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Class of methods for discrete-time system identification and parameter tracking of sampled-data systems Suryanarayanan, K.L. 1970

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A CLASS OF METHODS FOR DISCRETE-TIME SYSTEM IDENTIFICATION AND PARAMETER TRACKING OF SAMPLED-DATA SYSTEMS by K. L. SURYANARAYANAN B.E., Madras Uni v e r s i t y , India, 1960 M.Tech., I.I.T. Bombay, India, 1964 M.Sc.E., University of New Brunswick, 1966 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor • Members of the Committee ,,...« Acting Head of the Department ». Members of the Department of E l e c t r i c a l Engineering . •• THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1970 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t he r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT In t h i s work, methods for on-line i d e n t i f i c a t i o n of discrete-time systems and for parameter tracking of sampled-data systems are presented. These methods are s u i t a b l e f o r implementation using small computers. A cl a s s of methods f o r the i d e n t i f i c a t i o n of the c o e f f i c i e n t s of l i n e a r and nonlinear d i f f e r e n c e equations i s developed. The philosophy of i d e n t i f i c a t i o n i s divided into three parts based on the norm of the err o r to be minimized. Techniques are derived using a common framework of minimization of these e r r o r functions, incorporating uniqueness and s t a b i l i t y properties. Prac-t i c a l examples are included which demonstrate that among these proposed methods the i d e n t i f i c a t i o n error method solves the problem s u c c e s s f u l l y . Extensions of these methods to continuous systems are b r i e f l y o utlined. Methods are proposed for the generation of parameter s e n s i t i v i t y functions f o r sampled-data systems on a hybrid or a d i g i t a l computer. Both l i n e a r and nonlinear systems are considered, and f o r a class of l i n e a r , systems,, an economical approach for the generation i s developed, making extensive use of s i g n a l flow graph techniques. A new technique i s devised f o r s o l v i n g the problem of parameter tracking of l i n e a r and nonlinear sampled-data systems using the s e n s i t i v i t y functions. Examples are presented to demonstrate that the proposed techniques solve the problem. TABLE OF CONTENTS Page 1. INTRODUCTION . ... 1 1.1 Preliminary Considerations 1 1.2 Some D e f i n i t i o n s 2 1.3 Scope of Research 4 2. IDENTIFICATION OF THE COEFFICIENTS OF LINEAR DIFFERENCE EQUATIONS... 6 2.1 Introduction ... 6 t 2.2 I d e n t i f i c a t i o n Error Method 8 2.2.1 I d e n t i f i c a t i o n E r r o r Method f o r Discrete-Time Systems... 8 2.2.1.1 The Algorithm. ...'.V ......... 8 2.2.1.2 A Special Case. .. . . . . . . . . . . . . . . . . .. ... . . 1 6 2.2.1.3 Ap p l i c a t i o n , 18 2.2.2 I d e n t i f i c a t i o n Error Method f or Continuous Systems 18 2.3 Output E r r o r Method v . . 19 2.3.1 Output Error Method f o r Discrete-Time Systems 19 2.3.1.1 The Algorithm 19 2.3.1.2 P r a c t i c a l Considerations 26 2.3.1.3 Ap p l i c a t i o n 29 2.3.2 Output Error Method f o r Continuous Systems 30 2.4 Equation Er r o r Method. .. ."' . 31 2.4.1 Equation E r r o r Method f or Discrete-Time Systems 31 2.4.1.1 The Algorithm 32 2.4.1.2 P r a c t i c a l Considerations — . . . . . . . 35 r 2.4.1.3 Ap p l i c a t i o n . . . . . . . . ....... 37 2.4.2 Equation Error Method f o r Continuous Systems 37 2.5 Conclusions.... 38 3. IDENTIFICATION OF THE COEFFICIENTS OF NONLINEAR DIFFERENCE EQUATIONS 40 3.1 Introduction :. 40 Page 3.2 T h e o r e t i c a l Development of the I d e n t i f i c a t i o n Technique 40 3.2.1 Problem Formulation '.' 40 3.2.2 I d e n t i f i c a t i o n Error Method 41 3.2.3 Output Error Method ; 44 3.2.4 Equation Er r o r Method 46 3.2.5 Conclusions .. . ... 47 3.3 A p p l i c a t i o n of the I d e n t i f i c a t i o n Error Method 48 3.3.1 Example 3.1 48 3.3.2 Polynomial Form of Nonline a r i t y 49 3.3.3 Piecewise Linear Form of Nonlinearity 52 3.3.4 Multivalued N o n l i n e a r i t i e s . 69 3.4 Some P r a c t i c a l Considerations 76 3.4.1 Time-Varying C o e f f i c i e n t s 76 3.4.2 Unknown Form of Nonline a r i t y 78 . 3.5 I d e n t i f i c a t i o n E r r o r Method f o r Nonlinear D i f f e r e n t i a l Equations 79 3.6 Conclusions •••• 81 4. GENERATION OF PARAMETER SENSITIVITY FUNCTIONS ...... 83 4.1 Introduction 83 4.2 Extension of the Vuscovic and C i r i c Method f o r Sampled-data Systems 84 4.3 S e n s i t i v i t y Functions f o r Linear Sampled-data Control Systems.. 90 4.3.1 Preliminary... • ••• '•'« 90 4.3.2 The General Case ....... 94 4.3.3 Special Case 1 - Subgraph T g Has No Black Node 108 4.3.4 Sp e c i a l Case 2 - System Error i s Sampled 114 4.3.5 Sp e c i a l Case 3 - Completely Sampled System 117 4.4 Examples 119 4.5 Conclusions : 126 Page 5. PARAMETER TRACKING 127 5.1 Introduction 127 5.2 Identification Error Method. ... 128 5.2.1 The Algorithm 128 5.2.2 Practical Considerations.... 132 5.3 Output Error Method 133 5.4 Examples 135 5.5 Conclusions.... 141 6. CONCLUSIONS •.. . 142 6.1 Summary 142 6.2 Further Research 144 APPENDIX A Eventual Stability. 146 APPENDIX B Extension of the Vuscovic and Ci r i c Method for the Generation of Second Order Sensitivity Functions 148 REFERENCES.. 152 ACKNOWLEDGEMENT I wish to thank the National Research Council of Canada for awarding me with Studentship and Post-graduate Scholarships during my graduate program at the Univ e r s i t y of B r i t i s h Columbia. I am indebted to Dr. Bohn and Dr. Davies f o r t h e i r c r i t i c a l reviewing of the thesis and for the many i n t e r e s t i n g and useful discussions and suggestions. The typing of the thesis by Miss Beverly Harasymchuk and the patience she exercised throughout the long hours of proof-corrections are acknowledged with deep appreciation. Very s p e c i a l thanks are due to Dr. S.G. Rao whose help throughout the major part of the work was beyond measure. I am very g r a t e f u l to him for several highly f r u i t f u l discussions, for h i s great patience in- c r i t i c a l l y reviewing the work at various stages and for h i s constant i n s p i r a t i o n . His help was indeed invaluable. I wish to thank Mr. W.R. Robinson f o r the useful discussions during .the i n i t i a l stages of the work. I am very g r a t e f u l to Dr. A.C. Soudack, the thesis supervisor for h i s continued encouragement a l l through these years and f o r h i s c r i t i c a l reviewing of the t h e s i s . The help rendered by Dr. G.A. Bekey i n supplying a number of us e f u l references and i n putting the thesis i n the proper perspective i s acknowledged with deep gratitude. F i n a l l y , I want to thank the fellow graduate students and other friends fo r t h e i r help i n proof-reading the t h e s i s . This research was supported by NRC Grant No. 67-3138. 1 1. INTRODUCTION 1.1 Preliminary Considerations In many areas of co n t r o l theory, such as optimal and adaptive c o n t r o l , a need often arises to represent the system by a model. For t h i s purpose, a model may be defined as a representation of the e s s e n t i a l aspects of the system which provide information about the system''". The determination of the parameters 2 of the model i s generally c a l l e d the i d e n t i f i c a t i o n problem . An i d e n t i f i c a t i o n procedure, which uses the normal operating conditions of the system, and hence, 3 does not employ test s i g n a l s , i s termed on-line i d e n t i f i c a t i o n . Continuous-time systems, where some of the state variables are allowed to change only at d i s c r e t e instants of time, are known as sampled-data systems^. A d i s c r e t e -time system i s one i n which the input and a l l the state variables are defined 4 only at d i s c r e t e instants of time , and so i s governed by an ordinary difference equation. A number of methods e x i s t f o r the i d e n t i f i c a t i o n of continuous and sampled j t 1,2,5-29,58-67 w , , . 3 . c. , 3 9-11 data systems . Many standard i d e n t i f i c a t i o n methods employ schemes not s u i t a b l e for small computers. Work has been reported i n the l i t e r a -T -u A - u *.v 3,12-24,29,59,60,62-64 ture on n o n i t e r a t i v e , on-line methods ror both continuous 25—2861 and discrete-time ' systems and there are a number of apparently unrelated 12 17—19 23—29 algorithms a v a i l a b l e ' ' . However, no work has been reported on the i d e n t i f i c a t i o n or on the tracking of v a r i a b l e parameters of sampled-data systems by n o n i t e r a t i v e , on-line techniques. The object of t h i s work i s to develop methods for so l v i n g the i d e n t i -f i c a t i o n problem for discrete-time systems and the tracking of the v a r i a b l e parameters of sampled-data systems, using small s p e c i a l purpose computers on-l i n e . The memory and the speed requirements of the computer are considered to be l i m i t e d , thereby precluding such operations as the inv e r s i o n of large matrices and the s o l u t i o n of large number of simultaneous equations, as required i n many standard i d e n t i f i c a t i o n techniques Sequential methods of i d e n t i f i c a t i o n are investigated and d i f f e r e n t •on-line techniques are derived i n a common framework of erro r function minimiza-t i o n . With such an approach, i t i s possible to inves t i g a t e the r e l a t i v e merits of several d i f f e r e n t techniques of i d e n t i f i c a t i o n with ease. The general procedure adopted f o r s o l v i n g the i d e n t i f i c a t i o n problem f o r l i n e a r d i s c r e t e -time systems i s described. The i d e n t i f i c a t i o n procedure i s then extended to nonlinear dis"crete-time systems and applied to various types of n o n l i n e a r i t i e s . New developments on the generation of parameter s e n s i t i v i t y functions f o r sampled data systems are c a r r i e d out and they are applied to the problem of parameter tracking of sampled-data systems. ~~A~~few~ b a s i c concepts, used i n th i s work, are explained b r i e f l y here. The system to be i d e n t i f i e d , i s assumed noise - f r e e . A model of the system i s chosen, u t i l i z i n g a p r i o r i knowledge of the system. I t i s assumed that the sampling period i s uniform throughout, m both the system and the model. The model and the system are excited with the same input. Using measure-ments of the input and the output of the system, and knowledge of the model, the parameters of the model are computed and adjusted at each sampling in s t a n t . It i s assumed that the input and the output, at the sampling instants are the only measurable q u a n t i t i e s of the system. 1.2 Some D e f i n i t i o n s Below, a few terms used i n the thesis are defined. Though t h e i r usage .is not r e s t r i c t e d to l i n e a r systems, a l i n e a r , sampled-data system i s considered to f a c i l i t a t e the explanation. . " Let the l i n e a r system be represented by the dif f e r e n c e equations (hence for t h denoted as d.e.) r e l a t i n g the input r ( j ) at time inst a n t j to the output m n y (j) = E k r(j-i) - Z k y (j-i) (1.1) d i=o 1 i=i m + 1 d where the k^, called the system coefficients, are functions of the system parameters, p^(j), i.e., k^ = k^(P(j)) where P (j) is the vector of system parameters. Let the d.e. of the model be given by tn' . n' \ y(j) = E k r ( j - i ) - E k , y(j-i) (1.2) x=0 i=l where y(j) is the output, and the k_^  are the model coefficients and are functions of the parameters, p^(j) of the model, i.e., k_^  = k^,(P(j)) where P(j) is the vector of model parameters. Identification is the term applicable only when the system and the model d.e. are of such type that m'=m and n'=n in eqns. (1.1) and (1.2). Modelling is the term used when such is not the case, usually m' < m and/or n' < n. Identification of the parameters of the system refers to the process of making P(j) tend towards P(j). For time-invariant P(j), identification is achieved i f for any e > 0, there exists a number N(e) > 0 such that | |P.-P(j)|| < E ; Vj > N(e) Identification of the coefficients of the d.e. of the system refers to the process of making K(j) tend towards K(j). For time-invariant K(j), identification is achieved i f for any e > 0, there exists a number N ( E ) > > 0 such that | |K - K(j) J J < Vj > N;(E) In general, ||P-P(j)|| = 0 implies ||K-K(j)|| =0. Sensitivity function or parameter sensitivity function at instant j is defined as the partial derivative of the output, y^d) of the system with respect to a time-invariant parameter p of the system, i.e. } 4 A A 9 y d ( j ) w ( j ) = ( y , ) 1 - — p d p 8p In the case of the system parameters v a r y i n g w i t h time, i f the parameters are known i n i t i a l l y , then the process of t r a c k i n g t h e i r t i m e - v a r i a -t i o n s by a model i s given the name parameter t r a c k i n g . 1.3 Scope of the Research The f i r s t p a r t of t h i s work (chapters 2 and 3) deals w i t h the develop-ment of the i d e n t i f i c a t i o n methods and t h e i r a p p l i c a t i o n to l i n e a r and n o n l i n e a r d i s c r e t e - t i m e systems. In chapter 2, three i d e n t i f i c a t i o n p h i l o s o p h i e s f o r the i d e n t i f i c a t i o n of the c o e f f i c i e n t s of the d.e., d e s c r i b i n g l i n e a r processes, are discussed and 3 12—29 59 the i d e n t i f i c a t i o n algorithms are developed. U n l i k e past works ' . ' } the s t a b i l i t y of these i d e n t i f i c a t i o n methods i s proved without assuming time-i n v a r i a n t parameters of the system. C o n s i d e r a t i o n of the input to the system, i n the l i g h t of i d e n t i f i c a t i o n , i s a l s o d e a l t w i t h . Chapter 3 extends one of the p h i l o s o p h i e s of i d e n t i f i c a t i o n of chapter 2 to a wider c l a s s of systems, namely the ones w i t h n o n l i n e a r i t i e s . I d e n t i f i c a t i o n of the c o e f f i c i e n t s of the n o n l i n e a r d.e. d e s c r i b i n g the system, i s d iscussed i n t h i s chapter. A number of examples are presented i l l u s t r a t i n g the wide range of a p p l i c a b i l i t y of the method. A b r i e f d i s c u s s i o n of the continuous case i s a l s o given. Chapters 4 and 5 are devoted to the development of s e n s i t i v i t y f u n c t i o n s f o r sampled-data systems and t h e i r a p p l i c a t i o n to the parameter t r a c k i n g problem. The computer generation of parameter s e n s i t i v i t y f u n c t i o n s f o r sampled-data systems i s considered i n chapter 4. Both l i n e a r and n o n l i n e a r systems are d e a l t w i t h . S i g n a l flow graph techniques are used f o r economizing the s i m u l a t o r components or computation time, f o r a c l a s s of l i n e a r systems described i n the chapter. Chapter 5 deals with the parameter tracking of sampled-data systems Applications of the i d e n t i f i c a t i o n techniques derived i n chapter 2, and of th s e n s i t i v i t y functions developed i n chapter 4, are made use of and new r e s u l t s are presented f o r both l i n e a r and nonlinear systems. A summary and recommendations f o r further research are included i n chapter 6. 6 2. IDENTIFICATION OF THE COEFFICIENTS OF LINEAR DIFFERENCE EQUATIONS 2.1 Introduction The object of t h i s chapter i s to present a cla s s of methods f o r solv i n g the problem of the i d e n t i f i c a t i o n of the c o e f f i c i e n t s of l i n e a r d i f f e r e n c e equations by on-line sequential methods. Techniques are derived i n a common framework of minimization of d i f f e r e n t error functions. For t h i s purpose, the philosophy of i d e n t i f i c a t i o n i s c l a s s i f i e d into three d i v i s i o n s depending upon the s e l e c t i o n of the performance c r i t e r i a . Each of the three i s developed i n a separate s e c t i o n and i t s r e l a t i v e merits are discussed. A d e r i v a t i o n of each method i s c a r r i e d out and the s t a b i l i t y , of the proposed algorithms i s discussed. Extensions of the approach f o r continuous systems are o u t l i n e d . Problem Formulation Consider a n o i s e - f r e e , l i n e a r , s t a b l e , discrete-time dynamic system whose d i f f e r e n c e equation i s given by eqn. (1.1). Assume that the v a r i a t i o n of the k. with time i s slow. l Choose a model whose d.e. i s given by eqn. (1.2), where m'=m and n'=n. Adjust the k^ n o n i t e r a t i v e l y at each instant such that the i d e n t i f i -cation of the c o e f f i c i e n t s , k., i s achieved. i D e f i n i t i o n s and C l a s s i f i c a t i o n of I d e n t i f i c a t i o n Methods The three methods of approach to be discussed i n t h i s chapter are defined below. The k_^  are treated as time i n v a r i a n t s i n de r i v i n g the adjustment algorithms. Later, i n proving the convergence of these algorithms, t h i s condition i s relaxed. Let eqns. (1.1) and (1.2) be wr i t t e n compactly as - y d ( j ) = K T S ( j ) (2.1) and y ( j ) = K X ( j ) S ( j ) (2.2) where K(j) = t k o ( j ) k x ( j ) k m + n ( J ) ] T > < 2' 4 ) S(j) = [s ( j ) S l ( j ) . . . . 8 . ( j ) ] T o 1 m+n = t r ( j ) r ( j - l ) . . . r ( j - m ) - y / j - D - y d ( j - n ) ] T , (2.5) 3 1 1 ( 1 S(j) = [sQ(i) ^ ( j ) ... S m + n ( j ) ] T = [ r ( j ) r ( j - l ) ... r(j-m) - y ( j - l ) ... - y ( j - n ) J T (2.6) In the above representations, superscript T denotes transpose. The (m+n+1)-vectors, K and K}are c a l l e d the c o e f f i c i e n t vectors of the system and of the model r e s p e c t i v e l y and the (m+n+1)-vectors, S and S, are c a l l e d the s i g n a l vectors,of the system and of the model r e s p e c t i v e l y . D e f i n i t i o n 2.1 I d e n t i f i c a t i o n error E(j) i s defined to be the (n+m+1)-vector of the error i n the i d e n t i f i c a t i o n of the c o e f f i c i e n t s , i . e . E ( j ) = E = K(j) " K(j) (2.7) D e f i n i t i o n 2.2 Output error e(j) i s defined to be the error i n the outputs of system and model , i . e . , e ( j ) £ e = y d ( j ) - y ( j ) (2.8) D e f i n i t i o n 2.3 Equation error.£(j) i s defined to be the error i n the s a t i s f a c -t i o n of eqn. (2.1) i f the model c o e f f i c i e n t vector, K ( j ) , i s substituted f o r the system c o e f f i c i e n t vector, K^ , = e = y d ( j ) " K T ( j ) S ( j ) (2.9) D e f i n i t i o n 2.4 I d e n t i f i c a t i o n error method (output error method, equation error method) i s defined to be a method which chooses an e x p l i c i t function of E(e,e) as the performance c r i t e r i o n (henceforth c a l l e d PC) and develops an i d e n t i f i c a t i o n method by minimizing i t . 8 The output and the equation error methods have been investigated for ^ c i o \£i 16 22 2 A 2S 58 60 continuous systems by many authors » » ' » > ->» -.- For discrete-time 28 systems Mantey has discussed the. i d e n t i f i c a t i o n problem employing the equa-29 t i o n error methods with steepest descent. Braverman , and Richalet and 26 Gimonet used an algorithm which i s e s s e n t i a l l y the same as the one derived here f o r the i d e n t i f i c a t i o n error method. The approach described i n th i s chapter i s distinguished from the previous works i n that a l l the methods are derived i n a common framework of minimizing s u i t a b l e functions of the corresponding er r o r s . Furthermore the time-variations of the system c o e f f i c i e n t s are allowed i n proving the convergence of the proposed algorithms. 2.2 I d e n t i f i c a t i o n E r r o r Method This section i s divided into two subsections; the f i r s t one developing the technique f o r discrete-time systems; the second one b r i e f l y considering the continuous case. 2.2.1 I d e n t i f i c a t i o n Error Method for Discrete-Time Systems 2.2.1.1 The Algorithm 1 For the problem posed i n section 2.1, the method i s described i n the following steps. Step 1. Choice of performance c r i t e r i o n Since the approach i s n o n i t e r a t i v e , a s u i t a b l e choice i s an i n s t a n t -.. * 21 aneous c r i t e r i o n , V , ..at instant j , (2.10) minimum Any s u i t a b l e norm of E w i l l s a t i s f y the above requirements. Let V. be chosen as Vj = ||E|| = ETME (2.11) 9 where :M i s any (n+m+1 x n+m+1) p o s i t i v e d e f i n i t e symmetric matrix. Step 2. Choice of incrementing algorithm Choose a general algorithm f o r incrementing K at each instant such that the increment i s a measurable . quantity. Thus, AK(j) = K(j+1) - K(j) (2.12) =t7(r(i) , y d ( i ) , y ( i ) .k^^,^,.... ,k n + m) = hF (2.13) where h i s a sc a l a r and F i s an (n+m+1)-vector to be chosen l a t e r , and i < j , i e l , where I i s the set of p o s i t i v e i n t e gers. ""Step-"31 Choice of~TT~aad determinatiorTof A V ^ Determine AVJ. and minimize it__at__eachL.jLnstant by the proper choice o f h. From eqns. (2.7), (2.11) and (2.13), = -2hE T(j)MF + h 2F TMF (2.14) Minimizing AVj^j) with respect to h, M l n A « j ) - - < E < 1 ) M F ) ' (2.15) F MF and the corresponding h i s given by = .ET(,j)MF F TMF < 2- 1 6> Thus, from eqn. (2.15), AV.j.(j) i s obtained as a nonpositive quantity. Step 4. Determination of AK Determine the a c t u a l adjustment algorithm f o r AK(j) i n eqn. (2.13) such that i t i s a known function of only measurable q u a n t i t i e s . From eqns. (2.13) and (2.16), . F MF . (K T-K T(,i))MF F < 2 > 1 7 ) • ' • ." F MF . 1 0 In the above expression f or A K(j), the quantity K i s unknown. From eqn. ( 2 . 1 ) , i t i s seen that the measurable vector S(j) commutes with K to produce the measurable quantity y^Cj). Thus, to eliminate K i n eqn. ( 2 . 1 7 ) , l e t F be F = M _ 1 S ( j ) ( 2 . 1 8 ) Hence, eqn. ( 2 . 1 7 ) reduces to [ y H ( j ) - K ( j ) s ( j ) ] . A K ( j ) = M _ i S ( j ) ( 2 . 1 9 ) S T ( j ) M ^ ( j ) where, A K ( j ) i s s t r i c t l y a function of measurable q u a n t i t i e s . Thus f a r , K has been assumed time-invariant. In the proof of convergence of the algorithm that follows, t h i s r e s t r i c t i o n on K - i s avoided. Step 5 . Proof of Convergence Consideringjthe v a r i a t i o n i n K ( j ) as A K ( j ) , eqn. ( 2 . 1 9 ) becomes AE ( j ) = " V ^ 5 ^ M _ 1 S ( j ) + A K ( j ) S i ( j ) M X S ( j ) ( 2 . 2 0 ) This equation may be v i s u a l i z e d as the v e c t o r - d i f f e r e n c e equation of the ' i d e n t i f i c a t i o n system' ( F i g . 2 . 1 ) . To prove the convergence of the proposed algorithm, i t i s s u f f i c i e n t to prove the Liapunov asymptotic s t a b i l i t y of: - Q E AK AE AK -1 T v x -1 ; S M S F i g . 2 . 1 The ' I d e n t i f i c a t i o n System' 11 eqn. (2.20). Such a proof would however, be impossible since E = 0 i s not an 31 e q u i l i b r i u m state of eqn. (2.20), and also since the system i s nonautonomous '. 31 The concept of eventual s t a b i l i t y , which i s a ge n e r a l i z a t i o n of Liapunov s t a -b i l i t y i s appropriate f o r th i s s i t u a t i o n . The d e f i n i t i o n s of eventual s t a b i l i t y and the re l a t e d theorems, which are used here, are l i s t e d i n Appendix A. Theorem 2.1 Let the following assumptions be made concerning the v a r i a t i o n of the system c o e f f i c i e n t vector K: Assumption 2.1 | |K(j ) | | < 0 0 Vj (2.21) Assumption 2.2 AK(j) + 0 as j + « (2.22) Let the input r(j) be such that, i f the vector K were time-invariant, then f o r T a l l ||E|| > 0, there e x i s t s a J < ~ f o r each j , with |E ( i ) S ( i ) | >_a(j) (a(j) > 0), f o r some i , vj <_ i < J +j. Then the o r i g i n of the ' i d e n t i f i c a t i o n system', defined by eqn. (2.20) i s eventually asymptotically stable i n any bounded region. Proof F i r s t , consider the s t a b i l i t y of the ' i d e n t i f i c a t i o n system' with AK(j)=0 Vj. Then the d.e. of the system reduces to AE(j) = - fo)SW M ^ S Q ) (223) S i(j)M 1S(j) U " " J M I l s < 1 ) f < 1 ) E ( j) (2.24) Si(j)M"J"S(j) I t i s seen that E=0 i s an equilibrium state of t h i s system. Hence, the concept 31 of Liapunov s t a b i l i t y i s appl i c a b l e here . Choose V(E) = ETME (2.25) The t o t a l increment AV(E) i s given by AV ( E ) . - q s'i>> s o Now, def i n i n g a s c a l a r u ^ j ) for r J 0 as T 2 /•\ ( E i S ( j ) ) Z u (j) - sup .- -= ^ -r |E||>r S i ( j ) M X S ( j ) i t i s seen that 12 (2.26) (2.27) I u _ ( i ) |E]|>r S T ( i ) M 1 S ( i ) i=j (2.28) <_ - c^Cj) < 0. (2.29) where a^(j) i s a constant. Hence 3 I p (j) = -co-j=0 r f o r each r > 0. Furthermore, from eqns. (2.26) and (2.27), (2.30) AV(E) < v r ( j ) Now applying Theorem A . l of Appendix A, i t i s seen that the o r i g i n of the d.e. (2.26) i s uniformly asymptotically stable i n any bounded region Q . Consider now the ' i d e n t i f i c a t i o n system' of d.e. (2.20) with AK(j) term present. Since AK(j) -> 0 as j -> °° due to Assumption 2.2, i t i s seen that AK(j) s a t i s f i e s the condition (A.3) i n Theorem A.2 of Appendix A. Hence the.applica-t i o n of Theorem A.2 r e s u l t s i n the eventual asymptotic s t a b i l i t y of the o r i g i n . of the ' i d e n t i f i c a t i o n system' of d.e. (2.20) i n any bounded region . E Remark 2.1 Consider the matrix s o ( j ) S o ( j + D 8 l ( J + I ) . . S ^ C j - r l ) s o ( j + J . - l ) s 1 ( j + J j - l ) (2.31) 13 = t+oCj.Jj) ^ ( j . J j ) V n ( j ' J j ) ] ( 2 . 3 2 ) ' Let the input r(j) be chosen such that, i f the vector K were time-i n v a r i a n t , the rank of T t j j J . ) i s (m+n+1) f o r some J . and for a l l j . Then i t T can be shown that i f | | E | | > 0, i t i s not po s s i b l e to have E (i)S(i)=0 V i , j <_ i < J + J j • For, suppose E T ( i ) S ( i ) = S T ( i ) E ( i ) = 0 V i , j <_ i < j+J (2.33) with ||E||^0. Then, using eqn. (2.32), the following equation may be obtained: V ( j , J j ) E ( j ) = 0 (2.34) where 0 i s a J.-vector. Since the rank of ^ ( i j J . ) i s m+n+1, i t follows that 3 3 E must be zero - a c o n t r a d i c t i o n . Hence, i f E ^ O , there e x i s t s some i , j <_ i < J + J j such that E T ( i ) S ( i ) # ) . The input considered above, s a t i s f i e s p a r t i a l l y the requirement of the same given i n Theorem 2.1. However, i n the present case, an assumption T has to be made that |E ( i ) S ( i ) | admits a p o s i t i v e lower bound, ct(j), f o r some i , j <_ i < J + ^ j • Then t h i s input w i l l be a s u f f i c i e n t one to guarantee eventual asymptotic s t a b i l i t y of the o r i g i n of the i d e n t i f i c a t i o n system. A c l a s s of inputs which s a t i s f i e s the above condition on the rank of ^ ( j j J . ) i s described i n the following c o r o l l a r y . The approach followed i s 32 s i m i l a r to that of L i f f wherein he considered s i n u s o i d a l inputs for s o l v i n g the i d e n t i f i c a t i o n problem by the s o l u t i o n of a set of simultaneous equations. C o r o l l a r y 2.1 I f the input r(j) i s a sum of a f i n i t e number of sinusoids with at l e a s t (m+1) d i s t i n c t , d i s c r e t e , complex. Fourier components and i f the poles of the system defined by eqn. (1.1) - i n the z-plane - are d i s t i n c t , then the rank of 1'(j,J..) i s m+n+1, with K considered time-invariant. Proof Write the input r(j) as 14 N r ( j ) = E g.exp(t j ) 1=1 N 1=1 x 1 (2.35) and the output y ^ t j ) of the system from eqn. (1.1) as N . ri . m-n-N/2 y d ( j ) = y d j = z V i 3 + ^ q i Q x + . E n q - i 6 i 1=1 i = l 1=0 (2.36) where the l a s t summation does not e x i s t f o r m<n+N/2; and I f m>n+N/2, i t vanishes i n (m-n-N/2) i n s t a n t s , '6 being the Kronecker d e l t a . Let j>max(n,m). Substituting eqns. (2.35) and (2.36) i n eqn. (2.31), where D = * ( j , J j ) = CD C = 1 1 m J 1 T J T J 1 g l 1 g l 1 g 2T J 2 8N N 1 1 g 2T J 2 SN N 0 ,,0 0 0 •w2-TV. N v l -hi'1 V 2 " 1 V / 1 V r 1 n n J . - l -h,T 1 1 .-h„T 2 2 (2.37) (2.38) 15 C and D being (J^xn+N) and (n+Nxn+m+1) matrices,respectively. Write D = °1 D2 °3 D4 ; D =0; D. i s an (Nxm+1) matrix: D, i s an (nxn) matrix. (2.39) 3 1 4 33 Using Sylvester's i n e q u a l i t y , rank(C)+rank(D)-(n+N) <_ rank(¥( j , J_. ) ) <_ min (rank (C) , rank (D)) (2.40) It can be observed that rank(C) = min(J.., N+n) . (2.41) Choose J . > N+n. Hence J -rank(C) = N+n (2.42) Now, three cases a r i s e depending upon whether N i s equal to, greater than, or less than, m+1. Case 1 Let N=m+1 1 • • Now, matrices and are nonsingular and hence matrix D i s nonsingular, since |D| = \D \ |D |.. Hence from eqns. (2.40) and (2.42), n+m+1 <_ rank(f (j ,J_.)) <_ min (n+m+1, n+m+1) i . e . , ' *' r a i i k C P(j,J ) ) = n+m+1 'Case 2 Let N > m+1 An increase i n the number of rows of from (m+1) of ca s e ' l to N cannot decrease the rank of D^. Hence i t i s easy to see that rank(D) = n+m+1. Hence rank (j , )) = n+m+1 Case 3 Let N < m+1 From eqn. (2.40), rank OP (j ,J^)) <_ min (N+n, rank(D)) i . e . , rankCF (j ,J )) < n+m+1 From these three cases, i t i s seen that i f the input r ( j ) has at leas t (m+1) complex Fourier components, rank ( V ( j , J j ) ) i s m+n+1. Q.E.D. T V = E M E s s s s E = K -K , s s s K = s £ k o k i 16 2.2.1.2 A Special Case • Consider the case where some of the coefficients k. are known in 1 eqn. (1.1). The object is to extend the method of subsection 2.2.1.1 for this case. Without loss of generality, let k ,, k ......,k ,k „,k , k , pi pl+1 m m+p2 m+p2+l m+n be known (0 <_ pi <_ m, 1 <_ p2 <_ n) and the other coefficients be unknown. Then one can choose k =k. for a l l known k.. i i . i Select a PC, where s  T pl-1 km+l km+2 ''' km+p2-l] > K s = [ k o k p l - l km+l km+2 *•• km+p2-l] » and Mg is a (pl+p2-l x pl+p2-l) positive definite, symmetric matrix. Following the steps 1 through 4 of subsection 2.2.1.1, the adjustment algorithm is obtained as ' . ty,(j)-K„(j)So(j)- E k.s (j)- Z k.s (j)]M/S (j) . U i _ o . _ X X • A , X X S S AK (j) = : . T 1 = ^ : ( 2 . 1 9 s ) S s"(j)M XS (j) s s s E s ( J ) S s ( J ) -1 S'(j)M iS (j) S S s s s where S g(j) = [r(j) r(j-l) ... r(j-pl+l) -y d(j-l) ... yd(j-?2+l) ] T . The identification system,' corresponding to eqn. (2.20), will now be g i V 6 n b y E * < j ) S (j) , AE (j) = - , S M S (j) + AK (j) (2.20s) S^(j)M s 1S s(j) For the proof of convergence, the following theorem and corollary are applicable. 17 Theorem 2.1s Let the following assumptions be made: Assumption 2.1s | | K s ( j ) | | < 0 0 V J Assumption 2.2s AK^(j) -> 0 as j 0 0 Let the input r(j) be such that i f the vector K were time-invariant, then for T a l l I|E II > 0, there exists a J. < » for each j , with |E (i)S (i)| > a (j) l l g l l j - J » 1 s s 1 — s (a (j) > 0), for some i , j < i < j+J.. Then the origin of the identification s — j system' defined by eqn. (2.20s) is eventually asymptotically stable in any bounded region. Remark 2.Is Let the input be chosen such that, i f K were time-invariant, the rank of the matrix s o(j) S l ( j ) ... s p l _ 1 ( j ) s m + 1 ( j ) .... s ^ ^ C j ) 8 o(J+l) %^+JfV s ^ ^ a + y i ) is pl+p2-l for some J. < <*> and^for^all j . Then there exists some i , T i < i < i+J.,such that E (i)S (i) ^  0 i f E ^ 0. If now an assumption is J — 3 s s s T made that |E (i)S (i)| admits a positive lower bound, a(j), then this input s s r(j) will be a sufficient one to guarantee eventual asymptotic stability. Corollary 2.1s If the input r(j) is a sum of a finite number of sinusoids with at least (max(l,pl)) distinct, discrete, complex, Fourier components, and i f the poles of the system defined by eqn. (1.1) in the z-plane are distinct, then the rank of ¥ (j,J.) is pl+p2-l, with K considered time-invariant. s 3 The-proofs of the above theorem and corollary are similar to those of Theorem 2.1 and Corollary 2.2, respectively, and are not given here. 18 2.2.1.3 Application The technique was successfully applied to various examples on a digital computer. Since this method is extended in Chapter 3 for nonlinear d.e.'s and examples are included therein, no example is presented here. 2.2.2 Identification Error Method for Continuous Systems The approach presented in subsection 2.2.1.1 is now used to develop an identification method for continuous linear systems. Consider the identification of a linear system whose differential equation is given by m . n . 7 d(t) = [ E k.p ]r(t) - [ E k n r f ip 1]y d(t) i=0 1=1 or, compactly, by y d(t) = K TS(t) (2.43) i A d* where p = — r . The symbols used in this subsection have the same meaning dt 1 as in subsection 2.2.1.1 but defined for continuous time t. Let a model, represented by the differential aquation, y(t) = [ E k,(t)p i]r(t) - [ E k ( t ) p i ] y ( t ) i=0 1 i-1 nri"1' or compactly, by y(t) = K T(t)S(t) (2.44) be chosen. Following the steps outlined in subsection 2.2.1.1, let a positive definite PC be chosen; ' V I C=E TME -Choose a general adjustment algorithm for K(t), K(t) = hF 19 Hence. V r = -2hETMF assuming that the system c o e f f i c i e n t s are time-invariant. The s c a l a r h can be chosen i n many d i f f e r e n t ways to make V <^  0. A general function f or h i s XL* h = q|E (t)MF| sgn(E (t)MF), i e l , qeR, q > 0. Hence V I C = -2q|E T(t)MF| 1 < 0 To determine the actual adjustment algorithm, K(t), vector F must be so chosen that the unknown K i s eliminated from the algorithm. From eqn. (2.43), i t i s seen that a choice of MF = S(t) . , i s adequate. Thus, ' K(t) = q | y d ( t ) - K T ( t ) S ( t ) | 1 ~ 1 [ s g n ( y d ( t ) - K T ( t ) S ( t ) ) ] M _ 1 S ( t ) Since the i d e n t i f i c a t i o n error method i s extended to nonlinear con-tinuous systems i n section 3.5, no discussion of the above algorithm i s presented here. 2.3 Output Error Method 2.3.1 Output Error Method for Discrete-time Systems The d e r i v a t i o n of the method follows the general procedure of - subsection 2.2.1. The i d e n t i f i c a t i o n algorithm i s presented. 2.3.1.1 The Algorithm Step 1. Choice of performance c r i t e r i o n The PC to be chosen must have a unique minimum i n the space of K. A s u f f i c i e n t condition to ensure t h i s i s achieved i f the PC i s quadratic i n a l l the model c o e f f i c i e n t s . 20 Let eqns. (1.1) and (1.2) be written as m n y (j) = Z a r ( j - i ) - Z b y ( j - i ) • (2.45) i=0 1 i = l 1 ° m* n ' * . y ( j ) = Z a r ( j - i ) - z B . y ( j - i ) (2.46) i=0 i = l where, f o r the present, i t i s not assumed that the model and the system are o f the same order and the following developments are for the general problem, where n' and m' need not be the same as n and m. A theorem given below s p e c i f i e s the conditions f o r achieving the unique minimum. Theorem 2.2 Consider a system and i t s model described by eqns. (2.45) and (2.46). Let the following equations he v a l i d f o r future i n s t a n t s : m n Y^Cj+q) - E a r (j+q-i) - Z b y (j+q-i) (2.47) i=0 i = l 1 m' n ' y(j+q) = Z a . ( j ) r (j+q-i) - Z 3 , (j)y (j+q-i) (2.48) i=0 i = l where i t i s assumed that a(j+p) = a ( j ) ; 6(j+p) = 0 ( j ) ; 0 <_p ^ q ; p,qel (2.49) Let the PC be chosen as k V = Z e (j+q) q=0 Then there is a unique minimum of for the adjustments of the c o e f f i c i e n t s , @ a i > 0 i . 1 1 m ' ' a n d ei» k + 1 — 1 1 n ' i f k < n'» ai» 0 1 1 1 m * i f k >_ n'. o The case of 8 ^ , f o r k >_ n ' w i l l be considered subsequently (see Remark 2.3). 21 Proof Consider partial derivatives of V q with respect to a^Cj); 0 < i < m? 9V k o v 0 /. ,N 9e(j+q) k = -2 Z e(j+q)r(j+q-i) q=0 92V k 2° = 2 E r (j+q-i) (2.50) 9a i(j) q=0 which is a positive quantity, independent of a^(j). Hence V q is quadratic in ou(j). Thus there exists a unique minimum for the surface, V (a^) •• Now, let k < n'. Consider partial derivatives of V q with respect to 3V k ~ - E Q 2e(j+q)[y(j+q-i) + 6.(j) ^ f f i f f 3 , k+1 < i < n' Since, j+q-i < j for a l l 0 <_ q <_ k and for a l l k+1 <_ i <_ n', at the instant j , y(j+k-i) is a past value and so 90! (j) 0 k Hence 9V o i i 7 C j T = ^ 2e(j+q)y(j+q-i) a n d - 92V k , 2° - 2 Z y Z(j+q-i) (2.51) 96.. (j) q=0 which is a positive quantity, independent of 6^(j). Hence V q is quadratic in 3^(j), k+1 <_ i ^.n'. Thus there is a unique minimum for the adjustment of those 3.(j) also, i Remark 2.2, Since there is a unique minimum for the adjustment of the parameters 0^(0 <_ i <_ m'.) and 3.^  (k+1 <_ i <^n'), a method of direct minimization will lead to useful results. However, i t is important to note that the minimum need not be a.single point in the parameter space and i t could be a simply connected valley. 2 2 -Remark 2.3 Consider the variation of e ±(3) , 1 <_i < k, i f k <_n'; or, of 1 < i < n' i f k >_ n'. Since j+q-i >_ j for at least some 0 <_q <_k, in general, \ a y d + g - D . 0 • ' Hence, 3V k k r . , . \ - = 2 Z e(j+q)y(j+q-i) + 2 Z e(j+q)g (j) 9 7 K 2 q • q=0 q=0 1 3B 1(j) and 33 i(j) q=0 q=o i 2 The quantities ^ ^ ' t ^ " 1 ^ and 9 yli+q-1-) m a y b e expanded further using 1 ae^ Cj) . . • .  _ eqn. (2.48). , .- v 1 : : - • ' ' 3 2 V - -- • - ' .. It can be seen that — ~ is of variable sign and is not ae^ (j) independent of ^ (j). Hence, i t is not possible to conclude anything about the uniqueness of V for the variation of these 3,(j). In fact, even when o i ' n'=n and m'=m, i.e. when the minimum of V is zero, multiple minima in 6 28 have been known to occur . To avoid the problem of more than one minimum, one could choose k=0 in which case there is uniqueness of minimum for the variation of a l l the and B^. However, in general, i f the total number of and 8 is more than one, then there will be an infinite number of ways of achieving the minimum of V q at each j though a l l these minima con-stitute a simply connected valley. 23 For the present problem, a choice of k=0 i s made, and l a t e r the conditions f o r achieving the i d e n t i f i c a t i o n are discussed. Thus, V q = e 2 ( j ) , (2.53) Step 2. Choice of the incrementing algorithm Choose AK(j) = hF (2.13) Step 3. Choice of h and the determination of AV o The increment i n V q i s given by AV (j) = V (j+1) - V (j) o o o = [K TS(j+l)-(£(j)+hF) TS(j+l)] 2-[K TS(j)-K T(j)S(j)] (2.54) using eqns. (2.1), (2.2), (2.8) and (2.13). Minimizing A V Q ( j ) with respect to h, Min AV Q(j) = ,-e 2 ( j ) £ 0 (2.55) and ' , h = tK TS(1+l) - K T ( j ) S ( j ± l ) I ( 2 > 5 6 ) F S(j+1) Step 4.. Determination of AK From eqn. (2 .13) , " AK ( j) = CK TS ( 1+1) - K T ( i ) S ( i 4 - l ) ] F ( 2 > 5 7 ) F S(j+1) In the above expression, besides K, the q u a n t i t i e s , S(j+1) and S(j+1) are unknown, being q u a n t i t i e s of the future i n s t a n t . For the present, an assump-t i o n i s made regarding t h e i r values. Later, the p r a c t i c a l r e a l i z a t i o n of the assumption i s discussed. Assumption 2.3 The quantities S(j+1) and S(j+1) can be obtained exactly from t h e i r past values, and K^S(j+l) and S(j+1) are known functions of past measurements. 24 This assumption r e s u l t s i n AK(j) being s t r i c t l y a function of measurable q u a n t i t i e s , with the choice of F being free. Step 5. Proof of Convergence Once again convergence of the algorithm i s investigated without the r e s t r i c t i o n that K be time-invariant. Now Ae(j) i s given by Ae(j) = e(j+l) - e(j) = - e ( j ) + A K T ( j ) S ( j + l ) (2.58) using eqns. (2.1), (2.2), (2.8) and (2.57). I f now, the 'output tracking system' as described by eqn. (2.58) i s stable such that Ae(j) + 0 as j os, assuring s a t i s f a c t o r y output tracking. I t then remains to show that t h i s output tracking leads to i d e n t i f i c a t i o n . Now, the system as given by eqn. (2.58) i s nonautonomous and e=0 i s not an equilibrium s t a t e . Hence, as discussed i n subsection 2.2.1, the theorems on Liapunov s t a -b i l i t y are not a p p l i c a b l e , and use i s once again made of the theorems on eventual s t a b i l i t y . . Theorem 2.3 Under Assumptions 2.1, 2.2 and 2.3, the 'output tracking system' defined by the d.e. (2.58) i s eventually asymptotically stable. Proof F i r s t , consider the s t a b i l i t y of the d.e. (2.58) with AK(j) = 0 Vj Then the d.e. reduces to A e ( j ) = - e ( j ) (2.59) I t i s seen that e=0 i s an equilibrium state of the d.e. Hence, the concept o f Liapunov s t a b i l i t y i s applicable here. Choose V(e) = e 2 25 The t o t a l increment, V(e) i s given by AV(e) = - e 2 I t i s seen that V(e) i s p o s i t i v e d e f i n i t e and ^V(e) i s negative d e f i n i t e . The uniform asymptotic s t a b i l i t y of eqn. (2.59) i s e a s i l y ascertained. Now consider the 'output tracking system' of d.e. (2.58) without requiring that AK(j) be zero. Define h(j) & s u p [ A K T ( j ) S ( j + l ) ] 2 (2.60) ||S||<-Since the o r i g i n a l system (eqn. (2.1)) i s assumed to be stable, h(j)<°° f o r bounded inputs. Furthermore, as j-*- 0 0, h(j)->0, since AK(j)->0. .Hence, i t i s seen that a l l the conditions of Theorem A.2 are s a t i s f i e d and the o r i g i n of the 'output tracking system' i s eventually asymptotically s t a b l e . Remark 2.4 So f a r i n t h i s subsection, i t has not been assumed that n'=n and m'=m. Thus, i f Assumptions 2.1, 2.2 and 2.3 are s a t i s f i e d , the model output w i l l track.the system output eventually. To determine the a p p l i c a b i l i t y of the above method f o r the problem of i d e n t i f i c a t i o n , l e t n'=n and m'=m. To ascertain whether Theorem 2.3 leads to the i d e n t i f i c a t i o n of the o r i g i n a l system (eqn. 1.1), the following theorem i s a p p l i c a b l e . Theorem 2.4 I f a) AK(j)=0 Vj>some L, b) the 'output tracking system' i s such that e(j)=0 f o r a l l j>some M^(M^>L), c) the time-variation of S(j) i s slow enough to render S(j+1)=S(j), and d) the matrix ^ ( j , J ^ ) (eqn.(2.31)) has rank n+m+1 for j>M=M^ + max(n,m) and f o r some , then ||E(i)||=0 f o r a l l i>M. Proof From condition c ) , S(j+1) may be written as S(j+1) = S(j) + e A 1 ( j ) ; e, a r b i t r a r i l y . s m a l l , ° where A^(j) Is a bounded (n+m+1)-vector. Let j>M. Since, e ( j ) = 0 Vj>M^, i t i s seen that S(j)=S(j). Furthermore from eqn. (2.57), AK(j) = e — — — F - E A- (j ) , say. F [S (j )+eA(j ) ] 26 It is then easy to show that ... ., J J+i-l S T(j+i) E(j) = eS T(j+i) £ A„(k). , i e I Now using eqn. (2.31), and forming a m a t r i x , Y r ( j ) with the linearly indepen-dent rows of t(j,J.) one may'write E(j) = E f " 1 (j,J,)A„(j) where A^(j) is a vector with elements of the form S (j+i) E (k). T If now F is chosen such that F S(j)^0, then ase+ 0, EA2(J)->-0 and hence E(j)+0. A corollary of Theorem 2.4 for the case of a special class of inputs is as follows. Corollary 2.2 If a) K(j)=0 Vj>some L, b) e(j)=0 ¥j>some M^M^L), c) the time-variation of S(j) is slow enough to render S(j+l)=S(j), d) the input is the sum of a finite number of sinusoids with at least (m+1) distinct, discrete, complex Fourier components, and, e) the poles of the system of eqn. (1.1) in the z-plane are distinct, then ||E(j)|| =0 Vj > M=M^  + max(n,m). 2.3.1.2 Practical Considerations 1) Realization of Assumption 2.3 For sampled-data systems with uniform period T, the vector S(j+1) may . . 34 be represented by a power serxes expansion , 2 2 S(j+D = S ( j ) + | | T + ^ - | | 7 + . . . . (2.63) 3t z. where T = sampling period. Now, -^ 7 = (S(j) - S(j-1))/T (2.64) 2 9 — | = (S(j) - 2S(j-l) + S(j-2))/T 2 (2.65) 3t 27 and so on. ' ' Hence, S(j+1) - S(j)+(S(j)-S(j-l)) + (S(j)-2S(j-l) + S(j-2))/2 +. . . " (2.66) Thus, generation of S(j+1) is possible. Accuracy can be improved by including as many terms as necessary. However, i t is often impractical to include a large number of terms in the expansion. Below, two levels of approximations are discussed.. A l l these arguments are valid for S(j+1) also. Case (a) Zero order expansion Consider only the first term in the expansion (2.66). Then, S(j+1) = S(j) ; S(j+1) = S(j) ' (2.67) _This will^be a reasonable approximation i f a l l the signals vary slowly with respect to the sampling frequency. Applying eqn. (2.67) in eqn. (2.56) h , A ( 1 ) - K T ( i ) i ( , i ) . e i i i _ ( 2 W Case (b) First order expansion Considering up to the first derivative term in the expansion (2.66), S(j+1) = 2 S(j)-S(j-1) ; S(j+1) = 2S(j)-S(j-l) (2.69) Hence eqn. (2.56) becomes 2e(j)-y (j-1) + K T(j)S(j-l) h _ (2.70) F i(2S(j)-S(j-l)) 2) Increment step size Since the two expansions discussed above realize Assumption 2.3 only approximately, the optimum value of h, given by eqn. (2.56), is never realized. Often, the increment AK(j) has to be smaller than that given by eqn. (2.57), contingent with the requirement that the model output varies slowly. Hence i t is important to determine fir s t , the range of step-size which gives convergence of the algorithm. 28 Writing eqn. (2.57) as AK(j) • q K T S ( 1 + l ) - K T ( i ) S ( i + l ) g ( 2 . 7 1 ) where q i s a s c a l a r , and using eqn. (2.54), AV Q(j) = - e 2 ( j ) + ( l - q ) 2 ( K T S ( j + l ) - K T ( j ) S ( j + l ) ) 2 (2.72) Considering a zero order expansion and w r i t i n g S(j+1) = S(j) and S(j+1) = S ( j ) , AV o(j) = + e 2 ( j ) ( ( l - q ) 2 - l ) (2.73) Hence, the approximate range of q for eventual asymptotic s t a b i l i t y of the 'output tracking system' can e a s i l y be shown to be 0< q <2 (2. 74) For the f i r s t order expansion, AV Q(j) =.-e 2(j) + ( l - q ) 2 ( 2 e ( j ) - y d ( j - l ) + K T ( j ) S a - l ) ) 2 < 2' 7 5) I t i s not p o s s i b l e to evaluate the range of q. However i f the adjustment AK(j) i s small and i f a l l the si g n a l s vary slowly, then y^ C j-i) - y ^ ( j ) and S ( j - l ) =• S ( j ) . This-gives the same range f o r q as i n the zero order case. 3) Choice of F The choice of F i s u n r e s t r i c t e d i n the present method. Let F be chosen as F. = M - 1 S ( j ) (2.76) With t h i s choice the adjustment algorithms are given by AK(j) • s t f * ' 1 ^ (2.77) s'(j)M X S ( j ) f o r the zero order case, and (2e(j) - y . ( j - l ) + K T ( j ) S ( j - l ) ) M - 1 S ( j ) AK(j) = q — ^ z f r - ^ (2.78) 2S i(j)M X S ( j ) - S i ( j ) M ^ ( j - l ) f o r the f i r s t order case. . 29 Remark 2.5 It is of interest to compare the algorithm for the zero order case as given by eqn. (2.77), with that of the gradient method using an output 2 error PC. For a PC equal to e (j), the latter is given by AK(j) = -qV-e2(j)/2 (2.79) = qe(j)S(j) (2.80) Hence, the zero order case can be considered as a variable step-size descent technique. 2.3.1.3 Application Zero order, first order and gradient methods were investigated for the identification of the coefficients of various d.e.'s with time-invariant coefficients. Performance of these three methods with regard to the speed of convergence, sensitivity to the step size constant q, convergence from different starting values of the coefficients of the model and convergence for different inputs were studied. The input used for this study was a sum of sinusoids of randomly chosen frequencies within a specified range. The following comments are in order. In come of the examples attempted, the rate of reduction o f | J E | | was slow and was deemed unsatisfactory. For other examples, a fast convergence was obtained for eachmethod. If the input frequencies were low so that Assumption 2.3 was upheld closely for zero and first order cases, then the tracking of the output of the system by that of the model was obtained for a l l the three methods in a l l the examples attempted. Whenever a fast identification was obtained, i t was found that the zero order method was the least sensitive to the constant q and as such had the greatest region of convergence in q. The gradient method was particularly sensitive to q and was unreliable.-30 The f a s t e s t convergence was obtained for the f i r s t order case and t h i s held for widely d i f f e r e n t s t a r t i n g values. The gradient method was the l e a s t r e l i a b l e f o r changes i n s t a r t i n g values. From t h i s study, the following conclusions may be drawn: In i d e n t i f i c a t i o n problems, none of the three methods discussed i n t h i s subsection i s r e l i a b l e . An area i n which these methods may f i n d a p p l i c a t i o n i s the parameter tracking problem,'and the zero order case i s pre-f e r a b l e . This a p p l i c a t i o n i s discussed i n chapter 5. 2.3.2 Output Error Method for Continuous Systems The approach presented i n subsection 2.3.1.1 i s followed here f o r developing the output error method for continuous systems. This presentation i s very b r i e f . Consider a l i n e a r system whose d i f f e r e n t i a l equation i s given by eqn. (2.43). Let the model d i f f e r e n t i a l equation be given by m' . . n' „ y ( t ) = E k p V t ) - Z k , p X y ( t ) 1=0 i=l. or, compactly, by y ( t ) = K T ( t ) S ( t ) (2.81) Following the steps described i n subsection 2.3.1, l e t the PC be chosen as V n„ = e 2 ( t ) oc Choosing a general adjustment algorithm for K ( t ) , K(t) = hF V 2e(t)e(t) = 2 e ( t ) ( K i S ( t ) - K T ( t ) S ( t ) ) - 2 h e ( t ) S T ( t ) F (2.82) To make V < 0, s c a l a r h can be chosen as oc h = q | e ( t ) S T ( t ) F | 1 1 s g n ( e ( t ) S T ( t ) F ) , i e I (2.83) 31 where q i s given by, q l e C O S ^ O F l 1 > e ( t ) ( K T S ( t ) - K T ( t ) S ( t ) ) (2.84) I f S(t) i s not a v a i l a b l e , then, an approximation s i m i l a r to the zero order approximation of subsection 2.3.1 may be made. Thus, assume that S(t) = S(t) = 0. Then, from i n e q u a l i t y (2 . 8 4 ) , a value q > 0 w i l l s u f f i c e . However, a very large q may not be s u i t a b l e , since, i t may cause large values of S (t) thus v i o l a t i n g the assumption that S (t) = 0. Hence the choice of q may need care. Now, the parameter adjustment vector K(t) i s given by K(t) = q | e ( t ) S T ( t ) F | 1 _ 1 sgn(e(t ) S T(t ) F ) F (2.85) In t h i s expression f o r K ( t ) , the choice of F i s u n r e s t r i c t e d . One such choice w i l l be F = S(t) which gives K(t) = q ( S T ( t ) S ( t ) ) 1 _ 1 | e ( t ) I 1 " 1 sgn (e(t ) ) S(t) (2.86) S i m p l i f i e d forms w i l l be obtained f o r values of i = l and i=2. Thus K(t) = q sgn ( e ( t ) ) S ( t ) , for 1=1 (2.87) K(t) = q ; L e ( t ) S ( t ) , f o r i=2 (2.88) In the l a s t equation, the p o s i t i v e quantity, S ^ ( t ) S ( t ) , i s included i n q^. I t i s seen that t h i s expression i s s i m i l a r to the steepest descent algorithm f o r the PC of the square of the output error which has been investigated 20-22 ~ extensively by Bekey, Meissinger and McGhee 2.4 Equation E r r o r Method . . 2.4.1 Equation Error Method f o r Discrete-time Systems- - -.— . This d e r i v a t i o n of the method follows the general procedure of subsection 2.2.1 and i s described b r i e f l y . 32 2.4.1.1 The Algorithm Step 1. Choice of performance c r i t e r i o n A PC which has a unique minimum i n the space of K must be chosen. The following theorem deals with the q u a d r a t i c i t y of the PC i n a l l the model c o e f f i c i e n t s thereby ensuring uniqueness of the minimum. Theorem 2.5 Consider a system and i t s model described by eqns. (2.45) and (2.46) with m'=m and n'=n. Let eqns. (2.47) and (2.48) hold good f or future instants with the assumption of eqn. (2.49). Let the PC be given by k • V p = E £ Z(j+q) (2.89) q=0 Then there i s a unique minimum of V for the adjustments of the c o e f f i c i e n t s a., 0 < i < m and 6., 1 < i < n. l — — x — — -Proof Taking the p a r t i a l derivatives of V with respect to a ; ( j ) , E x 9VR k —-Try = - E 2e(j+q)r(j+q-i) , 0 < i < m x q=U e(3+q) = y „ ( j + q > - " * ± W W > + . ° e i ( j ) y d ( J + q - i ) (2.90) • i=0 x 92V k * - = E 2r (j+q-i) (2.91) sxnce 9 a 2 ( j ) q=0 Consider p a r t i a l d e r i v a t i v e s of V with respect to g . ( j ) . E X 3VF k = E 2 e(j+q)y.(j+q-i) 98. (J) . . V " 'd' 9 V k j = E 2y Z(j+q-i) (2.92) 3g^(-j) q=0 33 From eqns. (2.91) and (2.92), i t i s seen'that the second p a r t i a l d e r i v a t i v e s of V with respect to a (j) and $.(j) are p o s i t i v e constants, independent of i I a.(j) and $.(j) r e s p e c t i v e l y . Hence V (j) i s quadratic i n a l l the adjustable parameters and there e x i s t s a unique minimum f o r the surface V (a., 6.). E i i Remark 2.6 As discussed i n Remark 2.2, the minimum of the above theorem could be a simply connected v a l l e y instead of being a s i n g l e point i n the parameter space of a. and g. . Furthermore, the surface V i s time varying i i E and the minimum may thus be a d i f f e r e n t point or region at d i f f e r e n t i n s t a n t s . This may be inconsequential i n the present case as explained below. Unlike the i d e n t i f i c a t i o n error or the output e r r o r , the equation er r o r has no p h y s i c a l s i g n i f i c a n c e . In.the i d e n t i f i c a t i o n problem, the aim i s to minimize some norm of the i d e n t i f i c a t i o n error and- i n the modelling problem, since the i d e n t i f i c a t i o n e r r o r and the equation e r r o r are not defined, the aim w i l l be to minimize some norm of the output e r r o r . To u t i l i z e the equation e r r o r f o r i d e n t i f i c a t i o n , i t i s unimportant that the minimum of the PC i s a s i n g l e , f i x e d point f o r a l l time i n s t a n t s . I t w i l l s u f f i c e i f i t can be proved that the reduction of the PC leads to a unique minimum of a s u i t a b l e norm of the i d e n t i f i c a t i o n e r r o r . This i s shown below to be the case with a choice of ? V E = e ( j ) (2.93) Step 2. Choice of incrementing algorithm Let : AK(j) = hF (2.13) Step 3. Choice of h and the determination of V^ , ' The increment AV„ i s given by AV E(J) = vE(j+i) - V £ ( j ) = [K TS(j+l) - ( K ( j ) + h F ) T S ( j + l ) ] 2 - [ K T S ( j ) - K T ( j ) S ( j ) ] 2 (2.94) 34 using eqns. (2.1), (2.2), (2.9) and (2.13) Choosing h to minimize AV (j) £ and Min AVE(j) = -e 2(j) <_ 0 ^ (2.95) h = KTS(,i+l) - K T ( j ) S ( j + l ) ( 2 9 6 ) Step 4. Determination of AK From eqns. (2.13) and (2.96), AK(j) • K S(-1+1> - K/.i)S(,i+D F • ( 2 > 9 7 ) •T- - • n  - £ T F TS(j+l) In the above expression, the quantities K and S(j+1) are unknown. For the present, an assumption, similar to Assumption 2.3 is made regarding the value ( of S(j+1). The realization of this assumption is discussed later. Assumption 2.4 The quantity S(j+1) can be obtained exactly from its past T values and K S(j+1) is a known function of past measurements. Due to this assumption, AK(j) is strictly a known function of measur-able quantitias and the choice of F is free. Step 5. Proof of Convergence Considering the variation, AK(j), of the system coefficient vector eqn. (2.97) may be written as T AE(j) = - E <.i>S(.i+D F + A K ( j ) (2.98) F S (j+1) Observing the similarity of the above equation with eqn. (2.20), i t is possible to extend the discussion on eqn. (2.20) to eqn. (2.98). Thus, considering eqn. (2.98) as the vector-difference equation of the 'identification system', the concepts of eventual stability may be. used to investigate the stability of this system. A theorem, similar to Theorem 2.1, is stated below. 35 • Theorem 2.6 Let Assumptions 2.1, 2.2 and 2.4 be valid, and let a choice of F = M - 1S(j+l) (2.99) be made. Let the input r(j) be such that, i f the vector K were time-invariant, then for a l l ||E|| > 0, there exists a J. < °° for each j , with |ET(i)S(.i+l) I >_a(j)(a(j) > 0), for some i , j <_ i < j+J.. Then the'identifi-cation systemj defined by eqn. (2.98) is eventually asymptotically stable in any bounded region. The proof of this theorem differs very l i t t l e from that of Theorem 2.1 and is not presented here. It is important to note that the vector F as given in eqn. (2.99) is a known quantity owing to Assumption 2.4. Remark 2.1 and Corollary 2.1 of subsection 2.2.1 are applicable in the present case also, replacing f(j,Jj) by ¥(j+l,J^) and making minor changes in the proof of Corollary 2.1. 2.4.1.2 Practical Considerations 1) Realization of Assumption 2.4 The approach used for the realization of Assumption 2.3 in Practical Considerations of subsection 2.3.1 is applicable to the present case. Using the power series expansion of eqn. (2.63), an approximation of is made for the zero order expansion. Hence, from eqns. (2.97) and (2.99), S(j+1) = S(j) AK(j) = (y d(j) - K T(j)S(j))M ^ ( j ) (2.10C-: S T(j)M _ 1S(j) For the first order expansion, making S(j+1) = 2S(j) - S(j-l) the adjustment algorithm is obtained as 36 [ 2 y , ( j ) - y , ( j - D - K T ( j ) ( 2 s C i ) - s ( j - i ) ) ] M ~ 1 f 2 s ( j ) - s ( j - i ) ] AK(j) = S « : (2.10i: ( 2 S ( j ) - S ( j - l ) ) M ( 2 S ( j ) - S ( j - l ) ) Remark 2.7 Comparing eqn. (2.100) with eqn. (2.19) of the i d e n t i f i c a t i o n error method, i t i s evident that the adjustment algorithms are the same. Hence the zero order expansion approximation f o r the equation error method r e s u l t s i n the convergence properties discussed i n subsection 2.2.1 and a l l the theorems and remarks made therein are applicable i n the present case. I t i s important to note that Assumption 2.4 i s not invoked i n th i s s t a b i l i t y i n v e s t i g a t i o n . 2) Increment step s i z e Since the zero order and the f i r s t order expansions r e a l i z e Assumption 2.4 only approximately, the optimum value of h i s not r e a l i z e d and hence AK(j) may have to be smaller than i t s values given by eqns. (2.100) and (2.101). However, f o r the zero order expansion, as discussed i n Remark 2.7, the s t a b i l i t y can be proved without u t i l i z i n g Assumption 2.4 and hence, a reduction i n step s i z e i s unnecessary. For the f i r s t order expansion, w r i t i n g eqn. (2.97) as AK(j) = q K TSC 1 +1) - F C j + l ) S C j t l l F ( 2 > 1 0 2 F S (j+1) with S(j+1) = 2S(j) - S ( j - l ) and using eqn. (2.99), i t i s seen that AV E = - e 2 ( j ) + (l-q) 2[2£(j)-y d(j-l)+K T(j)S(j-l)] 2 (2-103 From t h i s , i t i s d i f f i c u l t to determine the range of the sca l a r q such that AV < 0. If, however, the adjustment AK(j) i s small, and i f a l l the signals E — vary slowly, one may l e t y d ( j - l ) ' - y d ( j ) and S ( j - l ) = S( j ) . : The range of q i s then found to be \ ' 0 < q < 2 (2.104 37 Remark 2.8 It i s of interest to compare the zero order expansion case with the steepest descent technique using an equation error PC. For the PC of 2 e ( j ) , the latter gives, AK(j) - - | V-e 2(j) - q e(j)S(j) (2.105) Comparing this algorithm with that of eqn. (2.100) i t is seen that the algorithm of the zero order expansion may be considered as a variable step size descent technique. The algorithm given in eqn. (2.105) i s the same as the one obtained 28 35 by Mantey and Belanger 2.4.1.3 Application From Remark 2.7, i t i s seen that unlike the f i r s t order expansion, the zero order expansion can be shown to result in the eventual asymptotic s t a b i l i t y of the 'identification system' without using Assumption 2.4. Since this assumption i s not usually satisfied by the f i r s t order expansion, no con-clusions may be drawn about the s t a b i l i t y . Thus the zero order method is intutively appealing. It is possible, however, that the f i r s t order method is equally powerful. For the application of the zero order expansion, subsection 2.2.1 may be referred to, since the algorithm of the zero order expansion (eqn. (2.100))is the same as that of the identification error method (eqn. (2.19)). 2.4.2 Equation Error Method for Continuous Systems A brief presentation i s given here on the development of the equation error method for systems described by the d.e. (2.43). Eet the model d.e. be given by eqn. (2.44). The PC is chosen to be V w„ - e 2(t). (2.106) 38 Choosing a general adjustment algorithm, K(t) = hF (2.107) V E C = 2 e(t)e(t) = 2 e(t)(E TS(t) - hF TS(t)) (2.108) For making V <_ 0, the scalar h may be chosen as h = q| e(t)F TS!(t)| 1" 1 sgn(e(t)F TS(t)) , i E I (2.109) where q is given by q| & ( t ) F ^ S ( t ) I 1 > e(t)E TS(t) (2.110) If the vector S ( t ) is not available, an approximation similar to that of the zero order expansion may be made. Assuming slow variation of S(t) with time, S(t) may be considered negligible, resulting in any value of q > 0 to be adequate. The parameter adjustment vector is given by K(t) = q | e ( t ) F T S ( t ) | i " 1 sgn( e(t)F TS(t))F (2.111) The choice of F is free in the above expression. Making a choice of F = S(t), eqn. (2.111) i s modified tt)> K(t) = q 1 | e ( t ) | i _ 1 sgn (e(t))S(t) (2.112) where q x * q ( S T ( t ) S ( t ) ) 1 ' 1 This expression for the adjustment algorithm i s essentially the same as that obtained in subsection 2.2.2 for the identification error method when M=I. Hence the discussion on the algorithm given there i s applicable in the present case also. 2.5 Conclusions An error functioh minimization approach to the identification of discrete-time systems has been considered in this chapter. Three categories of identification methods have been distinguished and treated: 1) identification -39 error method, 2) output error method, and 3) equation error method. Questions that a r i s e using these methods such as t h e i r s t a b i l i t y properties and the s i m p l i c i t y of t h e i r implementation, have also been treated. From the t h e o r e t i c a l i n v e s t i g a t i o n s of these three methods and from t h e i r a p p l i c a t i o n s , i t i s seen that f o r the i d e n t i f i c a t i o n problem considered, the i d e n t i f i c a t i o n e r r o r method is superior to the others. A novel feature of t h i s chapter i s that the i d e n t i f i c a t i o n procedure presented encompasses a class of on-line sequential methods, for which the algorithms i have been derived following a common procedure. Furthermore, new algorithms have been obtained for the output er r o r method. Another feature of t h i s chapter is the novel a p p l i c a t i o n of eventual s t a b i l i t y concepts to e s t a b l i s h convergence of the i d e n t i f i c a t i o n algorithms. 40 3. . IDENTIFICATION OF THE COEFFICIENTS OF NONLINEAR DIFFERENCE EQUATIONS 3.1 Introduction The purpose of t h i s chapter i s to extend the i d e n t i f i c a t i o n methods of chapter 2 to the problem of the i d e n t i f i c a t i o n of the c o e f f i c i e n t s of nonlinear d i f f e r e n c e equations. Attention i s devoted l a r g e l y to the i d e n t i f i c a t i o n e r r o r method with b r i e f references to the output error and equation error methods. I d e n t i f i c a t i o n techniques f o r d.e. have been investigated by Richalet 26 3 19 24 —and Gimonet and applied to l i n e a r systems. Others ' ' have studied the i d e n t i f i c a t i o n of nonlinear d i f f e r e n t i a l equations, using equation e r r o r methods. I t i s therefore desirable to apply the i d e n t i f i c a t i o n error method to nonlinear d.e. i n view of i t s s t a b i l i t y and s i m p l i c i t y . The proposed algorithms are also extended to the i d e n t i f i c a t i o n of nonlinear d i f f e r e n t i a l equations. To t e s t the u t i l i t y and l i m i t a t i o n s of the proposed method, i d e n t i f i -c a tion of various types of nonlinear d i f f e r e n c e equations has been considered. Included are systems with multivalued n o n l i n e a r i t i e s and time-varying c o e f f i -c i e n t s . 3.2 T h e o r e t i c a l Development of the I d e n t i f i c a t i o n Technique 3.2.1 Problem Formulation Consider a n o i s e - f r e e , discrete-time dynamic, system whose d.e. i s given by f ( K , S ( j ) , y d ( j ) ) = 0 (3.1) Let $ = $(j) = n-dimensional vector of known functions of measurable qua n t i t i e s of the system [0X 0 2 0 N 1T 0 ± = 0.(j) = 0 - ( r ( j ) , r ( j - l ) , . . . . , y d ( j ) , y d ( j - l ) , . . . ) 41 Consider the case where the unknown quantities, k^, enter as linear coefficients in the d.e. and let eqn. (3.1) be written as N E k.0 (j) =0 . (3.2) i=l 1 1 or, as K T$(j) = 0 (3.3) Choose a model whose d.e. is given by K T(j) $(j) = 0 (3.4) where T 4 = = [Q± $2 0 N] (3.5) 0 ± = 0.(r(j), r ( j - l ) , , y(j), y(j-l),....) (3.6) Adjust the k^ noniteratively at each instant such that the identification of the coefficients, k., is achieved. i As in the linear case, the system coefficients are considered time-invariant in the development of the algorithms. This restriction is relaxed later in the investigation of their stability. 3.2.2 Identification Error Method The approach described here essentially follows the same steps out-lined in subsection 2.2.1. Step 1. Choice of Performance Criterion Let a quadratic performance criterion, V^, be chosen, such that = ETME (3.7) where M is any (NxN) positive definite matrix, and E= [ ( k ^ ) (k 2-k 2) .... ( l c ^ - ^ ) ] 1 = K - K (3.8) Step 2. Choice of Incrementing Algorithm Choose a general algorithm AK(j) = hF (3.9) 42 Step 3. Choice of h and Determination of AV^. The increment on V can be determined to be I . AVjCj) =• VjCj+1) - Vj.(j) Choosing h to minimize AV^ , = -2hET(j)MF + h2FTMF (3.10) h = 4 p ^ (3.11) F MF Step 4. Determination of AK From eqns. (3.9), (3.11) and (3.8) £W . ssdw&i * (3.12) F MF Choose F such that AK(j) is a known function of measurable quantities. In the above expression for AK(j), the vector K is unknown. From eqn. (3.3), i t is seen that K commutes with $ yielding the scalar, 0. Thus, to' eliminate K in eqn. (3.12), a choice of F = M - 1$(j) (3.13) is made. Hence the expression for AK(j) is reduced to - AK(j) = I ^ W * ^ lfh(j) (3.14) • (j)M ^ ( j ) From eqns. (3.10), (3.11) and (3.13), the increment AV^. is found to be W - ( J ) • - - g T < 1 > * < 1 » 2 < 0 ( 3 . 1 5 ) 1 • ( j » f :(D Step 5. Proof of Convergence Now, let the system coefficient vector K vary with time. Eqn. (3.14) can be written as T (3.16) AE(j) = - y i W ^ MS(j) + AK(j) (j)M «(j) 43 * This equation may be interpreted as the vector-difference equation of the 'identification system'. The st a b i l i t y of this system is to be investigated. An explanation similar to the one following eqn. (2.20) in subsection 2.2.1 may be advanced here for employing the eventual s t a b i l i t y concepts, since eqn. (3.16) is nonautonomous and E=0 is not an equilibrium state. However, i t i s not possible to investigate the eventual s t a b i l i t y of the 'identification system' on the same lines of subsection 2.2.1 without making additional assump-tions. For, defining ^ ( j , J ) as 01<J> 0 1 ( j + D 0 2 ( j ) 0N(J) 0 2 ( j + l ) . . . . . . 0 N ( j + D (3.17) 0^+^-1) 0 2(j+JL-l) ... 0 N(j+J j-l) i t i s seen from eqn. (3.3) that the rank of ¥(j,J..) cannot be N for | |K| | ± 0. Hence observations, similar to the ones made in Remark 2.1, are not possible here. To obviate this d i f f i c u l t y , l e t an assumption be made. Assumption 3.1 At least one of the system coefficients i s known. Without loss of generality, assume that k^+^, kp+2»**«>k^ are known, 1 <_ p < N, pel. Choose k^=k^, 1 <_ i <_ p', i e l . Redefining (3.18) E = [(k.-k.) (k.-k.) ..... (k -k ) ] T 1 1 Z c P P and $ = [0 X 0 2 V (3.19) and following the general case, the adjustment algorithm i s given by eqn. (3.14) and the d.e. of the 'identification system', by eqn. (3.16), M being (pxp) matrix. The d.e. of the 'identification system' may also be expressed as • N „ Z k.(j)0.(j) AE(j) = — M - 1$(j) +• AK(j) $ T ( j ) M - 1 * ( j ) (3.20) 44 For the s t a b i l i t y of the above system, the following theorem i s app l i c a b l e . Theorem 3.1 Let Assumptions 2.1, 2.2 and 3.1 be v a l i d . Let the input be such that, i f K were time-invariant, then f o r a l l | |E| | > 0, there e x i s t s a J. < °° for T each j , with [E ( i ) $ ( i ) | >_ct(j) (a(j) > 0), for some i , j <_ i <_ j+J_. . Then the o r i g i n of the ' i d e n t i f i c a t i o n system', defined by eqn. (3.16), i s eventually asymptotically stable i n any bounded region. The proof of t h i s theorem i s s i m i l a r to that of Theorem 2.1 and i s therefore omitted. Remark 3.1 Consider the matrix 01(j> 0 1(j+l)' 02(j) 0p(j) 02(j+D 0p(j+D Q^j+J.-l) 02(3+^-1) 0 p ( j + J j - l ) (3.21) Let the input r ( j ) be chosen such that, i f K were time-invariant, the rank of ¥(j,J.) i n eqn. (3.21) i s p. Then i t can be shown that i f T | | E | | > 0, there e x i s t s some i , j <_ i < j+J.,such that E ( i ) $ ( i ) ^0. I f now an assumption i s made that |E ( i ) ' $ ( i ) | admits a p o s i t i v e lower bound a ( j ) , then t h i s input w i l l be a s u f f i c i e n t one to guarantee eventual asymptotic s t a b i l i t y of the o r i g i n of the ' i d e n t i f i c a t i o n system'. 3.2.3 Output Error Method I t i s p o s s i b l e to extend the output error method of subsection 2.3.1 to cover nonlinear d.e. Assume a d.e. -45 P T y , ( j ) = Z k.0.(j) = K $(j ) (3.22) i = l 1 f o r the system where $ ( j ) i s redefined excluding y^ C j ) from arguments jand a d.e. p i y d ( j ) = Z ^(3)0. (j) = K T ( j ) $ ( j ) (3.23) i = l for the model. Choosing a PC of V o = e 2 ( j ) (3.24) and following the procedure of subsection 2.3.1, the adjustment algorithm i s obtained as A K ( j ) = ( K T < i + l ) r K T ( i ) ^ ( i + l ) ) F ( 3 < 2 5 ) F *(j+l) , The 'output tracking system' i s given by the d.e. Ae(j) = -e ( j ) + AK T(j) $(j+l) (3.26) For the s t a b i l i t y of t h i s system an assumption concerning the measurability of $(j+l) and $(j+l) i s made. Assumption 3.2 The vectors $(j+l) and $(j+l) can be obtained from t h e i r T past values and K $(j+l) i s a known function of past measurements. Theorem 3.2 Under Assumptions2.1, 2.2 and 3.2, the 'output tracking system' defined by the d.e. (3.26) i s eventually asymptotically stable. Theorem 3.3 Let p'=p i n eqn. (3.23) and l e t n be the order of the d.e. (3.23). I f a) AK(j) = 0 Vj > some L, b) the 'output tracking system' i s such that e(j)=0 f o r a l l j >.. some M ^ ^ i > L) , c) i f the time-variation of $(j) i s slow enough to render *(j+l) = $ ( j ) , and ':d) the'matrix ?(j»Jj) has rank p for j > M=M^ +n and for some J\., then | |E(L)| | = 0 f o r a l l i > M. Remark 3.2 With minor modifications, the zero order expansion and the f i r s t 46 order expansion, described i n subsection 2.3.1 are applicable to the present case, for representing $(j+l) and $(j+l),' and a choice of F can be made. 3.2.4 Equation Error Method -An extension of the equation e r r o r method of subsection 2.4.1 i s made here. The uniqueness of the minimum of the PC i n the parameter c o e f f i c i e n t •space i s ascertained f i r s t . Theorem 3.4 Consider a system and i t s model described by eqns. (3.22) and (3.23) with p'=p. Let the following equations be v a l i d f o r future i n s t a n t s : P y.(j+q) = £ k.0.(j+q) (3.27) i = l 1 1 y(j+q) = Z k (j)0 (j+q) (3.28) i = l 1 1 where i t i s assumed that k ±(j+Jl) = k ± ( j ) , 0 _ < £ ^ q , £, qel Let the PC be given by k ' V = Z e 2(j+q) (3.29) E q=0 Then there i s a unique minimum of for the adjustments of the c o e f f i c i e n t s k.. l The proof of t h i s theorem i s s i m i l a r to that of Theorem 2.5 and i s not given here. Following the development of the algorithm i n subsection 2.4.1, a choice of • V E = e 2 ( j ) (3.30) i s made. The adjustment algorithm i s obtained as ; A K ( j ) = * V l + l ) - K T(jH(,t+D F ( 3 . 3 1 ) F $(j+l) 47 The 'identification system* i s given by the d.e. T AE(j) = -i E ( J > ^ J + 1 ) F + A K ( j ) (3.32) F 4 (j+1) For the sta b i l i t y of this system, the following assumption and theorem are stated. Assumption 3.3 The vector 4(j+l) can be obtained from i t s past values T and K 4(j+l) i s a known funbtion of past measurements. Theorem 3.5 Let Assumptions 2.1, 2.2 and 3.3 hold good and let the choice of t = M - 1$(j+l) (3.33) be made. Let the input r ( j ^ be such that i f the vector K were time-invariant, then for a l l ||E|| > 0, thefrre exists a < <=° for each j , with |E (i)$(i+l)| >_ a ( j ) (a(j) > 0) for some i , j <_"JL <_ j+J^. Then the 'identification system' defined by eqn. (3.32) is eventually asymptotically stable. Remark 3.3 The zero order expansion and the f i r s t order expansion are applicable to the present c£se, for representing 4(j+l). Remark 3.4 The zero ord£r expansion for the realization of 4(j+l) results in the same adjustment algorithm AK(j) as the one developed in subsection 3.2.2 for the identification errot method. Hence i t results in the same s t a b i l i t y properties of the identification error method, without Invoking Assumption 3.3. Since this assumption i s not usually satisfied by the f i r s t order approxima-tion, no conclusions may be drawn about s t a b i l i t y . Thus the zero order method is intuitively appealing. 3.2.5 Conclusions The three philosophies of identification discussed in chapter 2 have been extended to the identification of nonlinear d.e. From an investigation 48 of these methods, i t is seen that the identification error method is superior to the.other methods for the identification problem considered. The rest of the chapter is devoted to the application of the identi-fication error method in various types of practical situations. 3.3 Application of the Identification Error Method In this section, examples are presented for the application of the identification error method of subsection 3.2.2. Unless stated otherwise, the following comments are applicable in a l l the examples. The coefficients of the system d.e. are assumed to be time-invariant. Otherwise, no other knowledge of the system coefficients is assumed. Hence, the i n i t i a l values of a l l the coefficients k^(0) are arbi t r a r i l y chosen to be zero. For the same reason, the matrix M in eqn. (3.7) is selected to be the identity matrix. Hence, the PC reduces to P . 2 V =E (k.-k . r . (3.34) i=l 1 1 The forcing function r ( j ) is represented by a sum of sinusoids whose frequencies are chosen randomly within some range. In each example, a variety of inputs of different frequency ranges is considered to substantiate the applicability of the method for a wide range of input conditions. The adjustment of the model coefficients i s started after allowing time for the transients to die down, and the time on the respective plots i s counted from this instant. The examples are simulations carried out on a d i g i t a l computer. 3.3.1 Example 3.1 ' 36 This example is taken from Steiglitz and McBride in which the authors use an iterative method for identification. The system is given by y d ( j ) + k 1 r ( j ) + k 2 y d ( j - l ) + k 3 r ( j ) y d ( j - l ) = 0. (3.35) Let k^=-0.5, k2=-0.9, and k^=-1.0. The model d.e. i s given by y(j) + k^Cj) + k 2 y ( j - l ) + k 3 r ( j ) y ( j - l ) = 0 F i g . 3.1 shows the convergence of the model c o e f f i c i e n t s to t h e i r correct values. (The l i n e s j o i n i n g the points do not have any p h y s i c a l s i g n i f i c a n c e and are drawn only f o r the sake of c l a r i t y ) . The convergence of the model c o e f f i c i e n t s i s rapid and the PC reduces from 2.06 at the s t a r t to 7.0 x 10 5 at the 1 00 t h i n s t a n t . Since the same c h a r a c t e r i s t i c reduction of PC has been observed i n most of the examples solved, the p l o t of PC i s not shown i n the following examples. Any exceptional behaviour of the PC i s stated e x p l i c i t l y wherever applicable i n the text. 3.3.2 Polynomial Form of Nonlinearity Consider a function f^C^^) which, over a f i n i t e i n t e r v a l i n 0_^ , i s defined, everywhere single-valued, f i n i t e , s e c t i o n a l l y continuous and has bounded v a r i a t i o n s . Then t h i s function can be represented by a polynomial i n 0^  within a given nonzero error which can be made a r b i t r a r i l y small by 37 taking a s u f f i c i e n t l y large number of terms . Thus f (0 ) can be written LILI i as fML< 0i> = . \ V i ' ( 3 ' 3 6 ) Consider that the nonlinearity. i n the system d.e. can be represented without er r o r by a polynomial. The general case when t h i s i s not possible i s discussed i n subsection 3.4.2. I f the polynomial of the system n o n l i n e a r i t y has more terms than the polynomial of the model n o n l i n e a r i t y , then convergence problems of the type discussed i n subsection 3.4.2 w i l l r e s u l t . On the other hand, i t can be e a s i l y shown that the i n c l u s i o n of more terms i n the model polynomial than i n the system polynomial w i l l not present any d i f f i c u l t y since the extra terms of the model w i l l tend to zero f o r large time. 51 For a general case of n polynomial type nonlinearities, the system d.e. may be written as n, , n i L Z Z k 03 + Z k 0 = 0 ' (3.37) 1=1 j=0 1 J 1 i=n+l 1 1 which includes L-n other terms. The above equation can be transformed into the form of enq. (3.2) by reassigning the k!s and the 0^s and the identifi-cation procedure subsection 3.2.2 is applicable. Example 3.2 38 The following example is taken from Narendra and Gallman who consider an iterative approach for identification. The system.and the model are shown in fig. 3.2. r - k 5 ( - ) 2 f ( r ) fr. 1 +-0.7Z-"1 - 1.5z~2 y d -1 -2 -3 1 + k±z A + k 2z z + k 3z System • -r • k 4(.)-k 5(-) - k 6 ( 0 J ; A f ( r ) . 1 + 0.7z 1 - 1.5z~2 " - I * _? - 3 1 + k x z + k 2 z + k z Model Fig. 3.2 Example 3.2 52 Three terms are considered i n the model n o n l i n e a r i t y as compared to only the square term i n the system n o n l i n e a r i t y . The system and the model d.e. can be written as y d ( j ) + k 1 y d ( j - l ) + k 2 y d ( j - 2 ) + k 3 y d ( j - 3 ) + k 5 ( r 2 ( j ) + ^ = 0. (3.38) y( j ) + k i y ( j - l ) + k 2 y ( j - 2 ) + k 3 y ( j - 3 ) + k 4 ( r ( j ) + 0 . 7 r ( j - 1 ) - 1 . 5 r ( j - 2 ) ) + k 5 ( r 2 ( j ) + 0 . 7 r 2 ( j - l ) - 1 . 5 r 2 ( j - 2 ) ) + k g ( r 3 ( j ) + 0 . 7 r 3 ( j - l ) - 1 . 5 r 3 ( j - 2 ) ) = 0. (3.39) The c o e f f i c i e n t s k^,k 2 and k^ are chosen to be time-invariant constants with values of 0.9, 0.15 and 0.002,respectively. The c o e f f i c i e n t k,. i s assumed to have bounded but random v a r i a t i o n within +5 units up to an a r b i t r a r i l y chosen time of 50 instants and then i s assumed to be a time-invariant constant with a value of -1.5. F i g . 3.3 demonstrates the eventual s t a b i l i t y of the ' i d e n t i f i c a t i o n system'. For the sake of c l a r i t y , the v a r i a t i o n of the c o e f f i c i e n t s , k^, k 2, k^, k^ and k^ p r i o r to 60 i n s t a n t i s not shown. The PC reduces from 17.5 (at the s t a r t ) to 3.29 x 10~ 4 (at 2 0 0 t h instant) and to 8.1 x 10~ 7 (at 3 0 0 t h i n s t a n t ) . 3.3.3 Piecewise Linear Form of Nonl i n e a r i t y Let the n o n l i n e a r i t y be a piecewise l i n e a r function as shown i n 39 f i g . 3.4. I t can then be represented by a s e r i e s of the form q f . n (0 ) = E k . y .(0 -0 .) (3.40) NL n , n i K n i n n i i=-p where f o r k > 0, u n k A = 1 , 0 n > 0 n k > 0 = o , 0 n < 0 n k > 0, f o r k=0 u = 1 , 0 = 0 , and no no f o r k < 0 „ n k *= 1 , » n < 0 n k < 0 = ° • 0 n i « „ k < 0 53 Time Instants Fig. 3.3 Identification of the Coefficients in Example 3.2 rri(-2) y n(-l) slope k +k , no n l slope k +k ,+k . no n l n2 slope k slope k +k , ,. no n ( - l ) •nl *n2 F i g . 3.4 Piecewise Linear Form of Nonlinearity For the general case of P n o n l i n e a r i t i e s , each represented piecewise l i n e a r l y , the system d.e. i s given by P H i L E E k .u .(0 -0 .) + E k ,0 . = 0 „-i • n i / m n n i . i i n=l i=-p^ i=P+l (3.41) where the l a s t summation comprises other terms i n the d.e. For the s i m p l i c i t y of notation only one piecewise l i n e a r element i s considered i n the development below. Extension to more than one component i s straightforward. Let the d.e. be given by k o 0 o + «l\ + *5 4N = ° where k i s a known c o e f f i c i e n t , o L •K1.*- = E k .0 . L L , i 1 1 1=1 (3.42) (3.43) 55 arid E k .u ,(0 -0 ,) ni ni n ni i=-p (3.44) In conformity with the problem formulated in subsection 3.2.1, let the linear coefficients, k. and k , be the unknown quantities to be identified i ni and let the functions, 0 , $T and $ be measurable. This necessarily assumes o L N that 0 ., the break points of the piecewise linear element, are known. Thus, ni . for the piecewise linear element, the problem reduces to that of identifying the slopes, given the break points. The identification procedure described in subsection 3.2.2 may now be applied to the system given in eqn. (3.42) and and may be iden-tified. Define (3.45) E I [E* E*]* (3.46) and choose M to be a diagonal matrix given by \ 0 M = 0 ^ M "P M -p+1 *M -3J Then the adjustment algorithm for Ak^ may be written as A k rii -(k 0 +KT$) y (0T,"0„,) o o ni n ni T -1 $ M $ (3.47) (3.48) (3.49) Thus, for i > 0, i f 0 . < 0 < 0 a l l the coefficients, k ., i > j > 0, ni n — n(i+l) nj — J — will adjust. The condition on the rank of ¥(j,Jv) of eqn. (3.21) for eventual asymptotic stability is equivalent to the requirement that, over a finite . • • • 56 time i n t e r v a l , 0.. , 0„, 0T, 0 are l i n e a r l y independent and p(0 ) ^0, f o r x /. 1J no xi a l l 0 , . N-A < 0 < 0 +A ; A ,A > 0, where p(*) i s the p r o b a b i l i t y density n(-p) p — • n — nq q p q function. Let the above method be c a l l e d the 'simultaneous adjustment scheme'. When the above method i s applied to simulated problems, an undesirable i n t e r -action of the c o e f f i c i e n t s k . i s observed.. Referring to eqn. (3.44),. suppose n i that k f o r a l l 0 <_ j <_ i-1, have already been adjusted to t h e i r correct values. Now i f 0 > 0 ., a l l the above c o e f f i c i e n t s w i l l adjust and move away n n i J from t h e i r desired l e v e l s . This coupling of c o e f f i c i e n t s r e s u l t s i n slow con-3 vergence. To obviate a s i m i l a r problem i n continuous systems, Butler proposed an ' i n t e r v a l adjustment scheme' i n which only one of the c o e f f i c i e n t s k ^ i s adjusted at any i n s t a n t . A b r i e f discussion of t h i s method-for d i s c r e t e time systems i s given here followed by a q u a l i t a t i v e proof f o r convergence by induction. Let the system d.e. be given by eqn. (3.42). Define A = [X £ X (0 -0 , J..X 0 ...-X (0 -0 ) ] T (3.50) o L -p F n nC-p) o n q n nq where for k > 0 X. = 1 , 0 , < 0 < 0 , . k nk n — n(k+l) = 0 , otherwise f o r k = 0 X = 1 , 0 ,,v < 0 < 0 , o n ( - l ) — n — n l = 0 , otherwise, and f o r k < 0 X, = 1 , 0 ,v < 0 < 0 , k n ( k - l ) — n nk = 0 , otherwise. Let the following algorithm be chosen: (3.51) AK = . H'1 A- ^ (3.52) A M A Again, l e t t i n g M to be a diagonal matrix, Ak i s given by Ak . = 1 n n i (3.53) n l AM* A M. I t i s seen from t h i s equation that when 0 . < 0 <0 i > 0 , the c o e f f i c i e n t nx n — n(i+l) k . alone w i l l adjust, n i J Let the functions0 have the property that they are bounded and that n p(0 ) 4 0 for a l l 0 . s-A < 0 < 0 +A ; A ,A > 0. ^ ' V n(-p) p - n - n q q ' p ' q Consider a p o s i t i v e d e f i n i t e function V = ETME (3.54) From eqns. (3.44), (3.50) and (3.52), 1 q T AV = - 1 [ Z E u (0 -0 ) + E * ].' . l . — X. . n i n i n n i L L AM A i=-p [ j _ p E n i ( 2 X i ^ n i ) ( V 0 n i ) + E L $ L ( 2 X o - I ) ] < 3" 5 5> where E n ± = k n i - ^ n i - Defining V L and V ±(-p <_ i <_ q) as V^=E^M^E^ and V.=E2.M., i t i s seen that x n i x q V = VT + Z V. L . i i=-p Whenever 0n^_1j 1 ^ n 1 0nI» A V i s given by AV = AV + AV L o 1 [E 0 + E ^ $ T ] 2 < 0 (3.56) T -1 L no n L L J -A M A A A. Since k and K, ad-just only when 0 , , x < 0 < 0 ,, i t i s seen that V -> 0 no L J n ( - l ) — n — n l L and — u as 1 *>• °° • Hence E T and E -*• 0. o. J L no Now consider a l l the instants when 0 , < 0 < 0 n l n — nZ AV = AV 1 [E (0 -0 J + e J t E f0 -0 , ) - e 1 T -1, 1 n l n n l ' ^ l ' 1 n l n n l 1 A M A 58 where AM A e = E 0 + E $ 1 no n L L From eqn. (3.57), i t i s seen that V w i l l decrease whenever 1 n l 1 Eventually, when -*• 0 as j -> °°, i t i s seen that and E ^ -> 0. the instants when 0 , < 0 < 0 ,, ,, ^  nk n — n(k+l) AV = AV, (3.57) (3.58) (3.59) Now assume that the errors IE .1 -> 0, 0 < i < k-1 as i + ». Consider where / -I [E 2. (0 -0 , ) 2 - e 2 ] A M * A nk n nk k T e = S E . (0 -0 .) + E T a k . _ nx n nx L L 1=0 It i s evident from eqn. (3.60) that V^ w i l l decrease i f e. l E n k l < 0-0 n nk (3.60) (3.61) and since |E n^| 0, 0 <_ i <_ k-1, as j. -> «>, i t i s seen from eqn. (3.61) that -*• 0. This implies that V^ 0, eventually as i s evident from eqn. (3.60). A s i m i l a r i n d u c t i v e approach holds good f o r the convergence of the slopes of the piecewise l i n e a r element f o r negative values of 0^ . From the examples solved using the simultaneous adjustment scheme, i t has been observed that the convergence of the model c o e f f i c i e n t s to t h e i r correct values i s always good. In general, the i n t e r v a l adjustment scheme has a higher rate of convergence than the simultaneous adjustment scheme, though the l a t t e r i s found to be less s e n s i t i v e to the nature of the input than the former. 59 A typical example is presented below. Example 3.3 Consider the system given in fig. 3.5. -It is -0.5 0 0.5 -1 -2 -3 l+k1z +k2z +k3z Sys tem Model rd -0.5 1 ~ -1 ~ -2 " -3 l+k1z +k2z +k3z 0 0. 5 ' ~ V k6 Fig. 3.5 Example 3..3 required to identify the coefficients k^ of the linear part of the system and the slopes of the piecewise linear element whose break points are given. The system d.e. is given by y d ( j ) + k ^ C j - l ) + k 2y d(j-2) + k 3y d ( j-3) + k ^ 4 + k ^ + k ^ = 0 (3.62) where, 0 4 = r(j ) ; 0 5 = y r 5<r(j) - 0.5), \i^5 = 1 for r(j) > 0.5 and u r 5 = 0 otherwise; 0. = u ,(r ( j ) + 0.5), u , = 1 for r(j) < -0.5 and u , = 0 otherwise, o ro ro ro The values of k^,k2 and k 3 are chosen to be 0.9, 0.15 and 0.002 respectively. The nonlinearity is of the saturation type with values of k^ = -1, kj. = 1 and k^  = 1. The model d.e. is given by y(j) + ^ y C j - l ) + k 2y(j-2) + k 3y(j-3) + k ^ 4 + k ^ 5 + k ^ = 0 (3.63) Fig. 3.6 shows the convergence of the coefficients of the linear part of the system for both the schemes. Fig. 3.7 shows the identification of the is nonlinear part of the system. It is seen that the convergence i s f a i r l y rapid for both.the methods and that the interval adjustment scheme has a faster con-vergence than the simultaneous adjustment scheme. The values of PC are given below with the corresponding time instants in parantheses: Simultaneous adjustment scheme: 3.83(0), 5.79 x 10 - 1(80), 1.10 x 10 - 1(160). -1 -4 Interval adjustment scheme: 3.83(0), 1.93 x 10 (80), 9.88 x 10 (160). 3 Butler discusses some special cases and the problems encountered in the application of the above methods with reference to the identification of piecewise linear elements in continuous systems by the equation error method. Many of his results, with suitable modifications, are applicable to the discrete-time systems also. It i s of interest to note that there are cases where the interval ' J adjustment scheme i s inferior to the simultaneous adjustment scheme. The following two examples demonstrate this condition. Example 3.4 Consider the system given in f i g . 3.8 which is identical to that of Example 3.3 but for the dynamics in the numerator of the linear part of the system. As i n the previous example, i t i s required to identify the and the three slopes of the nonlinearity. The system d.e. is given by y d ( j ) + k ^ j - 1 ) + k 2y d(j-2) + + + k ^ + k ^ = 0 (3.64) where 0 4 = r(j)• + 0.7r(j-l) - 1.5r(j-2) 0 5 = (y 5 a ( r ( j ) - 0 . 5 ) + 0.7 ^'(r(j-l)-0.5) - 1.5y 5 c(r(j-2)-0.5)} 0 6 = {u 6 a(r(j)+0.5)+0.7y 6 b(r(j-l)+0.5> - 1.5y 6 c(r(j-2)+0.5)} F i g . 3.6 I d e n t i f i c a t i o n o f t h e L i n e a r P a r t o f t h e S y s t e m i n E x a m p l e 3.3 62 f ( r ) , f ( r ) S i m u l t a n e o u s A d j u s t m e n t Scheme M o d e l N o n l i n e a r i t y S y s t e m N o n l i n e a r i t y • • • • A t 40 i n s t a n t A t 8 0 t h i n s t a n t A t 1 2 0 t h i n s t a n t A t 1 6 0 t h i n s t a n t F i g . 3.7 I d e n t i f i c a t i o t i o f t h e N o n l i n e a r i t y i n E x a m p l e 3.3 63 -0 5 0 0.5 f(rV System -0.5 f'(r), 0 0.5 -1 -2 -3 1+k^ z +k2z +k3z J l+0.7z"1-1.5z"2 ~ -1 ' -2 * -3 l+k^z +k2z +k3z Model Fig. 3.8 Example 3.4 j 5 a = 1 for r(j) > 0.5; y 5 f a = 1 for r(j-l) > 0.5, y 5 c = 1 for r(j-2) > 0.5 =0, otherwise = 0, otherwise = 0, otherwise 1 for r(j) < -0.5; u,, = 1 for r(j-l)< -0.5; u, = 1 for r(j-2) < -0.5 OD DC J6a = 0, otherwise = 0, otherwise = 0, otherwise The model d.e. is given by y(j) + k i y ( j - l ) + k2y(j-2) + k 3y(j-3) + k ^ + k ^ + k g0 6 =0 (3.65) It is evident that the interval adjustment scheme cannot be applied directly in this case, since i t is difficult to define regions where k^, k__ or k, alone can be adjusted. To obviate this problem, the system d.e. may be 6 modified into the one given by y d(j) + K ^ C j - l ) + k 2y d(j-2) + k^Cj-3) + {k^rCj) +. k 5 ay 5 a(r(j)-0.5) + k6ay'6a(r(j)+0.5)} + 0.7 k 4 b r ( j - l ) + k 5 by 5 b(r(j-l)-0.5.) + k 6 by 6 b(r(j-l)+0.5} - 1.5 (k 4 cr(j-2) + . 3.9 Identification of the Linear Part of the System in Example 3.A 65 0.5 f ( r ) , f ( r ) * • * i — - ' — — ' ->1.0 -0.5 ft' • ** 0 0.5 ' ' r 1.0 # -0.5 Simultaneous Adjustment Scheme At 60^ instant At 120 t b instant th At 180 instant til At 240 instant System Nonlinearity Fig. 3.10 Identification of the Nonlinearity in Example 3.4 66 •k 5 cP 5 c(r(j-2)-0.5) •+ k 6 cy 6 c(r(j-2)+0.5} = 0 (3.66) The terms in each pair of braces may be treated as thos.e representing an individual nonlinearity. Then interval adjustment scheme is applicable, how-ever, at the expense of having to identify nine coefficients for the nonlinearity instead of three. The simultaneous adjustment scheme does not have the above-mentioned difficulty. For this scheme, fig. 3.9 and fig. 3.10 show the convergence of/ the coefficients of the linear part of the system and of the nonlinearity, respectively. The PC values are: 3.81 (instant 0), 1.75 x 10 _ 1 (instant 100), 1-.-4-2-X-10 (instant 200) . Example 3.5 The system is "identical "to that of "Example 3.4 except that the saturation type nonlinearity is replaced by the hard spring type nonlinearity of fig. 3.11. 0.5 i y /1+0.5 r o -0 .5 0 0.5 1+0.5 / -0.5 Fig. 3.11 Hard spring type nonlinearity For reasons mentioned in Example 3.4, the interval adjustment scheme is not directly applicable here. When the simultaneous adjustment scheme is used for identification, i t is found that a very fast convergence is obtained especially for large input levels. Figs. 3.12 and 3.13 illustrate -3 this rapid convergence. The-PC values are: 2.33 (instant 0), 2.47 x 10 (instant 80), 8.16 x 10~5(instant 160). Fig. 3.12 . I d e n t i f i c a t i o n of the Linear Part of the System In Example 3.5 68 Simultaneous Adjustment Scheme At 40 insta n t t t l - - - - - At 80 instant At 120 t h i n s t a n t At 160 insta n t System n o n l i n e a r i t y F i g . 3.13 I d e n t i f i c a t i o n of Nonline a r i t y i n Example 3.5 69 From Examples 3.4 and 3.5 and from the other examples solved, i t i s found that the convergence is slow i f the coefficients of the system non-linearity are not of the same sign. This may be explained by the fact that a l l the coefficients of the model nonlinearity i n i t i a l l y tend to adjust in the same direction, especially for large inputs, as can be seen from eqn. (3.49). If the coefficients of the linear part adjust nearly to their correct values, then, k Q0 Q + K $ =-E $ becomes very small, and correspondingly, the increments of the coefficients of the nonlinearity also become small, resulting in slow convergence. 3.3.4 Multivalued Nonlinearities Multivalued nonlinearities in the system d.e. may be identified i f the unknowns enter linearly in the d.e. as shown in eqn. (3.2). In this subsection, this i s demonstrated with an example. Since a general representation of a multivalued nonlinearity w i l l be highly cumbersome, such an attempt is not made here. Instead, just two special forms of the nonlinearity are discussed, and an example is presented for one of them. The underlying principles of solving similar problems, may be deduced from this discussion. 40 Consider the nonlinearity used by Gibson , and shown in f i g . 3.14. This is a generalized form of many types of two-valued nonlinearities including the relay type nonlinearity. Let the value of f(0) in the double-valued region be given by the history of 0. For 0 > 0, i f the signal 0 has entered the two-valued region from 0 < 2®±> t n e n w i l l be at the lower of the two levels. If, on the other hand, 0 has entered this region from 0 > t n e n f(0) w i l l assume the higher level. Similar conditions are valid for the two-valued region on the negative side of 0. The function f(0) is given by the following expression: X 2 k - l + 2 k - 2 _ k _ rC "T"_lc _™i" _ lc _ 1 - 1 2 - 1 2 - 2 2 - 3 ^ -k ,+„k ,+,k + k , 2 - 1 2 - 2 2 - 3 2 -4 70 ^k,+^k +^ k +„k, 2 1 2 2 2 3 2 4 , k =„k,+Jc„+Jc., 1 1 2 1 2 2 2 3 slope= 2k T+ 9k, slope= 2k 1 F i g . 3.14 A Two-valued Nonline a r i t y 4. where f(0) = J V. . P . ^ l V + "2.2 , 2 k i 2 V J i ( 0 - 2 0 i ) 1 1 x 1 i x=-4 x (3.67) « 1 = 1 for 20_± 1 0 1 2 0 1 = 1 for ' 0 < 0 <'203 a n d f o r 2^-1 > ^ > 2^-3 i f 1 3 e n t e r e d t h e s e regions from 0 < 20^ and 0 > 20 ^ res p e c t i v e l y . = 0 otherwise « 2 = 1 for U i 2 0 3 and for 0 1 2 0_ 3 1 for 0 . < 0 < 2 0 3 and f o r 2 0_ 1 > 0 > 2 0_ 3 i f 0 entered these regions from 0 > 2 0 3 and 0-< 20_-j resp e c t i v e l y . = 0 otherwise 71 and, for j=l, 2, and for 1 > 0 .u. = 1 i f 0 > .0 > 0 J i j i = 0 i f 0 < .0. > 0 - j I for i = 0 .y - 1 and .0 = 0 , and j o j o for i < 0 .u. = 1 i f 0 < .0, < 0 J i j i =0 i f 0 > .0. < 0 - J i Figure 3.15 shows the values of and at different regions of the nonlinearit a2=0 f(<|,) - 2 - i Fig. 3.15 Values of and Let the break points, .0. of the nonlinearity be known and let the J i slopes, be the quantities to be identified. This enables eqn. (3.67) conform to the requirements of subsection 3.2.1 that the unknown enter as linear coefficients in the system d.e. Eqn. (3.67) may be written as f(0) = KJt N (3.68) 72 where K„ is the vector with elements .k,, and $ is the measurable signal N j i N vector with elements c\. . In eqn. (3.67) there are 12 coefficients, jk_^. Some of them are either known or are related to the others by the following equations. ,k = k = 0 1 o 2 o 2k3 = l k l " 2 k l " 2k2 2-3 1-1 2-1 2-2 Using these relationships, eqn. (3.67) can be simplified to have only 8 unknown coefficients but s t i l l be of the form of eqn. (3.68). An example is solved below in which the nonlinearity in the system d.e. is composed of polynomials. Example 3.6 Let the system consist of a two-valued nonlinearity followed by linear dynamics as shown in fig. 3.16. The nonlinearity is an approximation to magnetic hysteresis with functions f^ and given by s ,2i-l f = E -k (r(j) - 0.4)' 1=1 (3.69) f, = E -k^. .(r(j) + 0.4) 2 i = 1 6+i 2i - l (3.70) f 2 • A*! / o /.4 Fig. 3.16 Example 3.6 73 and f by f = o 1 f 1 + a 2 f 2 (3.71) where k. = k =-0.5, k = k = +0.085, k, = k n =-0.006, and 4 7 . 3 o b y * = 0 i f r has reached the present value from a higher value = 1 otherwise a 2 = 0 i f r has reached the present value from a lower value = 1 otherwise Since a sampled quantity has zero value between samples, let i t be assumed that the input r(j) is passed through a zero order hold before being applied to the nonlinearity. Then, at any instant j , = 0 i f r(j) < r ( j - l ) , = 1 otherwise c-2 = 0 i f r(j) > r ( j - l ) , = 1 otherwise For values of r(j) within about +2.1 units, the approximation to hysteresis is reasonable as seen from fig. 3.17. Hence the input is chosen such that its magnitude is approximately within this range. The system d.e. is given by 3 2 i - l y d(j)+k i y d(j-l)+k 2y d(j-2)+k 3y d(j-3)+ Z k ^ a ^ r (j )-0.4) Z 1 > i=l 3 + Z k-.. a o(r(j)+0.4) 2 i - 1 =0 (3.72) 1=1 6 + 1 2 The values of k^, k 2, and k^ are chosen to be 0.9, 0.15 and 0.002 respectively. It is required to identify the k^ with the other quantities known or measurable. Hence the model d.e. is given by 3 \ y(j)+k1y(j-D+k2y(j-2)+k3y(j-3) + z ^ a^vQ)> 0.4) 2 i _ 1 3 + E a 9(r(j)+0.4) 2 i" 1 = 0 (3.73) i=l 6 + 1 2 Figs. 3.18 and 3.17 show the identification of the linear and the nonlinear parts of the system. The values of the PC are: 1.38 (instant 0), F i g . 3.17 I d e n t i f i c a t i o n of N o n l i n e a r i t y i n Exampl 75 -O . U . Fig. 3.18 Identification of the Linear Part of the System in Example 3.6 76 -3 -5 1.3 x 10 (instant 400), 3.7 x 10 (instant 800). From these values and from the plots, i t may appear that the convergence of the model coefficients is appreciably slow in comparison with the previous examples presented. However, i t can be explained by the fact that there are twice the number of coefficients to be identified in the present example as compared to the previous ones, and that only half of them are adjusted at any instant. Thus the speed of identification is reduced by a factor of four. 3.4 Some Practical Considerations 3.4.1 . Time-Varying Coefficients In a l l the examples presented so far, i t has been assumed that the variation of the system coefficients conform to Assumptions 2.1 and 2.2. If these assumptions are not upheld, then Theorem 3.1 which ensures the eventual asymptotic stability of the identification system is not applicable. An example is presented below in which some of the system coe-fficients are chosen to violate Assumption 2.2. Example 3.7 Consider the system shown, in fig. 3.2. Let and be assumed known and of values 0.15 and 0.002,respectively. The coefficient k^ is given by k^(j) = 0.6 + 0.4 sin 2trj/400 and the coefficient k^ is a trian-gular function with maximum and minimum values of -0.8 and -1.2^  respectively with a period of 300T where T is the sampling time. It is required to identify these two time-varying quantities. The model chosen is shown in fig. 3.2 in which a third degree poly-nomial is chosen to represent the nonlinearity. Fig. 3.19 shows the identifi-cation of the coefficients k^ and k,. by the corresponding model quantities. A. A It is of interest to note that the coefficients k. and V., remain close to 4 6 zero throughout the time of observation, after a short i n i t i a l period. Thus the model coefficients are observed to follow the system coefficients even when Fig. 3.19 Identification of Time-Varying Coefficients in Example 3.7 78 two superfluous terms, k.^  and k^, are included in the representation of the nonlinearity. When the frequencies of the sinusoid and the triangular wave of and k^ respectively are increased, i t is seen that the tracking progressively becomes poorer. Yet, i t can be concluded that in practice, an adequate iden-t i f i c a t i o n of the system may be possible, even i f the variations of the system coefficients do not conform to the assumptions of Theorem 3.1. 3.4.2 Unknown Form of Nonlinearity If the form of the nonlinearity in the system is not known, i t can be approximated by a polynomial as shown in subsection 3.3.2. However, i f the system nonlinearity does not exist in the form of a polynomial, then a prohibitively large numbfer of terms w i l l be needed to represent i t with negligible error. Use of bnly a few terms may cause problems in the conver-gence of the model coefficients since the identification of subsection 3.2.2 assumes the possibility of exact identification. An example i s presented below in which the system d.e. contains a nonlinearity which is not a smooth func-tion of i t s argument and hfence cannot be represented exactly by a power series with a f i n i t e number of terms. Example 3.8 Consider the system shown in f i g . 3.5 with k^ = 0;9, k^ = 0.15 anc * 0.002. The system noftlinearity Is known to be an odd function but i t i s assumed that no other information i s available about i t . Let the model be chosen with a nonlinear function given by f ( r ) = Z - k r 2 j _ 1 j=l ^ and let i t be followed by the same dynamics as given i n the model of f i g . 3.5. The value of J is chosen tip be 3, 4, 6 and 8 in four different runs. The identification i s terminated in each case a r b i t r a r i l y at the 300 t h instant and the nonlinear functions of the model at this instant are shown in fig. 3.20. It is seen that the model nonlinearities identify the system non-linearity only approximately. No major improvement is observed when the identification is continued for a longer time. From an examination of the model nonlinearities at different instants (not shown in figure), i t has been observed that they do not have a tendency to converge to any particular function. This is due to the fact that the adjustment algorithm is based on an instantaneous criterion and hence at each instant the model nonlinearity is adjusted in the direction of minimizing the PC at that instant. Since none of the three nonlinear functions chosen for the model can represent the system nonlinearity exactly at a l l instants, the "best" nonlinearity for the model is undefined. ' . Thus, this example shows that only approximate identification is possible i f the model is incapable of exact representation of the system. 3.5 Identification Error Method for Nonlinear Differential Equations The identification error method of subsection 3.2.2, derived for the discrete-time systems, is extended here to the case of continuous systems. Consider a continuous system described by the differential equation, N I k 0 (t) =0 (3.74) i=l 1 where k^ are the unknown coefficients to be identified and 0^(t) are the known or measurable quantities. Let eqn. (3.74) be written compactly as KT$(t) =0 (3.75) The symbols used in this section have the same meaning as in subsection 3.2.2 but defined for continuous time t. Let'a model with the following differential equation be chosen: E k.(t)0.(t) = K T(t)$(t) = 0 (3.76) i=l 1 1 o Fig. 3.20 Identification of an Unknown Form of Nonlinearity i n Example 3.8 81 Choosing a PC of VI(, j - E T(t)ME(t) (3.77) and following the procedure oif linear continuous systems given in subsection 2.2.2, the adjustment algorithm i s obtained as K(t) = -q[K T(t)$(t)| 1" 1sgn(K T(t)$(t))M~ 1$( t), i e l , qeR, q > 0 (3.78) A From this algorithm, i t i s seen that K(t) = 0 makes any further adjustment of A K(t) impossible. To obviate this spurious result, Assumption 3.1, which considers at least one of the system coefficients to be a known quantity, i s made. The A adjustment algorithm i s s t i l l the same as given by eqn. (3.78), but the k^ are chosen to be equal to the k^ for a l l known values of the system coefficients. -The-algorithm given in eqn.-(3,78)-is-essentially the same as that 3 24 of Butler and i s a generalization of that of Roberts , both of them having used an equation error method. A state variable f i l t e r i s used by Kohr1'' and HoberockX^ in a similar problem to avoid differentiators which would otherwise be-necessary. A detailed discussion of the practical applications of this algorithm 3 16 i s given by Butler and Hobertjck 3.6 Conclusions In this chapter, th© identification techniques developed i n chapter 2 have been extended to the case of nonlinear d.e. Identification of the unknown ^parameters of the system has Ijeen shown to be possible i f they enter as linear coefficients in the system d.e. The practical application of the proposed techniques has been illustrated by means of a number of examples. Whenever the forms of the nonlinearities in the system d.e. are known, model nonlinearities which are capable of exactly representing those in the system have been chosen. Then identification has been achieved for a wide range of nonlinear d.e.'s. Appro-ximate identification i s obtained i f the form of the system nonlinearity i s not 82 known and cannot be represented exactly i n the model. I d e n t i f i c a t i o n of time-varying c o e f f i c i e n t s of the system has been shown to be f e a s i b l e with reasonable accuracy, i f these v a r i a t i o n s are moderate. The p o s s i b i l i t y of extension of the i d e n t i f i c a t i o n technique to continuous systems has been b r i e f l y outlined. The new approach presented i n t h i s chapter f o r the i d e n t i f i c a t i o n of nonlinear d i f f e r e n c e equations i s a t t r a c t i v e due to i t s s i m p l i c i t y . Other novel features are that a l l the three i d e n t i f i c a t i o n methods have been derived i n a common framework of error function minimization and s t a b i l i t y has been proved allowing time-variations of system c o e f f i c i e n t s . I t i s of i n t e r e s t to note that many d i f f e r e n t types of n o n l i n e a r i t i e s can be i d e n t i f i e d by the propos i d e n t i f i c a t i o n error method. 83 4. GENERATION OF PARAMETER SENSITIVITY FUNCTIONS 4.1 Introduction This chapter is devoted to the development of methods for the generation of parameter sensitivity functions of sampled-data systems for a hybrid or a digital computer simulation. The sensitivity functions are necessary in the parameter tracking problem to be discussed in chapter 5. The parameters considered in this work do not include the sampling period of the samplers or the time delay, i f any, in the hold circuit. Though techniques are available for the generation of sensitivity 30 41-43 functions for continuous systems, ' l i t t l e work has been reported on 44 sampled-data systems. The investigations of Tomovic , et al. and 45 Rosenvasser , pertaining to the sensitivity in sampled-data systems, are not oriented towards the generation of sensitivity functions from the system structure. In this chapter both nonlinear and linear sampled-data systems are considered for the generation of sensitivity functions. A technique 43 developed by Vuscovic and Ciric for nonlinear continuous systems is extended in section 4.2 to nonlinear sampled-data systems. Though this method is very general, i t is seen that a separate sensitivity structure, which is similar in configuration to the system structure, is required for each sensitivity function. Thus the method is uneconomical in problems such as the 21 43 46 optimization of systems with many adjustable parameters ' ' , where the simultaneous generation of a l l the sensitivity functions is required. For linear continuous systems which belong to a certain structural form"^'^", i t is possible to generate a l l the sensitivity functions with but one sensitivity structure. A similar procedure is not directly applicable to sampled-data systems. Section 4.3 investigates this problem and discusses the modifications necessary to obtain economy of simulator components (for 84 hybrid computer generation), or of computational time (for d i g i t a l computer simulation) when many s e n s i t i v i t y functions are required simultaneously. Section 4.4 presents comprehensive examples i l l u s t r a t i n g the a p p l i c a b i l i t y of the method. The generation of second order s e n s i t i v i t y f u n c t i o n s 3 ^ u s e f u l 48 49 i n the optimization problems with s e n s i t i v i t y reduction ' , i s discussed i n Appendix B. 4.2 Extension of the Vuscovic and C i r i c Method for Sampled-data Systems 43 The method of Vuscovic and C i r i c f o r the generation of s e n s i t i v i t y functions i s a p p l i c a b l e to single-input single-output continuous systems. The system i s considered i n i t s s t r u c t u r a l form composed of l i n e a r , nonlinear and summing elements. This method i s . extended i n t h i s s ection to nonlinear multi-input multi-output sampled-data systems and an example i s presented. Consider a general sampled-data system whose structure i s given i n f i g . 4.1 where black nodes i n d i c a t e d i s c r e t e v a r i a b l e s and white nodes continuous v a r i a b l e s . Let u^ and v^ be the inputs and outputs, respectively, of the l i n e a r elements, and l e t s. and z. be those of the nonlinear elements l i i n the system. Let w. and x_ denote the inputs and the outputs, respectively, of the sampler elements and of the hold elements. I f and d_^  are the impulse responses of the l i n e a r continuous and l i n e a r d i s c r e t e elements, then v. ( t ) = / t g. ( T ) U.(t-x)dx (4.1) i o l x k v.(kT) = E d.(mT)u.(kT-mT) (4.2) 1 m=0 1 1 For the sampler p_^  i n f i g . 4.1, assuming an impulse modulator, 85 S Fig. 4.1 System Structure x ±(t) = W i(kT) , t = kT = 0 , otherwise, (4.3) .For a general hold element, x ±(t) = h ±fc (kT)), where tu is a hold operator. Or, in general, M . X j ( t ) = Z h. (t-kT)w (kT-mT) , (k+l)T > t > kT (4.4) 1 m=0 m ^ ~ where M is the order of the hold element. For example, for an ideal zero order hold, h^(t-kT) equals unity. For the nonlinear elements, the following equations are obtained: z i(t) = f.(s i(t)), (4.5) z±(kT) = n 1(s.(kT)), (4.6) where f and n_^  are the continuous and discrete nonlinear operators, res-pectively. It is assumed in the following development that the quantities, g i ' d i ' h i ' ^ i a n d n i » a r e differentiable functions of the system parameters. .Let {r^} be the set of inputs to the system. The coupling between various elements, shown in fig. 4.1, may be expressed by summing the inputs and outputs of the elements algebraically at each node. For a node j , 86 the following equation is obtained. R L S Z p..r. + Z (a.,u.+8..v.) + Z(v. .w.+<5. .x.) i=l X J 1 i=i X J 1 X J 1 i=i iJ i i j i N C +Z (e..s . + c j > . .z.) + Z i p..(y,).=0 (4.7) i=l ^ 1 X J 1 i=i X J d 1 where p. ., a. ., 8. . > v. ., 6. ., 9. . > cb. . and if). . take values of 0, 1 , or - 1 , i j 1 3 1 3 1 3 1 3 1 3 1 3 1 3 R •» total number of inputs to the system, L = total number of linear elements, S = total number of samplers and hold elements, N .= total number of nonlinear elements, ' . C = total number of outputs of the system, j - 1» 2 , p; p = total number of nodes, and (y^)^ are the system outputs. Taking the partial derivative of the coupling equation (4.7) with respect to an arbitrary parameter q of• the system and defining u^ = 3u^/8q, v' = 3v./3q, w! = 3w./3q, x. = 3x./3q, s! = 3s./3q, z! = 3z./3q, and iq l ^ iq l ^ iq l ^ iq l n xq x ( y d ) ] _ q = S(y i) i/3q, one obtains, L S Z (cx..u! +0..vl ) + Z (v.. w! +6..x! ) i=l X J lc* X J 1(* i=i X J i(* x 3 l c l N C + z ( e . .s! +<j>. .z! ) + z . ( y , ) ' = o (4.8) i=i 1 J lc* X J l c i i=i X J d l c i Thus eqn. (4.8) suggests a system configuration with the same internal coupling as that of eqn. (4.7) but with no r^ inputs and with the time functions replaced by their partial derivatives. Let this structure be known as the sensitivity structure (S in fig. 4.2). The outputs (y,)! of this structure are the sensi-q d xq tivities of the system outputs to the variation of the parameter q. 87 D u ! © u ! D w ! © w ! iq iq iq iq Fig. 4.2 Structural Configuration for the Generation of Sensitivity Functions To obtain the partial derivatives, differentiate eqn (4.1) through eqn (4.6) partially with respect to q. Thus the following equations are obtained: V i q ( t ) =  fQ 8j_q(T)ui(t-T)dT +/Jg.(t)u^q(t-T)dT (4.9) (where i t is assumed that g! and u! are continuous in q) iq iq v! (kT) = z d! (mT)u.(kT-mT) + E d.(mT)u! (kT-mT) 1(1 m=0 i q m=0 i iq (4.10) x' (t) = w! (kT) , t = kT iq iq = 0 , otherwise (4.11> 88 , . . " h! (t-kT)w. (kT-mT) x' (t) = I lmq 1 i q m=0 M + y h. (t-kT)w! (kT-mT) , (k+l)T > t > kT „ im iq •_ m=0 3f. iq 3 s ^ iq iq 3n. . z! (kT) = — —• • s! (kT) + n! iq 3s i iq iq (A.12) (4.13) (4.14) In the above equations, the g! , d! , h! , f ! , and n! will be zero i f the ^ & i q iq lmq iq . iq corresponding elements are not functions of q. The equations (4.9) through (4.14) provide coupling between the system structure and the sensitivity struc-ture as shown in fig. 4.2. Comparing the sensitivity and the system structures, i t is seen that the nonlinear elements in the system structure are replaced by multipliers in the sensitivity structure. Otherwise, the two structures are identical. Example 4.1 Consider the nonlinear system shown in fig. 4.3. The sensitivity functions of the output for variations of parameters a and b are generated in fig. 4.4. It is seen that corresponding to each sensitivity function, one + T + , -1 a 1-z T s n is disturbance noise Fig. 4.3 Block Diagram for Example 4.1 Fig. 4.4 Generation of Sensitivity Functions for Example 4.1 external structure is required. 4.3 Sensitivity Functions for Linear Sampled-data Control Systems In this section methods for the generation of sensitivity functions for a class of linear, single-input single-output sampled-data control systems are developed. Extensive use of signal flow graph algebra is made for this purpose. The symbols used, the classes of systems considered, and the flow graph techniques employed are discussed in subsection 4.3.1. A general deriva-tion for the generation of sensitivity functions is described in subsection 4.3.2, followed by a few special cases in the subsequent subsections. 4.3.1 Preliminary There are many methods available for analyzing the sampled-data 50 52 3A 53 systems using signal flow graph techniques ' ' . The method developed 53 by Sedlar and Bekey is a very powerful flow graph technique for linear sampled-data control systems. In this section, these methods are applied to the problem of generating sensitivity functions. The notation followed 53 is largely due to Sedlar and Bekey The following symbols and definitions are used in this section: A black coloured node, © , denotes the sampled sum of signals at the node, and a white coloured node, O , indicates that the output of the node is a continuous signal. If i t is-not known, or i f i t is immaterial whether signals at a node are continuous or discrete, then the node is denoted by a half-blackened circle, © . A sampled gain is defined as the transmittance from one sampler to .the same sampler or to another sampler, or from input to a sampler, or from a sampler to the output, keeping a l l the samplers open in each case. The sampled gain is the same as the transmittance of a segment. 91 All the signals, transfer functions, and sensitivity functions are in Laplace transform notation unless otherwise stated. The transform variable, s, is suppressed in a l l cases. Further, R, y^, G^ , and (y^)!^ denote the input, the output, the transfer function of an element, and the sensitivity function of y^ with respect to any parameter in the element G^ , respectively. In the transmittance expressions,'the. nature of the node (black, white, or half-blackened) is added as the subscript (b, w, or x) to the gain of the directed branches entering the node. Thus in fig. 4.5(a), the signal 1 R G, G. G, G. G- G G D -© 2 — O 3 — * - 0 ^ — ® — - — « - 0 — - — " G - 2 O y , d (a) G, G„ G G ., G ,» G ® — i - O 2*o - ~0-^~&£±*^F?^Q ^Q_5_^@ •> . ~ ' (b) Fig. 4.5 Signal Flow Graphs illustrating the Application of Symbols y^ is given by y d _ VlbG2wG3wG4bG5wG6bG7w = R*G*(G2G3G4)*(G5G6)*G? The above symbolism.lends i t s e l f to the following laws: G. + G_ = G ' + G lx 2x 2x lx G1 + (G0 +G„ ) = (G. +G„ ) + G, lx 2x 3x lx 2x 3x G, (G_ +G_ ) = G, G. '+ G, G, lx 2x 3x lx 2x lx 3x In general, Glx G2x ^ G2x Glx To determine the laws for a string of branches and nodes, consider fig. 4.5b and denote the sampled gains by U. and IL; A o A^ Glw G 2 w G n b UB = G(n+l)w G ( n + 2 ) w G m b Within UA, A G— •••• 0. G, •••• G , -~ G- G. G. •••• G . lw iw jw nb lw jw IW nb and G. .... G G , = G, .... G G,, lw iw nb lw nw ib For U A and Ufi, U A = UBUA However, ..G. . .G ,G. 1 1 N .«G, . .G , ^ G ..G, . .G ,G. 1 N . .G , lw iw nb (n+l)w kw mb lw kw nb (n+l)w iw mb It can be formulated that the functions within a sampled gain are commutative as long as the last function of the sampled gain carries the subscript b. Sampled gains, as i s , are commutable. However, two functions belonging to different sampled gains are not commutable. Two forms of the structure are considered in .this section. Structure 'type A' is shown in fig. 4.6(a). Let the input and output be considered sampled. Let there be samplers, X^ , , in the system at some, or a l l , of the small 32 3A circles of fig. 4.6(a). The sampled flow graph ' is shown in fig. 4.6(b) * where Gp^  represents the sampled gain from the node P to the node Q. If the directions of a l l the paths between the nodes are reversed, then * r* r* GpQ=G^ p where G^ p is the sampled gain from the node Q to the node P in the graph where the paths are reversed (fig. 4.7(a)). Thus for each path from X to X , R C there is an identical path from X to X in the reversed graph. In the system - C R structure this implies that the same transmittance will be obtained i f the directions of a l l the paths are reversed. The reversed path yields the structure shown in fig. 4.7(b) (called 'type B'). Since the individual paths are identical in 'type A' and 'type B' structures, their partial derivatives with respect to 93 F i g . 4.6(b) Sampled Flow Graph F i g . 4.7(a) Sampled Flow Graph for fo r 'Type A' Structure 'Type B' Structure any system parameter, and hence the s e n s i t i v i t y functions, w i l l also be i d e n t i c a l . The two structures are thus i d e n t i c a l . A s i m i l a r r e s u l t i s reported i n l i t -49 erature f o r continuous systems In the following subsections, the analysis i s r e s t r i c t e d to 'type B' .9.4 C 1 1 1 ' 0-*©-*»Q—-O <>-*<> 2m-2 '2m Fig. 4.7(b) 'Type B' Structure structure, and the sensitivity of 'type A' structure, i f needed, may be obtained by directly applying.the techniques for 'type B' structures. Furthermore, unless otherwise stated, samplers are considered only at the nodes of 'type B' structure graph (shown by small circles in fig. 4.7(b)) and the sampled output _.of the system is taken for generating sensitivity functions. 4.3.2 The General Case In the'type B'structure shown in fig. 4.8(a), let the first black node in the forward path of the system flow graph proceeding from the input, be at the output of the element G2n+1' Consider the structure as composed of a subgraph and a main graph as shown in fig. 4.8(b). Transmittance T of subgraph can be written as s T = s ? P .A . 1 sx si (4.15) and transmittance T of the entire system excluding the fictitious branch R at input can be written as Z p . A . J J J — J with m-1 ' (4.16) 1 R 1 Subgraph i si si 2n+l -r? ! i 6 2n+? J2m-2 12m, 9 5 ^ J < D y d Fig. 4.8 System Structure for the General Case J (a) (b) O - O-2i 2m Fig. 4.9 Generation of (-YA)^. , i > n 96 ra-1 N A = A ~{G. . + E t n G . . ]G / 0 M, 0. }[Zp .A . L s (2n+2)x N = n + 1 £ = n + i (2£+l)x (2N+2)x r s i s i b = A - [L Z p .A .], (4.17) s M . s i s i b 1 where stands f or mutual loops and m-1 N hi = { G ( 2 n + 2 ) x + [ £ = ; + 1 G ( 2 . i l + l ) x ] G ( 2 N + 2 ) x } ( A ' 1 8 ) Let =. 9T/9q (where q i s an element i n and l e t G!^  = dG±/dq> Hence, (y )! = RT! = ^y{A[R Z p.A.] ' - A ' f R Z p . A . ] } (4.19) A j J J j 3 J Eqn. (4.19) gives the general expression for a l l the s e n s i t i v i t y functions, a) Generation of ( y ^ ) ^ ' > n From eqn. (4.19), where Hence, 3 Z P j A j < V 2 i = " ^ C R ] P j A j ] (since - J ^ - = 0) =-A'RT/A, (4.20) A* = -LA(Z p .A .). M . s i s i b = ~GVo J.o^ (Z p .A .), , i=n+l (2n+2)x ^ r s i s i b _ r 1 Z 1 lG'nJ* P • A . ] , , i > n+1 (4.21) Jl=n+1 ^ J t r - L ' 1 ( V k = W ^ s M R T / A> 1 = 1 1 + 1 i - l = [0 " G ( 2 £ + l ) x l G 2 i x [ E P s i A s i ] b R T / A > 1 > n + 1 £=n+l I Since the output y^ i s sampled, T=T^ and y d = ( y ( j ^ ' Hence, %y2i * R T 2 i " R Tb G(2n +2)x ( f s i A s l V A ' i = n + 1 - R V „ ^ G(2£ +l)x ] G2ix ( EPsi Asi )b / A> 1 > n + 1 ( 4 ' 2 2 ) Fig. 4.9 gives the flow graph for the generation of (y^)^* T n e output of the system forms the input to a sensitivity structure which is identical to the system structure. It is seen that a l l the sensitivity functions, (y^)^, i > n, can be obtained with one sensitivity structure. However, in addition, a separate subgraph and an element are needed for each function ( V 2 i -b) Generation of (y^^i+l' 1 > n For m-1 > i > n+1, i t is seen that i-1 m-1 = " i w ^ W W ) * + "A l,J + 1 G(2«l) X 1 G(2» +2) X>-Lt F s i s i b (4.24) If i=n+l, the quantity [ II G,~ ] will be absent in the last two equations. £=n+l <-z*+i>x m-1 m-1 N If i-m-1, the terms . [ ^ n + i G ( 2 i + 1 ) x ] G ( 2 ! f f 2 ) x will be absent in eqns. (4.23) and (4.24) respectively. Let j and k be two integers such that j < i <_ k and that G(2j+l)x ~ G/o.>-i\w a n ^ G/m.TS = _,,», and further that there exists no element G,„ T 1- X, (2j+l)b (2k+l)x (2k+l)b (2L+l)b with j < L < k. Then, for j > n, eqn. (4.23) can be written as r 98 j If -j=n, then the factor [ n G-„ ] will be absent in the above equation. A=n+1 ^* + ±> x b In either case, k [REp A. ] ' = [ n G , 9 0 . 1 V ] ' [REp . A. ] 2 2 2 JM+1 ( 2 £ + 1 ) W b j J J - k [,,," + 1 G(" +i)„ 1b Define, A [REp A ]' _ [ n G, . 1' s - j J n - ^ b & = i + 1 ( 2 m)w b ( 4 > 2 5 ) A k [£ = j + 1G<2*+Dw] b . For j > n and i < k-1, eqn. (4.24) can be written as (y d ) b ( y d ) b i-1 k-1 N ~ ~ ~ A' A G(2£ + l)x ] Gbi+ l)x { G(2i+2)x + N = J + 1 [ £ =.; i G(2£ + l)x l G(2N+2)x }-[Ep .A .], ^ s i s i b ( y d ) b j k m-l N ~ ~ ^ J l G ( 2 * + 1 ) x l b ^ = j + l G ^ [EP .A .], ^ s i s i b = -P - Q (4.26) j If j=n, then the quantity [ II G(r)0.,v ], will be absent in Q. From eqn. (4.19) Jl=n+1 ^ z + l ) x b [REp A.]' (yH).A' (y d)! b = .1 3 J b _ =S + P + Q (4.27) A A If i=k, then P=0, and i f i=k-l, then (y d ) b i - i P = A ^ J . G(2£+l)xlG(2i+l)xG(2i+2)xtEpsiAsi]b Ji=n+1 l Consider f i g . 4.10 in which the signal y^from the output of the system structure i s applied at the black node at the output of &2n+l of the sensitivity structure as well as to the structure 100 k [£ = J + 1 G ( 2 £ + l ) w ] b / [ £ = J + 1 G ( 2 £ + l ) w ] b Output of the elements G , L=i+1, i+2, . k are divided into two groups -those ending in black nodes, and those ending in white nodes. The latter are sximmed and passed through an element G',_.,1N /G/0.,1N . For those ending in • (2i+l)w (2i+l)w black nodes, signals are taken just before, they are sampled, i.e., when they are in continuous form, summed, passed through an element G/,.,.* /G / 0. I 1 X ' ' r ° (2i+l)w (2i+l)w and then sampled. These two signals are then added and passed through a T g block as shown in the figure. .A verification that the sum of the signals, P, Q and S of fig. 4.10 yields the sensitivity function 'typ2"+l' * S s h o w n n e r e " * n fi§- 4.10, for i < k-1, t \ . , N [Ep .A .] £=n+l v ' N £=i (2i+l)w s w Nb G' ^ s i A s i ] b + tj < * G(2£ +l)w> G(2 V2)w * • A~ • • • V Nb l~X b G(2£+l)w b where and represent the white and the black nodes, respectively, among the nodes at the output of elements ^21,' ^  = i + ^ ' i+2,...,k. Further, the first summation (over N ) is carried out for a l l G„ elements which end in white nodes W ZL • and the second summation (over N^) is carried out for elements ending in black nodes. Simplifying, ( y d ) b i-1 k-1 N P = ~~A~ [ £ =^ + 1 G(2£+l)x ] G(2i+l)x { G(2i+2)x + N = ^ + 1 [ £ = J + 1 G(2£+l)x ] G(2N+2)x } [ J P s i A s i ] b (4.28) This agrees with the value of P in eqn. (4.26). The agreement for the cases when i=k-l and when i=k can be proved likewise but will not be attempted here. Consider the output S+Q of fig. 4.10. k s + Q - ( y d ) b — J + — [ i , G 1 r T I p i £=n+l £=j+l ( 2 A + 1 ) W b k T n-1 N [ T s ± " s i \ l \ n + 1 G ( 2 £ + D w ] b { G ( 2 k + 2 ) x + N ^ + / ^ 1 G ( 2 m ) x ^ ( 2 N + 2 ) x > L - T " ^ ' ' 101 k [ J G ( 2 £ + l ) w 1 (y ) j - ( y A - 1 ^ k + n G r n r i £=n+l ' b J5=j+1 b m-1 N (2k+2)x N = k + 1 £ = k + 1 (2£+l)x (2N+2)x . s i s i fc This value agrees with the s i g n a l S+Q given i n eqns. (4.25) and (4.26). Thus P+(0+S) i n f i g . 4.10 gives the required ( y d ) 2 i + 1 -In the r e s t of the sect i o n , such v e r i f i c a t i o n s are omitted i f they are straightforward. The structure of f i g . 4.10 needs two T blocks out of which one i s s » s common f o r a l l s e n s i t i v i t y functions (vd ) 2 i + p J* < 1 <_ k,whereas the othei T t block i s needed one f o r each s e n s i t i v i t y function ( y , ) 0 . , , . Further an element k , . k d 2 1 + 1 f n G . " ? o 4 . - i v J / [ N G / " > O J . I V J i s needed for each s e n s i t i v i t y function. £=j+l U £ + i ) w b / £=j+l ( 2 £ + 1 ) w b . A few s p e c i a l cases are discussed below. ( i ) In the s p e c i a l case when a l l the elements G ( 2 i + 2 ) ' G ( 2 i + 4 ) ' ""*' G , end i n black nodes, the configuration f o r the generation of s i g n a l P of 2k eqn. (4.26) can be modified so that only one T g block i s needed to generate p for a l l the s e n s i t i v i t y functions, ( y d ) 2 I j + 1 » i f. L < k as shown i n f i g . 4.11. ( i i ) I f there i s no black node i n the forward path a f t e r the element G . , i . e . , i f no G. ' .* element e x i s t s , then eqns. (4.25) and (4.26) modify (2i+l) (2k+l) to (for i < m-1) 1 0 2 + -Fig. 4.11 Generation of Signal P i f  G2±+2' G21+4'"*'*' 3 n <* G2k 6 n C* i n Black Nodes. m-1 'I I » ? . A i l i t 1 J 1 « « W ) . l 0 , ( 2 I + l ) w ^ + 1 6 ( 2 W l , . l b = s 103 ( y d ) b , -<y d) b i - l N ~~& A = ~ A ~ [ £ = n n h l G ( 2 £ + l ) ] G ( 2 i + D w { G(2i +2)x + N = . ^ 1 [ J l = J + 1 G(2£ +l)x ] G(2N +2)x }' (Zp .A .). ' ^ s i s i b = -P If i=m-l, -(y ) 1-1 " P - - ^ - [ f l I G(2£+l)x]G(2i+l)wG(2i+2)x ^ s i ^ A Fig. 4.12 gives the configuration which is a modified version of that of fig. 4.10 Here the sensitivity function ( y e q u a l s P + S. Fig. 4.12 Sensitivity Function Generation when Black Node X„ , Does Not Exist 104 ( i i i ) Suppose there are samplers in the feedback paths in places other than at the small circles of f i g . 4.7(b). Such feedback paths affect the development of only the signal P but not Q and S(, and they can be treated with ease. An example of this case is shown in f i g . 4.13 where the generation of signal P i s shown when the transmittance of one of the feedback paths is given by G , G G„,G„ . " 6 7 Ab Bw Cb Dw O-+O-- *o-Fig. 4.13 Generation of P when Feedback Path Samplers Are Not at the Assumed Places 105 -c) —Generation of C/^^i+l a n c ^ ^ yd^2i' * — n ' ° r ' ^ yd^s' q """S a n e-*- e m e n t °^ subgraph T Assume that the input R i s sampled before i t enters the subgraph. "From eqn. (4.19) A [REP.A.] A 3 3 -• i { [ V P s i A s i ] ' - V b [ As" LM (f s i A s i ) , ] b } (4.30) i b £=n+l i Before attempting to generate (y^) ,consider f i g . 4.14 where the technique of s e c t i o n 4.2 f o r the generation of s e n s i t i v i t y functions i s shown for subgraph T . The output of the s e n s i t i v i t y structure i s (R T ) . I f R i s a function s s s s of q, then R must also be given as input since R ^ 0. Then the output, s s t i t t -(R T ) = R T +R.T . But i f R i s not given as input to the s e n s i t i v i t y struc s s s s s s s ture, then i t s output w i l l be given by R T'+R'T -R'T , i . e . , by R T', since th s s s s s s s s system i s l i n e a r . subgraph structure T S e n s i t i v i t y structure for subgraph F i g . 4.14 S e n s i t i v i t y of Subgraph T by the Method of Section 4.2 s F i g . 4.15(a) gives a s t r u c t u r a l configuration f o r the generation t of (y^) '. Here two T blocks share the same subgraph s e n s i t i v i t y structure t v i a the element G\ The expression for ( y ( j ) g i n the f i g u r e i s given by, 106 F i g . 4.15(a) Generation of S e n s i t i v i t y Function f or an Element of the Subgraph Ep A . , . s i s i , T = r - 1 sb 1 A \ s b frA- + r V T a b sb t S p s i A s i ] K A i b s s A 2 * v s i s i , A i b s Hence m-1 , A t ? s i A s i ] h [ 5 " G ( 2 £ + l ) x ] h , v i b £=n+l b ( y d } s = — ~ A A s . s. [Zp .A .] + A \ . s i s i , i b • i(v? p8i*siy mj!wi)*i - V b [ A s - v \ p S i A s i ) b i b &=n+l b i i This expression for (y,) agrees with the one given by eqn. (4.30). In d s addition to the sensitivity structure, this configuration needs three T g blocks. In problems where the input to the system may be applied at any desired node in the -system, such as in the off-line generation of sensitivity ..functions ^ jthe conf iguration of fig. 4.15(b) may be used. This configuration eliminates the need for simulating the transmittance 1/1 ^ and thus may be simpler to use. Fig. 4.15(b) An Alternative Configuration of Fig. 4.15(a) 108 4.3.3 Special Case 1 - Subgraph T g Has No Black Node Though the equations and the graphs developed for the general case in section 4.3.2 can be followed for this special case, yet, in certain cases, i t is advantageous to develop new graphs which may be simpler than those for the general case. Only these variations are given below. In this case, i t is worthwhile to note that A =1, Ep .A .=T and &=l-l^JT , . s ^ sx sx s M S D a) Generation of (y^2±' 1 > n From eqns. (4.19) and (4.21), ( V 2 i " " R ? P j A j A J i-1 n £=n+l = t 1 1 W l ) x ] G 2 i x T s b ( V b / A ' ± > n + 1 ( 4 ' 3 1 2i+l In fig. 4.16 the black node on the forward path immediately preceding G is at the output of G2^+i a n d t b" e o n e i m m e d i a t e l y following &2±+l 1 S a t the output of G2\a+± a s defined in the previous section. Referring to the figure, the signal at the output of element G(2j+i)b 1 S P a s s e d through the subgraph T . The output of the subgraph is applied to a black node B and to s a white node W. Two sections of forward paths, G(2j+3)w t 0 G(2k 1) from W and B. If G^ of the system ends in a black node, then (y^2± 1 S obtained from section of forward paths from B. Should i t end in a white i node, (y^)2^ will be taken from the section from W. In either case, i t can t be easily proved that the signal (y^2± l n 4.16 corresponds to the one given by eqn. (4.31). ensue w 109 Fig. 4.16 Generation of Sensitivity Functions ij^^t 1 a n d ^ yd^2i+l' 1 > n ' when Subgraph Has No Samplers. 110 b) Generation of ( y ^ ^ i + l » x > n Following the same l i n e of development for the general case (eqns. (4.25), (4.26) and (4.27)), the qu a n t i t i e s P, Q, and S can be obtained as given below (for i < k-1). k / k " S = ( y A ' [ = y ( 2 . W / f 4 + 1 G ( 2 , + l ) w ] b ( 4 ' 3 2 ) ( y d ) b k , m-1 N Q = T ~ [ £ = I + l G ( 2 £ + 1 ) x l b f G ( 2 k + 2 ) x + N = k+ A = k + l G ( 2 £ + 1 ) x ] G ( 2 N + 2 ) x } T s b (4.33) ( y d ) b i - 1 , k-1 -N P = ~A [ a = n * 1 G ( 2 m ) x ] G ( 2 i + l ) x { G ( 2 i + 2 ^ (4.34) The q u a n t i t i e s P, Q, and S can be generated i n the same manner as for the general case. However, P can also be generated as shown i n f i g . 4.16 i f i t i s advantageous. In that f i g u r e , an element G^ L* 1 < 1 < k i s connected to e i t h e r the section of forward paths from W or that from B, depending upon the termination node of G^ element i n the system graph; a white node t e r -mination indicates connection to section from W, and a black node indicates connection to section from B. Though t h i s configuration requires many elements of the form G and G , i < L < k, yet i t has the advantage of req u i r i n g only two subgraphsT g for a l l the s e n s i t i v i t y functions ( y ^ ^ ^ a T l d ^ y d ^ 2 i + l ' j < i:. <_ k. For the case of i=k, P reduces to zero. » t i c) Generation of ( y , ) O J and (y,).,..,, i < n, or, (y,) , q i s an element of a 21 a Z I T I — d s  Subgraph T g. „ •From eqn. (4.19), with the assumption of input sampling, [R.ZP.A.1 I l l i m-1 T . [ II G, A A M sb (4.35) In f i g . 4.17(a), the output of the s e n s i t i v i t y structure passes through two blocks each l a b e l l e d 'Subgraph T g ' • These two blocks constitute the configuration f or simultaneous generation of a l l the s e n s i t i v i t y func-.36,41 tions" of the subgraph T . I f X and Z are the input and the output, res-p e c t i v e l y , of the subgraph T , then the outputs of the second T block, taken s s 30 1 from the s e n s i t i v i t y points through s u i t a b l e elements, w i l l be XT . ^ s Fig. 4.17(a) Generation of S e n s i t i v i t y Functions f o r Elements of Subgraph T When I t Has No Samplers s In f i g . 4.17(a), s i g n a l (y^) i s given by, I sb 1 » + - 1 — T } T , sb sb . - m-1 ( X J ) L , C . " G(2£+l)x ]b _ d P.. T T + RT Z - - i - T " " 4 T I s b 1 sb A * T , sb sb m-1 ( y H ) h , C. " G(2£+l)x ]b , " A V s b D A sb which agrees with eqn. (4.35). If the input to the system may be applied at any node, such as in off - l i n e work, then the configuration of f i g . 4.17(b), which eliminates the t Fig. 4.17(b). Off-line Generation of (y,) When Subgraph Has No Samplers 113 simulation o f . l / T , may be adopted, sb In problems where s e n s i t i v i t y functions of only the subgraph para-meters are needed, i t may be advantageous to use the configuration of f i g . 4.18(a), or of f i g . 4.18(b). The former i s f o r on-line generation, and the l a t t e r i s f o r o f f - l i n e work. . Fig. 4.18(a) An Alternative Configuration of Fig. 4.17(a). 114 Fig. 4.18(b) An Alternative Configuration of Fig. 4.17(b) 4.3.4 Special Case 2 - System Error is Sampled If the system error is sampled, i.e., i f there is a sampler immediately preceding G^ , then subgraph T g does not exist. In such cases, simplifications result as shown below. a) Generation of (y^)^! ' 1 > 0 From eqn. (4.19), since T g=l, A s=l, and A=l-L^, (y ) i-1 ^ 2 1 = -P [£Q G(2£+l)x]G2ib ( 4 . 3 6 ) This is shown in fig. 4.19(a) 115 k [ n G J ± L _ k [ n G j + l (2*+l)w ] b Q+S i (yd>2i+i OP '(21+1)  :(2i+l) 21 - ^ ( y d ) 2 i F i g . 4.19(a) S e n s i t i v i t y Functions f o r Error Sampled Case b) Generation of ( y ^ ^ i + l ' 1 > ^ Following the same l i n e of development of the general case, the qua n t i t i e s S, Q and P f o r i < k-1 are given by [ n G s - R T b • & = i + 1 ( 2 £ + l V b k (4.37) 116 (y,), j k , m-1 N ( y d ) b 1-1 , k-1 N P = — A ~ " [ £ f 0 G ( 2 m ) x ] G ( 2 i + l ) x { G ( 2 i + 2 ) x + N = ^ + 1 t j i = J + 1 G ( 2 £ + l ) x ] G ( 2 N + 2 ) b (4.38) } (4.39) Since the outputs of a l l the feedback elements are sampled at the err o r sampler, the nodes at the ends of the direc t e d branches G , 0 < L <_ m can be made white. Thus, a s i m p l i f i c a t i o n r e s u l t s for the generation of P. Signals S and Q are generated i n the same way as i n the previous cases. t F i g . 4.19(a) gives the generation of 0^21+1' 2 < i f_ k. I f there i s no black node i n the forward path a f t e r the element G2i+1' "*"*e"' ^ t h e e ? - e m e n t : G2k+1 d o e s n o t e x i s t > then the s e n s i t i v i t y function (^d^2i+l can be generated as shown i n f i g . 4.19(b). F i g . 4.19(b) Generation of ( y ^ ^ i + l ^ o r ^ r r o r Sampled Case When Black Node ^2k+l ^ o e s ^ot E x i s t 117 In the case where some of the feedback paths have samplers in places other than at the ends of ^ < ^ _^m, a modification is needed in the rea-lization of P. Fig. 4.20 shows the generation of P when an element G_ is given by G^G^. • / 21+1  '21+1 21+2 1 GA 71+7 '2k+2 2m P - P 1 + P 2 2i+l  ;2i+l Fig. 4.20 Error Sampled Case When Feedback Path Samplers Are Not at Assumed Places 4.3.5 Special Case 3 - Completely Sampled System In this case, the input and the output of each element is sampled. Thus effectively a l l the nodes are black. This offers considerable simplification as shown below. 118 a) Generation of (y,) 0. , i > 0 a 21 (y ) i-1 , ^ 2 1 = [ A G(2£ +l)b ] G2ib Jo—u • « This expression is very similar to that of eqn. (4.36). (4.40) b) Generation of (y^^i+l ' 1 > ^ Following the development for the general case, S, P and Q are given by S " R Tb G(2i+l)b / G(2i+l)b P = 0 (y d ) b i-1 » .'. m-1 N Q = " ~ A ~ C J l f 0 G ( 2 J o + l ) b ] G ( 2 i + l ) b { G ( 2 i + 2 ) b + N = J + 1 [ ^ + 1 G ( 2 i l + l ) b l G ( 2 N + 2 ) b } Fig. 4.21 shows the generation of ( y ^ ^ i a n d (yd^2i+l" Fig. 4.21 Completely Sampled Case - Sensitivity Functions A-slight modification to the configuration of fig. 4.21 is carried out in fig. 4.22. Though this figure affords l i t t l e extra simplicity, i t gives a structure which is similar to the one for continuous systems as developed by t W b *yd*2i+l Fig. 4.22. A Modified Set-up for Sensitivity Function Generation for Completely Sampled Case Kokotovic^'^ 1. The configuration of fig. 4.22 may be developed directly, using the approach of Kokotovic but considering the transfer functions of the elements to be in z-transform instead of in Laplace transform. '' 4.4 Examples -A few illustrative examples for the generation of sensitivity functions of linear sampled-data systems are considered. In each case, the sensitivities 120 for the parameters of a l l the elements in the system are generated. It is important to compare the structural configuration for sensitivity functions obtained by the methods of section 4.3 with that of section 4.2 in terms of economy of simulator components before making a choice. Example 4.2 The block diagram of a two loop sampled-data system is shown in f i g . 4.23. It i s seen that this example belongs to the error sampled case G(s) y d(s) H^s) D*(s) H2(s) Fig. 4.23 Block Diagram of Example 4.2 (subsection 4.3.4). Sensitivities of the sampled output are desired. The flow graph for the generation of a l l the sensitivity functions t is shown in fig. 4.24(a). The generation of (y,) n by the method of section U Car 4.2 is shown in fig. 4.24(b). If (y d)g alone is desired, this configuration uses fewer components than that of fig. 4.24(a). Example 4.3 Fig. 4.25 gives the block diagram of the system whose structure belongs to ''type A*. The flow graph of. this structure and its 'type B' equivalent are shown in fig. 4.26. ^ ( y d ) ' H &AK D' 1 d D Fig. 4.24(a) Configuration for the Generation of Sensitivity Functions for Example 4.2 4 4 1 1 „1 *^ CH—^—©•« Q Fig. 4.24(b) Generation of ( y , ) r by the Method of Section 4.2 122 R(s) F i g . 4.25 Block Diagram of Example 4.3 -H. 6 - 1 6 -H, 6 -H. F i g . 4.26 System Flow Graph and i t s 'Type B' Equivalent I t i s observed that the system structure belongs to the e r r o r sampled case of subsection 4.3.4. The configuration for s e n s i t i v i t y func-t i o n generation i s shown i n f i g . 4.27. A comparison of t h i s structure with the one obtained using the method of s e c t i o n 4.2 (not shown) i s made and i t i s found that the configuration of f i g . 4.27 needs l e s s components i f a l l the s e n s i t i v i t y functions are desired. 123 F i g . 4.27 Configuration f o r the Generation of S e n s i t i v i t y Functions f o r Example 4.3 Example 4.4 F i g . 4.28 shows the block diagram of a system which does not conform with e i t h e r the 'type A' or the 'type B' stru c t u r e . However, by flow graph reductions, t h i s structure can be modified to that shown i n f i g . 4.29(a). I f now the two forward paths with transmittances G„ and D, G_ (1/G ) are . 2w b 7w J b combined to form a s i n g l e path, the structure w i l l belong to 'type A'.' The. 'type B' equivalent of t h i s graph i s shown i n f i g . 4.29(b) and the transmittance G_G„, + G_,D, i s taken as a s i n g l e element. The 'type B' equivalent belongs ' 3 2b lb b to the s p e c i a l case discussed i n subsection 4.3.3. The s e n s i t i v i t y function generation i s shown i n f i g . 4.30. A few 124 Fig. 4.28 Block Diagram of Example 4.4 (a) (b) Fig. 4.29 Simplified Flow Graph of Example 4.4 and its 'Type Bf Equivalent G3 G2 (VG9 o—^ *-© 2 <7d>G, Fig. 4.30 Configuration for the Generation of Sensitivity Functions for Example 4.4 simplifications have been carried out in arriving at this graph. The elements G^  and G^  appear twice in the system structure of fig. 4.29(b). In the sensitivity generation each such appearance is treated as a different element t and the sensitivities are obtained and later they are added to obtain (y,).-, and (y^Q • For the presence of G, in the feedback path as GK/Gj, the sensi-5' 4' tivity function is to be taken from the node at the input of branch G^/G^ via i a transmittance [3 (GC/G. )/3G. ]G., , which reduces to -(GCG./G.), . Hence the signal may be taken from the output of G^ /G^  branch and passed through an i element -G., . 1 4b The sensitivity function generation shown in fig. 4.30 is found to be considerably simpler than the one obtained using the method of section 4.2 126 (not shown). The saving i n the number of elements i s about 50% for analog-hybrid simulation. For d i g i t a l simulation, the computational time f o r s e n s i -t i v i t y function generation i s halved. 4.5 Conclusions In t h i s chapter, methods have been presented f o r the generation of s e n s i t i v i t y functions for sampled-data systems. A s t r u c t u r a l approach has been developed f o r general nonlinear multi-input multi-output systems. For a class of l i n e a r systems, i t has been shown that the number of simulator com ponents or the computation time i s reduced i f many s e n s i t i v i t y functions are required simultaneously. Comprehensive examples are presented i l l u s t r a t i n g the a p p l i c a b i l i t y of the methods. Unlike the continuous case 3^' 4"'", the component reduction i n sampled data systems f o r s e n s i t i v i t y function generation i s very complex. For the l a t t e r , i t has been shown i n t h i s chapter that a number of d i f f e r e n t cases have to be treated depending upon the p o s i t i o n of the samplers i n the system flow graph. A few new modifications have been used i n the notation and a p p l i c a t i o n of sampled-data s i g n a l flow graphs. I t has also been shown that i n the simplest case, where the input and output of each element are sampled, the s e n s i t i v i t y function generator reduces to a structure s i m i l a r to the Kokotovic structure f o r continuous systems"^'4"'". The new methods presented i n t h i s chapter are to be used i n the "parameter t r a c k i n g " problem discussed i n the next chapter. 5. PARAMETER TRACKING . ' 1 5.1 Introduction In situations where the time varying parameters of a sampled-data system are known ini t i a l l y , i t may be possible to track their time-variations with those of a model as shown in this chapter. Both linear and nonlinear systems are considered for this and use is made of sensitivity functions discussed in the last chapter. With suitable assumptions regarding the variation of the system and model parameters, methods are devised so that the model parameters closely follow the time-variations of the corresponding system parameters. Little work has been reported in the literature for the parameter tracking problem in sampled-data systems. In this chapter, the proposed schemes are divided into two types: the identification error method and the output error method. The identification error method ensures that the system is closely tracked by its corresponding model parameters whereas in the output error method only the system output is tracked by the model output. Since in designing optimal control systems, i t is often imperative that the para-meters of the system be known, in such situations, the use of output error necessitates the assumption that output tracking ensures parameter tracking. . Examples are presented illustrating the application of the parameter tracking methods. Problem Formulation Given a noise-free sampled-data system, which may be linear or non-linear, and whose d.e. is given by ydj = f ( P j ' r j ' r j - l " " , r j - N ' y d ( j - l ) ' y d ( j - 2 ) " - -••• yd(j-M) ) ; J ' N ' M E I . ( 5 - 1 } 128 where P = P(j), the time-varying system parameter vector, y ^ = y^(i), and r 1 = r ( i ) . Choose a model described by the d.e., y. = f(P., r., r . ^ , . . . , ^ , y.^, y._2,... ,y._M) where P_. = P(j), and y^ = y(i). Given that in i t i a l l y , (5.2) P = P (5.3) o o and that the variations of P. are slow. J Adjust Pj such that the tracking of parameters is achieved. Assumptions The slow variation of P, given in Problem Formulation, may be in-corporated as an assumption given below. Assumption 5.1 The parameter vector, P, changes by a small amount at instant unity and remains constant at that value over a finite interval of time equal to the identification time, J.T , i.e., P = P(J-i) = P(l) f P(0) 0 .<__ i < J - l ; i , Jel The following assumption is made concerning the method of parameter tracking to be developed: Assumption 5.2 The rate of adjustment of model parameters is kept low. 5.2 Identification Error Method 5.2.1 The Algorithm Based on the assumptions made in section 5.1, an identification error method is developed below for solving the parameter tracking problem. Step 1. Choice of performance criterion Let a quadratic PC be chosen; V I(j) - E T ( j ) M E(j) = ETMEJ (5.4) where the identification error, E . is defined as 3 j j j Step 2. Choice of incrementing algorithm Let the algorithm be chosen as A = h.F. ] where h. is a scalar and F., a vector. J 3 Step 3. Choice of h^ and the determination of AV^(j) The increment on V^(j) is determined to be AVT(j) = Vx(j+1) - Vj(j) T T = E..- ME. - E . M E . j+1 j+l j 3 =2 ET'MAE. + A E T M A E . 3 3 3 3 E . = P - P. (5.5) where AE. - E... - E . 3 3 +1, 3 • - ( P-P. + i) - (P-P.) 3 making use of Assumption 5.1. From eqns. (5.6), (5.7) and (5.8), AVT(i) = -2h.E?MF. + 1I?FTMF. • < , I 3 3 3 3 3 3 Choosing to minimize AV^ .(j) at every instant, the optimum h^ is given by j j Hence from eqn. (5.9), h = - f — 1 (5.10) FTMF. (E; M F . ) Z AV (j) = 1 -1 <_0 (5.11) T F .MF. i 3 Step 4. Determination of AP. J If now F^  is chosen such that the vector, hjF^ is easily computable, then the minimization problem is solved. A Making use of Assumption 5.1, and eqns. (5.3) and (5.10), AP.. is given by - t . (P-P.JKF A P r - ^ - ^ F J F.MF. J 3 3 [(P-P ) - (P.-P )]TMF. 2 J _ ° 1 F T 1 ~ F^ MF. J j J where AP = P -P = P-P « o 1 o o [AP - (P.-P )]TMF. - ° T 1 ° L F . (5.12) FT MF. 2 3 3 Writing y ' and y. of eqns. (5.1) and (5.2) as y,.(P.) and y.(P.).. respectively, and expanding them about P and P , the following equations are obtained: o o = y^&J + [ V P y H i ( p n ) ] A P n + ( 5 - 1 3 ) dj j dj o P q dj o o y j ( P J ) = y j ( P o } + C VP y j ( P o ) ] ( P j " P o ) + * 5 1 ' 4 ) where 3 y d j ( P o ) dW?o) v y , , ( P N ) = [—rz r: ], (5.15) p d J ° 8 pio 9 p20 o ' . 3y.(P N) ay.(P ) V- y (P Q) = [ K ° 2\° ] (5.16) 9 pio 9 p20 V .'I 131 * th " . and p. , p. are the i elements of P and Y »respectively. r i o r i o o o Invoking Assumptions 5.1 and 5.2, eqns. (5.13) and (5.14) can be truncated at the first order terms to give t / W ~- V V + w T ( j ) A P o ( 5 a 7 ) y.&J - y.&0) + w T(j)(p J-P Q) (5.18) where W(j) and W(j) are the respective sensitivity function vectors, defined b y wT(j) = 7 y D , ( P 0 ) (5.19) ° . WT(j)= V- y (P q) (5.20) ° Noticing from eqn. (5.3) that P O=P q, and using Assumptions 5.1 and 5.2 for the slow variation of P and P, the following approximation is made: v v =vv (5-2l) Hence, using eqn. (5.3), i t is plausible to write — ^ - 2 - (5.22) 9 p i o 8 p i o which, from eqns. (5.19) and (5.20), leads to the following: W(j) = W(j) (5.23) Now, from eqns. (5.17) and (5.18), i t is seen that, y d j ( V ~ y j ( V = [ y d j ( p o ) - y j ( V ] . ' " t ^ O J A P ^ a x P j - ^ ) ] - W T(j)[AP -(P.-P )•] . (5.24) o 3 o by virtue of eqns. (5.21) and (5.23). Noticing that y ^ j ^ j ^ a n d y j ^ P j ^ a r e a v a i ± a D i e a s t n e outputs of the system and the model, respectively, and assuming for the present that W(j) is available from measurements, a logical choice for vector F. to render AP. 132 in eqn. (5.12) computable from measured quantities, is F = M"1 W(j) (5.25) Hence, AP. is obtained as J [ydjO\)-y (P ) ] - l -AP, = -AT 1 ^ 1 M W(j) (5.26) 3 Wi(j)M~±W(j) 5.2.2. Practical Considerations 1) Sensitivity functions: Assumption 5.2 concerning the slow variation of P.., permits the following approximation: a y , ( P J 3y . (P.) 3p. ap. . Hence from eqns. (5.16) and (5.20), the sensitivity function vector W(j ) is given b y ~r .• 9 y i ( P V ay^V W'(j) = hr1- 1 (5.28) ; 9 P l j 9 P2j Recall that the generation of sensitivity functions is discussed in chapter 4 under the assumption of time-invariant parameters.. Though the parameter vector P in the present case is incremented at each instant, Assump-tion 5.2 validates the use of the methods of chapter 4 for sensitivity function generation. It may be seen that in the present case, the sensitivity structure is to be excited by signals from the model and not from the system and that the parameters of the.sensitivity structure and the model are incremented by the same amount each instant. 2) Increment Step Size: Since Assumptions 5.1 and 5.2 and eqns. (5.23) and (5.28) are realized only approximately in practice, the optimum value of h given by eqn. (5. is never atta.ined. The increment, AP^  , may have to be smaller than the value given by eqn. (5.26), contingent with Assumption 5.2. Writing eqn. (5.26) as AP = q — ^ r - ^ M W(j) (5.29) J WX(j) M_J- WX(j) where q is a scalar which determines the step size, and using eqns. (5.7), « y (5.8), and (5.24), AVT(j) = q ^ 2q = - T — [Ejw(j)] 1 V ( j ) M A W(j) W"(j) M X W(j) J 2 - q(q-2) (5.30) A m "1 ^ wx(j) M"x W(j) The range of q for making AVj.(j) nonpositive is given by 0 <_ q <_ 2. For obtaining an increment smaller than that given by eqn. (5.26) for P_., the range of scalar q is seen to be 0 < q < 1. 5.3 Output Error Method Step 1. Choice of performance criterion The PC to be minimized at instant j is chosen to be V (j) = e 2 (5.31) where the output error, e^  is defined as ^ O ' 1 W - W . < 5 - 3 2 > Step 2. Choice of incrementing algorithm Choose AP = h.F. (5.6) j J J Step 3. Choice of h. and the determination of AV (j) 3 ° The increment AVQ(j) is given by AVo(j) = V o(j+l) - V Q(j) - . - t y d ( j + 1 / p j + 1 ) - y j + 1 ( p j + 1 ) ] 2 - t y ^ C P ^ - y ^ V 1 ' ( 5 - 3 3 ) Now, using a power series expansion for y ,(P. j+1 j+1 = y j + 1(P j) + [v p y j + 1(p.)]AP. + .... (5.34) Invoking Assumption 5.2, and retaining only terms to first order in the above expansion, eqn. (5.33) may be written as AVJ> - [ y d ( J + i ) ( p ) - y J + i ( v - ( v ^ y i + i ( v ) h J F j ] 2 s, 3 - t y d j ( p ) _ y j ( V l 2 ( 5 - 3 5 ) Minimizing AVQ(j) with respect to h, the optimum h is obtained as 3 [ 7? 4.' j +i< V»j Hence, from eqn. (5.35), A V O ( J ) = - t y D J ( P ) - y j ^ j ) ! 2 (5.36) = -e 2 <. 0 . (5.37) Step 4. Determination of AP. J From eqns. (5.6) and (5.36), AP.. is given by ~ [ yd(1+l) ( P ) ~ y i + l ( P i ) ] AP. 9IJ+i2 _ 1 + 1 .1— F. (5.38) J [ v / i + i ( v ] . y 3 To obtain AP.. in terms of known quantities, the following simplifications are in order. Expanding y d(j+i) a n d y j + ^ ( P j ) i n power series about the time instant j , / T > x yd^+D^j^ = ydj ( p j ) + 8 t ~ ~ T + <5-39> 135 9y,(p.) y j + i ( V = W + at T + ( 5 ' 4 0 ) Various orders of expansions can be used on the lines of subsection 2.3.1. However, from a practical point of view, the computation of partial derivatives is difficult. A zero order expansion would eliminate the need for these parital derivatives. Accordingly, * d 0 + D < V - W ( 5 - 4 1 ) w v - v v ( 5-4 2 > Hence from eqn. (5.38), [ yd J( V - V V ] F j AP. = : (5.43) J [Vp y.(P.)]F j Using eqn. (5.28), AP_. may be written as AP, = ^ r — 1 — F 2 W i(j)F j 2 Any measurable function can be chosen for F to render AP^  easily computable. On such choice is F = M_1W(j) (5.25) Hence AP, is given by , A 2 . e M_iW(j) AP. = V YT. (5.45) 2 Wi(j)M"XW(j) Remark 5.1 A comparison of eqns. (5.26) and (5.43) shows that the incrementing algorithms of the identification error method and the output error method are the same. However, i t is observed that no i n i t i a l knowledge of the parameters is assumed in the output error method and that P(0) and P(0) need not be equal. Thus under the additional assumption that initially the parameter values of the model and the system are the same, i t may be inferred that the output tracking, obtained from output error method, would imply parameter tracking also. • • . • ..-V'..' . '136- • If however, ini t i a l l y the model and the system parameter vectors are not equal, then output tracking may not lead to parameter tracking, even i f first and higher order expansions are used. « J • • 5.4 Examples • ' e In this section, examples are presented illustrating the application of the parameter tracking method developed in sections 5.2 and 5.3. Unless stated, otherwise, the following comments are valid for a l l the examples. j Assumption 5.1 is considerably relaxed, and the rate of time varia-tion of the system parameter vector P is made significant. The.input r to the system and to the model is chosen as a sum of sinusoids with random frequencies. Each example is solved for many values of q (eqn. (5.29)) and i t is found that the parameter tracking algorithm is stable for a wide range of step sizes. Adjustment of model parameters is begun after the transients die down, and the time on the respective plots is counted from this instant. The Continuous System Modelling Program (CSMP) has been used on an I.B.M. 360/67 computer for simulating the examples. Example 5.1 Consider the linear system shown in fig. 5.1. The parameter p^ is a staircase function as shown in fig. 5.2, and the parameter p 2 is given by r p = 0.75 + 0.5sin(2irj/300) ZJ , It is desired to track the time-variations of p^ and p 2 by p^ and p 2 respectively. Fig.. 5.1 shows that the system belongs to the error sampled case of subsection 4.3.4 and hence the method detailed there is used for generating I . I the sensitivity functions, y^ and y" . The matrix M in eqn. 5.29 is chosen Pi P2 to be the identity matrix. The parameter tracking is shown in fig. 5.2. Notice that i n i t i a l l y , System Fig. 5.1 Block Diagrams of System and Mbdel for Example 5.1' p^p^ which is a severe restriction as compared to the assumption in the problem formulation. It is observed from .the figure, that the tracking is. b I good throughout the range of observation, except where p^ has step changes in magnitude. \ Example 5.2 A single loop nonlinear system shown in fig. 5.3 is considered in this example. The nonlinearity in the system and the model are given by f = P l e z + p e* t f =_Plez + p 2e z (5.46) (5.47) Parameter p^ is considered time-invariant with a value of one unit and parameters p 2 and p^ are staircase functions as shown in fig. 5.4. The i n i t i a l values 1. -J OQ <U 'O rH tO > 0) u -s-to 100 200 300 400 500 Time in sees. Fig. 5.2 Parameter Tracking for Example 5.1 System T=lsec Zero Order Hold f y 0 e 2 Model Fig. 5.3 Block Diagrams of System and Model for Example 5.2 of none of the parameters is assumed known and PQ^PQ. The matrix M is chosen to be the identity matrix. Sensitivity i f ? functions, y" , y" and y" are generated on the computer using the method of p l P 2 P 3 • . -section 4.2. It is seen from fig. 5.4 that the parameter tracking is adequate. After each step change of p^, the model parameters settle down to tracking the corresponding system parameters. Example 5.3 Consider the system shown in fig. 5.3. Let a l l the three system parameters vary with time as shown in fig. 5.5. Initially, P3Q =P3Q> B U T PIQ and P 2Q are assumed to be unknown. The model is considered to be purely linear 140 1.6 P< i»4f • 1.2 1.0 w - >, •5 0.81 0.6 A A ' 0.4 I 0.2 ^ 1 I f . .r . / v . v or -0.24 r I V ~~l I J '300 i ' \/L00 200 400 i Time in sees, Fig. 5.4 Parameter Tracking for Example 5.2 at start with p = 1 and p 2^ = 0. Fig. 5.5 shows the parameter tracking. The variations of p^ and p are shown at 10 times the scale of the others. It is seen that p and p closely follow the variations of the corresponding system parameters, but in comparison the tracking of p 2 is inferior. However, i t is to be noted that the level of input magnitude r^ .was kept low from considerations of stability of the system and hence the contribution of the cubic term in eqn. (5.46) is small. Therefore, with large disparity in the values of p 2j and p2_., i t is possible to have small error e., with the result, the increment, Ap2. (eqns. (5.29) and (5.45)) is small and ineffective. 5.5 Conclusions New methods have, been developed in this chapter for tracking system parameters. It has been shown that both the identification error method and the output error method result in the same incrementing algorithm. The output error method does not require the i n i t i a l values of the system parameters to be known but parameter tracking is not guaranteed without additional assumptions. The examples indicate that the parameter tracking method is applicable to a wider class of parameter variations and under more severe conditions than has been assumed in the theory. 143 6. CONCLUSIONS 6.1- Summary f This work has been concerned with the identification of discrete-time systems and parameter tracking of sampled-data systems, by methods suitable for small computers. The identification problem has been formulated for linear and nonlinear discrete-time systems in chapters 2 and 3 respectively and methods of solution have been proposed. The" philosophy of identification has been broadly divided into three approaches, using either identification error, output error, or equation error methods. A common error function minimization approach, incor-porating uniqueness and stability properties, has been applied to each method and algorithms have been obtained which are attractive in their simplicity. It has been shown theoretically and demonstrated with practical examples that the proposed identification error method solves the problem successfully. The generation of sensitivity functions for sampled-data systems has been considered in chapter 4 and their application to the parameter tracking problem in chapter 5. A structural approach has been developed for the generation of sensitivity functions for nonlinear sampled-data systems. It has been shown that for the simultaneous generation of many sensitivity functions in a class of linear sampled-data systems, the number of simulator components or the computation time is reduced. Examples have been presented illustrating the applicability of the sensitivity function generation methods. The parameter tracking problem has been formulated in chapter 5 and the identification error and output error methods have been applied for solving i t . Use has also been made of the sensitivity function generation discussed in chapter 4. It has been shown in theory that under suitable assumptions, para-meter tracking is possible, and.the example indicate that the method may be--.' V;- S'V.-.'.-'-v •; .!:.7:\'':: ::;'V:'-'. .:-' .V- ^ •;v:-\"-:-K--"-^ ;-[ 144 -applicable to a wider class of problems, under more severe conditions, than has been assumed in the theory. The work presented in this thesis has the following novel features. The procedure presented in chapters 2 and 3 encompasses a class of on-line sequential methods for solving the identification problem. Thus many algorithms proposed in the past both for discrete-time and continuous time systems occur as particular cases of the general approach presented here. Besides, many new algorithms have also been obtained, particularly in the identification of nonlinear systems and in the output error method for the linear case. Further-more, unlike the past works, the stability of the proposed algorithms has been investigated allowing the time-variation of the system coefficients and making use of the concepts of eventual stability. Extensions of the existing method for continuous systems, has been made to the generation of sensitivity functions for nonlinear sampled-data systems. For a particular class of continuous, linear systems, i t has been previously shown that the sensitivity functions may be generated very simply. In this thesis, an attempt has been made to find a corresponding class of systems in the sampled-data case. It has been shown that such a class exists, but that i t is restricted and factors such as the location of samplers can sometimes cause difficulty. A few new modifications have been made in the notation and in the application of signal flow graphs for sensitivity function generation. .The methods presented in chapter.5 for the solution of the parameter tracking problem for linear and nonlinear sampled-data systems are new. In Appendix B, a new extension for the generation of the second order sensitivity functions for nonlinear sampled-data systems is presented. •sJ.45 6.2 Further Research Methods such as quasilinearization and invariant imbedding have been found to be useful in practical situations. The proposed techniques may be of value in cases where limited computational facilities preclude the use of the above methods. However, modifications as suggested below are required to make the method practically attractive. The conditions on the input to the system, set forth in chapters 2 and 3, warrant further investigation, since in optimal control situations they may not be satisfied. It is seen from subsection 3.4.2, that when the form of nonlinearity in the system difference equation is unknown, only approximate iden-tification results. The instantaneous error criterion used in this work, is intended for systems whose difference equations can be written in the form of eqn. (3.2). Hence the situations, where the forms of the nonlinearities in the system are unknown or signals are corrupted by noise, need further investigation. 25 It is felt that a quasi-iterative approach would be adequate for such problems. If the order of the system difference equation is unknown, or i f i t is desired to model a higher order system with a lower order model, then the problem is to be formulated giving consideration to the input and to the type of performance 54 function, since the "best" model involves both. Further, i t is also seen that stability constraints are needed for the model since i t is possible that the best model may be unstable at some instants. The methods proposed for the generation of sensitivity functions for sampled-data systems suggest numerous applications. Parameter optimization for system synthesis and adaptive control of systems are two such areas where the works proposed for continuous systems^'^'"'^, may be extended to sampled-data systems. -Generation of second order sensitivity functions, proposed in Appendix B, may be applied to sensitivity reduction . in sampled-data systems 48 49 on the lines of corresponding work for continuous systems . ' Another area where further research is warranted is the identification of the parameters of sampled-data systems. As observed from chapter 1 , the coefficients k. of the difference equation describing the system, are functions of the system parameters, p^. The method of identifying the coefficients k^ and then solving for p_^  is often impractical, since, in general i t involves the solution of a set of simultaneous nonlinear equations. The parameter tracking schemes for sampled-data systems suggest the possibility of developing similar methods for adaptive control"'7. APPENDIX A Eventual Stability The following definitions and theorems are essentially those of 31 LaSalle and Rath , modified to suit the stability of discrete-time systems. Consider the fundamental system whose d.e. is given by Ax(j) = f(j,x) (A.l) In some region ft , i t is assumed that j > 0 and - ( 4 + x 2 + ) 1 / 2 < \ Let x(j,Jo,x°) denote the solution of eqn. (A.l) that starts at instant i at x°, i.e. J o a , oN o .j >x ) = X o o Definition A.l Eventual stability. The origin is said to be eventually stable i f , given e. >0, there exist numbers 6 and J such that ||x°|| < 6 implies I I x(j >J0,x°) || < e for a l l j >_ j Q >_ J where j ,jQ£l. Definition k-.2 Eventual asymptotic stability. The origin of the system of equations (A.l) is said to be eventually asymptotically stable, i f (i) i t is eventually stable and (ii) there is an r > 0 and a J q such that | |x°| | < r and j ±.JQ implies x(j,j Q,x 0) ->• 0 as j -> °°. Theorem A.1 Let the system of equations (A.l) be autonomous or let the origin be an equilibrium state of eqn. (A.l). Let V(j,x) be positive definite in n and have the property that V(j ,x) •> 0 as |jx|| -> 0 uniformly in j , j e l . Let AV(j,x) <_ 0 for a l l 0 <_ | |x| | < R and for a l l j >_ j ; j , j el. Furthermore, . X """"" o o let AV(j,x) £ u (J) for r < | |x| | < Rx where for each r > 0 , • * . C O I \x(5) = -~ (A.2) Then the origin of the system of equations (A.1) is uniformly asymptotically stable in. £2 . If R = °°, then the origin is said to have uniform asymptotic stability in the large. ^ Theorem A.2 Assume that the system Ax(j) = X(j,x) has a uniformly asymptotically stable origin. Then the system Ax(j) = X(j,x) + p(j,x) is eventually asymptotically stable i f |p(j,x)| <_h(j) for | |x| | <_ r Q(r Q> 0), where h(j) satisfies the condition - i j i e J E e h(i) •+ 0 as j + » (A.3) i=0 Note Condition (A.3) is satisfied i f |h(j)| is bounded, and h(j) •>• 0 as j + M. APPENDIX B Extension of the Vuscovic and Ciric Method for the Generation of Second Order Sensitivity Functions The second order sensitivity function of a system is defined to be A A 9 2 y d ( j ) W ( i ) = (v ) " = — qp V j ; ''Vqp 8p8q where p and q are constant parameters. Consider a multi-input multi-output nonlinear sampled-data system (fig. 4.1) whose structural equation is given by eqn. (4.7). Taking the first partial derivative of eqn. (4.7) with respect to q, eqn. (4.8) is obtained. Again, taking the partial derivative of eqn. (4.8) with respect to p, the following equation is obtained: L • S Z (a..u'.' +g..v'.' )+ Z (v..wV +6..xV ) I = 1 I J iqp I J iqp i = 1 I J iqp 1 3 iqp ... N C + Z (e..sV +<{>..z'.' ) + Z .. (y,) V =0 (B.l) 1 = 1 I J iqp I J iqp i = 1 1 3 d iqp Eqn. (B.l) suggests that a system structure with the same internal couplings as those of eqns. (4.7) and (4.8) will generate the required second order sensitivity functions (y.)V . Let this structure be known as the second d iqp order sensitivity structure, S . The coupling between this structure and the qp structures S^ , S^ , and S is obtained by determining the relationship between each pair of input-output quantities of each element in the second order structure. Thus, taking partial derivatives with respect to p of equations (4.9) through (4.14) the following equations are obtained! V V (t) = fZ g'.' (T)u.(t-T)dT+/ t g! (T)U| (t-T)dT+/ t g! ( T)u! (t-l)dT iqp o iqp i o °iq lp o °ip iq + ! t g.OOuV ('t-T)dT (B.2) o l iqp "150' — k - - k v" (kT) = r dV (mT)u.(kT-mT) + £ d'. (mT)u'. (kT-mT) + E d I p ( m T ) u l q ( k T _ m T ) iqP ! m= 0 i q p 1 m=0 i q 1 P m=0 +—:Z-d7-(mT)u':--(-kT-mT) ' ' / r(B.3) m=0 1 i q P ' _JE? (t) = w'.' (kT) , t = kT iqp xqp ="0 , otherwise (B.4) M M ' n - ^ n*.' (t-kT)w. (kT-mT) + E h ! (t-kT)w! (kT-mT) X i q p ( t ) = E i m q P 1 m=0 i m q . 1 P n r m=0 ' ' . . _ M M " +"""E"h!" (t-kT)w! (kT-mT) + E h. (t-kT)w" (kT-mT.) m=0 m p i q m=0 m i q p (k+l)T > t >_kT (B.5) ... « _ i + l - . s ! + i s! -s! + _ ^ i g . . s + f •. - Z i q P ( t ) 3s, S i q p + - 3 P 3 8 , ^ " ^ — X - q i p i q P ( B' 6 )-S . 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