]
for fc, = 0 , 1 , . . . , 19. W i t h the recursive formulae and m = 5, the system i n Eq.4.33 is analysed v i a block-pulse f u n c t i o n s , shifted Tchebycheff p o l y n o m i a l s o f the first and second k i n d s , shifted Legendre p o l y n o m i a l s , shifted Laguerre p o l y n o m i a l s and shifted H e r m i t e p o l y n o m i a l s and the response x(t) w i t h t h e exact response x(t),
i n each case is p l o t t e d for the purpose of comparison see Fig.4.5. I n Fig.4.5 i t is seen t h a t the response
x(t)
o b t a i n e d v i a o r t h o g o n a l p o l y n o m i a l s agrees excellently w i t h the a c t u a l response. A l t h o u g h the block-pulse s o l u t i o n is not s m o o t h , i t is i n close agreement w i t h the exact response
x(t).
I n t h e last three examples i t is observed t h a t the time-partition
method
produces
excellent results and hence this m e t h o d should be used for the efficient analysis of t i m e - d e l a y systems. T h e first basis f u n c t i o n i n a l l the classes of o r t h o g o n a l p o l y n o m i a l s or sine-cosine functions is a u n i t pulse on an a p p r o p r i a t e i n t e r v a l of o r t h o g o n a l i t y . S i m i l a r l y , the first t e r m i n the systems of a l l piecewise constant basis functions ( P C B F ) happens to be a u n i t pulse.
For ffi = 1 , the block-pulse s p e c t r u m , Haar s p e c t r u m , W a l s h
s p e c t r u m , Legendre s p e c t r u m , or Fourier s p e c t r u m of any square-integrable f u n c t i o n
152
Chapter
4: Analysis
of Time-Delay
Systems
4.7
153
Problems
/ ( < ) is s i m p l y an average value o f f(t)
over the i n t e r v a l i n w h i c h / ( i ) is square-
i n t e g r a b l e . M o r e o v e r , for m = 1 the i n t e g r a t i o n o p e r a t i o n a l matrices of block-pulse f u n c t i o n s , Legendre p o l y n o m i a l s a n d sine-cosine functions are e x a c t l y i d e n t i c a l , i.e., E
B
= E
P
= E
F
= (tj — i ) / 2 .
T h i s is t r u e even for W a l s h functions and
0
Haar
f u n c t i o n s . So the single-term Legendre p o l y n o m i a l or sine-cosine f u n c t i o n approach is e x a c t l y the same as the single-term P C B F approach [257]. Therefore, the t i m e p a r t i t i o n m e t h o d v i a the first t e r m of shifted Legendre p o l y n o m i a l s or
sine-cosine
functions produces the same results as the single-term P C B F approach w h e n a p p l i e d to the analysis of d y n a m i c a l systems, i n general, and delay systems, i n p a r t i c u l a r . I n t e r e s t i n g l y , i t has one a d d i t i o n a l advantage t h a t i t involves no t i m e scaling and normalization.
4.7
Problems
1. F i n d a s o l u t i o n t o the state e q u a t i o n x(t) w i t h x(t)
= 1.0 and u(t)
= - 1 0 x ( i ) - x(t - 0.25) +
= 0.0 for t < 0, and u(t)
2u(t)
= 1.0 for t > 0. C o m p a r e the
c o m p u t e d s o l u t i o n w i t h the exact s o l u t i o n given by 0.1 + 0 . 9 e x p ( - 1 0 t ) , 0.19 + (0.135 - 0 . 9 t ) e x p [ - 1 0 ( i - 0.25)]
if0<<<0.25;
+ 0.9exT»(-10<),
i f 0.25 < t < 0.5; 2
0.181 + (0.0765 - 0.36* + 0 . 4 5 t ) e x p [ - 1 0 ( i - 0.5)] + (0.135 - 0.9t) x(i) = \
e x p [ - 1 0 ( i - 0.25)] + 0 . 9 e x p ( - 1 0 t ) , 0.1819 + (0.04381875 2
i f 0.5 < i < 0.75;
0.194625<+
3
0.2925< - 0 . 1 5 i ) e x p [ - 1 0 ( i - 0.75)] + 2
(0.0765 - 0.36i + 0 . 4 5 t ) e x p [ - l 0 ( t - 0.5)] + ( 0 . 1 3 5 - 0.9t)exj>[-10(* - 0.25)] +0.9exp(-10t),
i f 0.75 < t < 1.0.
2. O b t a i n a s o l u t i o n t o the state e q u a t i o n x(t)
= - 5 x ( i ) - 5 x [ i - t (t)\ x
+ 2u(<)
subject t o the conditions
t (t ^' ~ T
f 0,
0 < i < 0.8;
I
0.3,
0.8 < t < 1.4;
j 0.6,
1.4 < t < 1.7;
I
1.7 < t < 2.0.
0.9,
154
Chapter
4: Analysis
of Time-Delay
Systems
and w i t h x{t) = 1.0 , u{t) = 0.0 for t < 0, a n d u(t) = 1.0 for t > 0. C o m p a r e the c o m p u t e d s o l u t i o n w i t h t h e exact solution given by if 0 < t < 0.8
10
0.2 + 0 . 8 e - ' , 3
4
3
i f 0.8 < t < 1.1
5 i
0.2 + 0 . 8 e - " " + 0 . 8 e " [ l - e ] e - , 6
1 0
2 5
3
0.2 + 0 . 8 e - ' - 4 e - [ l 3
+0.8(1 - e ][6.5e9
2
5
5
e ]t - ' e
if 1.1
5
+ e^e" ',
10<
1
3
5 <
- 4e" [l - e ]i = • 0.2 + 0 . 8 e +0.8[1 - e ][8e-' - 0 . 5 e - + e " ] e - ' ,
x(t)
e
3
2 5
1 2
1 0
4
0 5
3
0.2 + 0 . 8 e - ' - 4 e [ l - e ] i e 3
+ 0 . 8 [ 1 - e ][9.5e 2
5
4
0
5
- O^e"
5
if 1.4 < t < 1.7
5 <
1
5
-0.5e" - + e- ]e- ',
if 1.7 < * < 2.0
3. O b t a i n a s o l u t i o n to t h e scalar delay-differential e q u a t i o n y(t) i.e., o b t a i n y{t),0 -r
= y{t) + u(t -
T)
< t < 1 subject t o t h e conditions y ( 0 ) = 1 . 0 a n d y(t) = 0 for
< t < 0 when r = 0 . 0 , 0 . 1 2 5 , 0 . 2 5 , 0 . 3 7 5 , 0 . 5 , 0 . 6 2 5 , 0 . 7 5 , 0 . 8 7 5 a n d 1.0. Compare
the n u m e r i c a l l y calculated solution w i t h t h e exact s o l u t i o n given b y y(t)
= exp(t)
+ [exp(t
— r ) - 1]M(I - r )
4. F i n d t h e s o l u t i o n o f = y(t) + y(t - r ) + u(t)
y{t)
subject t o t h e conditions y(0) = 0.0 a n d y(t) = exp(t) for 0 < t < 1 a n d
T
for — r < t < 0 w h e n u(t) = 1
= 0.25.
5. For the system described by ' •
(t)'
X1
1 ' ' x,(t) '
' 0
. 0 0 .
+
. x (t) 2
.
+
0 -0.3
* , ( t - 1) x (t - 1) 2
0 u(t
1) + 0.3r(r - 1)
'^(0) . * (0) . 2
0 ' . 0 .
where u(t— 1) a n d r(t — 1) are u n i t step and r a m p functions, respectively, each delayed by 1.0 sec, o b t a i n t h e s o l u t i o n over 6.0 sees.
4.7
155
Problems Consider the second order system
i,(l)
1
0
»x(*-j)
. -25
-5 .
X i(0) ' i x (0) J
" 0
x (t) 2
-
for
. 0
2
i)
0 '
+
. 1 .
u(t)
< t < 0
where u ( r ) = 1.0 is a u n i t step. F i n d x ( t ) for t € [ 0 , 1 ] .
7. Consider the f o l l o w i n g v e c t o r - m a t r i x differential e q u a t i o n w i t h delay i n state:
r o
f*i(oi
i (i) 1 L i (0 J
I
=
s
0
L —2
3
[ j
+
-
i
o i l
0
l j j x (t) 2
—5
0 0
0 0
[ -2
-3
—3
i I
J L x (t) J 3
0 ] [ 1,(1-0.1) ] [ 0 ] O i l x (t - 0.1) I + I 0 | u(t)
L x (t 2
-2
J
3
- 0.1)
J
L1J
f «i(0 1 | x (t)
[ 0 ] I = j 0 j for t < 0
L x (t)
\
2
3
i/(i) = [ 2
3
L0 J
2 ][*,(*-0.1)
C o m p u t e y{t) for 0 < t < 2.0.
x (t-0.1) 2
x (<-0.1)] 3
T
Chapter
5
Identification Parameter
of
L u m p e d
Systems
T h e m e t h o d s t o estimate the parameters and wherever possible, the i n i t i a l c o n d i t i o n s , of linear t i m e - i n v a r i a n t systems f r o m measurements of the i n p u t s and o u t p u t s
are
o u t l i n e d i n t h i s chapter using o r t h o g o n a l functions. F i r s t , s i n g l e - i n p u t single-output systems are considered and t h e n m u l t i - i n p u t and m u l t i - o u t p u t systems whose m o d e l is assumed i n the f o r m of a transfer f u n c t i o n m a t r i x . To i m p r o v e the accuracy, one shot o p e r a t i o n a l m a t r i x for repeated integrations of the o r t h o g o n a l functions is i n t r o d u c e d .
5.1
Introduction
T h e l a n d m a r k paper by C o r r i n g t o n , 1973 [75] for the s o l u t i o n of differential a n d i n t e g r a l equations v i a W a l s h functions p r o v i d e d the i m p e t u s t o Chen and Hsiao, 1975 [34] for a p p l y i n g these functions t o e s t i m a t i n g the parameters of linear t i m e i n v a r i a n t systems w i t h zero i n i t i a l c o n d i t i o n s . I n the same year Rao and S i v a k u m a r , 1975 [264] also successfully investigated the same p r o b l e m v i a W a l s h functions, considering the effect o f i n i t i a l c o n d i t i o n s .
A f t e r five years, Palanisamy and B h a t t a c h a r y y a ,
1981
[233] i n t r o d u c e d a recursive approach using the block-pulse functions for the same i d e n t i f i c a t i o n p r o b l e m . B o h n , 1982 [7] developed a parameter i d e n t i f i c a t i o n m e t h o d by expressing i n t e g r a l functions i n terms of periodic i n p u t - o u t p u t d a t a . T h i s m e t h o d was represented i n a f o r m suitable for i m p l e m e n t a t i o n on a microprocessor w i t h the help o f W a l s h f u n c t i o n s . A l l the above p u b l i c a t i o n s d i d n o t discuss the system i d e n t i f i c a t i o n i n a noisy e n v i r o n m e n t . T h i s issue was taken up by H w a n g and S h i h , 1982 [136] by using Laguerre polynomials.
T o estimate the parameters, they [136] i n t r o d u c e d t w o i d e n t i f i c a t i o n
algorithms.
One, for the zero i n i t i a l conditions and the other, for nonzero i n i t i a l
conditions.
I n the same year, Cheng and H s u , 1982 [54] investigated the l u m p e d
p a r a m e t e r system i d e n t i f i c a t i o n i n a noisy e n v i r o n m e n t by using block-pulse functions.
Rao a n d S i v a k u m a r , 1982 [262] presented a W a l s h f u n c t i o n m e t h o d for the
158
Chapter
5: Identification
of Lumped
Parameter
Systems
simultaneous identification of the order and the parameters o f a single-input singleo u t p u t system. I n this c o n t e x t , they stated t h a t the W a l s h spectra were i m m u n e to zero-mean a d d i t i v e noise to some extent, and so was the i d e n t i f i c a t i o n a l g o r i t h m e m p l o y i n g t h e m . T o estimate the parameters v i a Legendre p o l y n o m i a l s C h a n g and W a n g , 1982 [19] adopted the same i d e n t i f i c a t i o n a l g o r i t h m of H w a n g and Shih [136] w h i c h was n o t valid for higher t h a n first-order systems. L a t e r , by considering zero i n i t i a l conditions K u n g and Lee,1982 [167] investigated the i d e n t i f i c a t i o n of a u n i t y feedback system v i a Laguerre p o l y n o m i a l s .
I n order
to reduce the error i n the repeated integrations of a W a l s h basis vector, Rao and Palanisamy, 1983 [258] i n t r o d u c e d one shot o p r a t i o n a l m a t r i x for repeated integrations ( O S O M R I ) and d e m o n s t r a t e d its s u p e r i o r i t y over the conventional i n t e g r a t i o n o p e r a t i o n a l m a t r i x i n the p r o b l e m of i d e n t i f i c a t i o n . T h i s concept of O S O M R I was not new and was, i n fact, taken into consideration by C o r r i n g t o n for the solution of differential and integral equations [75]. Paraskevopoulos, 1983 [237] considered a different m o d e l c o n t a i n i n g no t i m e derivative terms i n the forcing functions to estim a t e b o t h the parameters and the i n i t i a l conditions v i a Tchebycheff p o l y n o m i a l s of the first k i n d .
He suggested a linear t r a n s f o r m a t i o n to m a t c h the i n t e r v a l of mea-
surements w i t h the i n t e r v a l of o r t h o g o n a l i t y of Tchebycheff p o l y n o m i a l s of the
first
k i n d and redefined the o r i g i n a l m o d e l of the system accordingly. L a t e r , for the same m o d e l Paraskevopoulos and Kekkeris, 1983 [240] used H e r m i t e p o l y n o m i a l s . d i d not discuss the p r a c t i c a l l i m i t a t i o n s of this approach.
They
M o u r o u t s o s and Sparis,
1985 [218] presented a Taylor series-based m e t h o d , as i t had considerable computat i o n a l advantages due to the simple s t r u c t u r e of the i n t e g r a t i o n o p e r a t i o n a l m a t r i x of T a y l o r p o l y n o m i a l s over o r t h o g o n a l p o l y n o m i a l s when the i n p u t a n d o u t p u t signals were a n a l y t i c functions of t.
W a n g and M a r l e a u , 1985 [323] i n t r o d u c e d a system
i d e n t i f i c a t i o n m a t r i x v i a generalized block-pulse o p e r a t i o n a l matrices to i d e n t i f y the coefficients of a transfer f u n c t i o n from the d a t a of a u n i t step response. These publications [167,258,237,240,218,154]
again d i d not discuss the i d e n t i f i c a t i o n in a noisy
environment. W i t h a view to m a k i n g the identification process more general, W a n g et a l , 1987 [334] proposed a generalized
o r t h o g o n a l p o l y n o m i a l approach using the identifica-
t i o n a l g o r i t h m of H w a n g and Shih [136]. W i t h the same o b j e c t i v e , they [335] again proposed the same generalized o r t h o g o n a l p o l y n o m i a l approach w h i c h , this t i m e , rel y i n g on the derivative o p e r a t i o n a l m a t r i x as was used i n [334]. T h e y stated t h a t the parameter e s t i m a t i o n must start at zero t i m e i f one wished to use the i n t e g r a t i o n operational m a t r i x i n the i d e n t i f i c a t i o n process. T h i s disadvantage, i n t h e i r v i e w , could be prevented by using the derivative o p e r a t i o n a l m a t r i x for w h i c h the knowledge of i n i t i a l conditions was not required. T h i s approach based on successive differentiation is sensitive to noise and is therefore not acceptable. I t is o n l y t o avoid t h i s undesirable s i t u a t i o n , all the i d e n t i f i c a t i o n techniques using m e t h o d f u n c t i o n s , m o d u l a t i n g functions, Poisson m o m e n t functionals, direct i n t e g r a t i o n as i n the present t e x t , etc.
are
all based on the p r i n c i p l e of repeated integrations. Secondly, the knowledge of i n i t i a l
5.2 One Shot
Operational
Matrix
for Repeated
159
Integrations
c o n d i t i o n s is not at a l l r e q u i r e d for the i d e n t i f i c a t i o n as the augmented
parameter
vector (see Section 3.3) includes terms due t o these i n i t i a l c o n d i t i o n s . C h u n g , 1987 [70] suggested the Fourier series approach for system i d e n t i f i c a t i o n . He considered the m o d e l of Paraskevopoulos,
1983 [237] for this purpose.
Recently, by assuming
zero i n i t i a l c o n d i t i o n s L i o u and Chou,1987 [188] studied the system parameter i d e n t i fication
v i a piecewise linear p o l y n o m i a l functions, w h i c h were i n fact, n o n o r t h o g o n a l
functions.
T h e preceding p u b l i c a t i o n s [335,70,188] again d i d not discuss p a r a m e t e r
e s t i m a t i o n i n a noisy e n v i r o n m e n t . I n this chapter, an a l g o r i t h m is developed w h i c h is different f r o m t h a t given i n [136] a n d is applicable v i a o r t h o g o n a l p o l y n o m i a l s a n d sine-cosine functions i n a noisy e n v i r o m n e n t . T h e concept of O S O M R I is extended t o a l l o r t h o g o n a l p o l y n o m i a l s and sine-cosine functions a n d a unified i d e n t i f i c a t i o n approach is developed so t h a t the same can be a p p l i e d v i a any class of o r t h o g o n a l systems t o the system p a r a m e t e r i d e n t i f i c a t i o n . F i n a l l y , the results o b t a i n e d v i a a l l o r t h o g o n a l functions are compared to f i n d out the systems of o r t h o g o n a l p o l y n o m i a l s w h i c h are suitable for the purpose of i d e n t i f i c a t i o n . A p a r t of the w o r k i n c l u d e d below m a y be f o u n d i n [214,213].
5.2
O n e S h o t O p e r a t i o n a l M a t r i x for R e p e a t e d Integrations
(OSOMRI)
To develop the i n t e g r a t i o n o p e r a t i o n a l m a t r i x E, of an o r t h o g o n a l basis vector r
il>{r),
s
every element of V > ( ) i i n t e g r a t e d once w i t h respect t o r and the result is expressed i n t e r m s of the o r i g i n a l set of basis functions, see Section 3.4. I n other words T
l where E
i,(r)dr
T
x
= E,. I f / ( r ) « f ip{r) f
f(r)dT T
=
V
.(r)« £
l
¥
(r)
(5.1)
= /*(r),then = i
T
J\
(r)dr
T
» t Erf
(r)
(5.2)
TO
0
T h e error due t o a p p r o x i m a t i o n i n the i n t e g r a t i o n process is given by ei(r) = f
T
f\r)dr-i E^{T). TO
N o w , let the r e l a t i o n i n Eq.5.2 be i n t e g r a t e d once more w i t h respect t o r , t h e n we have f r o m Eq.5.1
fP TO
f'(h)dt
2
TD
dt, «
T
f E,J
T
y> ( * , ) * !
Kt E E 4,(T) l
l
TO T
2
= f £ V(r)
(5.3)
160
Chapter
5: Identification
of Lumped
Parameter
Systems
T h e error i n the process o f two i n t e g r a t i o n s i n Eq.5.3 is given by
T
0
To
A l t e r n a t i v e l y , i t is also possible t o w r i t e t h a t J TO
= ^ j ( r ) R*
J ' ^(t^dt^dt, To
E rl>{r) 2
so t h a t f
T
X
f
T
f'(t )dhdt rV 2
t E i>(r)
1
(5.4)
i
T h e error i n the process of i n t e g r a t i o n of Eq.5.4 is given by
e
r
2 < )
=
/
/ TO
' f ' ^ )
d
t
*
d t
i
~
{
T
e
^
r
( )
T
0
T h e expression £ ( r ) is the accumulated error at the end of t w o i n t e g r a t i o n s , each 2
one w i t h an error of £ i ( r ) . B u t , a l t h o u g h £ ( r ) also represents t h e error at the end of 2
two i n t e g r a t i o n s , this error is due t o the a p p r o x i m a t i o n i n t r o d u c e d at t h e final stage of t h e i n t e g r a t i o n only. N a t u r a l l y £ ( r ) is greater t h a n e j ( r ) . 2
I n general, for k times repeated integrations of V ( T ) we have
(5.5) where E
k
is called one shot o p e r a t i o n a l m a t r i x for repeated i n t e g r a t i o n s ( O S O M R I ) .
T h e expression for O S O M R I for a l l classes of piecewise constant o r t h o g o n a l functions is available in the l i t e r a t u r e [258]. T h i s concept of O S O M R I w i l l now be extended to a l l classes of o r t h o g o n a l p o l y n o m i a l s and sine-cosine f u n c t i o n s .
For t h e sake of
completeness, we include below the O S O M R I of block-pulse f u n c t i o n s .
5.2.1
O S O M R I
The O S O M R I E
k
E
k
=
of Block-Pulse
Functions
for block-pulse functions can be represented as [258] (Tf ~ r„)*
i n w h i c h I is an m x m i d e n t i t y m a t r i x and
0
:
fy»-i)x(™-.ij
A = L 0
0'
5.2 One Shot
Operational
Matrix
for Repeated
161
Integrations
For m — 4 a n d k = 2 we have
E
=
2
E x a m p l e 5.1 O S O M R I
16
1/4
1
2
3
0
1/4
1
2
0
0
1/4
0
0
0
1 1/4
1/6
1
2
0 0
1/6 0
1 1/6
1
0
0
0
1/6
3 2
of Block-Pulse Functions
/ ( T ) = 1,0 < T < 1 twice with respect to r from
It is required
to
2
T h e exact value o f the given f u n c t i o n u p o n twice i n t e g r a t i o n is g(r) m = 4,r
0
integrate
zero to r. = T / 2 . For
= 0 and Tj = 1, we have t h e block-pulse spectra for / ( r ) and g(r)
as
f = [1,1,1,1,f g = [0.0104166,0.0729166,0.1979166,0.3854166] g
T
1
v i a E] = fE]
= [0.015625,0.078125,0.203125,0.390625] "
and g
via E
2
= fE
= [0.0104166,0.0729166,0.1979166,0.3854166]
2
5
I t m a y be observed t h a t the spectral vector g o b t a i n e d v i a O S O M R I E
2
is exactly
t h e same as t h e a c t u a l g .
5.2.2
O S O M R I First
of Shifted
Tchebycheff
Polynomials
Kind:
F r o m t h e development o f Section 3.4 we have, 2
( r - r ) [ 0 . 7 5 V o ( r ) + t/, (r) + /
/
f
*
Mt )dt dt 2
2
l
0
1
0.25t/> (r)]/4 2
/
J>,(i )di
(*7 - r « ) ' [ - W O / 3 - 3 ^ , ( 0 / 8 +
2
To To
0 (r)/24]/4 3
f
J
%l> (t )dt dt 2
2
2
1
2
(T-/ " r ) [ - 3 V o ( r ) / 1 6 - V r ( r ) / 3 0
^ ( r ) / 6 + ^ ('-)/48]/4 2
4
of the
Chapter
162
5: Identification
of Lamped
Parameter
Systems
and
/
J '
i/>,{t )dt dt 2
2
i
2
=
2
(*/ - r „ ) { ( - i y + V o ( r ) / ( i - 4 ) , +
V (r)/(i
2
+
(-l)
-
0 , - ( r ) / [ 2 ( i * - 1)] + V
1
- 1) + 0 , _ ( r ) / [ 4 ( i - i ) ( < - 2)] 2
i + 2
( r ) / [ 4 ( i + l ) ( i + 2)]}/4
for i = 3 , 4 , 5 , . . . A l l the above expressions may be c o m p a c t l y w r i t t e n i n a vectorm a t r i x form as
J
^(t^diidij
J T
=
[£ ,e,e,] I 0 ( r ) 2
m
T
°°
L 0 l(T) J m+
F V>(r)
(5.6)
2
Now,for m = 4 we have
r )
I" 3/4 I -1/3
2
0
-3/16
L 5/24 - r„)
2
1/4 0
-1/3
-1/6
1/4
3/4 {r,
1 -3/8
0
1
-1/3 -3/16 L 1/5
0 1/24 0 -1/24
1/4 0
-3/8 -1/3 1/4
0
1/24 -1/6 0 0 -1/16 E x a m p l e 5.2 O S O M R I o f T c h e b y c h e f f P o l y n o m i a l s o f t h e F i r s t K i n d It is required
to integrate
the
function / ( r ) = 1 + r + 2r
2
3
+ 3r , - 1 < r < 1
twice with respcet to r from T = —1 to r . T h e exact value of the above f u n c t i o n u p o n twice i n t e g r a t i o n is g(r) . W i t h m = 4, r / ( r ) and g(r) as
2
3
5
= ( 1 + 5 r ) / 1 2 + r / 2 + (1 + r ) r / 6 + 3 r / 2 0
= — 1 and T/ = l , w e have the Tchebycheff first k i n d spectra for
0
=
[2, 3.25, 1, 0 . 7 5 ] ;
g
=
[0.3791666, 0.6354166, 1/3,
T
g v i a E] = f E] t
g via E
2
= i E
T
f
2
r
0.0885416) ; 7
=
[0.3854166, 0.6354166, 1/3, 0.1041666] ";
=
[0.3791666, 0.6354166, 1/3,
r
0.0885416) .
T h e spectral vector g obtained v i a O S O M R I is exactly the same as t h a t of the actual
5.2 One Shot 5.2.3
Operational
Matrix
O S O M R I Second
for Repeated
of Shifted
163
Integrations
Tchebycheff
Polynomials
of the
Kind:
F r o m Section 3.4, we can easily w r i t e / +
To
V o ( * ) dh dt, = {T, - r ) [ 0 . 6 2 5 V o ( i " ) 2
/
2
T
O.SMT) /
0
0
(5.7)
+ 0.125V> (?-)]/4, 2
/
' Mh)dt dt 2
=
l
TQ
TO
{TJ
- r ) [ - 2 t M r ) / 3 - 5V (r)/12V (r)/24]/4,
y
y
(5.8)
2
0
l
xb {t )dt dt i
(-1)'
2
2
1
. ,
,
,
3
= (rj - r ) { 0
i{i + 2)
0I-.(T)
2 ( i + 1)
4i(t + l )
2 i ( t + 2)
+ 4(i
+ l ) ( i + 2) (5.9)
for i = 2 , 3 , 4 , ••• . As usual, Eq.5.8- Eq.5.9 can be c o m p a c t l y w r i t t e n i n the f o r m of Eq.5.6. For m = 4 we have -
E.
=
4 .
5/8
1/2
-2/3 5/12
-5/12
1/8 0
-7/24
• 5/8 (r,
-2/3
2
~
r) 0
5/12
4 . E x a m p l e 5.3 is required for
-1/16 0
1/2
1/8
-5/12
0
1/6
-1/16
-5/48
0
0 -1/48 . 0 ' 1/24 0 -1/30 .
O S O M R I of Tchebycheff Polynomials of the Second K i n d
to compute
the function
-4/15
1/6 -5/48
0 ' 1/24
the spectra
/ ( r ) considered
of the Tchebycheff in Example
polynomials
of the second
5.2.
For the same f u n c t i o n considered i n E x a m p l e 5.2, and w i t h m = 4, r Tf = 1, we have the Tchebycheff second k i n d spectra for / ( r ) and g(r)
T
g
v i a E] = i E
g
via E
2
= f
= — 1 , and
0
as
T
f
=
[1.5, 1.25, 0.5, 0 . 3 7 5 , ] ;
g
=
[0.2125, 0.2734375, 0.15625, 0.0395833] ;
2
=
[0.203125, 0.2734375, 0.15625, 0.0442708] ;
E
2
It
kind
T
7-
[0.2125, 0.2734375, 0.15625, 0.0395833]
T h e d e r i v a t i o n of the O S O M R I i.e., E
k
.
for any value of k is not a t t e m p t e d for
shifted Tchebycheff p o l y n o m i a l s of the first and the second k i n d .
Chapter
164 5.2.4
O S O M R I
5: Identification
of Lumped
of Shifted Legendre
Parameter
Systems
Polynomials
T h e k times repeated integrations of an i t h degree shifted Legendre p o l y n o m i a l is given by [214]
+
(-iy **C,-0,- - (r)
k
To) A
+aj
t
(5.10)
<) + 1 ] and *_ (r) = -#,-i(r)
(5.11)
f
for a l l i . Eq.5.11 is called the superimposing equation for the reason explained below. For the sake of i l l u s t r a t i o n , let us consider m = 5 and k = 3 a n d p u t the various elements generated from Eq.5.10 and Eq.5.11 i n a t w o - d i m e n s i o n a l array as showm in F i g . 5 . 1 . T h e desired O S O M R I can be o b t a i n e d by s u p e r i m p o s i n g p a r t one on part two and r e m o v i n g p a r t three. I n this process, a l l the elements of p a r t one take their opposite signs i n d i c a t i n g the negative sign preceding i / \ _ i ( r ) i n E q . 5 . 1 1 . T h i s means t h a t 1/3 (Tf -
T)
3
0
1/3 -1/7
1/15
3/35
0
-1/45
0
1/63
0
-1/315
0
2/405
3/5
1/3 -1/7
1/15 0
0
-1/45
1/63 0
0
3/5 -1/3
-1/5 1/15 -1/105 0
0
0
1/105 0 -2/315 0
while
(T, -
Tf 0
1/3 -1/5
-1/3 3/35
1/15 -1/105 0
0 -1/315
0 1/105 0 -3/385
1/165
0
E x a m p l e 5.4 O S O M R I o f L e g e n d r e P o l y n o m i a l s It is required spectra
of the Legendre
polynomials
For the function f(r)
for the function
in Example
considered i n E x a m p l e 5.2, and w i t h m = 4 , r
Tf = 1 , We have the Legendre spectra for f(r)
g v i a E, g via E
/ ( r ) considered
to compute
2
[5/3, 14/5, 4/3, 6 / 5 ] ;
g
=
[4/15, 6 1 / 1 0 5 , 3 / 7 , 2 / 1 5 , ] ;
=
[4/15, 6 1 / 1 0 5 , 3 / 7 , 1 6 / 1 0 5 ] ;
=
= — 1 and
T
=
2
0
as
f
~E] E
and g(r)
the 5.2.
r
T
[4/15, 6 1 / 1 0 5 , 3 / 7 , 2 / 1 5 ]
T
I n this example again the spectral vector g o b t a i n e d v i a O S O M R I is e x a c t l y the same as t h a t of the a c t u a l g . T h u s O S O M R I is found t o be superior.
5.2 One Shot
I/>_ (T) 3
Operational
Mr)
t/>- (r) 2
-1/3
Matrix
Mr)
165
Mr)
Mr)
Mr)
Mr)
1/15
0
0
0
0
-1/7
0
1/105
0
0
0
3/35
0
-1/45
0
1/315
0
0
-1/105
0
1/63
0
-3/385
0
1/693
0
0
-1/315
0
1/165
0
-1/273
0
1/1287
3/5 -1/5
-1/15
Part 1
Mr)
Integrations
Mr)
-1/3 1/3
for Repeated
Part 2
Part 3
F i g u r e 5 . 1 : D e r i v a t i o n o f O S O M R I for shifted Legendre p o l y n o m i a l s w h e n m — 5 and k = 3
166
Chapter
5.2.5
O S O M R I
5: Identification
of Laguerre
of Lumped
and Hermite
Parameter
Systems
Polynomials k
For these i n f i n i t e range o r t h o g o n a l p o l y n o m i a l s , i t is observed t h a t E
= E.
k
Hence,
t
the i n f i n i t e range o r t h o g o n a l p o l y n o m i a l systems have no O S O M R I . 5.2.6
O S O M R I
of Sine-Cosine
Functions:
I t is interesting to note t h a t for the sine-cosine functions
the elements of E
k
are
recursively related by the following easy-to-derive expressions: , 2 , • • • , ( m - 1) and k = 2 , 3 , • • • ,
=
[T/(k
=
-(T/2j7r)e ,
l,m+j,k
=
[(-iy T/jw]e _,
e«+i,i,t
=
(T/2i7r)e
6:1,1,*
e
e
e
t
+ (T/2j7r)e
m + 1
(r/2i7r)e (T/2iV)e
, , _ 1
,
m + 1
,
m +
t
i m +
1
j, _
1
t
t
(T/2i7r)[(-l)'e , , _
m
=
l
(T/2i7r)[(-l)'e
l i i + 1
,
t
1
, _
J + 1
1
m + i,j + l,t
1
1Atk
e +,,i,t e
_
m + > i t
+i
=
i + l,m+j k
t_i
l t l )
1
=
i+i,j+i,t
+ l)]e
( :
l i J + 1
-
1
, _ t
e, ,i _,] + 1
-
1
e
i t
< + 1
,j_ , _ ] M
i l
1
and e-m+i, +j,k = ( X / 2 i * ) [ ( — l ) ' e i m
where T = (tj — t ).
2
[ l/4-l/(2* ) E]
=
(«/ - « „ )
2
-1/(2TT) L [
E
2
=
2
_i — e
(tf-t )
i + l t r n + J i k
2
-l/(2* ) -1/(2TT)
_,]
-l/(2*)
]
0
I; 2
-3/(2vr) J
-l/(2* ) 2
1/(2*)
2
0 2
1/6 -l/(2*)
0
2
-1/(4*)
L -1/(4*) E x a m p l e 5.5
i m + J | f c
For k = 2 and m = 2 we have
0
1/(2*)
2
]
0 2
0
-3/(2vr) J
O S O M R I of Sine-Cosine Functions
/ ( < ) = 1 + cos 2-rrt — sin 2 * t , 0 < t < 1 twice with respect
It is required to t from
to
integrate
t = 0 to t.
I n t e g r a t i n g the above f u n c t i o n twice, its exact value is g(t) = 1/4*
2
W i t h m = 2 , « = 0 and t 0
g viaE gvia£
2
T
= fE
f
= 1, we have the Fourier spectra of f(t)
f
=
[ 1 , - 1 , l -, f j
=
[1/6 - 1 / 4 * + 1 / 4 * , - 1 / 4 * , 1/2TT -
=
[ 1 / 4 - 1/4* - 1 / 4 * , - 1 / 4 * , 1/2* -
2
= f F
2
g
T
2
2
- t / 2 * + O.St + (sin 2 * i - cos 27ri)/47r .
2
=
2
2
2
and g(t)
2
3/4* ]
2
3/4* ]
2
2
2
2
[1/6 - 1/4* + 1 / 4 * , - 1 / 4 * , 1/2* - 3 / 4 * ]
T
r
r
as
5.3 Identification
of Lumped
Parameter
Systems
Here also O S O M R I t u r n e d o u t to be superior. It is because of this superiority of E to E we use OSOMRI in the identification schemes of this chapter as well as in Chapter 6 k
k
5.3
Identification of L u m p e d P a r a m e t e r Systems
L e t us consider a linear t i m e - i n v a r i a n t single-input single-output system m o d e l l e d by the differential e q u a t i o n [214] :
( n
( I )
+ O n - i 2 / ~ ° ( r ) + • • • + a , 2 / ( r ) + a y{r)
=
a
f> u
( m )
m
(r) + 6 _,u
( m
-
m
i n w h i c h U(T) and y(r)
1 )
w
( r ) + • • • + b,u (r)
+ 6 u(r)
(5.12)
0
are respectively the i n p u t and o u t p u t of the system assumed
to be available over an a r b i t r a r y b u t active p e r i o d r e [ r , r ^ ] ; n is the order of the 0
sysem assumed to be k n o w n ; a _ n
l t
a „ _ , • • • , a ,6 , t> _i, • • • , b 2
0
TO
m
a
are the parameters
of the system t o be identified and m < n. Let the i n i t i a l conditions of Eq.5.12 be = Pi Although a
=
(i)
y {To), «
( , )
i = 0 , 1 , 2 , - • • ,n - 1 )
(TO),
i = 0 , 1 , 2 , - • ,m - 1
/'
and /J are a c t u a l l y k n o w n from the o u t p u t and i n p u t records t h e y w i l l
0
0
be t r e a t e d as u n k n o w n s along w i t h the other i n i t i a l conditions i n the i d e n t i f i c a t i o n process as t h e y m a y n o t represent t r u e values i n the noisy s i t u a t i o n [258]. T h e i n p u t and o u t p u t signals are now represented i n a finite series o f o r t h o g o n a l functions as: t-i
u(r)
«
r
T,u,Mr)
= u V(r)
(5.13)
j= 0
t-1
y(r)
as
Let us i n t e g r a t e Eq.5.12 f r o m r
0
£ y ^ ( r ) j=o
T
= y ^l>(r)
(5.14)
t o r successively rt times w i t h respect t o r to get
an i n t e g r a l e q u a t i o n , i n t r o d u c e Eq.5.13 and Eq.5.14 i n t o the i n t e g r a l e q u a t i o n , make use o f Eq.5.5 for o r t h o g o n a l p o l y n o m i a l s and simplify t o o b t a i n Qp = y
(5.15)
where Q
=
[ - E * y \ - E < _
m
+ 1
T 2
y \ - - - \ - E
u | ••• | £ £ _ i " I E I M
n
^ y \ - E
T
I e | Eje
n
y \ E l _
m
u \
T
| •• • | E_e 2
| ^ . e ] ,
Chapter
168 P
5: Identification
[o»-i,n»-2i' ••
=
Parameter
Systems
,a ,a ,b ,b . , l
/o • / l
I " ' " 1 f n - 2 , / n - 1f
[1,
f ^ J ) ]
7
of Lumped
0
m
m 1
,
"
( £ - 1 ) term* Ao -
B/3
[/(J,
' " ' I /TI-2»
/ l i
« 0 , O h
f
/?m-2, /?m-
f
' 1 « n - l
„
/ n - l ]
• • • >« n - 2 . « n - l
[A>,y3„
fl
(5.16)
—2
a?
• • ,
0 1 u
0 0
• • • •
0 0
0 0
1
••
0
0
3
a
4
••
1
2
a
3
••
0 1
n
- I
. «i
a a
' 0
0
... o
0
0
0 K
... o ... o ... o
0 0 0
&2
b
• •
b
. 6,
b
•••
b_i
0
3
2
0
m
b
m
m
.
T h e m a t r i x A is always nonsingular. T h e m a t r i x B does n o t exist i f b alone is present i n Eq.5.12. Since Q is an I x ( 2 n + m + 1) m a t r i x , t m u s t be at least equal to (2n + m + 1) to solve the linear algebraic system i n Eq.5.15. T h e least squares estimate of the augmented parameter vector p is given by 0
P =
T
1
T
[Q Qr [Q y]
(5.17)
T h u s , i t is always possible to estimate a l l the parameters o f the system. Once the system parameters are e s t i m a t e d , the i n i t i a l conditions a, t o o can be e s t i m a t e d from Eq.5.17, p r o v i d e d fl is a n u l l m a t r i x . Otherwise, i n i t i a l c o n d i t i o n i d e n t i f i c a t i o n is not possible.
This
polynomials, Remark:
algorithm
can be applied
Walsh functions
or Haar
in conjuction
with
any
class
of
orthogonal
functions.
T h e model considered by H w a n g and S h i h , 1982 [136] can be obtained
f r o m Eq.5.12 by p u t t i n g m = n — 1, For this m o d e l , Q a n d p i n Eq.5.15 are shown t o be [136]: Q
=
l-Efy
| -Ely
T
| • • • | -E y n
| Efu | £
r 2
u | ••• | £
r n
u | e |
5.3 Identification
of Lumped
< e
Parameter
Systems
r
I --- I g - g j
~£, e | -£
n — l cola P
*7n-l , Co> Cl I t m a y be observed
e | •• • |- E ^ e ]
n— 1 e o i «
," n - j r' ' .08(fra-l,*n-2i
=
r 2
i ' ' ' i d - 2 ]
T
' ' ' fi j il O i
'
1 1
>
-
t h a t t h e m a t r i x Q does not have f u l l r a n k as t h e last ( n — 1)
columns i n i t are repeated i n t h e ( n — l ) columns preceding t h e m and so t h e inverse T
of [Q Q]
does n o t exist.
Therefore, t h e i r a l g o r i t h m i n this f o r m cannot be a p p l i e d
for t h e i d e n t i f i c a t i o n of parameters and i n i t i a l c o n d i t i o n s for systems o f order higher t h a n one. i) B l o c k - P u l s e F u n c t i o n A p p r o a c h : I n t h e a l g o r i t h m proposed above, w i t h e=
[1,
T
1 ^ _ M (*-l) iefma
we can also a p p l y the block-pulse functions for the i d e n t i f i c a t i o n of systems defined by
Eq.5.12. i i ) S i n e - C o s i n e F u n c t i o n A p p r o a c h [213]: I n t h e case of sine-cosine f u n c t i o n s , we can also d i r e c t l y a p p l y t h e proposed algo-
rithm with e=
T
[ 1 , 0, • • • 0 ] ,
a (2£ — 1)— vector. I n t h e present case, u and y are also (2£ — 1)— vectors.
Therefore,
to solve Eq.5.17, (2£ — 1) m u s t be at least equal t o (2rt + m + 1).
5.3.1
Examples
of L u m p e d
Parameter
System
Identifica-
tion T h e a p p l i c a b i l i t y o f the i d e n t i f i c a t i o n a l g o r i t h m v i a different classes o f o r t h o g o n a l systems is now d e m o n s t r a t e d
i n d e t a i l by considering a few n u m e r i c a l examples.
I n each case, t h e p r a c t i c a l difficulties t h a t are n o r m a l l y encountered i n p a r a m e t e r e s t i m a t i o n are discussed. E x a m p l e 5.6
L u m p e d P a r a m e t e r System Identification V i a Orthogonal Sys-
tems It is required y ( l ) and y ( l )
to estimate
of the
the parameters
H>{.T) + a,y(r) from
its unit
the system
ramp
are a
x
a i , a o , and b as also the initial Q
conditions
system
response
+ a y{r)
data available
= 3 , a „ = 2 and b = 1. B
0
= b u(r) 0
(5.18)
over 1 < T < 2. The actual parameters
of
170
Chapter
5: Identification
of Lumped
Parameter
T h i s example, once again, emphasizes the i m p o r t a n c e of shifting orthogonal polynomials.
Systems
i n i n f i n i t e range
I n order to proceed for the i d e n t i f i c a t i o n of the system i n Eq.5.18, first the u n i t r a m p response d a t a is generated by s i m u l a t i n g Eq.5.18 w i t h U(T)
= T applied at
r = 0 and zero i n i t i a l c o n d i t i o n s . F r o m this d a t a , the i n i t i a l conditions are f o u n d to be i / ( l ) = 0.0840456 and y(l)
= 0.1997882.
Since there are altogether five u n k n o w n s to be e s t i m a t e d , to start w i t h we take £ = 5 for block-pulse functions and o r t h o g o n a l p o l y n o m i a l s b u t I — 3 for the sinecosine functions. For all finite range o r t h o g o n a l functions, we take r and a p p l y the i d e n t i f i c a t i o n a l g o r i t h m .
0
I n each case the e s t i m a t e d
= 1 and Tj = 2 results are as
shown in Table 5 . 1 . I t may be seen from this table, t h a t the e s t i m a t e d results are i n close agreement w i t h the a c t u a l results and i t is more so as the value o f £ is increased. T h e system i d e n t i f i c a t i o n v i a the infinite range o r t h o g o n a l p o l y n o m i a l s is not as t r i v i a l as i t m a y appear. T h e i d e n t i f i c a t i o n v i a Laguerre and H e r m i t e polynomials w i t h r = 1 and Tj = 2 w i l l not be successful i f carried o u t i n the usual manner. I t is because, the signal characterization over r € [1,2] w i t h a finite n u m b e r of infinite range o r t h o g o n a l p o l y n o m i a l s is not posible as discussed i n C h a p t e r 3. Unless the signal characterization is accurate, one cannot proceed w i t h the i d e n t i f i c a t i o n . The signal characterization v i a the infinite range o r t h o g o n a l p o l y n o m i a l s is possible only when the signal i n f o r m a t i o n is available over the whole i n t e r v a l of o r t h o g o n a l i t y . For the Laguerre p o l y n o m i a l s , i t is not a serious m a t t e r as the system under identification is excited for T > 0. B u t for H e r m i t e p o l y n o m i a l s , the signal characterization w i t h a finite n u m b e r of p o l y n o m i a l s is difficult as the i n p u t and the o u t p u t signals are simply zero for t < 0. T h i s p r a c t i c a l difficulty, makes the system i d e n t i f i c a t i o n v i a Hermite p o l y n o m i a l s not reliable. I t is because of t h i s difficulty, the H e r m i t e polynomials for the system i d e n t i f i c a t i o n are not pursued a l t h o u g h i t has the advantages of (i) a simple s t r u c t u r e o f the i n t e g r a t i o n o p e r a t i o n a l m a t r i x and ( i i ) the nonexistence of OSOMRI. 0
However, to study the c a p a b i l i t y of the i d e n t i f i c a t i o n a l g o r i t h m v i a a l l classes of o r t h o g o n a l functions i n c l u d i n g the i n f i n i t e range o r t h o g o n a l p o l y n o m i a l s , i t is assumed t h a t the i n p u t signal « ( r ) = r and the o u t p u t signal j / ( r ) = —0.75 + 0.5r + e x p ( — r ) — 0 . 2 5 e x p ( — 2 r ) are k n o w n i n their a n a l y t i c a l f o r m so t h a t a l l the pract i c a l difficulties in signal characterization are removed.
T h i s a s s u m p t i o n makes the
spectra evaluation of the i n p u t and the o u t p u t signals faster and q u i t e accurate as we employ Gaussian open q u a d r a t u r e formulae for this purpose. I n order to estimate the i n i t i a l conditons along w i t h the system parameters the shifted m i a l s , o r t h o n o r m a l over r € [ l , o o ] and the shifted
Laguerre polyno-
H e r m i t e p o l y n o m i a l s , orthogonal
over T € [—oo, oo] w i t h the o r i g i n shifted from zero to one are employed to o b t a i n the spectra of the i n p u t and the o u t p u t signals. W i t h t h i s m o d i f i c a t i o n , the identification via the Laguerre and H e r m i t e p o l y n o m i a l s is carried o u t and i n each case, the estim a t e d results, seem to be excellent, are as shown i n Table 5 . 1 . T h i s means t h a t the proposed i d e n t i f i c a t i o n a l g o r i t h m works efficiently v i a the i n f i n i t e range o r t h o g o n a l
5.3 Identification
of Lumped
Parameter
Systems
171
T a b l e 5 . 1 : Parameter estimates i n E x a m p l e 5.6. Appr-
t
OQ
6o
3
2
1
0.0840456
0.1997882
5 6
3.0284429
2.0085034
1.0068576
0.0860669
0.1940037
3.0199119
2.0059606
1.0048031
0.0854443
0.1957967
7
3.0147001
2.0044038
1.0035470
0.0850710
0.1968671
8
3.0112901
2.0033839
1.0027247
0.0848296
0.1975574
5
3.0193772
2.0032223
1.0043256
0.0840459
0.1997680
6
2.9999753
1.9999920
0.9999939
0.0840455
0.1997883
5
3.0127620
2.0021398
1.0028555
0.0840470
0.1997688
6
3.0000380
2.0000062
1.0000085
0.0840456
0.1997880
5
2.0026052
1.0034843
0.0840465
0.1997682
6
3.0155869 3.0000532
2.0000085
1.0000118
0.0840456
0.1997881
LaP
5
3.0000369
2.0000226
1.0000115
0.0840459
0.1997859
HeP
5
2.9999999
1.9999999
0.9999999
0.0730514
0.2229334
6
»?
11
»»
0.0871496
0.1935069
11
)i
>>
)) 3.0002314
)» 2.0000388
>) 1.0000518
0.0833944
0.2010942
0.1018564
0.1382217
2.0000490
1.0000659
0.0968457
0.1555389
8
3.0002945 3.0004122
2.0000654
1.0000917
0.0900502
0.1790287
16
3.0004404
2.0000454
1.0000945
0.0869543
0.1897318
32
3.0001227
1.9997955
0.9999955
0.0854767
0.1948439
2/(1)
oach I actual - * BPF
TPl TP2 LeP
7
1 8 SCF
3 4
systems also, p r o v i d e d the signal c h a r a c t e r i z a t i o n is done accurately. I n t h e present and also i n t h e subsequent examples, t h e signal c h a r a c t e r i z a t i o n is made possible by t h e above a s s u m p t i o n w i t h a view t o s t u d y i n g t h e c a p a b i l i t y of t h e i d e n t i f i c a t i o n algorithm via Hermite polynomials. E x a m p l e 5.7 L u m p e d P a r a m e t e r S y s t e m I d e n t i f i c a t i o n v i a O r t h o g o n a l S y s t e m s i n P r e s e n c e of N o i s e w i t h Z e r o I n i t i a l C o n d i t i o n s T h i s e x a m p l e shows the parameter e s t i m a t i o n i n noisy e n v i r o n m e n t . I n C h a p t e r 3 i t is already seen t h a t t h e o r t h o g o n a l functions have inherent f i l t e r i n g properties. T h e r e f o r e , i n t h i s example we estimate the parameters a i , a
0
and b
given b y Eq.5.18 f r o m its u n i t step response j / * ( r ) , defined as y'(r)
0
of t h e system = y(r)
+ n(r)
w h e r e j / ( f ) is t h e a c t u a l response of the system a n d r)(r) is the w h i t e noise w i t h a c e r t a i n a m o u n t of noise-to-signal r a t i o ( N S R ) , available over r 6 [ 0 , 2 0 ] . D u e t o t h e
172
Chapter
5: Identification
of Lumped
Table 5.2: Parameter estimates i n E x a m p l e Approach J
No of
a
NSR
0
Parameter
Systems
5.7 bo
terms t A c t u a l —>
BPF
3 4
0 0
8
0
16 32
0
TP2
1
1
1.2791912
-0.5093126
3.2877246 2.0734912
1.2331763
1.5240077
1.0387510
1.0953028
1.8613561 1.8149477
1.0090934
1.0215492
1.0022410 1.0020818
1.0052606 1.0038849 1.0030674
32
0 0.05
32
0.10
1.8065789
32
0.15
1.8019165
1.0019513 1.0018504
32
0.20 0.25
1.7969300 1.7916158
1.0017798 1.0017402
1.0021621
3 4
0
11.7549262
-8.7944016
3.6787687
0 0
2.2477140 1.7992371
0.9414090 1.0012397
1.1184758
5 3 4
0 0
0.9868161
1.1596184
0.7883851
1.7562783
0.9789322
0.9795571
5 6 6
0 0
1.7790485 1.7977851
1.0002769
0.9934729
1.0000240
0.9993069
1.7970406 1.7993084
0.9989611 0.9964522
0.9994610
6
0.05 0.10
6
0.15
1.8015234
0.9939547
1.0008901
6
0.20
1.8036743
1.0015852
6
0.25
1.8057898
0.9914690 0.9889954
32 TPl
1.8 -3.6632494
1.8109213
1.0046156
1.0011679
1.0002749
1.0001819
1.0022670
c o m p u t a t i o n a l difficulties associated w i t h the evaluation of t h e Tchebycheff first kind s p e c t r u m of a noisy signal y'(r),
see C h a p t e r 3, the parameter e s t i m a t i o n i n noisy
e n v i r o n m e n t v i a the Tchebycheff p o l y n o m i a l s of the first k i n d is n o t carried out in this and in the next example. S i m i l a r l y for the same p r a c t i c a l difficulties discussed in the previous example i n connection w t i h the e v a l u a t i o n of the H e r m i t e s p e c t r u m of the i n p u t and o u t p u t signals, the parameter e s t i m a t i o n v i a the H e r m i t e polynomials i n a noisy e n v i r o n m e n t w i l l not be considered. B u t , the noise-free case is studied i n the same way as i n the last example, as the system response y(r)
= 1 -
2.2941573 x e x p ( - 0 . 9 r ) sin (0.4358898r + 0.4510268)
in the present example is assumed to be k n o w n in its a n a l y t i c a l f r o m over t h e whole i n t e r v a l of o r t h o g o n a l i t y i.e., T
€
[—00,00].
T h e Laguerre approach w o r k e d here
5.4 Transfer
Function
Matrix
Identification
173
effectively for t h e obvious reason t h a t the i n p u t - o u t p u t d a t a is collected over r
€
[ 0 , 2 0 ] . T h e p a r a m e t e r estimates o b t a i n e d v i a each system of o r t h o g o n a l functions is as s h o w n i n Tables 5.2 and 5.3. T h e t r u e parameters are a, = 1.8, o
0
= 1 and 6 = 1. 0
Here no a t t e m p t is made t o estimate the i n i t i a l conditions as t h e y are s i m p l y zero. I t is clear f r o m t h e Tables 5.2 and 5.3 t h a t a l l the parameter estimates o b t a i n e d v i a t h e finite range o r t h o g o n a l p o l y n o m i a l s are very near t o the a c t u a l parameters. T h e estimates based on t h e Laguerre p o l y n o m i a l s are also q u i t e satisfactory. B u t due to t h e p o o r representation of y(r)
or y * ( r ) v i a sine-cosine functions, t h e parameter
estimates are not as a t t r a c t i v e as t h a t o b t a i n e d v i a o r t h o g o n a l p o l y n o m i a l s .
Of
course, t h e result w i l l i m p r o v e , i f more n u m b e r o f terms i n t h e series expansion of sine-cosine f u n c t i o n are i n c l u d e d .
E x a m p l e 5.8 L u m p e d P a r a m e t e r S y s t e m I d e n t i f i c a t i o n v i a O r t h o g o n a l S y s t e m s i n P r e s e n c e of N o i s e w i t h N o n - z e r o I n i t i a l C o n d i t i o n s .
I n the last e x a m p l e , zero i n i t i a l conditions were considered and the parameters were only estimated.
B u t , now t h e i n i t i a l c o n d i t i o n s along w i t h t h e parameters w i l l be
e s t i m a t e d . T o do t h i s , a system described by V(T)
+ a y(r) 0
=
b u(r) 0
w i t h an i n i t i a l c o n d i t i o n 2/(0) is considered. T h e a i m is t o estimate the parameters and b a n d t h e i n i t i a l c o n d i t i o n s 2/(0) f r o m its u n i t step response y'(r) 0
*" 6 [ 0 , 1 8 ] , where y'(r) i n i t i a l c o n d i t i o n are: a are as s h o w n i n Tables
a
0
available over
is as defined i n the last example. T h e t r u e parameters and t h e 0
= b = 2 and y(0) = 0.25 respectively. T h e e s t i m a t e d results 0
5.4 and 5.5. T h e system response y(r)
= 1 — 0.75 exp( — 2 r )
is assumed t o be k n o w n i n its a n a l y t i c a l f o r m to carry o u t t h e system i d e n t i f i c a t i o n v i a H e r m i t e p o l y n o m i a l s . T h e e s t i m a t e d results i n each case are f o u n d t o be q u i t e satisfactory.
5.4
Transfer Function Matrix
Identification
Let t h e M I M O system be described by
(
E«,rf, "(t) = E E *=0
j' = l
b uf\t),i=\,2,...,p ijk
(5.19)
*=0
where r is t h e n u m b e r o f i n p u t s and p the n u m b e r of o u t p u t s of t h e system. I n terms of transfer f u n c t i o n m a t r i x , t h e system described by Eq.5.19 can be w r i t t e n as Y(s)
=
G(s)U(s)
(5.20)
C h a p t e r 5: Identification
174
of Lumped
Parameter
Table 5.3: Parameter estimates i n E x a m p l e 5.7 A p p r o a c h J.
No of
a
NSR
0
bo
Terms I 3 4 5 6 LeP
LaP
SCF
1.3677901
0.9917605
0.8758327
0 0
2.1181219
1.0349396
1.7887488 1.7965880
0.9971963
1.1022000 0.9957780
0.9995188 0.9989527
0.9988587 0.9994654
6
0 0.05
6
0.10
1.7975398 1.7983462
6
0.15
1.7990034
6 6
0.20 0.25
3 3 3 3 3 3 HeP
0
0.9983741
1.0000241 1.0005334
1.7995076
0.9977758 0.9971574
1.7998550
0.9965185
1.0013988
0 0.05
1.8000040
1.0000011 0.9978397
0.10
1.7935757 1.7903232 1.8156052
0.15 0.20 0.25
1.7986808
1.0091820 1.0137390
1.0009920 1.0000015 0.9970551 1.0060926 1.0091220 1.0004232
1.8195439
0.9997469 0.9996819
1.6511276
0.1063884
0.5342819
0
1.7931580
1.0665975
1.0272499
0
1.9448625
1.2995514
1.1505024
0
1.8015404
1.1165289
1.0415885
8
0
1.8056741
1.0429993
1.0182453
8
0.05
1.8105321
1.0345909
1.0180840
8
1.7930093
1.0428776
1.0134564
8
0.10 0.15
1.7853223
1.0435507
1.0108116
8
0.20
1.8207498
1.0641069
1.0277333
8
0.25
1.8239079
1.0697146
1.0299966
3 4
0
2 4
1.0005337
Systems
5.4 Transfer
Function
Matrix
Identification
175
Table 5.4: Estimates i n E x a m p l e 5.8 Approach
[
No of
NSR
a
0
bo
y(0)
Terms t A c t u a l —»
BPF
TPl
TP2
LeP
2
2
0.25
3 4
0
1.9816849
1.9816849
0.2568681
0
1.9896480
1.9896480
0.2538819
8
0
1.9973998
0.2509750
8 8
0.05 0.10
1.9895927 1.9817949
1.9973998 1.9946382
8
0.15
8
0.20
8
0.25
3
1.9918770
0.2512235 0.2514749
1.9740068
1.9891162
0.2517204
1.9662286
1.9863561
0.2519689
1.9584605
1.9835968
0.2522173
2.0000140
2.0000044
0.2499999
3 4
0
1.8527362
1.8733323
0.2844763
0
1.9907627
1.9920546
0.2537868
5
0
1.9996903
0.2504322
5
1.9945020
0.2496433
5
0.05 0.10
1.9996400 1.9938018 2.0021133
2.0032929
0.2529298
5
0.15
2.0075617
2.0034957
0.2484985
5
0.20
1.9584288
1.9612405
5
0.25
2.0403448
2.0384496
0.2536887 0.2483324
3 4
0
1.8055322
1.8343827
0.2761781
0
1.9841179
1.9864740
0.2521379
5
0
1.9992390
1.9993519
0.2501024
5
0.05
1.9999312
2.0010270
0.2503351
5
0.10
2.0005978
2.0026808
0.2505713
5
0.15
2.0012385
2.0043133
0.2508109
5
0.20
2.0018536
2.0059245
0.2510539
5
0.25
2.0024428
2.0075142
0.2513004
176
Chapter
5: Identification
of Lumped
Parameter
Systems
Table 5.5: Estimates i n E x a m p l e 5 . 8 Approach j
No of Terms
NSR
h
f(0)
I 2
2
0.25
3 3 3
0 0.05
1.9999963 1.9927142
1.9999965 1.9961582
0.2500006 0.2482502
0.10
1.9854071
1.9922835
0.2465185
3
1.9782191 1.9711474
1.9885047
0.2447809
3
0.15 0.20
1.9848194
3
0.25
1.9641890
1.9812253
0.2430375 0.2412885
3 4
0
2 2
1.9999999 1.9999999
2
1.9999999
-0.0193556 0.2354832
A c t u a l —»
LaP
HeP
0 0
8 12
0
2
1.9999999
0.2496409
2
0
1.9999968
1.9999966
0.2548170
4
0 0
1.9999983 1.9999981 2.0011819 2.0023524
0.2521212
0.05 0.10
1.9999959 1.9999955 1.9999049 1.9997821
8
0.15
1.9996274
2.0035097
0.2512858 0.2514314
8
0.20
1.9994408
2.0046537
0.2515772
8
0.25
1.9992226
2.0057845
0.2517232
8 8 8
SCF
0.9999999
0.2509951 0.2511404
where Di(.)
G(s)
£> (.)
D,(S)
D (.)
D,(.)
2
= I
R
£>,(«) *^(.) D (.) 2
'•"
D„(.) J
n,
Di(s)
=
5Z«i*A«lin. =
M
=
l,2,...,p
t = 0 n , - l
=
E
= 1,2,....p,j = 1 , 2 , . . . , r
k=0 Di(s) is the least c o m m o n d e n o m i n a t o r of the i t h row o f G ( s ) h a v i n g t h e degree n,-, and the Laplace t r a n s f o r m of a t i m e function / ( < ) is represented by t h e corresponding c a p i t a l l e t t e r F(s).
5.4 Transfer To
Function
Matrix
Identification
177
get E q . 5 . 2 0 f r o m E q . 5 . 1 9 by t a k i n g the Lapalace t r a n s f o r m o f b o t h sides o f
E q . 5 . 1 9 , we have considered the i n i t i a l c o n d i t i o n s t o be zero. I n fact, i f we consider any r e c o r d o f t h e o u t p u t due t o a given i n p u t , the response s h o u l d also have c o m p o n e n t s due t o these i n i t i a l c o n d i t i o n s . Therefore the system i d e n t i f i c a t i o n requires t h e determ i n a t i o n of t h e parameters { o , * , 6 , 7 * } , * = 1,2,...,p;j
= l,2,...,r;fc = 0,l,...,n,-
a n d t h e i n i t i a l c o n d i t i o n s f r o m a t r a n s i e n t record o f t h e system i n p u t s a n d o u t p u t s . I n t e g r a t i n g Eq.5.19 rt, times f r o m t
"i
t o t we have
0
f
t=o
- E
hnk {
i=u=o
vrXto)-———
,=0
l
= E E
1
k-l
E m {in.-kivm
(n<-fe +« ) l j
/,,-tMO] - E
uyXt )
,=0
I
where ( n , — k) times i n t e g r a t i o n of a f u n c t i o n f(t) io
T h e coefficients of t denoted by a
i k
k
io
}
**'
B
(5.21)
(".•-* + ?)!; is denoted by
io
i n t h e above expression of Eq.5.21 are g r o u p e d together
and
. So Eq.5.21 takes t h e f o r m n,—1
-
E
r
n,-l
a 7„._ [j/ (i)] + E i t
t
E
i
jfc = 0
&i;*7„,._ Mi)] t
j = l jfc = 0 71,-1
+ E where we have used the fact t h a t o , The outputs y , ( t ) , i = 0,1,.
=
y;(t)
(5.22)
= 1.
n i
1,2,. . . , p ; the i n p u t s Uj(t),j
k
=
1,2, . . . , r , a n d t ,k
=
. . , n,; — 1 are now e x p a n d e d i n t o an m - t e r m shifted o r t h o g o n a l series g i v i n g us:
= Uj
(t)
k
t
J/,O0o(l) +
=
ufv(0
=
"iOtAoW +
=
=
!/ilV'l(«)+--- +
"jlV>l ( < ) + • • • +
S/i,m-l'/'m-l(«)
«j,m-lV'm-l(<)
(5.23)
(5.24)
T ? > ( < )
T M*) + T Mt) + • • • + Tk, ~lll> -l(t) kB
kl
m
m
(5.25)
W e s h a l l , however, use unshifted p o l y n o m i a l s i f they are Laguerre a n d H e r m i t e . I n v i e w o f Eq.5.5 we n o t e t h a t ( r i ; — fc)-times i n t e g r a l o f J / , ( I ) a n d Uj{t) can be w r i t t e as n
f»,-*lft(0]
=
yr^. '"*V-(i)
(5.26)
Kt-kW),
=
uj
(5.27)
(0
Chapter
178
5: Identification
of Lumped
Parameter
S u b s t i t u t i n g Eq.5.24-Eq.5.27 i n t o Eq.5.22 and equating coefficients o f
Systems
t/\(t),i = 0 , 1 ,
. . . , m — 1 from b o t h sides we get
-
k
E
a y?E?<-
+ E
ik
fc = 0
E
b
i
j
ujEr
k
k
+
k=0
J = l
n,—1
+ E
dkrl
=
yf
(5.28)
t = 0
Let us denote the u n k n o w n parameter vector of the i t h row o f G(s) by x< t h e n
x, = [ a f b* bf .-.bf
r a
2
f f
where a,
=
[a,
0
b,i
=
[6
O 0
a,
=
O i l • • • <*;„,.-i]
T
T
&y -"WiJ
. J = 1,2.--r
l
["io
• • ' <*•„;-i]
T
Therefore Eq.5.28 can be w r i t t e n i n the following v e c t o r - m a t r i x f o r m :
=
y,,i
= 1,2,...,p
(5.29)
where Qi
=
T
[-(E^r-y, Sfu,
n
| ~(E r'-Vi T
| . . . | {E ru
I ••• I ( - £ ? > < | ( £ j ) ' u I ... I £?u
r
1
| ••• |
I r , | T , | -•- | r . , ^ ]
r
I n Eq.5.29 x ; is an ( r + 2)rt,-vector, y< an m-vector
and Qi an m x ( r + 2)n;
m a t r i x . I f we denote the set of a l l u n k n o w n parameters by the vector x : X
=
r T [Xj
x T • • • xT i T] 2
p
t h e n Eq.5.29 can be w r i t t e n i n the compact f o r m Qx
=
y
where Q
=
y
=
diag[Q ,Q ,...,Q ] 1
a
J 1
[yfyr..y,T
Since there are ( r + 2)rt u n k n o w n parameters i n x where n = i t is necessary t o choose m so t h a t nip > ( r + 2 ) n for a least squares s o l u t i o n .
+ n
2
+ ••• +
n, r
5.5
Conclusion
179
E x a m p l e 5.9 M I M O S y s t e m I d e n t i f i c a t i o n It is required eters of the model of a two-input y(t)
+ a y(t)
+ a y(t)
n
one-output = b u,(t)
10
nl
MIMO
system
to estimate
described
+ 6 o"i(<) + b iU (t) U
l2
+
2
the
param-
by b u (t) 120
2
3
W i t h t h e i n p u t functions u,(t)
= t and u ( < ) = t , t h e o u t p u t d a t a are generated 2
w i t h zero i n i t i a l c o n d i t i o n s t o estimate the parameters whose t r u e values are &iu
=
1.0,6
1 1 0
= 0.5,6
1 2
i = 0,6120 =
1.0, a
u
=
1.5, a
1 0
=
0.5
T h e estimates of the parameters and the i n i t i a l conditions o b t a i n e d using m = 8 are p r o v i d e d i n Table 5 . 6 . T h e H e r m i t e and Laguerre p o l y n o m i a l s are not i n c l u d e d , as the estimates offered by t h e m are not satisfactory.
Table 5.6: Parameter estimates of a t w o - i n p u t a n d o n e - o u t p u t system Parameters
Actuals
Legendre
Tcheby I
Tcheby I I
10
0.5
0.50081791
0.50090608
0.50072220
«ii
1.5
1.49894343
1.49884961
1.49906578
&110
0.5
0.49894343
bm
1.0
0.49905633 1.00037574
bi o
1.0
1.00000060 0.99953655
0.45686483 1.02019722 0.99949612
0.99958958
bin
0.0
0.00033206
0.00102923
0.00029379
2/(0) 2/(0) Parameters
0.0
0.00000000
0.00000000
0.0
-0.00000002
0.00002079
-0.00000003 0.00000021
a
2
Actuals
Gegenbauer
sine-cosine
PMF
0.5
0.50081791
0.46648337
an
1.5
1.49894343
1.44885305
0.49595748 1.49830691
bno
0.5
0.49894343
bm
1.0
1.00000060
0.46648337 0.96590204
&120
1.0
0.99953655
0.93296674
1.00000629
&121
0.0
0.00033206
0.00000000
-0.00000059
2/(0) 2/(0)
0.0
0.00000000 -0.00000002
0.00207060
0.00000000
0.0
-0.00037316
0.00000000
a
5.5
10
0.49703136 0.99826129
Conclusion
T h e i d e n t i f i c a t i o n a l g o r i t h m developed i n the Section 5.3 works efficiently v i a any k i n d of o r t h o g o n a l systems.
For a successful e s t i m a t i o n of t h e parameters a precise
signal c h a r a c t e r i z a t i o n is essentially needed. Therefore, t h e reconstructed signal, see
180
Chapter
5: Identification
of Lumped
Parameter
Systems
Chapter 3, must be very close t o the o r i g i n a l signal for o b t a i n i n g a g o o d measure of accuracy i n the estimates of the parameters. Since the response o f a t i m e - i n v a r i a n t continuous-time system is always s m o o t h , the block-pulse series representation of this signal is r e l a t i v e l y p o o r w h e n compared w i t h t h a t of the f i n i t e range o r t h o g o n a l p o l y nomials. T h e generalized block-pulse functions may be advantageous i n t h i s respect. For an accurate parameter e s t i m a t i o n v i a Laguerre p o l y n o m i a l s , the i n p u t and the o u t p u t signals of the system are required over a r e l a t i v e l y large i n t e r v a l . A l t h o u g h i n a l l numerical examples, parameter e s t i m a t i o n was shown possible v i a the H e r m i t e p o l y n o m i a l s , this approach is c o m p u t a t i o n a l l y not a t t r a c t i v e and p r a c t i c a b l e for the reasons explained i n the E x a m p l e 5.6. I t is shown t h a t the system i d e n t i f i c a t i o n v i a sine-cosine functions is quite possible. B u t the Fourier series approach becomes c o m p u t a t i o n a l l y laborious for accurate parameter e s t i m a t i o n due t o its p o o r represent a t i o n of the nonsinusoidal signals w h i c h are very c o m m o n i n i d e n t i f i c a t i o n . Keeping all the aforementioned considerations in m i n d , the finite range o r t h o g o n a l p o l y n o m i als are found t o be superior for system i d e n t i f i c a t i o n . T h e i d e n t i f i c a t i o n a l g o r i t h m as o u t l i n e d above is not recursive i n n a t u r e and therefore on-line i m p l e m e n t a t i o n is not d i r e c t l y possible w i t h i t . However, the extension o f this a l g o r i t h m t o the i d e n t i f i c a t i o n of a M I M O system is s t r a i g h t f o r w a r d as shown i n Section 5.4.
5.6
Problems
1. F i n d the transfer f u n c t i o n of a first order system whose u n i t step response is given by
y{t)
0
8
8
8
8
8
8
8
0.394
0.632
0.789
0.865
0.918
0.950
0.970
A n s w e r : y ( i ) -I- 4y(t)
=
0.982
4u(t)
2. For the system y(t) + a y(t) 0
=
b u{t) 0
estimate the parameters a and 6 when u ( i ) = 1 + t and y(t) = t, for t S [ 0 , 1 ] . 0
0
Answer:a = b = 1 0
0
3. Suppose a gate f u n c t i o n defined as 1 0
for 0 < t < elsewhere
7
5.6
181
Problems
is a p p l i e d t o a second order linear system and the o u t p u t is recorded as follows: /
O
—
y{t)
0
I
JL
1
J L
16
8
16
4
16
0.0624
0.1244
0.214
0.205
0.1855
0.1962
0.245
0.1866
3
.
J
8
0.241
0.1768
0.1669
L
16
0.236
I 2
0.230
iS. 16 0.1569
0.222 1 0.1469
F i n d the transfer f u n c t i o n of the system.
Answer: y(t) + 1y{t) + 2y(t)
= ii(t) + 2u{t)
4. Consider the system described by y(t)
+ a,y(t)
+ a y(t)
=
0
x(t)
Generate o u t p u t u p t o 1.0 sec w i t h zero i n i t i a l c o n d i t i o n s , a, = 3, a
0
= 2 and x(t)
=
u n i t step f u n c t i o n . E s t i m a t e the parameters.
5. Consider t h e linear system described by a y(t)
+ a,y(t)
2
w i t h zero i n i t i a l c o n d i t i o n s and a
0
r/(t)
and
+ a y(t)
= u(t)
0
= 2, a, = 3, a
2
= 1 a n d u(<) = 10 . A s u u m e t h a t
are t w o independent zero mean w h i t e noises w i t h variances r a n d q i n
the measurement of u(t)
and y{t),
respectively. E s t i m a t e the parameters by t a k i n g
r = q = 0.333
6. Consider a system described by the second order differential e q u a t i o n y(t) E s t i m a t e the parameters a
n
+ d » ( < ) + a y(t) 0
=
0
0
2
and 2/(0 0
= 1,6
0
0
a , b a n d the i n i t i a l conditions 3/(0), y ( 0 ) , w h e n u(<) = 2 i + t + 3
A n s w e r ^ ! = 0.25, o
b u(t)
2
= 2< - 1
= 1,2/(0) = - 1 , 2 / ( 0 ) = 0
182
Chapter
5: Identification
of Lumped
Parameter
Systems
7. Consider a u n i t y feedback system w i t h feedforward transfer f u n c t i o n 2
G(s)
b s + b,s + 6 = — 3 + as + as + a 2
0
2
2
x
0
T h e u n i t step response of this feedback system w i t h zero i n i t i a l c o n d i t i o n s is given by y(t)
- 2
= 0.5 - e~' + (0.5 + 2 i ) e '
E s t i m a t e the parameters of G ( s ) .
C hapter
6
Identification Parameter
of
Distributed
Systems
T h e a i m of this chapter is to develop a general and i n t e g r a l f r a m e w o r k for the estim a t i o n of p a r a m e t e r s , and i n i t i a l and b o u n d a r y conditions of first- a n d second-order linear t i m e - i n v a r i a n t single-input single-output continuous-time d i s t r i b u t e d parameter systems f r o m measurements o f the i n p u t and o u t p u t
6.1
signals.
Introduction
I t appears f r o m the l i t e r a t u r e t h a t Paraskevopoulos and Bounas, 1978 [239] were the first investigators to study the i d e n t i f i c a t i o n of d i s t r i b u t e d p a r a m e t e r systems v i a ort h o g o n a l functions.
T h e y used W a l s h functions.
i d e n t i f i c a t i o n a l g o r i t h m s , one for
first-order
systems. T h e i r a l g o r i t h m for the i d e n t i f i c a t i o n of perfect, and provides excellent results.
I n this paper t h e y i n t r o d u c e d t w o
systems and the other for second-order first-order
systems is f o u n d t o be
T h e m a i n drawback of t h e W a l s h f u n c t i o n
approach [239] is the selection of W a l s h functions on the basis o f 2* where k is any p o s i t i v e integer. For a m o d e r a t e l y large value of k this approach becomes c o m p u t a tionally laborious. I n the same year Tzafestas, 1978 [313] also investigated the general d i s t r i b u t e d p a r a m e t e r system i d e n t i f i c a t i o n by first t r a n s f o r m i n g the m o d e l of a d i s t r i b u t e d par a m e t e r system i n t o its equivalent l u m p e d f o r m by using G a l e r k i n expansion and t h e n by using t h e m e t h o d e x i s t i n g for l u m p e d parameter system i d e n t i f i c a t i o n v i a W a l s h functions.
A f t e r five years, the same p r o b l e m of Paraskevopoulos and Bounas [239]
was r e v i s i t e d by Paraskevopoulos and Kekkeris, 1983 [241]. T h i s t i m e t h e y employed Tchebycheff p o l y n o m i a l s o f the first k i n d and restricted t h e i r investigations t o o n l y first-order systems. I n t h i s paper, they suggested a linear t r a n s f o r m a t i o n such t h a t the i n t e r v a l of i n p u t - o u t p u t d a t a (for each independent variable i.e., space x and t i m e i) d i r e c t l y coincides w i t h the i n t e r v a l of o r t h o g o n a l i t y of Tchebycheff p o l y n o m i a l s . T h e p u b l i c a t i o n s [239,313,241] considered the parameter e s t i m a t i o n w i t h o u t noisy
Chapter
184 environment.
6: Identification
of Distributed
T h i s aspect was later taken up by R a n g a n a t h a n
Parameter
Systems
et a l , 1984(252] v i a
Laguerre p o l y n o m i a l s . Jha and Z a m a n , 1985 [146] made an a t t e m p t on t h e firstorder system identification v i a Laguerre p o l y n o m i a l s , i n b o t h noisy and environments.
noise-free
As i t is always convenient to use shifted TchebychefT p o l y n o m i a l s of
the f i r s t - k i n d defined over [0,1] rather than redefining the system m o d e l by m a t c h i n g the i n t e r v a l of measurement
w i t h t h a t of o r t h o g o n a l i t y of Tchebycheff p o l y n o m i a l s
i.e., the i n t e r v a l [ - 1 , 1], Horng et al, 1986 [97] reexamined the p r o b l e m of the firstorder system identification studied by Paraskevopoulos and Kekkeris [241]. Since the basic approach is essentially the same as t h a t of [241], no new results of identification were o b t a i n e d . To avoid the use of one class of o r t h o g o n a l system at a t i m e , Lee and C h a n g , 1986 [175] and then Lahouaoula, 1987 [172] suggested a generalized o r t h o g o n a l p o l y n o m i a l approach for the identification of first-order [175] and parabolic systems [172]. A g a i n , these publications [97,175,172] did not consider the i d e n t i f i c a t i o n i n noisy environment. In none of the above publications, the identification of d i s t r i b u t e d parameter systems was studied v i a block-pulse functions, Tchebycheff p o l y n o m i a l s of the secondk i n d , Legendre p o l y n o m i a l s and sine-cosine functions. Moreover no general,
flexible
and systematic approach for the identification of the d i s t r i b u t e d parameter systems is available t h a t one can use v i a any k i n d of o r t h o g o n a l system. Therefore, in order to b r i n g all the o r t h o g o n a l fuction approaches to a c o m m o n p l a t f o r m , a unified identification approach is very much needed.
In a d d i t i o n to this, a c o m p a r a t i v e study
of a l l o r t h o g o n a l function approaches is also very much needed to assess the relative merits and demerits of each approach in the identification of d i s t r i b u t e d parameter systems. I n this chapter, by t a k i n g all the aforementioned points i n t o consideration, an a t t e m p t is made to e l i m i n a t e the lacunae of all e x i s t i n g m e t h o d s by i n t r o d u c i n g a unified approach for the problem of identification of d i s t r i b u t e d parameter systems of the first and the second order. Some i m p o r t a n t results of this chapter are: ( a ) development of a unified identification approach, different from t h a t of Paraskevopoulos and Bounas [239], for the e s t i m a t i o n of parameters as well as i n i t i a l and b o u n d a r y conditions of the most general second-order linear t i m e - i n v a r i a n t dist r i b u t e d parameter systems v i a any t y p e of o r t h o g o n a l functions; ( b ) development of three sets of identification equations d i r e c t l y f r o m (a) for the identification of three classes of hyperbolic systems; ( c ) discussion on p r a c t i c a l l i m i t a t i o n s of the proposed i d e n t i f i c a t i o n applied v i a block-pulse fuctions;
approach
( d ) d e m o n s t r a t i o n of the v a l i d i t y and a c o m p a r a t i v e s t u d y of a l l o r t h o g o n a l funct i o n appraoches by four different i l l u s t r a t i v e examples
two for
first-order
systems
and two for second-order systems, the latter s t i l l not available in the l i t e r a t u r e ; and ( e ) discussion on difficulties n o r m a l l y encountered i n the i d e n t i f i c a t i o n process and c r i t i c a l comments on the work of the earlier researchers.
6.2 Unified
Approach
for
185
Identification
A p a r t of this w o r k is available i n [214,213,210].
6.2
U n i f i e d A p p r o a c h for I d e n t i f i c a t i o n
Consider t h e m o d e l of a linear t i m e - i n v a r i a n t d i s t r i b u t e d parameter system described by
2
2
d y(x,t)
(in
2
d y(xj)
ha
2
2
dt dy(x,t)
a,
dx dy(xj)
ha
x
dt
w i t h i n i t i a l c o n d i t i o n s f(x)
d y(x,t)
\- a^t
XT
f{x)
= u(x,t)
(0-1)
and b o u n d a r y c o n d i t i o n s q(t) and r(t)
=
9(x)
=
q(t)
=
r L t )
1- ay(x,t)
dx
and g(x)
+
dxdt
y(x,t ) 0
dy(x, t) —~ at t = t dt y(x ,t)
0
0
d
=
defined
^ l dx
S
L
i
x
=
Xo
To identify the system given by E q . 6 . 1 , the i n p u t u(x,t)
and t h e o u t p u t
of the system are assumed t o be available over the region x € [x ,xj],t 0
T h e a i m is to estimate the system parameters u , u it
x x
, a, xt
y(x,t)
G ['<)>*/]•
u , a , a and the i n i t i a l and Y
x
boundary conditions. 2
I t may be noted t h a t depending u p o n the value of £ = a
It
— 4a a , tl
t h e system
xx
described by Eq.6.1 t u r n s o u t t o be i)
an e l l i p t i c system for £ < 0 ;
ii)
a parabolic system for £ = 0 and
iii)
a h y p e r b o l i c system for £ > 0.
I n t e g r a t i n g Eq.6.1 t w i c e w i t h respect to x and twice w i t h respect t o t we o b t a i n the integral equation
"(( / pi a
+ *t J to ft
+ a J x
ft
J
y(xA)dx
+ «»/
px
J
ft
y(x,T)dxdr
+ a, J ft
y(x,T)d\dr
fx
2
+ aJ
ft
J
J
XQ
2
y(x,r)dx dr
XQ
fX
J
y(x,r)dT to
fx
J
to
XQ
2
/ to
XO
FX
J
7
/ XQ
fX
J
2
2
y(x,T)dx dr
Chapter
186
6: Identification
XQ
x
XO
Parameter
Systems
<0
to
XQ
/'tO f d dr -tof f f XQ
of Distributed
h{ )dx dr - f
f f s{r)d dT = /' /' f f u( ,T)d dT 2
X
X
XQ
tQ
to
XQ
X
XQ
X
2
2
2
(6.2) where
c
=
a y{x ,t ) xt
0
0
h(x)
=
a g(x)
+
s(t)
=
a „ r ( t ) + a ^-ydt
lt
+ a f{x)
(6.3)
+ a q(t)
(6.4)
t
dx lt
x
w h i c h are i n t r o d u c e d i n the process of successive i n t e g r a t i o n . A p p r o x i m a t i n g a l l the k n o w n and u n k n o w n functions in x a n d / o r t i n terms of a f i n i t e set o f ( o r t h o g o n a l ) basis functions, i n t r o d u c i n g t h e m i n Eq.6.2 and s i m p l i f y i n g w i t h the help of
/ /' /" I' ^ (x)F4>{r)dx dr ~ (x)E FE
' *0
2
T
2
i2
x2
*0
XQ
"
XQ
we get a set of linear simultaneous equations represented by M p = v.
(6.5)
T h i s means t h a t the original system described by Eq.6.1 is reduced to a simple algebraic system i n Eq.6.5 i n which
M
=
T
T
{vec(E[' Y)
| v e c ( y £ ) | v e c ( £ j y - £ ) | vec{E Y
x2
( 2
TT
vec(E YE ) vec{E x
(
E)
x2
t
| vec(£j y-£ ) | -vec(£j A,,) | ••• |
i2
2
-vec(Ej A ) 2
| -vec(A £
o l
r
-vec(f; A x
1 1
i 2
u
i 2
2
) | ••• | - v e c ( A
£
( 2
) |
£ ) | -vec(£j A f; ) | • i
2
u
<
-vec(£j A £;<) | - v e c ( £ j A £ 2
I / 3
7 l
n
1 2
) |
-vec^jA^E,,)] (6.6)
P
=
[o-u,a ,a ,a ,a ,a, xx
xt
t
9 / 9 - 1 , c, h , 0
v
=
vec{El UE ) 2
t2
x
• • •
f ,-• 0
s, 0
• , f _ q ,-• a
u
•,
0
• • •,«s-i]
r
(6.7)
(6.8)
where A y is an m x n m a t r i x h a v i n g ( i , j ) t h element u n i t y and a l l o t h e r elements zero,
6.2 Unified
Approach
for Identification
187
T
u(x,t)
«
y (x)U(t),
y(x,t)
«
1, (x)Y4(t),
fi
T
=
(6.9)
i = 0 , 1 , ••-,<* — 1 c*-l
/(x)
ft!
T
V (x)£
/,A
i + I i l
*(«),«< m
1= 0
h(x)
Qj
=
« . r 9 , , j = 0, 1, - • • , / ? - 1
It
m
i> (x)
c
=
V> (x)cA 0(<)
«
T
J2 A {t), qj
1J+l
(6.10)
0 < n
T
(6.11)
1 1
T
tl> {x)
-r-i Yl hiA ^,{t),
7 < m
i+1A
i=0 S-1
«
T
V (x)IZ^Ai
i J + 1
0 ( t ) , (5 < n
j=o and for any m a t r i x A of order m x n the vector valued f u n c t i o n is defined [173] as &i a vec(^)
{ m n x l )
2
=
a„
w i t h a, being t h e zth c o l u m n of A. Eq.6.5 is a linear algebraic system whose s o l u t i o n given by T
p = [M M] d i r e c t l y gives a l l t h e system parameters.
_ 1
T
[M v],
(6.12)
Since M is an m n x (7 + a + 0-\-j
+ 6)
m a t r i x , t h e r a n k of M m u s t be equal to (7 + a + 0 + 7 + S) t o d e t e r m i n e p f r o m Eq.6.12, o t h e r w i s e the system i d e n t i f i c a t i o n is not possible. are o b t a i n e d , t h e i n i t i a l c o n d i t i o n f(x)
Once t h e
parameters
and the b o u n d a r y c o n d i t i o n q(t) can also be
o b t a i n e d f r o m Eq.6.9 and Eq.6.10 respectively. For finding the other i n i t i a l c o n d i t i o n g(x),
we i n t e g r a t e Eq.6.3 once w i t h respect t o x t o o b t a i n an i n t e g r a l e q u a t i o n and
t h e n a p p r o x i m a t e a l l functions i n x i n terms o f a finite set of basis functions a n d i n t r o d u c e t h e same i n t h e i n t e g r a l e q u a t i o n , and simplify t o get
Chapter
188
6: Identification
« TP (X){[II t
g(x)
of Distributed
Parameter
1
- «,f] + o « ( £ ? J ) - [ c e , - f ] } / o 1
Systems
(6.13)
( <
where e
= [1,
x
0,--,0
( m - l )
T
]
(6.14)
d m . . ) .
I n a similar way, the other b o u n d a r y c o n d i t i o n r(<) i n Eq.6.4 may be o b t a i n e d from r
r
p(t) ss { [ s -
a i
1
q ] + a [cef
- q "]^,
xl
- 1
(6-15)
where e, = [ 1 ,
0, ••• ,0
(n-l)
]
T
(6.16)
elements
T h u s the above identification a l g o r i t h m is capable of e s t i m a t i n g the parameters as also the i n i t i a l and b o u n d a r y conditions of the m o d e l of any system defined by Eq.6.1. For other second-order systems whose models do not c o n t a i n a l l the parameters shown in the general m o d e l of E q . 6 . 1 , the following three models may be considered so t h a t any second-order system described by one of these models can be identified. These three models and the corresponding sets of identification equations may be directly o b t a i n e d from those of the general m o d e l in E q . 6 . 1 . M o d e l O n e : Models w i t h a the identification equations are:
= 0 i n E q . 6 . 1 . For this class of h y p e r b o l i c systems
t<
M
=
[vec(YE,)
| v e c ( F / y ) | vec(E Y)
\ vec{E YE )
x2
x
vec(£j y£ ) | -vec(AnB,5 | ••-] -vec(A 2
(
|
t
1 / 3
F )| (
- v e c ( E j ' A ) | - v e c ( £ j A ) | ••• | - v e c ( F j A 1 1
2
n
2
T l
) |
- v e c ( £ j A £ ; ) | ..- | - v e c t E j A ^ F , ) ] u
P
=
v
= =
(6.17)
0
/ l
h(x)
(
[ « « , O i l , i ( , « i , « , 9o> • • • , 9/s-i, c, h , . . . , 7
_ ! , 3
0
,
• • •
,«5-l]
(6.18)
T
vec(E UE ) x2
df(x) a —- — xt
7
t
+ a f{x)
(6.19)
i
Since there are altogether (6 + 0 + 7 + 6) unknowns in t h e present s i t u a t i o n , the rank of the m a t r i x M must be equal to (6 + ft + 7 + <5) t o o b t a i n p f r o m Eq.6.12. For finding
the i n i t i a l c o n d i t i o n f{x) f(x)
where I
x
we take the help of Eq.6.19 and o b t a i n
T
w j, (x)[a I
xt x
+ a Fj][Fjh + cej (
(6.20)
is an i d e n t i t y m a t r i x of order m.
M o d e l T w o : Models w i t h a the identificaion equations are
IX
= 0 in E q . 6 . 1 . For this class of h y p e r b o l i c systems
6.2 Unified
Approach
M
for Identification
T
=
\vec{E Y)
| vec(YEt)
vec(ElYE ) -veciA^E,)
x
=
( 2
) | ••• | -vec(A
a « , a , / o , . - . . . <, 7
t 2
7
n
. . . , / l
| vec(YE ) \
t
| - v e c ^ j A u E ) | ••• | - v e c ( E j A , E , ) |
-vec(A F
v
\ vec(E YE )
| - v e c ( £ j A „ ) | ••• | - v e c ^ A ^ ) |
n
P
189
_ l , S
0
, . . . ,
=
vec(£jt/£ )
=
a ^ - ^ at
3 { _ i ]
1 {
(6.21)
F )] i 2
8» re , 0
(6.22)
T
i 2
and S
(t)
+ a^i)
(6.23)
I n this case the rank of the m a t r i x M must be equal to (6 + a + 7 + <5) for successful i d e n t i f i c a t i o n . T h e b o u n d a r y c o n d i t i o n q(t) may be o b t a i n e d from
T
- l
q(t) « [ s £ , + c e f H a ^ i , + o » £ ] ^ ( t )
(6.24)
(
w h i c h i n t u r n m a y be o b t a i n e d from Eq.6.23. I n Eq.6.24 I order
is an i d e n t i t y m a t r i x of
t
n.
M o d e l T h r e e : Models with a
u
= a
XI
= 0 i n E q . 6 . 1 . For this class of h y p e r b o l i c
systems the i d e n t i f i c a t i o n equations are:
M
=
[vec(y) | v e c ( F j y ) | v e c ( y £ ) | v e c ( £ ; J y £ ) | (
-vec(A ) | -vec(FjA u
u
(
| ••• | - v e c ( F j A
- v e c ( A „ F , ) | ••• | - v e c ( A F ) ] l i
P
=
v
=
[a*!, «(,
i,c,
h ,. 0
T
0
) | (6.25)
i
. ., / i _ ! , s ,.
7 l
T
. . , s_] s
1
(6.26)
T
vec(E YE ) t
I n this case, the rank of m a t r i x M must be equal to (5 + 7 + <5). I t is interesting to note t h a t w h e n a the
first-order
xi
= 0 is set in this m o d e l , the same set of i d e n t i f i c a t i o n equations of
systems, proposed by Paraskevopoulos and Bounas [239], is o b t a i n e d .
T h e proposed procedure for the i d e n t i f i c a t i o n of d i s t r i b u t e d p a r a m e t e r m o d e l described by Eq.6.1 can be d i r e c t l y used v i a Haar functions, W a l s h functions, a n d all classes o f o r t h o g o n a l p o l y n o m i a l s .
I n order to use this m e t h o d v i a block-pulse
functions and sine-cosine functions, certain terms in the above a l g o r i t h m i c procedure must be redefined as o u t l i n e d i n the following t w o sections. One drawback w i t h W a l s h functions a n d Haar functions is t h a t the n u m b e r of basis functions have to be adjusted to 2* where k is any positive integer.
Chapter
190 6.2.1
Identification
6: Identification
of Distributed
via Block-Pulse
Parameter
Systems
Functions
W h i l e a p p l y i n g the unified approach t o t h e identification of d i s t r i b u t e d parameter systems by using block-pulse functions, the following m o d i f i c a t i o n s are t o be incorporated. For
block-pulse functions, ct,0,j
and 6 always take t h e i r m a x i m u m values i.e.,
ct = 7 = m a n d 0 = 6 = n. T h e m a t r i x A
f l
, i = 1,2,. . . , m m u s t be replaced by A
i r
w h i c h is an m x n m a t r i x c o n t a i n i n g a l l the i t h row elements e q u a l t o u n i t y and all other elements zero. S i m i l a r l y , A
V )
, j = 1, 2 , . . . , n m u s t be replaced by A
; c
w h i c h is
again an m x n m a t r i x b u t this t i m e h a v i n g a l l j t h c o l u m n elements equal to u n i t y and
a l l other elements zero. O n l y in E q . 6 . 1 1 , A n m u s t be replaced by A * „ w h i c h is
an m x n m a t r i x c o n t a i n i n g a l l the elements equal to u n i t y . T h e vectors e , i n Eq.6.14 and
e i n Eq.6.16 must be considered as (
(6.27)
(6.28)
To d e t e r m i n e p f r o m Eq.6.12, the rank of m a t r i x M must be equal t o ( i ) 7 + 2 m + 2 n for t h e m o d e l E q . 6 . 1 , ( i i ) 6 + m 4- 2n for the m o d e l one, ( i i i ) 6 + 2 m + n for t h e m o d e l two and ( i v ) 5 + m + n for the m o d e l three. T h e dependence o f the size of augmented parameter vector p on m and n may be clearly seen from ( i ) t o ( i v ) above i n this approach w h i c h is i n contrast w i t h the other o r t h o g o n a l f u n c t i o n approaches i n which the size of p depends on ct,0,-y a n d 6. T h i s p o i n t may be considered as one of the drawbacks of block-pulse approach as the parameter vector size always increases for every increment i n m a n d / o r n i.e., the n u m b e r o f block-pulses used i n signal characterization.
6.2.2
Identification
via Sine-Cosine
Functions
I n order t o apply the proposed identification procedure v i a sine-cosine f u n c t i o n s , all m — a n d n— vectors and m x m and n x n matrices m u s t be changed t o ( 2 m — 1) — and ( 2 n - 1 ) - vectors and ( 2 m - 1) x ( 2 m - 1) a n d (2ra - 1) x (2n - 1) matrices, respectively. For sine-cosine functions, the rank of m a t r i x M i n Eq.6.5 m u s t be equal t o (i)
3 + 2 ( Q + 0. + 7 + (5) for the m o d e l E q . 6 . 1 ,
(ii)
3 + 2(0 + 7 + f5) for the m o d e l one,
(iii)
3 + 2 ( Q + 7 + <5) for the m o d e l t w o , a n d
(iv)
3 + 2 ( 7 + 6) for the m o d e l three,
6.3 Practical
Limitations
of
Identification
assuming ce = /3 = y = 6 ^ 0 .
191
I t m a y be seen t h a t for a l l t h e models, t h e order
of t h e a u g m e n t e d parameter vector p does not depend on m and n as long as t h e basis f u n c t i o n s employed for the purpose of i d e n t i f i c a t i o n are Haar f u n c t i o n s , W a l s h functions, sine-cosine functions and any class of o r t h o g o n a l p o l y n o m i a l s - a s u p e r i o r i t y of t h e unified approach v i a these basis functions. T h e m a i n drawback w i t h t h e sinecosine functions is t h a t i t is not c o m p u t a t i o n a l l y as a t t r a c t i v e as other basis functions as the i d e n t i f i a t i o n procedure involves vectors and matrices w h i c h are almost double t h e size o f those encountered w i t h other functions.
6.3
P r a c t i c a l Limitations of Identification
I t was already to the n u m b e r as i n i t i a l and circumstances.
stated i n t h e last section t h a t t h e r a n k of m a t r i x M m u s t be equal of u n k n o w n s for t h e successful i d e n t i f i c a t i o n of t h e parameters as well b o u n d a r y c o n d i t i o n s . T h i s c o n d i t i o n sometimes fails under certain To investigate this i m p o r t a n t aspect we r e w r i t e E q . 6 . 5 as
where [A:r?] = M,
the m a t r i x A contains a l l the columns corresponding t o system
parameters a n d t h e m a t r i x B contains a l l the r e m a i n i n g columns of M. Now t h e r a n k of M is always less t h a n the n u m b e r of columns in i t i f there exists at least t w o l i n e a r l y dependent columns of A or of B . Since t h e l i n e a r l y
dependent
columns of A depends e n t i r e l y on Y, i t must be tested essentially w i t h respect t o t h e o u t p u t sinal y(x,t).
No general conclusions can be made i n this c o n t e x t .
Coming
to t h e case of l i n e a r l y dependent columns of B, the complete s t r u c t u r e of B must be k n o w n first.
T h i s depends on the i n t e g r a t i o n o p e r a t i o n a l matrices E
OSOMRI's E
and E
x2
x
i2
and
E, t
and the values of a, / ? , 7 , and <5. Since B is c o m p l e t e l y
independent of the i n p u t and o u t p u t signals, for the chosen o r t h o g o n a l system a n d a, (3,7 a n d 6 values, m a t r i x B must be tested separately before proceeding w i t h the i d e n t i f i c a t i o n .
A n extensive c o m p u t a t i o n a l e x p e r i m e n t a t i o n has revealed t h a t
there are m a n y possible c o m b i n a t i o n s of a
<
m,/J <
n,j
< m and 6 < n for
w h i c h m a t r i x B and thereby m a t r i x M t u r n o u t t o have less t h a n its full c o l u m n rank m a k i n g the i d e n t i f i c a t i o n not possible.
Therefore, t h e i d e n t i f i c a t i o n v i a Haar
f u n c t i o n s , W a l s h f u n c t i o n s , or any class of o r t h o g o n a l p o l y n o m i a l s m u s t be preceded by a linear independence test on the m a t r i x B. However, since a,/3,f
and 6 always take t h e i r m a x i m u m values for block-pulse
f u i n c t i o n s , i t is q u i t e possible t o say s o m e t h i n g about the l i n e a r l y independent columns of B. W h e n t h e l i n e a r l y independence test is conducted on B, o u t of a l l possible comb i n a t i o n s o f a, (3, 7 and 6, the f o l l o w i n g four c o m b i n a t i o n s only, see Table 6 . 1 , t u r n e d out
t o be favourable for a successful i d e n t i f i c a t i o n .
m e t h o d o f block-pulse functions fails.
For a l l other c o m b i n a t i o n s , t h e
192
Chapter
6: Identification
of Distributed
Parameter
Systems
Table 6.1: C o n d i t i o n s for i d e n t i f i c a t i o n w i t h block-pulse Functions. Model
Eqn.No.
c
a
6
6.6,6.17
0
0
0 0
7
All
0
0
0
0
0
m
0
0
0
0
0 n
6.21,6.25 General
6.6
One
6.17
0
0
0
0
n
Two
6.21
0
0
0
m
0
I t appears from Table 6.1 t h a t the i d e n t i f i c a t i o n m e t h o d w i t h block-pulse functions works only when ( i ) a l l the i n i t i a l and b o u n d a r y conditions are zero, ( i i ) the i n i t i a l c o n d i t i o n g(x)
alone is present, or ( i i i ) the b o u n d a r y c o n d i t i o n r(t)
alone is present.
T h i s m e t h o d , however, fails for the m o d e l three. T h e four conditions in Table 6.1 can be easily proved. A l l o p e r a t i o n a l matrices E ,E x
t
including OSOMRFs E
l 2
,E
t2
are always nonsin-
gular matrices. Hence a l l the columns corresponding t o a or 0 or 7 or 6 are always linearly
independent.
Whenever the i n i t i a l c o n d i t i o n f{x) c = y(x ,t ) 0
0
or the b o u n d a r y c o n d i t i o n q(t)
is present,
also comes i n t o picture and makes the columns (c, a) or (c,0)
linearly
dependent. S i m i l a r l y , the simultaneous presence of the i n i t i a l c o n d i t i o n g(x) and the b o u n d a r y c o n d i t i o n r ( i ) makes the columns ( 7 , i5) l i n e a r l y dependent. As a check, one may f o r m the columns of the m a t r i x B i n each case and determine the rank by a p p l y i n g a numerically reliable m e t h o d . T h e Haar and Walsh functions can be expressed as a linear c o m b i n a t i o n of blockpulse functions [257].
Therefore the i d e n t i f i c a t i o n v i a Haar or W a l s h functions is
not possible under the conditions when the m e t h o d of block-pulse functions fails. I n this c o n t e x t , i t may be recalled t h a t the block-pulse functions are applicable only if a = 7 = m and 0 = 6 = n and the above comparison is made w i t h these conditions in m i n d .
6.4
Illustrative
Examples
I n order t o d e m o n s t r a t e the a p p l i c a b i l i t y of the proposed i d e n t i f i c a t i o n scheme v i a all classes of o r t h o g o n a l functions, a few n u m e r i c a l examples are considered i n this section. E x a m p l e 6 . 1 It is required [239]
to estimate
the parameters
of the system
modelled
by
6.4 Illustrative
Examples
193
dy(x,t)
dy{x,t) + a.
~~dt with zero initial output
y(x,t)
to estimate a random
and boundary
are assumed having
conditions
of Eq.6.39
noise-to-signal
+ ay(x,t)
=
(6.29)
u(x,t)
when the input u(x,t)
to be available
the parameters noise
dx
= 4x + 2t + xt and the
over x 6 [ 0 , 1 ] , / € [ 0 , 2 ] . It is also
when the output ratio (NSR)
signal y{x,t)
required
is corrupted
7 = 0.05,0.10,0.15,0.20
with
or 0.25.
Since there are altogether three parameters to be e s t i m a t e d , let us take m = n = 2 for the signal characterization and a p p l y the i d e n t i f i c a t i o n scheme of m o d e l three w i t h a = 0. T h e p a r a m e t e r estimates are shown in Table 6.2. l t
I t may be seen f r o m the Table 6.2 t h a t the parameter estimates o b t a i n e d v i a each class of o r t h o g o n a l functions are in good agreement. T h e i m p o r t a n t observations
that
are made i n this study are: ( a ) T h e e v a l u a t i o n of the spectrun for the signals using block-pulse functions is quite simple and fast. ( b ) T h e p a r a m e t e r e s t i m a t i o n in a noise-free s i t u a t i o n is very fast and accurate if the Gaussian q u a d r a t u r e formulae of the open t y p e are used i n the evaluation of s p e c t r u m for the signals.
To a p p l y these formulae, the signals must be e x p l i c i t l y
k n o w n . T h e o u t p u t signal i n the present s i t u a t i o n is xt (see [239]). ( c ) T h e accuracy of the parameter estimates depends on the accurate c o m p u t a t i o n of the spectra for the signals. As p o i n t e d out earlier, i n C h a p t e r 3, s p e c t r u m evalua t i o n can be done i n t w o ways i.e., using ( i ) the closed q u a d r a t u r e f o r m u l a and ( i i ) the open q u a d r a t u r e f o r m u l a . T h e former, a l t h o u g h very m u c h useful i n c o m p u t i n g the spectra of noisy signals, produces less accurate results at the cost o f t i m e . T h i s consideration is very i m p o r t a n t i f the weight f u n c t i o n w(x,t)
has a " c u p " shape as i n
the case of Tchebycheff p o l y n o m i a l s of the second k i n d . These p o l y n o m i a l s , as m a y be seen in the Table 6.2, produce two different sets of estimates; the first set, m a r k e d ( - ) , corresponds to the open q u a d r a t u r e f o r m u l a while the second set corresponds to the closed q u a d r a t u r e f o r m u l a . ( d ) T h e parameter e s t i m a t i o n v i a Laguerre p o l y n o m i a l s ( L a P ) requires the i n p u t and o u t p u t signals over a wide region as these p o l y n o m i a l s are defined over an i n f i n i t e range. I t is t r u e t h a t recursive relations are possible (see [252]) due to the b i d i a g o n a l s t r u c t u r e of E
x
and E . t
B u t , an on-line i m p l e m e n t a t i o n using these p o l y n o m i a l s is
not a t t r a c t i v e as ( i ) the evaluation of the Laguerre s p e c t r u m is n o t an easy task and ( i i ) the past i n f o r m a t i o n of the signals cannot be lost for t a k i n g the effect of i n i t i a l and b o u n d a r y
conditions i n the i d e n t i f i c a t i o n .
Due to the disjoint n a t u r e of the
block-pulse functions and t h e i r s u i t a b i l i t y for the real t i m e c o m p u t a t i o n of spectra of signals, these functions are more convenient for on-line i m p l e m e n t a t i o n p r o v i d e d b o t h the i n i t i a l and b o u n d a r y conditions are not present simultaneously. T h i s parameter e s t i m a t i o n p r o b l e m is successfully solved v i a Laguerre p o l y n o m i a l s b y assuming
t h a t the signals are available over x € [ 0 , 1 1 ] , / € [ 0 , 1 2 ] .
T h e next
Chapter
194
6: Identification
of Distributed
Parameter
example is concerned w i t h the e s t i m a t i o n of the i n i t i a l and b o u n d a r y
Systems conditions
along w i t h parameters. E x a m p l e 6.2 It is required conditions
of a system
a, when the input
u(x,t)
It is also required with a random
to estimate
modeled
the parameters
dy(x,t) at
dy(x,t)
1- <x»
dx
= t + 1 and the output
to estimate
and the initial
and
boundary
by [146]
= u{x,t),
y(x,t)
are available
the same again when the output
noise having NSR=0.05,
(0.30)
0.10, 0.15, 0.20, or
y(x,t)
over x,t is
€ [0,1).
contaminated
0.25.
Since the m o d e l i n Eq.6.30 is a special case of m o d e l three i n Section 6.2, we apply the i d e n t i f i c a t i o n equations for m o d e l three w i t h a
It
= a = 0 , m = 2 , n = 4,7 = 2
and <5 = 3. T h e estimated results are shown i n Tables 6.3,6.4 and 6.5. As expected the m e t h o d using block-pulse functions has failed i n t h i s example as the columns of B m a t r i x , see Section 6.3, are n o t l i n e a r l y i n d e p e n d e n t . For zero NSR, i t is seen from Tables 6.3 and 6.4, how the parameter estimates using the Tchebycheff p o l y n o m i a l s of the second k i n d depend on the m e t h o d of s p e c t r u m e v a l u a t i o n . The (-) m a r k e d estimates are obtained v i a the Gauss open q u a d r a t r u e f o r m u l a (y(x,t) = x + 0.5t , see [146]) while the subsequent sets of estimates are o b t a i n e d v i a the closed quadrature formula. 2
T h i s example is also studied v i a Laguerre p o l y n o m i a l s by considering m — 2, n = 4 (just sufficient for the signal characterization and t o make the algebraic system given by Eq.6.5 o v e r d e t e r m i n e d ) , 7 = 2,6 = 3 and the i n p u t and o u t p u t signals over the region x,t G [0,15] to estimate the parameters and the spectra of the i n i t i a l and b o u n d a r y condtions. T h i s estimated result seems t o be q u i t e satisfactory. The estimates o b t a i n e d v i a sine-cosine functions w i t h m = 2, a n d n = 4 are shown i n Table 6.5. I n this case, the estimates can be f u r t h e r i m p r o v e d by considering more terms i n the series expansion. T h e next example shows the parameter e s t i m a t i o n for the most general secondorder system. E x a m p l e 6.3 It is required to estimate the parameters of the system described by Eq.6.1 when its input u(x,t) = (x + t + 2xt) and its output y(x,t) are available over i , l 6 [0,16] and the initial and boundary conditions are zero. 2
Since there are altogether six unknowns t o be i d e n t i f i e d , let us take m = n = 3 and apply the proposed i d e n t i f i c a t i o n approach using a l l the systems o f o r t h o g o n a l functions. T h e parameter estimates are shown i n Tables 6.6 and 6.7. T h e first set of parameter estimates o b t a i n e d using sine-cosine functions are nowhere near the actual parameters.
However, w i t h m = n = 15, these functions have p r o d u c e d excellent
results as is evident from the second set of SCF results i n Tables 6.6 a n d 6.7.
6.4 Illustrative
Examples
195
Table 6.2: D i s t r i b u t e d parameter system i d e n t i f i c a t i o n w i t h noisy d a t a i n E x a m p l e 6.1. A p p r o a c h J.
NSR
a, 4
2
1
0.00
2.0000000
1.0000000
0.05
4.0000000 4.0096739
1.9980807
1.0104679
0.10
4.0193903
1.9961488
1.0209641
0.15
4.0291495
1.9942042
1.0314885
0.20
4.0389516
1.9922469
1.0420413
0.25
4.0487969
1.9902769
1.0526223
TPl
0.00
4.0000000
2.0000000
1.0000000
(-)
0.00
4.0000010 3.9835752
2.0000005
0.9999975
1.9917876
1.0524816
1.9973049 2.0028184
1.0431734
0.10
3.9751867 3.9667872
0.15
3.9583765
2.0083280
1.0245487
0.20
3.9499549
1.0152323
0.25
3.9415223
2.0138337 2.0193354
0.00
4.0000000
2.0000000
0.05
4.0001419
1.9947943
1.0000000 0.9960242
0.10
4.0002889
1.9895830
0.15
4.0004413
1.9843661
0.9920936 0.9882074
Or
a
A c t u a l parameters —•
BPF
0.00 0.05 TP2
LeP
LaP
SCF
1.0338624
1.0059132
0.20
4.0005998
1.9791438
0.9843644
0.25
4.0007647
1.9739162
0.9805639
0.00
3.9975947
1.9968840
1.0063314
0.05
4.0015651
1.9985769
1.0043083
0.10
4.0055413
2.0002716
1.0022811
0.15
4.0095231
1.0002499
0.20
4.0023083
2.0019681 1.9923182
0.25
4.0175040
2.0053665
0.9961753
0.00
3.9999971
1.9999974
0.05
1.9998878
1.0000107 0.9879680
0.10
3.9998316 3.9995152 3.9990592
1.9997253 1.9995154
0.9762400
0.15 0.20
3.9984672
1.9992588
0.9536276
0.25
3.9977429
1.9989562
0.9427323
1.0044197
0.9647951
Chapter
196
6: Identification
of Distributed
Parameter
Systems
Table 6.3: D i s t r i b u t e d parameter system identification w i t h noisy d a t a i n E x a m p l e 6.2. A p p r o a c h J.
NSR
BPF A c t u a l —• TPl
1 0.00
TP2
fi
0.9999999
1 1
0.5 0.4999999
0.5 0.5000002
1
0.5
0.25
1
0.9999999
0.',
0.2499998
0.00
1.0221032 1.0265152
0.9903777 0.9878044
0.5087825 0.5108183
0.2551218
0.05 0.10
1.0309655
0.9852117
0.5128717
0.15 0.20
1.0354546 1.0399829
0.9825993
0.5149430 0.5170324
0.2586161
0.25
1.0445510
0.5191401
0.2609968
1
0.9799670 0.9773144
0.2562765 0.2574412 0.2599012
0.5
0.5
0.5 0.4968681
0.5000007
0.4937746
0.4944353
0.4907095 0.4876724
0.4916969
0.4846629
0.4862948
1
1 1
0.10
0.9953449 0.9907555
1.0004516 1.0008974
0.15 0.20
0.9862083 0.9817025
1.0013488
0.25
0.9772378
1.0018055 1.0022677
1
I
1
-1
0.00
1.0007593
1.0009469
1.0043015
-1.0021571
0.05
1.0016818
0.9939931
0.9966519
-1.0030547
0.10
1.0026061
0.9870260
0.9889881
-1.0039539
0.15 0.20
1.0035320 1.0044597
0.9800457
0.9813101
0.9730520
0.9736180
-1.0048547 -1.0057572
0.25
1.0053891
0.9660450
0.9659116
-1.0066614
0.00 0.05
Actual -»
LaP
fo
1
A c t u a l —>
LeP
<2x
0.00
A c t u a l —> (-)
a, failed
0.4971990
0.4889835
6.4 Illustrative
197
Examples
Table 6.4: D i s t r i b u t e d parameter system identification w i t h noisy d a t a i n E x a m p l e 6.2. Approach |
NSR
BPF Actual — TPl (-)
TP2
92
0.1875
0.25
0.0625
0.1874999
0.25
0.0624999
0.15625 0.1562498
0.125
0.00 0.00
0.1550907
0.125 0.1236015
0.03125 0.0312499 0.0302321
0.05
0.1548968
0.1233190
0.0300239
0.10
0.1546986
0.1230340
0.0298159
0.15
0.1227463
0.20
0.1544963 0.1542897
0.1224558
0.0296079 0.0294000
0.25
0.1540789
0.1221627
0.0291922
0.1666666
0.25
0.00 0.05
0.1666666 0.1674404
0.2499996 0.2515340
0.0833333 0.0833332
0.10
0.1682138
0.2530676
0.0841863
0.15
0.1689888
0.2546040
0.0846126
0.20
0.1697653 0.1705434
0.2561431
0.0850398
0.2576849
0.0854677
1
-2
1
0.9999942
-1.9994344
1.0015849
0.05
0.9905868
-1.9906511
0.9937106
0.10
0.98120006
0.9858359
0.15
0.9718358
-1.9817698 -1.9727902
0.20
0.9624923
-1.9637119
0.9700853
0.25
0.9531703
-1.9545348
0.9622093
Actual —
LeP
0.25
-» 0.00
LaP
9i
0.00
A c t u a l ->
Actual
9o
failed
0.0837608
0.9779608
Chapter
198
6: Identification
of Distributed
Parameter
Systems
Table 6.5: D i s t r i b u t e d parameter system i d e n t i f i c a t i o n w i t h noisy d a t a v i a sine-cosine functions i n E x a m p l e 6.2. NSR |
a,
Actual—•
1
o» 1
fo
/.
fx
0
0.3183098
0.00
0.9893760
1.0038594
0.5 0.4743052
0.0126766
0.3149706
0.05
0.9787617
1.0100963
0.4687277
0.0128956
0.3102211
0.10
0.9688667
1.0151062
0.0188288
0.3057458
0.15
0.9565849
1.0214645
0.4603366 0.4508572
0.0249483
0.3005180
0.25
0.9256266
1.0374357
0.4290567
0.0375418
0.2880553
9o
9i
92
9i
92
-0.0506605 -0.0516791
0.0126651
0.10
0.1756470
-0.0533830 -0.0610552
0.0136418 0.0198537
0.1591549 0.1580141 0.1596782
-0.0795774
0.05
0.1666666 0.1670388 0.1701241
0.1611883
-0.0846907
0.15
0.1813010 0.1870572
-0.0686443
0.0259743 0.0319794
0.1624854
-0.0873374
0.1635789
-0.0898234
0.0378483
0.1644817
-0.0921453
Actual 0.00
0.20 0.25
0.1928857
-0.0761143 -0.0834327
0.0124050
Table 6.6: Parameter estimates a ,a„ it
A p p r o a c h J.
a
Actual—*
0.5
tt
and a
xi
a-xx 0.5
-0.0784735 -0.0818898
i n E x a m p l e 6.3. a
xt
0.5
BPF
0.5
TPl
0.4994722
0.4999999 0.4994722
0.5000644
TP2
0.5000648
0.5000648
0.5000944
LeP
0.4998888
0.4998888
0.5000051
LaP
0.5001439
0.5001439
0.4992119
SCF
1.3831396
1.3831396
0.0502094
0.5076560
0.5076560
0.5159741
0.5
6.4 Illustrative
199
Examples
Table 6.7: Parameter estimates a ,a t
and a i n E x a m p l e 6.3
x
A p p r o a c h J.
a,
Actual—•
2
2
a 4
BPF
2
2
TPl
2.0042473
TP2
2.0042473 1.9990813
3.9999999 3.9771482
1.9990813
4.0045014
LeP
2.0009978
2.0009978
3.9943470
LaP
1.9981776
1.9981776
4.0035550
SCF
-0.7739838
-0.7739838
9.5803031
2.0092280
2.0092280
4.0217935
In the last e x a m p l e , an estimate of the parameters in presence o f i n i t i a l and b o u n d a r y c o n d i t i o n s is made for a h y p e r b o l i c system as i n m o d e l three. E x a m p l e 6 . 4 It is required boundary
conditions 2
input u(x,t)
= (xt)
to estimate
of the system
the parameters
described
by model
+ (xt + \ )(x + t) and the output
as well as the initial
three in Section y(x,t)
and
6.2 when
are available
the
over x,t
€
[0,16]. For t h e purpose of s i m u l a t i o n , the t r u e parameters of the system taken are a
xl
— 0.5, a, = a
x
=
= 0.5 and a = 1 w h i l e the i n i t i a l and b o u n d a r y conditions for the given
i n p u t are j / ( x , 0 ) = 3/(0,2) = 1. N o w , by e m p l o y i n g the set of i d e n t i f i c a t i o n equations for m o d e l three w i t h m = n — 3 a n d
boundary
conditions are estimated using all the classes of o r t h o g o n a l functions. T h e estimated results are as shown i n Tables 6.8 and 6.9. I t is observed f r o m these tables t h a t the estimates are very much i m p r o v e d when the values of m and n are increased f r o m 3 to 12 for sine-cosine functions. T h e first set of results for S C F corresponds to m = n = 3 w h i l e the second corresponds to m = n = 12. I n view of the results for the above four different d i s t r i b u t e d p a r a m e t e r systems using a l l classes o f o r t h o g o n a l functions i t appears t h a t the m e t h o d using block-pulse functions is the simplest as i t is very fast and so c o m p u t a t i o n a l l y very a t t r a c t i v e . U n f o r t u n a t e l y , i t has got its o w n p r a c t i c a l l i m i t a t i o n s discussed i n Section 6.3. Disreg a r d i n g the c o m p u t a t i o n a l effort, a l l the o r t h o g o n a l p o l y n o m i a l s seem to be powerful. O w i n g t o the poorer convergence rate of Fourier series, the sine-cosine functions seem to be t o o laborious c o m p u t a t i o n a l l y .
Chapter
200
6: Identification
Table 6.8: Parameter estimates a ,a xt
Approach 1 Actual—* BPF TPl TP2 LeP LaP SCF
Oil -0.5 failed -0.5000030 -0.5000039 -0.4999999 -0.4994676 -0.5006890 -0.5004474
t
of Distributed
a
Actual—*
1
BPF TPl TP2 LeP LaP SCF
failed 1.0000079 1.0000089 0.9999978 1.0006630 1.0046791 1.0000251
Systems
and a, in E x a m p l e 6.4. a,
.
0.5 0.4999987 0.4999990 0.5000002 0.4993234 0.4980849 0.4999774
y(*,0) l
a 0.5 x
0.4999987 0.4999990 0.5000002 0.4993234 0.4980849 0.4999774
Table 6.9: Parameter estimate a, and i n i t i a l c o n d i t i o n y(x,0) i/(0, t) in E x a m p l e 6.4. Approach I
Parameter
and b o u n d a r y condition
y(o,t) 1
l l 1.0000002
1 1 1.0000002
0.9982402 1.1350370 1.0279839
0.9982402 1.1350370 1.0279839
6.5
Problems
6.5
201
Problems
1. Consider t h e linear d i s t r i b u t e d parameter system described by
giKjMj , a
dy(x,t)
ha
h a y(x,t) = u(x,t) dx dt w i t h i n i t i a l c o n d i t i o n y ( x , 0 ) = 0 and b o u n d a r y c o n d i t i o n s y(0,t) 3
2
1
E s t i m a t e the parameters when the i n p u t u(x,t) y(x,t)
= 0,3/(1,*) =
t.
= xt + Ax + 2t and the o u t p u t
= xt are c o r r u p t e d w i t h measurement noises i ? ( x , / ) and { ( x , t ) ,
independent
zero-mean w h i t e noises, w i t h variances r = 0.0833 and q = 0.0833 respectively.
A n s w e r : a = 2,a 3
2
= 4, a, = 1
2. Consider the one-dimensional diffusion e q u a t i o n 2
dy(x,t)
d y(x,t) 0 , x € [0,1]
2
dt
dx
y ( 0 , r ) = 0,
dy(l,t) *\ dx
= 0,y(x,0) = 1
E s t i m a t e the p a r a m e t e r c.
[Hint:
For t h e purpopse of e s t i m a t i o n of c, s i m u l a t e the system w i t h c = 0.9 and
generate the d a t a y ( x , i ) . [
3.Consider t h e second order p a r t i a l differential e q u a t i o n dy(x,t)
2
d y(x,t) dy(x,t) = a — — — + b— dx dx L
2
dt
cy(x,t),x
6 [0, 1]
w i t h i n i t i a l c o n d i t i o n y ( x , 0 ) = 0 and b o u n d a r y c o n d i t i o n s dy(0,t) dx Estimate the
dy(l,t) dx
= 0.
parameters.
[ H i n t : Generate t h e d a t a y(x,t) c = 0.986]
^ = or;
by s i m u l a t i n g the system w i t h a = 0.1,6 = 0.2 a n d
Chapter
7
Identification
of
Linear
T i me-Var y in g a n d Distributed
Nonlinear
Parameter
Systems
A n i n t e g r a l approach for the i d e n t i f i c a t i o n o f parameters, i n i t i a l and b o u n d a r y c o n d i tions o f linear t i m e - v a r y i n g , and nonlinear single-input single-output c o n t i n u o u s - t i m e d i s t r i b u t e d p a r a m e t e r systems is presented i n this chapter w h e n the i n p u t and o u t p u t signals of the system are available.
7.1
Introduction
I t appears f r o m the state of affairs i n the field of system onal functions,
identification
via
the p r o b l e m of i d e n t i f i c a t i o n of linear t i m e - v a r y i n g and
orthog-
nonlinear
d i s t r i b u t e d p a r a m e t e r systems was n o t p a i d due a t t e n t i o n t i l l the end of eighties. As these problems are i m p o r t a n t , they are taken up for i n v e s t i g a t i o n a n d systematic i d e n t i f i c a t i o n a l g o r i t h m s for b o t h t i m e - v a r y i n g and nonlinear d i s t r i b u t e d p a r a m e t e r systems are developed i n this chapter. Some i m p o r t a n t results of this w o r k are now available i n M o h a n and D a t t a , 1989 [212], 1990 [211]. T h e required
mathematical
b a c k g r o u n d is p r o v i d e d below before presenting the i d e n t i f i c a t i o n a l g o r i t h m s .
7.2
Mathematical Preliminaries
T h e j ' t h degree p o l y n o m i a l j(t) i n a system of o r t h o g o n a l p o l y n o m i a l s can be expressed i n t e r m s of t h e T a y l o r p o l y n o m i a l s {?<(<)}
a
s
i (7.1) 1=0
Chapter
204
7: Identification
of LTV
and Nonlinear
DP
Systems
where q (t)
= (t -
t
for C = 0 , 1 , . . . ,j
q(0 For j
t )', 0
and T
=
[qo(t),q (t),...,q (t)] . 1
]
= n the n o n o r t h o g o n a l n-dimensional basis vector q ( 0 has the i n t e g r a t i o n
operational property / where
q(T)dr
&
E,q(i)
is the i n t e g r a t i o n o p e r a t i o n a l m a t r i x of order n and its general s t r u c t u r e is
1
0
0
2
0
"0 0
0
0
0
0
0
0
0
- 0
0
0
n-i
fit)
~
c
0
0
0
•
0
0 0
0
• •
0
0
• •
0
1 3
E,
I f a f u n c t i o n f{t) defined for t € [io>*/i p o l y n o m i a l s , then
• • • •
a
n
D
e
1
0
expanded as a series of o r t h o g o n a l
J T
E
= f 5q(t)
and
I t can be easily verified t h a t
W
V'q(t)
O
or,
where
Now let us consider a two-variable f u n c t i o n g(x, t) in the region x £ [ x , x ^ ] , t 6 [ i , t \ 0
0
f
and assume t h a t i t can be expanded as a T a y l o r series v i a o r t h o g o n a l p o l y n o m i a l s as g{x,t)
as =
T
i> (x)G4>(t) T
T
p (x)R GSci(t)
7.2 Mathematical
205
Preliminaries
where p {x) k
= (x - x ) * 0
for A; = 0 , 1 , . . . , ( m — 1) and = i n w h i c h t/>(x) a n d p ( x )
Rp(x)
are m - d i m e n s i o n a l vectors, and G and R are m x n a n d
m x m matrices respectively. Now it is n o t difficult to verify t h a t
/
/
f{T)g(x,T)dxdT
io
~
x
T
0
T
E
T
p {x)E R GS x
E Vv"K q(0
/j
(
and W
/ ' /*
f (.r)g(x,r)d dT x
T
ss p ( i ) £ V G 5 L / E ^ V " ' £ q W I
J
j=l
l
<=1
w h i c h play an i m p o r t a n t role i n the study o f t i m e - v a r y i n g systems. L e m m a 7 . 1 [llSJFor
block-pulse
functions,
c(x,t) can be expressed
T
L e m m a 7.2
=
product
a(x,t)b(x,t)
as r/, (x)C(t)
where the ijth
the
element
[llSJFor
T
=
T
T
il> (x)A(t)4> (t)B iJ>(x)
of C is
block-pulse
functions,
f {x,t)
a
k
if
V (x)F 0(O T
t
then S
3
n/*(*,«)
«
n
k=l
L e m m a 7.3
[llSJFor
F (t). k
k=l
block-pulse
functions,
f{x,t)
if
« V (x)F0(«) r
(nen /*(*,*)
«
V (x)F>(i). T
206
7.3
Chapter
7: Identification
of LTV
and Nonlinear
Identification of T i m e - V a r y i n g
DP
Systems
Systems
Let a linear t i m e - v a r y i n g d i s t r i b u t e d parameter system be m o d e l l e d by «(«)
dv(x,t) ' - + «<«) dt
dv(x,t) a
+ c(t)y(x,t)
ox
=
u(x,t).
(7.2)
I n this section, our objective is to estimate the parameters a(t), b(t) and c(t) as also the i n i t i a l c o n d i t i o n j / ( x , t ) and the b o u n d a r y c o n d i t i o n y{x ,t) f r o m measurements of the i n p u t u(x,t) and the o u t p u t y(x,t) over the region x € [ x , x ^ ] , r £ [ < , * / ] - To accomplish this, we integrate Eq.7.2 once w i t h respect to x as also w i t h respect to t to o b t a i n an integral equation. I n this integral equation we a p p r o x i m a t e a l l the k n o w n and u n k n o w n functions i n x a n d / o r t i n terms of T a y l o r series v i a o r t h o g o n a l p o l y n o m i a l s and make use of the i n t e g r a t i o n o p e r a t i o n a l m a t r i x p r o p e r t y of the Taylor p o l y n o m i a l s to get 0
0
0
T
ElR YS
E
T
«j E
T
- E R YS
E d E j=l 1=1 7 - 1 j E Cj E j=a i=o
1=0
j=0
j E b, E SjiV'E, j=o t=o
0-1 T
+
R YS
-
ER
T
X
T
E
I s ^ ^ E ,
l
s VE jt
t
u-l
,1-1 T
T
+ E R YS
0
/iA,
+ I
l
T
S - R
J 2
»;A,
J + 1
S£
(
=
T
T
E R USE
t
(7.3) where /,
=
a(«o)/i,
(7.4)
j(i)
=
b(t)g(t),
(7.5)
/(x) =
y(x,t ),
g(t) =
y(x ,t),
0
0
fi < m, u < n and A , j is an m x n m a t r i x whose ijth. element is u n i t y and a l l other elements are zero. By t a k i n g the vector valued f u n c t i o n of each t e r m i n Eq.7.3, i t can be r e w r i t t e n i n the f o r m of Mz
=
(2a + p +
7 +
v.
(7.6)
Since there are altogether u + v - 1)
unknowns i n mn equations, ran must be greater t h a n or equal to the n u m b e r of unknowns and the columns of the m a t r i x M must be linearly independent for the solution to exist, otherwise identification is not possible. Once t h e parameters a(t) and b(t) are k n o w n , the i n i t i a l c o n d i t i o n f(x) and the b o u n d a r y c o n d i t i o n g(t) can also be estimated f r o m Eq.7.4 and Eq.7.5 provided a(t ) ^ 0 a n d b(t) ^ 0. 0
7.4 Identification
of Nonlinear
E x a m p l e 7 . 1 Consider excited
with u(x,t)
Systems
207
the model described
= x and the output y'{x,t)
in which x is the maximum
= y{x,t)\l
output
distributed
by (7.2)
is expressed + xpiv
without
noise,
The system
is
~ 0.5)]
p is the noise to signal
ratio
(NSR)
and
rj is a uniformly
The
i d e n t i f i c a t i o n of t h e system is carried o u t w i t h shifted Legendre p o l y n o m i a l s ,
o r t h o g o n a l over x,t
random
with c(t) = 0.
by
€ [ 0 , 1 ] , a = 1,0
noise over 0 to 1.
= 2, p. = 3,v
= 6 , m = 4, a n d n = 4.
The
spectra of the estimates shown i n Table 7.1 are i n close agreement w i t h the t r u e values.
Table 7 . 1 : E s t i m a t e s of parameters, i n i t i a l a n d b o u n d a r y conditions i n E x a m p l e 7.1 Spectra a
7.4
Actual
values
Estimates
with
NSR=0.00
NSR=0.05
NSR=0.10 1.1053658 1.1466051
a
1
0.9999998
K
1
0.9999998
1.0638739 1.0346248
6,
1
0.9999998
1.0834387
fo
1.3333333
1.3333330
1.4201172
1.4771779
h
0.5
0.4999998
0.5280226
0.5445491
h
0.1666666
0.1666666
0.1794746
0.1887209
go
1.0666665
1.1072764
1.1330226
9i
1.0666667 1.1809524
1.1809522
1.2748253
1.3457113
£72
0.2142857
0.2142857
0.2250725
0.2320823
93
0.1444444
0.1444444
0.1549457
0.1630824
9*
0.0523809
0.0523809
0.0572331
0.0611967
9s
0.0079365
0.0079365
0.0089779
0.0098981
Identification of Nonlinear
1.0555318
Systems
Let us consider a nonlinear t i m e - i n v a r i a n t d i s t r i b u t e d parameter system described by the f o l l o w i n g second-order p a r t i a l differential e q u a t i o n
2
p
2
d y '(x,t)
—
a«
dt dy {x,t) ~dt
2
d'y»(x,t) ~ ~ dxdt
r
p
Pt
a,-
d y"*(x,t) — 1" «*< dx dy >(x,t) + ay (x,t) adx
1" " i i
2
+
re
x
=
p,
u (x,t)
(7.7)
7: Identification
Chapter
208
of LTV
D P
and Nonlinear
Systems
where Pi,i = 1 , 2 , . . . , 7 are integers. I n order to estimate the parameters a, , a , a , aa and a; the i n i t i a l c o n d i t i o n y(x,t ) and the b o u n d a r y c o n d i t i o n y(x , t) f r o m the measurements of the i n p u t u(x,t) and the o u t p u t y(x,t) we first integrate Eq.7.7 twice w i t h respect to x and twice w i t h respect to t to o b t a i n the integral e q u a t i o n t
0
a„ f
pi
f" y (x,
t)d
x
a
/
xx
/' f 'o
'0
f
p
F X
'O
xo
r
X
J io
ru (x,T)d dr 2
xl
f
P3
/ " y (, X
(
0
r)
*Q
p
2
y >{x,T)d dT X
+
Xo
0
f(x)d
7
-
X
XO
0
J rt J fi j
F7
x l
dxdr — X
'O
+ a
J
f XO
—cJ 2.
r to
- a„ f
2
V (x,r)d dT 2
2
f
x
t 2
2
io
+ a f
XQ
n
q(T)dr
io
r)dr
*0
XQ
Q
J
xo
y»U, 'O
2
X
f
f
X
h{ )dx drio
f
xx
fy '(x.T)d dT Xo
<0
u
+ a
'0
+ a, / ' f"
d dr
2 X
XO
XQ
x x
0
io
s(T)d dr
7
X
x
0
2
(7.8)
X
XQ
XO
vhere c
n
=
a y (x ,t ) xt
0
0
p
, . 9W
dy '(x,t) =
f(x)
=
h(x)
=
<=<„ at y(x,t ) 0
a g(x)
d
+ a
tt
x
f
{
" Ox
i
x
)
p
+ a,f *(x)
(7.9)
p2
(t)
=
9(0
=
dy (xj) a ox !/(*o,<)
s(t)
=
a. r(t)
r
lx=x„
P3
+ a
xx
dq (t) xt
p
"'+a q *(t) x
at
As usual, we now a p p r o x i m a t e p
y '{x,t),i
P7
= 1,2,. . . , 6 ;
f(x),q(t),c,h(x),s{t),u (x,i)
in terms of the orthogonal functions, s u b s t i t u t e the a p p r o x i m a t e d functions i n Eq.7.8 and make use of the O S O M R I of orthogonal functions, see Section 5.2, to o b t a i n a (Ej )y, + a YE ( l
2
xx
2
+ a ElY E,
l2
xl
+ ^E^Y^E,
3
+
a - l T
a E Y E, x
b
T
2
+ aE Y E 2
e
t2
- a E^ tt
2
£
/,A,
+ M
-
lt
x
7.5 Practical
Limitations
of the Identification
Algorithm
B-l
209
n-1
H
«,A
l l i + 1
£
l s
- cElA E hl
T
- E
t
£
x2
j=0
hiA
E
i + l i l
t
-
1=0
£
Sj&ij+iEv
1=0
= E^U E 7
(7.10)
t2
Now Eq.7.10 m a y be r e w r i t t e n i n the f o r m of Eq.7.6 a n d o b t a i n z using the leastsquares technique. I t m a y be n o t e d t h a t the n u m b e r of u n k n o w n s is [7 + ct + 0 + u. + v).
7.4.1
Identification
via Block-Pulse
Functions
W h i l e a p p l y i n g t h e unified approach for the i d e n t i f i c a t i o n of d i s t r i b u t e d p a r a m e t e r systems by using block-pulse functions the following m o d i f i c a t i o n s are i n c o r p o r a t e d . I n this a p p r o a c h a, /?, //and v always take t h e i r m a x i m u m values i.e., a = p = m and (3 = v = n. replaced by A
> r
T h e matrices A
, i = l , 2 , . . . , m i n the above approach m u s t
t l
and a l l o t h e r elements zero.
S i m i l a r l y , A ,j
= 1,2,.... , n m u s t be replaced by A j , .
l}
which is again an m x n m a t r i x b u t at this t i m e h a v i n g a l l jth u n i t y a n d a l l other elements zero. replaced by A
xy
O n l y i n Eq.7.10, A
c o l u m n elements
associated w i t h c m u s t be
n
w h i c h is an m x n m a t r i x c o n t a i n i n g a l l elements u n i t y . T h e n u m b e r
of u n k n o w n s is 7 + 2 m -I-
7.5
be
w h i c h are m x n matrices c o n t a i n i n g a l l t h e t t h row elements u n i t y
In.
P r a c t i c a l Limitations of the Identification Algorithm
A l t h o u g h the present p r o b l e m is concerned
w i t h nonlinear d i s t r i b u t e d parameter
systems, i t is i n t e r e s t i n g t o see t h a t the terms corresponding t o i n i t i a l a n d b o u n d a r y conditions i n Eq.7.10 are e x a c t l y i d e n t i c a l w i t h those of linear d i s t r i b u t e d parameter systems given i n Eq.6.12. Hence, this a l g o r i t h m also suffers f r o m t h e same p r a c t i c a l l i m i t a t i o n s discussed i n Section 6.3. E x a m p l e 7.2 initial
condition
It is required
to estimate
2
dt
For u(x,t) j / ( x , 0 ) = x, y(x,t)
= x(l
^'')
i t
x l
system
2
and also
the
by
at
and the output
data y(x,t)
2
+ xt ),a
ti
= a
xi
are
available.
= l , a , = 0.5,initial conditions
= 0 at i = 0, and b o u n d a r y c o n d i t i o n y(0,t)
+ < ).
t
described
dy (x,t)
axat
= 2(x + t) + 2xt(l a a
a , a , and a
parameter 2
d y(x,t)
2
data u(x,t)
distributed
2
d y(x,t)
when the input
the parameters
j / ( x , 0 ) of the nonlinear
= 0 t h e o u t p u t is
Chapter
210
7: Identification
of LTV
and Nonlinear
DP
Systems
Now f r o m Eq.7.9 and Eq.7.10 we have h(x)
=
1 + 0.5a;
2
a«Fjy
+ a Y E,
a—l 1
xl
T
+ a,E Y E,
3
4
T
- aE u
x
£
/,-A
i + l l l
-
i= 0
i= 0
=
EUE x
7
t2
Since p, = p = l Y = Y = Y. S i m i l a r l y , as p = 2 , Y = Y w h i c h m a y be easily c o m p u t e d f r o m the knowledge of Y by using L e m m a 7.3 o f Hsu and Cheng, 1982 [112] for the block-pulse approach and by a d o p t i n g the procedure o f Jha et al, 1986 [147] for the Laguerre approach. U = U. 3
t
t
3
4
4
2
7
N o w by t a k i n g m = 3,rt = 5,o: = 3 block-pulses/ 2 Laguerre p o l y n o m i a l s , p, = 3 and i , r e [ 0 , 1 ] ( o n l y for block-pulse approach) the parameters and the spectra of f(x) and h(x) are estimated. T h e estimates as shown i n Table 7.2 are q u i t e satisfactory.
Table 7.2: Estimates of parameters and spectra of f(x) Parameters
i n E x a m p l e 7.2
LaP
Actual
Estimated
Actual
a„
1
0.9960058
1
1
a
1
1.0140268 0.5055392
1
0.9999999
0.1659659
0.5 1
0.5 1
-1
-0.9999999
xi
a, fo A h ho h
7.6
BPF
and h(x)
2
0.5 0.1666666
Estimated
0.5
0.4980795
0.8333333
0.8300756
1.0185185 1.1296296
1.0282110
2
1.9999999
1.1396815
1.3518519
1.3639630
-2 1
0.9999999
-2
Conclusion
T h e a l g o r i t h m for t i m e - v a r y i n g systems presented above has the m e r i t t h a t i t can be d i r e c t l y applied w i t h the help of any class of o r t h o g o n a l p o l y n o m i a l s and a p p r o p r i a t e matrices i i and S.
the
C r i t i c a l l y e x a m i n i n g the a l g o r i t h m , one w i l l n o t fail
to notice t h a t the approach followed i n the development o f the i d e n t i f i c a t i o n scheme is not completely based on the o r t h o g o n a l functions b u t r a t h e r on b o t h o r t h o g o n a l and T a y l o r ( n o n o r t h o g o n a l ) p o l y n o m i a l s . Since the signal c h a r a c t e r i z a t i o n is done
7.7
Problems
211
w i t h o r t h o g o n a l p o l y n o m i a l s and the i n t e g r a t i o n is a p p r o x i m a t e l y p e r f o r m e d w i t h T a y l o r p o l y n o m i a l s , this approach m a y be t e r m e d as the hybrid
approach.
I t is not
i m p e r a t i v e t o use this scheme w i t h the help of block-pulse and sine-cosine functions as series expansion i n Eq.7.1 is n o t
finite.
U n l i k e of t i m e - v a r y i n g systems, the i d e n t i f i c a t i o n approach developed for nonlinear systems is c o m p l e t e l y based on o r t h o g o n a l functions and hence i t m a y be used v i a any class o f o r t h o g o n a l f u n c t i o n s .
7.7
Problems
1. Consider the nonlinear d i s t r i b u t e d parameter system described by %(*.») . dy\x,t) « 3 — 1- <*2 — + aiy(x,t) = u(r,t) dx at w i t h i n i t i a l c o n d i t i o n j / ( x , 0 ) = 0 and b o u n d a r y conditions y(0,t) 2
E s t i m a t e the parameters w h e n the i n p u t u(x,t) y(x, t) =
= 4x t
= 0,{/(l,t) =
t.
+ It + xt and the o u t p u t
xt.
Answers = l,a
2
= 2,a
3
= 2.
2. Repeat P r o b . l i f the i n p u t and the o u t p u t d a t a are c o r r u p t e d w i t h
independent
zero-mean w h i t e noises, each w i t h a variance of 0.001.
3. Consider the system described by the f o l l o w i n g nonlinear p a r t i a l differential equation o,
dy(x,t)
1
+ a,
cV(*,i)
dx
=
u(x,tj
at
w i t h i n i t i a l c o n d i t i o n t / ( x , 0 ) = 0 and b o u n d a r y conditions j / ( 0 , t ) = 0,2/(1, t) = t. 2
E s t i m a t e the parameters when the i n p u t u ( x , i ) = 2 x i + 2 r ; and the o u t p u t y(x,t)
Answenoj = 2,a
2
= 1,
4. For the f o l l o w i n g nonlinear p a r t i a l differential e q u a t i o n
^ ) at
=
, ^ ) ,
I
dx
6 i
[
0
)
1
]
= xt
212
Chapter
7: Identification
of LTV
and Nonlinear
DP
Systems
with initial condition v(x,0)
-{
1
and b o u n d a r y conditions y(0,t) initial condition.
20x,
0 < x < 0.5
20(1 - x),
0.5 < x < 1.0
= y{\,t)
= 0, estimate the p a r a m e t e r /? and
[ H i n t : Simulate the system w i t h fi = 0.0001 and proceed for e s t i m a t i o n . ]
the
Chapter
8
Optimal
Control
of
Linear
Systems
T h e use o f o r t h o g o n a l p o l y n o m i a l s , block-pulse and sine-cosine functions to determine the t i m e - v a r y i n g gain for state variable feedback i n a linear c o n t r o l system m i n i m i z ing a q u a d r a t i c performance index is o u t l i n e d i n this chapter. T h i s t i m e - v a r y i n g gain K(t) can, i n fact, be expressed i n terms of the solution of a m a t r i x R i c c a t i differential equation. T h e r e are t w o methods to c o m p u t e this t i m e - v a r y i n g g a i n . T h e first is a direct m e t h o d w h i c h does not require the c o m p u t a t i o n of the state t r a n s i t i o n m a t r i x . O n the c o n t r a r y , the second m e t h o d is based on first evaluating the state t r a n s i t i o n m a t r i x and t h e n the t i m e - v a r y i n g gain is expressed i n terms of the elements of this m a t r i x . T h e o p t i m a l c o n t r o l law can be expressed by u*(r.) = K(t)x(t). I f , however, a l l the states are not available, the estimated state x ( t ) must be generated v i a an observer and the corresponding c o n t r o l law can be c o m p u t e d v i a the system of o r t h o g o n a l functions.
8.1
Introduction
A linear t i m e i n v a r i a n t system considered i n this chapter is described by x(t)
=
m
Ax(i) +
n
flu(0,x(i ) 0
= x„,
(8.1)
n x m
where x € R " , u € R ,A G R * " and B G R . T h e c o n t r o l u ( i ) is an admissible c o n t r o l , i f i t is piecewise continuous i n t , for each t, its values belong to a given closed subset U(t) of R T h e i n p u t u ( t ) is derived by m i n i m i z i n g the q u a d r a t i c performance index m
J
=
1 -x. (t )Sx.(t ) 2 T
}
f
1
If* + - J (x Qx 2 to T
T
+ u Ru)dt,
(8.2)
where S and Q are s y m m e t r i c positive semi-definite and R is s y m m e t r i c positive definite. T h e r e are t w o ways, v i z . , Pontryagin's m a x i m u m p r i n c i p l e and H a m i l t o n -
Chapter
214
8: Optimal
Control
of Linear
Systems
Jacobi e q u a t i o n t o solve this p r o b l e m . T a k i n g t h e help of t h e f o r m e r , we define the H a m i l t o n i a n as ff(x,u,A,<)
T
=
T
^(x Qx
T
+ u Ru)
+ \ (Ax
+ Bu),
(8.3)
where A € R " is k n o w n as the costate variable. T h e o p t i m a l c o n t r o l by t h e m a x i m u m p r i n c i p l e is o b t a i n e d when u ( i ) is not subjected t o any constraints b y s o l v i n g
dH
T
=
du
(8.4)
0 = Ru + B A ,
where A is a s o l u t i o n of the adjoint equation A
dH
=
~ = -Qx — A A
dx
(8.5)
w i t h the terminal condition \(t )
=
f
(8.6)
Sx(t ). f
T h u s i t follows f r o m Eq.8.4 t h a t the o p t i m a l c o n t r o l law is u*(i)
(8.7)
T
=
-R~'B X(t),
where \ (t) is the solution of the H a m i l t o n i a n system A
1
-BR'
-Q
B
- A
(8.8)
T
w h i c h follows from Eq.8.1 w i t h u s u b s t i t u t e d f r o m Eq.8.7, and Eq.8.5. T h e o p t i m a l c o n t r o l can be i m p l e m e n t e d as a closed-loop c o n t r o l i f t h e s o l u t i o n t o t h e adjoint e q u a t i o n Eq.8.5 is assumed like Eq.8.6 as a linear f u n c t i o n o f t h e states i n t h e f o r m
\(t)
=
P(t)x(t),P(t,)
= S.
(8.9)
B y v i r t u e of Eq.8.5,Eq.8.8 and Eq.8.9 we have 1
x(t)
=
Ax(t)
-
A(t)
=
F(t)x(i) + P(I)X(I)
=
[P(t) + P(t)A
T
BR- B P(t)x(t),
1
-
T
P(t)BR- B P(t)]x(t)
T
=
l-Q-A P(t)]x(t),
where t h e first e q u a l i t y follows f r o m Eq.8.9 and t h e second f r o m Eq.8.8. Hence, [ P ( 0 + P(t)A
T
+ A P(t)
i
T
+ Q - P(t)BR- B P(t)]x(t)
= O.
8.2 Solution
of Riccati
Equation
Without
STM
215
Since t h e above e q u a t i o n m u s t h o l d for a l l nonzero x ( i ) , P(t)
m u s t satisfy t h e m a t r i x
Riccati equation -P(i)
=
P{t)A
P(t )
=
S.
f
T
1
+ A P(t)
T
+ Q - P(t)BR- B P(t),
(8.10)
I t follows f r o m the above development t h a t the o p t i m a l c o n t r o l law is given by, see Eq.8.7 a n d Eq.8.9, u*(«) a n d , as shown i n [3], P{t)
l
=
T
-R- B P(t)x{t)
(8.11)
can be c o m p u t e d using the f o l l o w i n g r e l a t i o n P(t)
=
l
W{t)V~ {t),
(8.12)
where \
V(t)
L
W{t)
1 J
-BR~
. W(t\
V(t,) Two
V(t)
B
=
I,W(t,)
.
= S.
(8.13)
(8.14)
m e t h o d s are described below t o solve the m a t r i x R i c c a t i e q u a t i o n t a k i n g the
advantage of o r t h o g o n a l f u n c t i o n s . T h e second m e t h o d requires the c o m p u t a t i o n of the s t a t e - t r a n s i t i o n m a t r i x ( S T M ) w h i l e the first does n o t .
8.2
Solution of R i c c a t i Equation W i t h o u t S T M
As the t e r m i n a l c o n d i t i o n s are g i v e n , the equations Eq.8.13 and Eq.8.14 have to be solved b a c k w a r d i n t i m e s t a r t i n g f r o m tj.
To get a r o u n d this inconvenience, we use
a new t i m e variable r defined by T = tf — t i n terms o f w h i c h Eq.8.13 is t r a n s f o r m e d into -A
' V"(r) • . W'(T)
where V(T)
: = V(t
s
.
'
BR^B* T
Q
A
— r ) , W(T) := W(t
s
V(T)
'
. W(T)
.
(8.15)
— r ) , a n d the p r i m e denotes derivatives w i t h
respect t o r . I f we set
Z(r)
=
V(T)
M
W(T)
=
-A
1
BR'
B (8.16)
T
Q
A
t h e n Eq.8.16 can be w r i t t e n as Z'(r)
=
MZ(T),Z(0)
= [I
T
S f,
(8.17)
Chapter
216
8: Optimal
Control
of Linear
Systems
where Z(T) is a 2 n x n m a t r i x . W e recall t h a t i f {<£,(r)} is a c o m p l e t e set o f o r t h o g o n a l functions over the i n t e r v a l [ r = 0,r 0
= tj — t ] a n d i f / ( r ) is an a r b i t r a r y f u n c t i o n
f
0
square-integrable over the same i n t e r v a l , t h e n TT1-1
OO
f(r)
=
£ / , 0 , ( r ) fa £ fiMr) i=0 ;=o
T
= i (r),
(8.18)
where
f
/,
[ / o / i - - - /
=
=
- f 7; 0
m
- , f ,
W(t)f(t)4>;(t)dt,
J
w(t)
is t h e w e i g h t i n g f u n c t i o n of t h e o r t h o g o n a l system under c o n s i d e r a t i o n , 7 given (
by Eq.2.1 is a constant, different for different systems a n d t h e set { / ; } is t h e set of Fourier coefficients or the spectrum of / ( r ) w i t h respect t o { < £ , ( T ) } . M o r e o v e r
J
E(T)
{t)dt as
TO
where E is an i n t e g r a t i o n o p e r a t i o n a l m a t r i x of order m , see Section 3.4. Suppose T
t h a t / * ( r ) = i {T).
Then /(r)
f
=
f'(t)dt TO
=
f
T
I J
T
T
*
f F0(r).
(8.19)
I n view of t h e a p p r o x i m a t i o n of a function as given by Eq.8.18, t h e s o l u t i o n t o Eq.8.17 s i m i l a r l y a p p r o x i m a t e d can be w r i t t e n as m - 1
Z(T)
=
E
ZiMr)
i= 0
=
[Z
0
Z ---Z _ ]F{r), L
m
(8.20)
l
where Z('s are 2n x n spectral matrices of Z(T) a n d ^o(r)/ F(r)
=
n
=
[ * ( T ) ® / „
]
(8.21)
8.2 Solution
of Riccati
Equation
Without
STM
217
is an mn x n m a t r i x . I n view of Eq.8.19- E q . 8 . 2 1 , i n t e g r a t i n g Eq.8.17 and s i m p l i f y i n g , see [173], we have Z ( r ) - Z(0)
=
M [
=
M[Z
Z
=
M[Z
Z ---Z _ ][E4,(T)®
T
Z{t)dt
0
0
T
--Z - \f [{t)®I \dt
l
m 1
1
m
n
/„],
1
w h i c h by m a k i n g use of the r e l a t i o n [ F ® 7 „ ] [ 0 ( r ) ® / „ ] = [ F 0 ( r ) ® /„] can be w r i t t e n as [Z„-Z(0), =
Z ,---,Z _ )[(r)® i
m
M[Z
/„]
1
Z ---Z _ ][E®I ][^(r)®I }.
0
l
m
l
n
n
F r o m t h e above e q u a t i o n i t follows t h a t [Z
Z, • • • Z _ ]
0
m
1
=
M[Z
+
[ Z ( 0 ) O ••-
0
Z , •••Z _ [E® m
/„]
1
(8.22)
which can also be represented as [/ „ 2m
2
- (E ® I )
T
® M]vec(Z)
n
=
vec(Z )
=
vec(Zo),
0
t h a t is, [/
r
2 m n 2
-(F ®/„)®M]vec(Z)
where Z Z
0
vec(£»)
—
=
[ / w
0
m
[Z(0)O---O],
=
[Of £> ---I>r] ,
r
r
2
D's are the columns of the m a t r i x D. vec ( Z )
[ Z Z ! ••• Z _ i ] ,
=
Hence T
- E
® (/„ ® M ) ]
_ 1
vec ( Z ) 0
S o l v i n g t h e set of simultaneous equations given by Eq.8.22, Z ( i ) defined by Eq.8.20 is c o m p u t e d and hence the R i c c a t i gain is given by F(r)
=
iy(r)V-'(r).
T h e m a t r i x E i n Eq.8.22 is the i n t e g r a t i o n o p e r a t i o n a l m a t r i x corresponding t o the system of t h e o r t h o g o n a l functions, o r t h o g o n a l p o l y n o m i a l s and sine-cosine functions w i t h w h i c h t h e a p p r o x i m a t i o n i n Eq.8.19 is made. Therefore for each system of t h e o r t h o g o n a l f u n c t i o n s , o r t h o g o n a l p o l y n o m i a l s and sine-cosine f u n c t i o n s , t h e s o l u t i o n 2
of Z ( T ) for 2n m
n u m b e r of its elements is o b t a i n e d f r o m Eq.8.22.
Chapter
218 8.2.1
Laguerre
Polynomials
8: Optimal
Control
and Recursive
of Linear
Systems
Computation
We note t h a t w i t h i n t e g r a t i o n o p e r a t i o n a l m a t r i x E of Laguerre p o l y n o m i a l s given i n Section 3.4, we have /„
-/„
o E ® I„
0
/„ -/„
0
0
/„
_ 0
0
0
=
• • I,
Therefore Eq.8.22 becomes [Z„ Zi • • • Z -t]
= [Z(0) + MZ ,
m
M(Z,
0
- Z ), • • •, M ( Z _ ! 0
m
Z _ )], m
2
whence Z
0
=
(/
Zt
=
-(/
Zi
Z _! m
-
2 n
2n
=
~(I
=
(7
M)-'Z(O), - M)-'MZ - M)'
7n
-
2
- M)-'M(/ „ -
2 n
2
M)"'MZ _
M)-'Z(O),
=
-(/ „ -
=
( - 1 ) " - ' ( J „ - M ) ~ ' M •••(/;„ - M ) " ' M
2
M)-'Z(O),
MZ,,
- M)-'M(/
2 n
= -(/ „ -
0
1
m
2
2
(m-1) (/ „ 2
times
M)-'Z(O),
and consequently Z,'s can be c o m p u t e d recursively. 8.2.2
Block-Pulse
Functions and Recursive
T h e i n t e g r a t i o n o p e r a t i o n a l m a t r i x for block-pulse functions is 1/2
1
1
0
1/2
1
0
0
E = A 1/2 J
where A = («/ -
t )/m. 0
Computation
8.3 Riccati
Equation
Solution
with
STM
219
Hence
E ® I
n
= A
0
i / ,
0
0
For block-pulse f u n c t i o n s , t h e last t e r m i n Eq.8.22 should be \Z(0)
Z(0)
•••
3(0)]
and therefore i t follows f r o m Eq.8.22 t h a t X
Z
=
Z,
=
A M
[z
Z _!
=
AM
[z
0
m
-AMZ
+
0
Z(0),
+ -Z,\
0
+ Z
a
l
+ Z(0),
+ --- + Z _ m
+ \z _\
1
+
m
Z(0).
Consequently, we have Z _, m
and solving for Z , k
= Z _ + -AM m
[Z _
2
m
2
+
Z _i] m
there results the r e l a t i o n Z
k
= [/ „ - V
]
2
[/ „ + ^ A A f ] 2
Z_ k
x
for k = 1 , 2 , 3 , . . . ,TO— l . T h e first r e l a t i o n gives z
0
8.3
=
[/ „ 2
Z(0)
^ A M ]
Riccati Equation Solution with S T M
As shown i n [193], the s o l u t i o n of P(t)
can also be o b t a i n e d by first c o m p u t i n g
t h e state t r a n s i t i o n m a t r i x G(t)
0
: = $ ( i , i ) associated w i t h M w h i c h satisfies t h e
differential e q u a t i o n G(t)
=
MG(t),G(t )
= U
0
where G(t) is a 2n x 2 n m a t r i x and M
A
1
-BR- B - A
T
T
(8.23)
Chapter
220
8: Optimal
T a k i n g the help o f Eq.8.18, the s o l u t i o n of G{t)
Control
of Linear
Systems
approximated by the orthogonal
functions 0 , ( t ) ' s is o b t a i n e d by s e t t i n g
G(t)
=
£
G,<M«)
1= 0
=
(8.24)
[G G,---G _ ][^(*)®/ „], 0
m
1
2
where G.'s are 2ra x 2rt matrices s t i l l u n k n o w n . Hence i n t e g r a t i n g Eq.8.23 f r o m t
0
to
t a n d in view of Eq.8.19 we have
G(t)
- G(i ) 0
Observing that G ( i ) = 7 0
=
M f
=
M[G
G,
••• G _ i ] /
=
M[G
G,
••• G _ ! ] [ £ ; 0 ( i ) ® 7 „].
0
0
m
[4>{T) ® 7 ] a Y 2n
m
2
and m a k i n g use o f Eq.8.24 we can w r i t e
2 n
[G -7 0
=
G(r)dT
M[G
0
,
2 n
Gi,-,G»-i][*(t)8 7 ] 2 n
G,
•••
G -A\E
® 7 ][<£(i) ® 7 „ ] .
m
2n
2
where we have used the r e l a t i o n [E ® /
2
n
] = [E ® 7 ][tf, ® 7 2n
2 n
],
see [173]: F r o m the above result i t follows t h a t [G„G ---G _ ] 1
m
1
=
M[G„G ---G _ ][F®7
+
[/ „
1
O
2
m
1
2 n
]
••• O].
(8.25)
T h e state t r a n s i t i o n m a t r i x G(t) of order 2 n associated w i t h M is now w r i t t e n in the p a r t i t i o n e d f o r m as
G(t)
where a l l the matrices gij{t),i,j
9u(t)
<7i (<)
02i(<)
<7 (<)
2
22
= 1,2 are o f order n and since G(t ) 0
S n ( ' o ) = S ( « o ) = 7„; <7i (t ) = S i ( < o ) = O. 2 2
2
0
2
T h e s o l u t i o n of Eq.8.8 for x ( ( ) a n d A ( t ) therefore takes t h e f o r m x(t) X(t)
G(i)
x(t„) M
= 7 „, 2
8.4
Examples
221
from w h i c h we get *(t)
=
9u(t)x(t )
+
g (t)X(t ),
M*)
=
9 i(t)x(t )
+
g (t)X(t ).
0
12
0
(8.26) 2
0
22
0
A t i = t , as a result of Eq.8.9 we have A ( t / ) = Sx(t ) f
a n d hence
f
,
S [<7n(t/)x(i ) +
g (t )\(t )]
0
x
=
12
f
0
<
<72i(
Consequently, solving for A ( i ) we get 0
A(t ) = # x ( t ) , 0
0
where _I
= [Sg (t ) 12
- <722(0)] [5'2i(
}
SW*/)].
S u b s t i t u t i n g t h i s value o f A ( t ) i n t o Eq.8.26 we have 0
x(<)
=
[<7n(0 + < 7 ( t ) # ] x ( i ) ,
=
[<7 i(t) +
12
0
g (t)H]x(t ).
2
22
B
Therefore A(i) =
P(t)x(t),
where 1
P(t) = [<72iM + i/22(*)^][
T h i s approach w h i c h is used i n the l i t e r a t u r e requires 4n m 0 , 1 , • • • , m — 1 to be d e t e r m i n e d for g e t t i n g the R i c c a t i g a i n
8.4
u n k n o w n s i n G.'s i = P(t).
Examples
T h r e e examples are now considered in w h i c h the linear state variable feedback gain m a t r i x is o b t a i n e d by solving the m a t r i x R i c c a t i equation and represented by K(t)
1
T
= R- B W{t)V-\t)
1
T
= R- B P(t),t
€
E x a m p l e 8.1 A s i n g l e - i n p u t s c a l a r s y s t e m T h e first e x a m p l e is a single-input scalar system described by x(r) = -2x(<) +
u(t)
[t ,tf] 0
222
Chapter
8: Optimal
Control
of Linear
Systems
with l
l
J = -x\l)+ -j\x\t)
+
u\t)]dt
for w h i c h the state variable feedback gain o b t a i n e d by solving t h e R i c c a t i e q u a t i o n is A cosh \(tf *(*)
A cosh \{tj
— t) — sinh \(tf
— t)
- t) + 3 sinh \(t
-
f
t)
w i t h tt = 1 , and A = y/E. E x a m p l e 8.2 A s i n g l e - i n p u t s e c o n d o r d e r s y s t e m T h e second example is a single-input second order system described by 0 1
x(<)
0 0
u(t)
x(0"
with 0 0
0 4
x(t) +
w h i c h on solving gives the gain m a t r i x K(t)
=
\ki,k ):
r
x (t)
dt
2
sinh (TT — 2t) — sin (TT — 2t)
K(t)
2
2
cosh (7r - 2 t ) / 2 + cos (?r - 2 t ) / 2 cosh (TT — 2t) — cos (TT — 2t) cosh (n - 2t)/2 +
2i)/2J
2
COS (TT -
E x a m p l e 8.3 A s i n g l e - i n p u t s e c o n d o r d e r s y s t e m T h e t h i r d example is a single-input second order system described by
x(t)
=
0
1
. 0 0 .
0
x(t)+
1
. 1 J
"(*)
w i t h the following q u a d r a t i c performance index h a v i n g a t e r m i n a l cost described by
J
=
1 -x (3) 2 J
1 0
0 2
x(3) +
w h i c h on solving gives the gain m a t r i x K(t) K(t)
= 2[0
T
x (i)
=
2
1
1
4
[k,,k ]: 2
x
\]W(t)V- (t)
x(i) +
dt,
8.4
Examples
223
where a
- 4a
4(a
2
0
V{t)
a
W(t)
0
- ( a , + 4ct + 8 a )
3
2
+ a )
+ 2 a ! 4- 4 a 2
2
2
a, + 2 a
3
2(a! + a ) + 16a A
3
c
3
3
2a
0
+ 32a
3
2
+ 4aj + 16a + 3 0 a 2
3
2
cosh A ( « - t) - A cosh A i ( t / - *) 2
/
2
A - A
2
'
A? sinh A ( i / - t) - A sinh A ^ t , - i ) 2
2
2
AjA (Aj — A ) 2
cosh A ^ t / — t ) — cosh A ( i ^ — i ) 2
A
^1
2
A sinh A ^ f y — t) — A[ sinh A ( i ^ — t) 2
2
2
2
AiA (A — A ) 2
2.7320508, A = 0.7320508. 2
Table 8 . 1 : O p t i m a l Feedback G a i n K i n E x a m p l e 8.1
time sec 1.000 0.875 0.75 0.625 0.500 0.375 0.250 0.125 0.000
TPl 1.02281 0.63510 0.45712 0.36087 0.30669 0.27624 0.25914 0.24930 0.24331
TP2 1.05800 0.64324 0.45946 0.36176 0.30716 0.27649 0.25920 0.24923 0.24313
LeP
LaP
HeP
SCF
Actual
1.04210 0.63947 0.45831 0.36129 0.30691 0.27635 0.25916 0.24925 0.24320
0.18478 0.17584 0.16056 0.12834 0.00144 0.95430 0.34769 0.28835 0.26570
1.00000 0.60478 0.18280 -3.16330 1.26666 0.78253 0.60190 0.49375 0.41667
0.28926 0.93532 0.36985 0.33825 0.28926 0.26390 0.25744 0.24064 0.28926
1.00000 0.64309 0.46007 0.36145 0.30691 0.27630 0.25898 0.24914 0.24353
time sec 1.0-.75 .75-.50 .50-.25 .25-.00
BPF 0.63636 0.35568 0.27332 0.24781
224
Chapter
8: Optimal
Table 8.2: O p t i m a l Feedback G a i n k,(t)
time sec 1.5708 1.3745 1.1781 0.98175 0.7854 0.58905 0.3927 0.19635 0.00000
Systems
i n E x a m p l e 8.2
TP2
LeP
LaP
HeP
SCF
Actual
-0.00369 0.01173 0.07995 0.26256 0.57724 0.97904 1.3713 1.6635 1.83
-0.00825 0.00914 0.07882 0.26207 0.57756 0.97983 1.3708 1.6604 1.8254
-0.00634 0.01026 0.07927 0.26213 0.57717 0.97916 1.3707 1.6616 1.8273
0.82532 0.53302 0.21611 0.76522 2.1755 2.4706 2.3743 2.2441 2.1288
0.00000 0.00834 0.07228 0.2696 0.67462 1.2177 1.6462 1.8405 1.8969
0.64650 0.05179 -0.01842 0.1243 0.46506 0.94128 1.2012 1.7505 0.6465
0.00000 0.01008 0.08014 0.26259 0.57718 0.97837 1.3667 1.6586 1.8343
2
sec 1.5708 1.3745 1.1781 0.98175 0.7854 0.58905 0.3927 0.19635 0.00000
of Linear
TPl
Table 8.3: O p t i m a l Feedback G a i n k {t)
time
Control
i n E x a m p l e 8.2
TPl
TP2
LeP
LaP
HeP
SCF
Actual
0.01116 0.06533 0.29566 0.66646 1.1105 1.5277 1.8222 1.964 2.0035
0.02681 0.07779 0.30422 0.67281 1.1163 1.5328 1.8258 1.9671 2.0085
0.01994 0.07205 0.29995 0.66936 1.113 1.53 1.8239 1.9656 2.0063
0.48769 0.52457 1.2525 3.9797 5.2623 4.2094 3.3514 2.8221 2.4783
0.11111 0.17385 0.371 0.71488 1.1605 1.5397 1.7198 1.8128 2.0077
2.7297 -0.15667 0.55492 0.68882 1.1677 1.6561 1.8681 1.9259 2.7297
0.00000 0.07707 0.30632 0.67073 1.1129 1.5296 1.8229 1.9659 2.00000
time i n
sec
1.57081.17810.78540.3927-
1.1781 0.7854 0.3927 0.0000
BPF k,(t) 0.00000 0.23525 0.95639 1.6558
BPF k (t) 0.15421 0.7497 1.6068 2.0131 2
8.4
Examples
225
Table 8.4: O p t i m a l Feedback G a i n k,(t)
i n E x a m p l e 8.3
time
TPl
TP2
LeP
LaP
HeP
SCF
Actual
sec 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
0.05241 0.76717 1.2822 1.5805 1.7676 1.86200 1.9146 1.945 1.964 1.9759 1.9838 1.9886 1.9922
0.14177 0.76884 1.2788 1.5814 1.7722 1.86470 1.9157 1.9456 1.9643 1.9761 1.9837 1.9887 1.9923
0.10009 0.76778 1.2802 1.5811 1.7705 1.86360 1.9153 1.9453 1.9644 1.976 1.9836 1.9886 1.9922
1.9404 2.0198 1.7277 1.9256 1.8954 2.0451 2.1372 1.8316 2.0099 2.0186 2.0538 1.9278 1.9757
0.00000 -3.987 -0.68499 1.7953 1.5051 -0.87263 -28.928 3.1731 3.5001 -3.1527 -0.24079 1.5559 0.85525
1.6494 0.15297 0.95056 5.8178 2.761 1.2977 -1.1465 2.4033 2.2312 2.1788 2.1034 1.9804 1.6494
0.00000 0.77295 1.27600 1.5853 1.7612 1.8585 1.9129 1.9445 1.9636 1.9757 1.9834 1.9887 1.9922
T h e gain matrices K{t)
o b t a i n e d using o r t h o g o n a l p o l y n o m i a l s , and block-pulse
and sine-cosine functions for the above three examples are listed i n Tables 8.1-8.5 respectively w i t h the exact s o l u t i o n for comparison.
T h e value of m is taken t o
be 4 except for H e r m i t e p o l y n o m i a l s i n E x a m p l e 8.3 i n w h i c h case m = 5 because otherwise the i n v e r t i b l e m a t r i x i n vec (Z)
becomes singular.
I n E x a m p l e 8.3 the
o r t h o g o n a l f u n c t i o n a p p r o x i m a t e s o l u t i o n is o b t a i n e d by considering t h e i n t e r v a l [0, 3] as three subintervals each o f u n i t l e n g t h , for each s u b i n t e r v a l [j,j + 1], j = 0 , 1 , 2 t h e s o l u t i o n for Eq.8.15 is o b t a i n e d for sine-cosine functions.
T h e solutions offered by
t h e p o l y n o m i a l s T P l , T P 2 and LeP are i n good agreement w i t h t h e exact s o l u t i o n . T h e sine-cosine functions provide also a good agreement except at the end points where on account of d i s c o n t i n u i t y these functions a p p r o x i m a t e w i t h average values. A l t h o u g h block-pulse f u n c t i o n a p p r o x i m a t e s o l u t i o n is piecewise constant and may be acceptable, the solutions offered by the infinite range p o l y n o m i a l s are far from satisfactory.
Chapter
226
8: Optimal
Table 8.5: O p t i m a l Feedback G a i n k (t) 2
Control
of Linear
i n E x a m p l e 8.3
time
TPl
TP2
LeP
LaP
HeP
SCF
Actual
sec 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
4.1452 3.1970 3.1970 3.2890 3.3615 3.4058 3.4290 3.4423 3.4502 3.4550 3.4580 3.4599 3.4612
4.3987 3.2011 3.1969 3.2887 3.3646 3.4061 3.4292 3.4424 3.4500 3.4550 3.4580 3.4599 3.4613
4.2797 3.1974 3.1970 3.2889 3.3604 3.4060 3.4292 3.4424 3.4502 3.4550 3.4580 3.4599 3.4612
3.3348 3.3566 3.1995 3.3957 3.3373 3.5249 3.6522 3.2343 3.4780 3.4895 3.5375 3.3653 3.4318
4.0000 -9.8209 0.94407 3.6488 1.6538 -0.91294 -31.8830 4.9439 7.2115 -4.0131 0.79226 2.9463 1.5133
3.2236 1.6574 3.3008 9.2084 4.6960 2.6513 -0.84653 4.0307 3.8398 3.7428 3.5841 3.4626 3.2236
4.0000 3.2023 3.1970 3.2895 3.3613 3.4043 3.4286 3.4421 3.4501 3.4549 3.4579 3.4599 3.4612
time in
sec
BPF
3.002.752.502.252.001.751.501.251.000.750.500.25-
2.75 2.50 2.25 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00
BPF fej(t) 0.3750 1.0424 1.4575 1.6943 1.8231 1.8932 1.9329 1.9564 1.9710 1.9804 1.9867 1.9906
fc (t) 3.375 3.164 3.2472 3.3348 3.3898 3.4205 3.4376 3.4473 3.4531 3.4568 3.4591 3.4607 2
Systems
8.5 Estimated
8.5
State
Feedback
Using An
Observer
227
Estimated State Feedback Using A n Observer
Let us consider a linear t i m e - i n v a r i a n t system described by
where x € R " , u € R
r
x
=
Ax + Bu,
(8.27)
y
=
Cx,
(8.28)
p
and y € R , A, B, and C are matrices of c o m p a t i b l e dimension
w i t h C f u l l r a n k . Since a l l the states may not be available at the o u t p u t , for the state variable feedback design, i t is very often required to estimate the states w i t h the help of an observer. T h e observer is an (n — p ) t h order d y n a m i c a l system represented by z where z € R
n
_
p
=
Fz + Gy+Hu,
(8.29)
, F , G and H are matrices of order (n — p) x (n — p), (n — p) x p a n d
(n — p) x r respectively w h i c h along w i t h the o u t p u t y = Cx can give an e s t i m a t e of the state o f Eq.8.27. To u n d e r s t a n d
the basic p r i n c i p l e , suppose t h a t z ( i ) is an estimate of Tx(t)
for
some (rt — p) x n constant m a t r i x T i n the sense t h a t e(t)
:=
z ( i ) - Tx(t)
(8.30)
tends t o zero as t tends to i n f i n i t y for any x ( 0 ) , z ( 0 ) and u ( i ) . Now e
=
z - Tx = F z + Gy + Hu - TAx
=
F e + ( F T - TA + GC)x
-
TBu
+ (H - TB)u
(8.31)
I f we choose ( a ) F h a v i n g a prescribed set of eigenvalues w i t h negative real p a r t s different f r o m those of A and (b)G so t h a t ( F , G) is controllable and i f T and H are o b t a i n e d satisfying (c)
TA - FT
=
GC,
(d)
H
=
TB,
t h e n i t follows from Eq.8.31 t h a t e Fi
and e ( i ) = e e(0)
=
Fe
tends to zero as t tends to i n f i n i t y .
(8.32) I t can be shown t h a t
the
L y a p u n o v e q u a t i o n (c) has a unique s o l u t i o n for T , given A, F, G , and C i f and o n l y i f the eigenvalues o f A are different f r o m those of F. Also i f A and F have no c o m m o n eigenvalues, necessary conditions for the existence of a s o l u t i o n of (c) for T w h i c h is of f u l l r a n k such t h a t P
=
°
T
(8.33)
Chapter
228 is nonnegative are (i){A,
8: Optimal
Control
of Linear
Systems
C} is observable and ( n ) { F , G} c o n t r o l l a b l e . For the single-
i n p u t case these conditions are also sufficient. I t follows f r o m Eq.8.28 and Eq.8.30 that y z —e
1
P-
(8.34)
and t a k i n g i n t o consideration t h a t e ( r ) tends t o 0 as / tends t o i n f i n i t y , the estimate x of x can be represented by x where P
- 1
=
My
+ Nz, _ 1
= [M N] and consequently f r o m P MC
+ NT
P
(8.35) = / „ we get
= /„
. T h e gain m a t r i x K for a linear state variable feedback can be designed b y m i n i m i z i n g a q u a d r a t i c performance index, from the consideration of a prescribed eigenstructure, or decoupling the system. I f the estimated state x is used t o i m p l e m e n t the c o n t r o l law u*(<)
=
Ki(t),
t h e n e m p l o y i n g Eq.8.30 and Eq.8.35 i t can be w r i t t e n as u*(<)
=
K[My(t)
+
Nz(t)}
=
A ' [ M C x ( « ) + NTx.{t)
=
/ f [ x ( t ) + iVe(<)].
+ /Ve(i)] (8.36)
S u b s t i t u t i n g Eq.8.36 i n t o Eq.8.27 we get x(0
=
Ax(t)
+ Be(t)
(8.37)
where A
:=
A +
B
:=
BK
BK N
T h e solutions of Eq.8.32 and Eq.8.37 for e ( t ) a n d x(t) are therefore necessary t o c o m p u t e the c o n t r o l law defined by Eq.8.36. I n the n e x t section we shall see h o w this c o n t r o l law can be o b t a i n e d using o r t h o g o n a l functions.
8.6 Orthogonal
8.6
Function
Approach
229
Orthogonal Function
T h e state vector x(t)
Approach
a n d t h e error vector e ( i ) can be a p p r o x i m a t e d by a set of m
n u m b e r of o r t h o g o n a l basis functions { (r), i{t), • • • , < / > _ i ( i ) } . T h u s we get 0
m
x(i)
«
22 Xi<j>;(t) =
s(i)
«
£
X4>{t)
(8.38)
and
where X a n d E are nxm
e i
0 i ( t ) = E(t)
(8.39)
a n d (w — />) x m matrices w i t h columns x< and e, respectively
w h i c h are s t i l l u n k n o w n , and the m - v e c t o r 4>(t) has 0 j ( t ) ' s as its elements. Once X and E are k n o w n , t h e desired c o n t r o l law u(t)
i n Eq.8.36 can be a p p r o x i m a t e d as a
series of o r t h o g o n a l basis functions { # , ( ( ) } g i v i n g us u(<)
w
K(X
+
NE)(t).
I n t e g r a t i n g b o t h sides of Eq.8.32 over the i n t e r v a l [0,t ] f
(8.40) and using (6.19) a n d Eq.8.39
we get E
- E
FEE(t),
=
0
(8.41)
where E is an m x m i n t e g r a t i o n o p e r a t i o n a l m a t r i x for t h e basis vector <^>(r) and EQ = [e o e i 0
• • • eom-i|-
0
Hence i t follows t h a t E - FEE
=
E
0
a n d solving for E we get
=
[l C„- ) m
- E
r
T
(8.42)
®F\
I n t e g r a t i n g b o t h sides o f Eq.8.37 a n d a p p r o x i m a t i n g x ( i ) a n d e(t) by Eq.8.38 a n d Eq.8.39 we get X
- X <j>{t)
=
0
AXE<)>{t)
+
where •^0
—
x
[ 00
x
0 1
' ' '
x
0 m - l ]
BEE(t),
Chapter
230
8: Optimal
Control
of Linear
Systems
Therefore X — AXE
=
X
+ BEE
0
= W
Since E is already determined by Eq.8.41 we have for X ' x
0
(8.43) X l
-
X
m
_ !
where the n-vectors W; are the columns o f the n x m m a t r i x X
0
+ BEE.
Suppose,
the linear state variable feedback gain m a t r i x K is already designed and the m a t r i x N is d e t e r m i n e d from Eq.8.33.
T h e n k n o w i n g E and X from Eq.8.41 and
Eq.8.43
respectively we can c o m p u t e the c o n t r o l law w i t h the help of Eq.8.40 using various o r t h o g o n a l functions. E x a m i n i n g Eq.8.41 and Eq.8.43 we find t h a t the m a j o r c o m p u t a t i o n a l b u r d e n is the inversion of matrices of order as h i g h as nm and m(n — p) w h i c h for a given system depends on the number of basis functions {4>i{t)}.
T h i s c o m p u t a t i o n a l procedure can,
however, be simplified i f the basis functions are block-pulse functions a n d Laguerre polynomials. 8.6.1
Laguerre
Polynomials
Using the i n t e g r a t i o n o p e r a t i o n a l m a t r i x for Laguerre p o l y n o m i a l s given i n Section 3.4, we can w r i t e Eq.8.42 as /„_„ -
O
F
O
O
...
O
o
o
o
F
" e
0
=
/„_„ - F J
- m-1 e
-
• e(0) • 0
. 0
where I is an ( n — p) x (rt — p) i d e n t i t y m a t r i x . Hence (/„_„ - F ) e Fe, + ( / „ _ , - F ) e
0
=
e(0)
i + 1
=
0
i = 0 , 1 , . . . , m — 2. T h e vectors e,'s c o n s t i t u t i n g the columns of E can therefore be c o m p u t e d recursively from the relations e e
0
i + 1
= ( / „ - , - F)-'e(O), =
-(/„_„-F)-'Fe,.,
8.6 Orthogonal
Function
Approach
231
i = 0 , 1 , . . . , m — 2, where now an inversion of an ( n — p) order m a t r i x is required. I t is evident f r o m Eq.8.43 t h a t x;'s f o r m i n g the columns of X can also be c o m p u t e d recursively f r o m t h e relations x
= (/„ -
0
Xi = ( / „ - i )
_ 1
A)
_ 1
w , 0
(w,- -
iixj-x),
i = l , 2 , . . . m — 1, where t h e n-vectors w , are t h e columns of t h e n x m m a t r i x X
0
+
8.6.2
BEE. Block-Pulse
Functions
For b l o c k pulse f u n c t i o n s , the i n t e g r a t i o n o p e r a t i o n a l m a t r i x is given by i n Section 8.2.2 and consequently, Eq.8.42 can be w r i t t e n as
O — AF
IAF
-AF
-AF
O
O
O
O
-AF
/„_„ - i A F
e(0) e(0)
L e(0) J where / „ _ , , is an i d e n t i t y m a t r i x of order (rt — p).
Hence
1 (/„_„- -AF)e 2 - AF(e„ + d
+ • - • + e,_i) + ( / „ _ , -
0
^AF)e,
=
e(0) e(0),
i = 1 , 2 , . . . , m - 1. As a result the vectors e.'s c o n s t i t u t i n g the columns of E can be c o m p u t e d recursively f r o m the relations e
= (/„_„ -
0
e< = ( / „ - , - j A F ) " i=
1
[AF(e
^AF)-'e(O),
0
+ e, + • • • + e,_,) + e(0)] ,
1,2,. . . , m — 1, where now an inversion of o n l y an (rt — p) order m a t r i x is r e q u i r e d .
I t is evident f r o m Eq.8.43 t h a t x^'s f o r m i n g t h e columns of X can also be c o m p u t e d recursively f r o m the relations x
0
=
(I„
-
1 -AA)
, w , 0
Chapter
232
X,
i = 1,2,. X
0
+
=
(4
-
1
^ A i )
. .m — 1, where the
[w,-
+
8: Optimal
A A ( X
ra-vectors
+
0
Xl
+
Control
• • • +
of Linear
X ; _ ! ) ]
Systems
,
w , are the columns of t h e rt x m m a t r i x
BEE.
T h e direct m e t h o d of d e t e r m i n i n g the t i m e - v a r y i n g g a i n is c o m p u t a t i o n a l l y advantageous as i t requires the c o m p u t a t i o n of a 2n x n m a t r i x i n comparison
with
the m e t h o d using the state t r a n s i t i o n m a t r i x i n w h i c h a m a t r i x of order 2rt is to be c o m p u t e d . T h e c o m p u t a t i o n of this gain i n the case of a f u l l state variable
feedback
as also the estimated state feedback can be done recursively for block-pulse functions and Laguerre p o l y n o m i a l s .
8.7
Problems
1. F i n d the feedback gain for the system x(«) = A x ( i ) + B u ( ( ) , x ( 0 ) = x , y ( < ) = C x ( r ) 0
w h e n the performance index specified is 1 (h ft J = - J (x Qx + T
T
u Ru)dt
where [ - 1 0 A = l
L
0 ]
[ 2 ]
[ 2 - 2 0 ]
0
0 - 2 j ; B = j 2 l ; < 3 = l - 2
0
2
L
oj
1 J
[
2 0
0 1 ; O o J
R = 2, and t, = 2.
2. For the linear system x ( i ) = A x ( i ) + Bu(t),y(t) find the o p t i m a l c o n t r o l law
= Cx(i)
w h i c h m i n i m i z e s the q u a d r a t i c cost f u n c t i o n a l 1 J
= ^ J
f°° T
(x Qx
+ u
T
Ru)dt
under the constraint t h a t the o u t p u t y ( t ) is o n l y available for c o n t r o l where 0 1 A = 0 -2 ;c= [ 1 0
J;
4 Q =
0
. 0 1 .
*i(0)
" l
. * (0) .
. 0
'
; f l = [i];
2
8.7
Problems
233
3. F i n d the o p t i m a ] feedback gain for the c o n t r o l system described by
X =
J
=
0
0 "
. 1
0 .
0 0
1 f>, T rJ (* 2 o T
J
1 x +
0 4
. 0 .
x + u . l .u)dt
where t,
Answer:The o p t i m a l feedback gains are: <7! = [sinh(TT
<7 = 2
\cosh{ir
- 2t) - sin(n
- 2t)]/[cosh (^
- t) + cos {^
- 2t) - cos{ir
-
-
2
2
2
2t)]/[cosh {^
t) +
co
- t)]
2
« ( ^ - 0]
4.For angular velocity s t a b i l i z a t i o n p r o b l e m find the o p t i m a l c o n t r o l u*(t) m i z i n g the performance index 2
2
J = [ ( x ( i ) + 100u (t))
= -Q.5x(t)
+ 150u(t)
x ( 0 ) = 100 A n s w e r : O p t i m a l C o n t r o l is u*(t)
= —
x(t)/\0
5.For the system x ( i ) = -x(t)
+ u(t),0
< t < 1
x(0) = 1 find the o p t i m a l c o n t r o l m i n i m i s i n g the performance index 1
2
J = -J\x (t) 2 •'o
+
2
u (t))dt
Answer: x(t) u(t)
= cosh(V2t)
= (1 + V2/3)cosh(V2t)
+
f3sinh(V2t)
+ (V2 +
0)sinh(V2t)
by m i n i -
Chapter
234
8: Optimal
Control
of Linear
where 0 = -[cosh(V2)
/
/
/
/
+ V 2sirt'i(V 2)]/[v 2cos/i(v 2r) +
sinh(V2)}
6. For the linear system
x(t)
0
1
1
0
*(<) +
y(t) find
0 -1
= I 1
-0.6
u(t),x(0)
0.35
0 j x(i)
the linear o p t i m a l c o n t r o l law such t h a t u*(t) = Kx.(t),t
€ (0,3).
7.For the system x(t)
= -0.5x(t)
+ u(t),0
x(0)
< i < 1
= 1
find the o p t i m a l c o n t r o l m i n i m i s i n g the performance index 1 J = 5x (l) + - J 2
f
l
2
(2x (t) +
2
u {t))dt
Answer: x(<) = [3exp(1.5t) - 4exp(3 - 1.51)] / [ 3 -
4exp(3)]
u(«) = [6exp(1.5i) + 4exp(3 - 1.5*)] / [ 3 -
4exp(3)]
-0.'
Systems
List
of
Figures
2.1
A set of eight H a a r functions
2.2
A set of four Rademacher functions
33
2.3
A set o f four W a l s h functions
34
2.4
A set o f four o r t h o n o r m a l block-pulse functions
35
3.1
32
A p p r o x i m a t i o n o f exp( — t) by o r t h o g o n a l functions and the residual error
3.2 3.3
98
A p p r o x i m a t i o n of exp t by o r t h o g o n a l functions and the residual error. 100 A p p r o x i m a t i o n of exp( —r) by shifted Laguerre a n d H e r m i t e p o l y n o m i als and the residual error
3.4
101
Schematic d i a g r a m t o s t u d y the inherent
filtering
properties of o r t h o g -
o n a l functions 3.5
102
F i l t e r i n g properties of p o l y n o m i a l s T P 2 . noisy signal, s(t)
s ( i ) = a c t u a l signal, i ( 2 ) =
r e c o n s t r u c t e d signal, e(t) = residual error
3.6
Residual error i n signal r e c o n s t r u c t i o n by T P 2
3.7
F i l t e r i n g properties of p o l y n o m i a l s LeP. i ( i ) = a c t u a l signal, s(t)= signal, s(t)
104 105 noisy
r e c o n s t r u c t e d signal, e(J) = residual error
3.8
Residual error i n signal r e c o n s t r u c t i o n by LeP
3.9
F i l t e r i n g properties of S C F . .s(*)=actual signal, i ( i ) = noisy signal, 5 ( i ) r e c o n s t r u c t e d signal, e(t) = residual error
106 107 108
3.10 Residual error i n signal reconstruction by S C F
109
4.1
Response o f x(t)
— 4x(t — 0.25) v i a o r t h o g o n a l functions
138
4.2
Response of x(t)
= -x(t)
and H e r m i t e p o l y n o m i a l s
140
4.3
Response o f x(t)
= x(t — 1) + u(t) v i a block-pulse functions
145
4.4
Response o f x(t)
= x(t — 0.35) + x{t — 0.7) -I- u(t) v i a block-pulse functions 149
4.5
Response of x(t)
= x[t — a(t)\ + u(t) v i a different o r t h o g o n a l functions
5.1
- 2x(t - 0.25) + 2u(< - 0.25) v i a Laguerre
152
D e r i v a t i o n of O S O M R I for shifted Legendre p o l y n o m i a l s w h e n m = 5 and k = 3
165
This page is intentionally left blank
List
of
Tables
2.1
Legendre P o l y n o m i a l s
2.2
Laguerre P o l y n o m i a l s
45
2.3
Hermite Polynomials
50
2.4
Tchebycheff P o l y n o m i a l s of F i r s t K i n d
53
2.5
Tchebycheff P o l y n o m i a l s of Second K i n d
55
2.6
42
Coefficients o f T h r e e - t e r m Recurrence F o r m u l a of O r t h o g o n a l Polynomials:
2.7
64
Coefficients a ,0 ,'y r
r
r
and r
r
of O r t h o g o n a l P o l y n o m i a l s , see
Eq.2.29
and Eq.2.35 2.8
68
Parameters a, 6, c i n Rodrigues' F o r m u l a , N o r m a l i z a t i o n Factor d
T
and
W e i g h t i n g F u n c t i o n w{t) of O r t h o g o n a l P o l y n o m i a l s 2.9
Parameters u,S ,r/ r
r
and F u n c t i o n g(t)
74 75
2.10 Coefficients of Differential Recurrence R e l a t i o n : r = 0
82
2.11 Coefficients of Differential Recurrence R e l a t i o n s > 1
82
3.1
W e i g h t i n g Functions for Shifted O r t h o g o n a l P o l y n o m i a l s
90
3.2
Spectra of e x p ( - t )
96
3.3
Spectra of exp t
97
4.1
T h e response x(t)
4.2
T h e response x(t) i n E x a m p l e 4.1 o b t a i n e d v i a o r t h o g o n a l p o l y n o m i a l s
i n E x a m p l e 4.1 v i a block-pulse functions
4.3
T h e response x(t)
4.4
Response x(t)
in E x a m p l e 4.3 v i a block-pulse functions
144
4.5
Response x(t)
in E x a m p l e 4.4 obtained v i a block-pulse functions.
148
5.1
P a r a m e t e r estimates in E x a m p l e 5.6
171
5.2
P a r a m e t e r estimates in E x a m p l e 5.7
172
5.3
P a r a m e t e r estimates i n E x a m p l e 5.7
174
5.4
Estimates i n E x a m p l e 5.8
175
5.5
Estimates i n E x a m p l e 5.8
176
5.6
P a r a m e t e r estimates of a t w o - i n p u t and one-output system
179
6.1
C o n d i t i o n s for identification w i t h block-pulse Functions
192
and sine-cosine functions
137 137
in E x a m p l e 4.2 v i a Laguerre and H e r m i t e p o l y n o m i a l s 139
238
List 6.2
of
D i s t r i b u t e d parameter system identification w i t h noisy d a t a i n E x a m ple 6.1
6.3
195
D i s t r i b u t e d parameter system i d e n t i f i c a t i o n w i t h noisy d a t a i n E x a m ple 6.2
6.4
196
D i s t r i b u t e d parameter system i d e n t i f i c a t i o n w i t h noisy d a t a i n E x a m ple 6.2
197
6.5
D i s t r i b u t e d parameter system identification w i t h noisy d a t a v i a sine-
6.6
Parameter estimates u ,a
6.7
Parameter estimates a ,a
6.8
Parameter estimates a ,a
cosine functions in E x a m p l e 6.2
6.9
Tables
it
t
xt
198
and a
xx
xt
i n E x a m p l e 6.3
a n d a i n E x a m p l e 6.3
x
and a
t
x
in E x a m p l e 6.4
198 199 200
Parameter estimate a, a n d i n i t i a l c o n d i t i o n t / ( x , 0 ) a n d b o u n d a r y cond i t i o n 3/(0, t ) i n E x a m p l e 6.4
200
7.1
Estimates o f parameters, i n i t i a l a n d b o u n d a r y conditions i n E x a m p l e
7.2
7.1 Estimates o f parameters and spectra o f f(x)
8.1
O p t i m a l Feedback G a i n K in E x a m p l e 8.1
223
8.2
O p t i m a l Feedback Gain k^t)
i n E x a m p l e 8.2
224
8.3
O p t i m a l Feedback Gain fc (i) i n E x a m p l e 8.2
224
8.4
O p t i m a l Feedback G a i n k,(t)
i n E x a m p l e 8.3
225
8.5
O p t i m a l Feedback G a i n k (t)
i n E x a m p l e 8.3
226
2
2
and h(x) i n E x a m p l e 7.2
207 210
List
of
Abbreviations
BPF
Block Pulse Functions
GP
Gegenbauer P o l y n o m i a l s
HP
H e r m i t e Polynomials
HF
Haar Functions
JP
Jacobi P o l y n o m i a l s
LaP
Laguerre P o l y n o m i a l s
LeP
Legendre P o l y n o m i a l s
LQG
L i n e a r Q u a d r a t i c Gaussian Multi-Input Multi-Output
MIMO MIS
M i n i m u m I n t e g r a l Square
NSR
Noise to Signal R a t i o
OF
O r t h o g o n a l Functions Orthogonal Polynomials
OP PCBF
Piecewise Constant Basis Functions
PLP
Piecewise L i n e a r Polynomials
PMF
Poisson M o m e n t Functionals
SCF
Sine-Cosine
SISO
Single I n p u t Single O u t p u t
TPl
Tchebycheff Polynomials of the F i r s t K i n d
TP2
Tchebycheff Polynomials of the Second K i n d
TS UP
T a y l o r Series U l t r a s p h e r i c a l Polynomials
WF
W a l s h Functions
Functions
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Author
Index
A
Chen, C. M . , 249
A h m e d , N . , 241 A l b r e c h t , R. W . , 2 4 1 , 262 A n d e r s o n , B . D . 0 . , 241 Annoussis, J., 265 A r a b s h a h i , A . , 262 A r d e k a n i , B . A . , 15, 241 A r u n a c h a l a m , V . P., 9, 258 A s m a h , N . , 247 Auslander, D . M . , 264
C h e n , K . C , 128, 130, 242 Chen, K . S., 8, 266
Chen, C. T . , 15, 249
Chen, M . Y . , 9, 1 1 , 12, 128, 249, 250 Chen, S. Y . , 253 Chen, W . L . , 5, 6, 7, 8, 10, 14, 128, 244, 245, 260 Cheng, B . , 6, 7, 157, 210, 245, 249 Cheng, D . K . , 3, 245 C h i a , W . K . , 6, 128, 263 C h o u , C. K . , 8, 242, 266
B
C h o u , J. H . , 9, 10, 1 1 , 12, 13, 15, 128,
Balachandran, K . , 15, 258
B o h n , E . V . , 157, 241
248 C h o u , Y . S., 14, 159, 254, 255 C h u n g , C. Y . , 244 C h u n g , H . Y . , 14, 15, 128, 159, 246 C o r r i n g t o n , M . S., 2, 3, 7, 157, 158, 246 C h r i s t o d o u l o u , M . A . , 16, 254 Chrysochoides, N . , 265
Bounas, A . C , 5, 183, 184, 189, 259 B r i s t r i z , Y . , 6, 241
D
130, 137, 184, 245, 246, 247, B a r n e t t , S., 5, 255 Belanger, P.R., 267 B e l l , E . T . , 241 Berger, B . S., 241 B h a t t a c h a r y a , D . K . , 6, 157, 258
D a l t o n , O. N . , 246 C
D a t t a , K . B . , 15, 203, 256, 257
C a m e r o n , R. G . , 6, 2 4 1 , 247, 253 C a m p b e l l , S. L . , 241 Chang, R. Y . , 7, 8, 9, 12, 14, 15, 128, 130, 158, 242, 243, 265, 266 Chang, Y . F . , 1 1 , 12, 13, 128, 137, 184, 243, 254 Chao, C. M . , 1 1 , 262
D e e k s h a t u l u , B . L . , 5, 247 D i n g , X . , 16, 246 D o o r e n , R. V . , 15, 265 D u m o n t , G . A . , 267 D u r r a n i , T . S., 253 D w o l a t z k y , B . , 246 E
Chen, C. F . , 4, 5, 6, 157, 243, 249, 253 Chen, C. K . , 9, 14, 128, 244, 264
E n d o w , Y . , 246
Author
Index
269
F
Jha, A . N . , 9, 10, 1 1 , 12, 184, 210, 2 5 1 , 252, 260 J i a n g , Z. H . , 12, 14, 252 Jie, Z. Q., 8, 263
Fourier, J . , 1 Frank, P. M . , 16, 246 Frick, P. A . , 252
K G K a l a t , J., 252 Georgiou, G . C h . , 16, 259
K a r a n a m , V . R., 252
G o p a l s a m i , N . , 5, 247
K a w a j i , S., 8, 10, 252
G o u , T . Y . , 8, 250
Kekkeris, G. T h . , 9, 14, 158, 183, 184, 252, 259
H
K e y h a n i , A . , 15, 241
Haar, A . , 2, 30, 247
K i n g , R. E., 253
Hadzer, C. M . , 247
K l i n e , M . , 253
Hale, J. K . , 247
Kobayashi, Y . , 7, 257
H a n , J. Y . , 5, 263
K o l l a , S. R., 253
H a r m u t h , H . F . , 247
K o u v a r i t a k i s , B . , 6, 247, 253
H e r m i t e , C., 1
K u n g , F. C , 7, 8, 9, 10, 1 1 , 12, 128,
Ho, S. G . , 10, 1 1 , 13, 14, 248, 249
130, 131, 158, 250, 2 5 1 , 253,
H o r n g , I . Ft., 9, 10, 1 1 , 12, 13, 14, 15,
254, 262, 264
128, 130,
137,
184, 245,
246,
Kurosawa, Y . , 253 K w o n g , C. P., 6, 253
247, 248, 249, 254 H o u t t e , N . V . , 3, 265
L
Hsiao, C. H . , 4, 157, 243, 249 H s u , C. H . , 244
Lacoume, J. L . , 253
H s u , C. S., 244
Laguerre, E . , 1
H s u , N . S., 6, 7, 157, 210, 245, 249
Lahouaoula, A . , 184, 253
H w a n g , C., 6, 7, 8, 9, 1 1 , 12, 14, 15, 128,
157,
158, 168, 249,
2 5 1 , 263 H w a n g , R. Y . , 8, 1 1 , 251
250,
Lancaster, P., 253 Langholz, G., 6, 241 Lee, C. L . , 7, 128, 244 Lee, H . , 7, 8, 128, 158, 253 Lee, L . , 128, 130, 131, 254
I I i j i m a , T . , 253 Ionue, M . , 7, 257 J
Lee, T . T . , 1 1 , 12, 13, 14, 128, 137, 139, 184, 243, 254, 264 Legendre, A . M . , 1 Lewis, F. L . , 16, 254, 256 L i o u , C. T . , 14, 159, 254, 255 L i u , C. C., 8, 10, 255, 263
Jan, Y . G., 6
L i u , J. J., 3, 245
Jan, Y . J., 128, 2 5 1 , 266
L u , Y . J., 15, 267
Jaw, Y . G . , 9, 251 Jeng, B . S., 6, 128, 244
Author
270
Index
M
R a n g a n a t h a n , V . , 9, 1 1 , 12, 184, 210,
M a h a p a t r a , G . B . , 255
Rao, G . P., 4, 5, 6, 7, 8, 15, 127, 128,
252, 260 M a i o n e , B . , 255
157, 158, 258, 259, 260, 261
M a q u s i , M . , 4, 255
Rao, K . R., 2 4 1 , 261
M a r l e a u , Ft. S., 158, 265
Rao, V . P., 261
M a r o u l a s , J., 5, 255
Razzaghi, M . , 15, 2 6 1 , 262
Marszalek, W . , 256 M c l n n i s , B . C., 262
S
M e r t z i o s , B . G . , 9, 16, 254, 256 M o h a n , B . M . , 15, 203, 256, 257 M o h l e r , Ft. Ft., 252 M o o r e , J . B . , 241 Mossaheb, S., 6, 247 M o u l d e n , T . H . , 257 M o u r o u t s o s , S. G., 9, 12, 128, 158, 257, 263
Saleh, Y . , 247 S a n n u t i , P., 4, 262 Sansone, G . , 262 Schaufelberger, W . , 252 Scott, M . A . , 257 Seed, T . J., 262 Shieh, L . S., 262 Shih, D . H . , 10, 1 1 , 12, 250, 253, 262
N a v a r r o , J. M . , 262
Shih, Y . P., 5, 6, 7, 8, 9, 10, 1 1 , 15, 128, 157, 158, 168, 244, 249, 250, 2 5 1 , 255, 263, 265
Nurges, Y . , 6, 257
S h i o t s u k i , T . , 10, 252
O
Shyu, K . K . , 14, 15, 2 5 1 , 263 Sinha, A . K . , 6, 7, 263
O h k i t a , M . , 7, 15, 128, 257
Sinha, M . S. P., 6, 7, 263
N
Sinha, N . K . , 8, 263 S i v a k u m a r , L . , 4, 6, 7, 157, 260, 261 S i v a r a m a k r i s h n a n , A . Y . , 263
P Palanisamy, K . Ft., 6, 7, 8, 9, 15, 128, 157, 158, 257, 258, 260, 264 Papastergiou, C , 265 Paraskevopoulos, P. N . , 5, 8, 9, 16, 158, 159, 183, 184, 189, 252, 253, 258, 259
Sklavounos, P. G . , 16, 259 Sparis, P. D . , 9, 12, 128, 158, 257, 263 Srinivasan, T . , 5, 6, 8, 127, 260, 261 Srisailam, M . C., 263 S t a v r o u l a k i s , P., 4, 5, 263 Stoiza, R., 253
P a t r a , A . , 15, 259
S u b b a y y a n , R., 264
Perng, M . H . , 12, 13, 259, 260
Sun, Y . Y . , 14, 15, 128, 246 Szczepaniak, P. S., 264 Szego, G . , 264
R Rabins, M . J., 264
T
Rademacher, H . A . , 2, 3 1 , 260 R a j a m a n i , V . S., 6, 7, 9, 1 1 , 12, 184, 260, 263 Ramasamy, R. S., 15, 258
T a b a t a b a i , M . , 241 Tada, R. I . , 252 T a h a i , A . , 262
Author
Index
T a k a h a s h i , Y . , 264
271 Z
T a m , L . D . C , 3, 265 Tchebycheff, P. L . , 2
Zakariah, K . M . , 264
T i s m e n e t s k y , M . , 253
Z a m a n , S., 10, 184, 210, 2 5 1 , 252, 266
Trzaska, Z . , 264
Zervos, C., 267
T s a i , C. H . , 1 1 , 12, 184, 248
Z h u , J. M . , 15, 267
T s a i , M . J., 9, 264 T s a i , R. Y . , 13 Tsay, S. C , 12, 14, 128, 139, 254, 264 Tsay, Y . F . , 12, 13, 254, 264 Tsay, Y . T . , 4, 5, 243 T u r c h i a n o , B . , 255 Tzafestas, S. G . , 4, 5, 12, 183, 263, 264, 265 V Vachtsevanos, G . , 16, 254, 256 V a n , T . L . , 3, 265 Vlassenbroeck, J., 15, 265 W W a l s h , J. L . , 2, 3 1 , 265 W a n g , C. H . , 158, 265 W a n g , L . F . , 262 W a n g , M . L . , 7, 8, 9, 12, 14, 15, 128, 130, 158, 242, 243, 265, 266 W o n g , K . M . , 6, 251 W r i g h t , S. A . , 241 W u , S. G . , 10, 14, 244, 245 W u , T . T . , 4, 5, 243 Y Yaaksoo, Y . , 6, 257 Y a n g , C . Y . , 14, 128, 244 Y a n g , S. Y . , 12, 14, 15, 128, 158, 242, 243, 266 Y a n g , T . W . , 13, 248 Yates, R. E . , 262 Y e u n g , C. K . , 262
Subject
Index
delayed u n i t step functions 8
A
d e l a y - i n t e g r a t i o n o p e r a t i o n a l m a t r i x 130, analysis o f time-delay systems 132
131 delay-operational m a t r i x 129
B
delay o p e r a t i o n a l m a t r i x 127
basis functions 3 basis vector 110 Bessel functions 16 Bessel's i n e q u a l i t y 30 bilinear system 5 block-pulse approach for time-delay systems 133 block-pulse functions 2, 3 1 , 94, 205 block-pulse spectral vector 95 block-pulse basis vector 95
for block-pulse f u n c t i o n s , 129 for o r t h o g o n a l p o l y n o m i a l s , 130 for sine-cosine f u n c t i o n s , 130 derivative o p e r a t i o n a l m a t r i x 118 design of d i g i t a l P I D c o n t r o l l e r 15 d i s t r i b u t e d p a r a m e t e r o p t i m a l controllers and filters 5 observers and filters i n o p t i m a l cont r o l systems 4 observers and filters for time-delay
C
systems 5
carry-over s p e c t r u m 142 closed loop q u a d r a t u r e f o r m u l a 89 complete o r t h o g o n a l functions 29 complete set of functions 30 complete system 88 completeness block-pulse functions 5 connection between B P F m e t h o d and trapezoidal method 5 c o n t r o l l a b i l i t y and observability properties 16 converges i n the mean 88 convolution i n t e g r a l 10 c r y s t a l l i z a t i o n process of breakage m o d els 8 D delay-differential equations 129
feedback c o n t r o l systems 8 differential e q u a t i o n for Tchebycheff p o l y n o m i a l s of first k i n d 53 Tchebycheff p o l y n o m i a l s of second k i n d 55 differential recurrence r e l a t i o n 74, 114 D i r i c h l e t ' s conditions 27 discrete d y a d i c - i n v a r i a n t system 3 discrete Laguerre series 10 discrete Legendre p o l y n o m i a l s 11 discrete pulse o r t h o g o n a l functions 13 discrete scaled m a t r i x 13 discrete Tchebycheff p o l y n o m i a l s 8 discrete Tchebycheff series 10
Subject
Index
E
273 identification v i a sine-cosine functions 190
e l l i p t i c system 185
integration operational matrix E
error i n an a p p r o x i m a t i o n 27
r
for
sine-cosine functions 129 impulse response i d e n t i f i c a t i o n 6
F
i n t e g r a l equations 12 filtering
properties of o r t h o g o n a l functions 102
filtering
i n t e g r a t i o n o p e r a t i o n a l m a t r i x 83 i n t e g r a t i o n o p e r a t i o n a l m a t r i x of Fourier series 118
properties of o r t h o g o n a l functions 99
i n t e g r a t i o n o p e r a t i o n a l m a t r i x of shifted o r t h o g o n a l functions 114
Fourier basis vector 94,112 Fourier series 27,93
i n t e g r a t i o n o p e r a t i o n a l m a t r i x for shifted o r t h o g o n a l p o l y n o m i a l s 115
F r e d h o l m i n t e g r a l 10
integral square error 27 G
integral-squared-error
95
i n t e g r a t i o n o p e r a t i o n a l m a t r i x of TayG a l e r k i n ' s m e t h o d 130
lor p o l y n o m i a l s 204
G a m m a and B e t a functions 58 Gauss-Tchebycheff open q u a d r a t u r e for-
J
m u l a 89, 102, 113 Gegenbauer p o l y n o m i a l s 67
Jacobi differential equation 66
general h y b r i d o r t h o g o n a l functions 15
Jacobi p o l y n o m i a l s 56,61
general o r t h o g o n a l p o l y n o m i a l s 11 generalised discrete o r t h o g o n a l p o l y n o mials 13
L Laguerre p o l y n o m i a l s 43
generalized o r t h o g o n a l p o l y n o m i a l s 25
Laguerre differential e q u a t i o n 47
H
least squares a p p r o x i m a t i o n 25
Haar functions 2, 30
Legendre differential e q u a t i o n 43, 66
H a h n p o l y n o m i a l s 15
Legendre p o l y n o m i a l s 40
Laguerre stretch o p e r a t i o n a l m a t r i x 7
H a m m e r s t e i n m o d e l 10
L e i b n i t z f o r m u l a 42
H e r m i t e differential e q u a t i o n 51
linear t i m e v a r y i n g d i s t r i b u t e d p a r a m eter system 206
H e r m i t e p o l y n o m i a l s 47 h y b r i d approach 211
linear t i m e - v a r y i n g systems 203
h y p e r b o l i c system 185
l u m p e d parameter system i d e n t i f i c a t i o n via block-pulse f u n c t i o n 169
h y p e r g e o m e t r i c f u n c t i o n 59
l u m p e d parameter system i d e n t i f i c a t i o n I
via sine-cosine functions 169
i d e n t i f i c a t i o n of l u m p e d parameter sys-
M
tems 167 i d e n t i f i c a t i o n v i a block-pulse functions 190
m i n i m u m energy c o n t r o l 15 model reduction 6
Subject
274 m u l t i - d e l a y system 147
power-law system 4
m u l t i v a r i a b l e discrete system 6
p r e d i c t i o n of l i m i t cycles 6
Index
p r o d u c t m a t r i x for W a l s h functions 5 N
properties of G a m m a functions 70
nonlinear d i s t r i b u t e d parameter systems 203, 207
pulse w i d t h m o d u l a t e d systems 14 R
n o r m a l i z a t i o n factor 29, 74 n o r m a l i z a t i o n factor o f Gegenbauer p o l y n o m i a l s , 72
Rademacher functions 2, 31 recursive a l g o r i t h m ( v i a block-pulse func-
o r t h o g o n a l functions, 87
t i o n s ) for the analysis of t i m e -
o r t h o g o n a l p o l y n o m i a l s , 89 n u m e r i c a l inverse Laplace transform 6
delay systems 134 representation of noisy signals 99 residual 27
O
residual error 95
o p e r a t i o n a l m a t r i x for backward inte-
Rodrigues' f o r m u l a 37, 74 for Legendre p o l y n o m i a l s , 40
g r a t i o n of Walsh vector 5
for Laguerre p o l y n o m i a l s , 44
o p e r a t i o n a l m a t r i x for stretch 6 o r d i n a r y differential equations 78 o r t h o g o n a l p o l y n o m i a l approach for t i m e delay systems 135 o r t h o g o n a l i t y relation 87 o r t h o g o n a l i t y r e l a t i o n for shifted p o l y nomials 89 o r t h o n o r m a l set 26 O S O M R I 7, 159 of block-pulse functions 160 of shifted Tchebycheff p o l y n o m i a l s 161 of shifted Legendre polynomials 164 of sine-cosine functions 166 o u t p u t sensitivity 9 P
S sampled d a t a c o n t r o l systems 10 scaled system 7 self-tuning c o n t r o l 16 shift W a l s h m a t r i x 5 shifted H e r m i t e p o l y n o m i a l s 92 Laguerre p o l y n o m i a l s 91 o r t h o g o n a l p o l y n o m i a l s 88 signal processing 87 signal reconstructor
99
sine-cosine functions 30, 93 sine-cosine f u n c t i o n approach for t i m e delay systems 136 single-term m e t h o d 6
Pade a p p r o x i m a t i o n 14, 139
singular system 8
parabolic system 185
spectral vector 89, 94
Parseval's i d e n t i t y 88
s p e c t r u m of a f u n c t i o n 88
piecewise constant basis functions 2
square integrable functions 30
piecewise constant delay system 150
state e s t i m a t i o n 12
piecewise linear p o l y n o m i a l functions 14
stochastic system 5
piecewise linear system 7
stretched system 6
p o p u l a t i o n balance equations 7
system order e s t i m a t i o n 7
Subject
Index
275
T
w e i g h t i n g f u n c t i o n of shifted orthogo-
table of
Walsh basis vector 99
nal p o l y n o m i a l s 89 H e r m i t e p o l y n o m i a l s 50
W a l s h functions 2, 31
Laguerre p o l y n o m i a l s 45
Walsh-Galerkin method 5
Legendre p o l y n o m i a l s 41
Weirstrass t h e o r e m 26
Tchebycheff p o l y n o m i a l s of F i r s t K i n d 52
Z
Tchebycheff p o l y n o m i a l s of second k i n d 55 T a y l o r p o l y n o m i a l s 203 T a y l o r series 48, 204 Tchebycheff differential e q u a t i o n of the first k i n d 66 of the second k i n d 66 Tchebycheff p o l y n o m i a l s of first k i n d 51 of second k i n d 54 t h r e e - t e r m recurrence f o r m u l a 37 for Laguerre p o l y n o m i a l s 45 for Legendre p o l y n o m i a l s 41 for H e r m i t e p o l y n o m i a l s 49 for Tchebycheff p o l y n o m i a l s 52 t h r e e - t e r m recurrence r e l a t i o n 38,114 t i m e - p a r t i t i o n m e t h o d 129, 141 two d i m e n s i o n a l square-integrable functions 103 t w o - p o i n t b o u n d a r y value p r o b l e m 12 transfer f u n c t i o n m a t r i x i d e n t i f i c a t i o n 173 U unified approach for i d e n t i f i c a t i o n 185 V V a n der Pol's oscillator 16 variational problem 7 V o l t e r r a i n t e g a r l 10 W w e i g h t i n g f u n c t i o n 74
zeros of o r t h o g o n a l p o l y n o m i a l s 39
