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Implementation of an adaptive controller on a TI TM 990 microcomputer Tooba, Bahram 1982

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IMPLEMENTATION OF AN ADAPTIVE CONTROLLER ON A TI TM 990 MICROCOMPUTER by Bahram Tooba B.Sc. Purdue U n i v e r s i t y , 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the department of E l e c t r i c a l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March 1982 © Bahram Tooba, 1982 In 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 of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the 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 study. I f u r t h e r agree 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 copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood 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 not be allowed without my w r i t t e n p e r m i s s i o n . Bahram Tooba Department of E l e c t r i c a l Engineering The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 D a t e March 11 1982 D F - f i f ? / 7 9 i A B S T R A C T The t h e s i s t e s t s the implementation of an a d a p t i v e i d e n t i f i e r and c o n t r o l l e r on a Texas Instrument TM 990 microcomputer system. The a d a p t i v e a l g o r i t h m was proposed by Martin-Sanchez [20], and was s e l e c t e d because o f the small amount of computation needed i n comparison with methods such as r e c u r s i v e l e a s t squares e s t i m a t i o n . The c o n t r o l l e r i s s i m i l a r to a s i m p l i f i e d minimum v a r i a n c e c o n t r o l l e r used i n s e l f - t u n i n g r e g u l a t o r s . The t h e s i s d e s c r i b e s the a d a p t i v e a l g o r i t h m and the hardware and software used i n the microcomputer implementation of t h i s method. The r e s u l t s show o p e r a t i o n of the a l g o r i t h m s f o r f i r s t - o r d e r and second-order p r o c e s s e s . TABLE OF CONTENTS TABLE OF CONTENTS . . i i LI ST OF TABLES i v LIST OF ILLUSTRATIONS . . .v ACKNOWLEDGEMENTS ..... . . . . v i i I . ADAPTIVE CONTROL 1 1. I n t r o d u c t i o n ..... 1 2. Adaptive Parameter E s t i m a t i o n and C o n t r o l ..........4 3. Computer S i m u l a t i o n 9 4. The O u t l i n e of T h i s T h e s i s .10 11. IDENTIFICATION AND CONTROL ALGORITHM ..... 12 1 . Ove rv i ew 12 2'. D i s c r e t e - T i m e Systems « ....... v .... 1 2 3. An A l g o r i t h m f o r Model Reference Adaptive Systems ..13 3.1. Formulation . ......16 3.2. H y p e r s t a b i l i t y of the C o n t r o l System ............20 3.3. R e l a t i o n Between e(k) and s(k) ...23 3.4. Convergence of the C o n t r o l Block Parameters ...25 3.5. C o n c l u s i o n s .27 I I I . HARDWARE AND SOFTWARE 28 1. Overview ......28 2. The Hardware 28 2.1. TM 990/101M- 1 Microcomputer 28 2.2. RTI-1241-S Board ....29 3. The Software .......30 3.1. The F i r s t - O r d e r Systems .30 3.2. The Second-Order Systems 34 3.3. A r i t h m e t i c Operations 35 3.4. The Sampling Rate 37 3.5. The Computer Programme 39 - i i i -IV.. MICROCOMPUTER EVALUATION OF THE ADAPTIVE IDENTIFIER AND CONTROLLER ............... .......... 42 1. Overview .....42 2. F i r s t - O r d e r Processes ....43 2.1. I d e n t i f i c a t i o n 47 2.2. C o n t r o l 51 3. Second-Order Processes 58 3.1. I d e n t i f i c a t i o n .58 3.2. C o n t r o l ..62 V. CONCLUSION .67 1 . Summmary 67 APPENDIX A THE SYSTEM HARDWARE . . . 70 A . l . TM 990/101M-1 Microcomputer .......70 A. 2. RTI-1241-S Board 71 A.3. TI 745 Terminal ...73 A. 4. TM 990/510 Card Cage . ...73 A.5. The Power Supply .". 73 REFERENCES 75 - i v -LIST OF TABLES Table 3.1 The sampling r a t e s and t h e i r c o r r e s p o n d i n g hexadec imal coded data. 38 -v-LIST OF ILLUSTRATIONS F i g - 1.1 The Block Diagram of (a) RLS Adaptive I d e n t i f i e r (b) S e l f - T u n i n g Regulator (STR). ......5 F i g . 1.2 Basi c C o n f i g u r a t i o n of a Model Reference Adaptive system (MRAS) 8 F i g . 2.1 C o n f i g u r a t i o n of the Adaptive System i n : (a) I d e n t i f i c a t i o n Mode, (b) C o n t r o l Mode. .......14 F i g . 2.2 General C o n f i g u r a t i o n of an A s y s m p t o t i c a l l y H y p e r s t a b l e Feedback Autonomous System. ..........19 F i g . 2.3a The N o n l i n e a r A s y p t o t i c a l l y H y p e r s t a b l e Feedback Autonomous Systems, .21 F i g . 2 = 3b The E q u i v a l e n t Autonomous System .......21 F i g . 3.1 Flowchart f o r the I d e n t i f i c a t i o n of a F i r s t - O r d e r System ....32 F i g . 3.2 Flowchart f o r the C o n t r o l of a F i r s t - O r d e r System 33 F i g . 3.3 Flowchart f o r two's Complement M u l t i p l i c a t i o n . ' ...36 F i g . 4.1 The Block Diagram of the Adaptive System i n I d e n t i f i c a t i o n Mode 4-4 F i g . 4.2 The Block Diagram of the Adaptive System i n the C o n t r o l Mode 45 F i g . 4.3a The Estimated Parameters, .48 F i g . 4.3b The Process Output and The P r e d i c r e d Output 48 F i g . 4.4a The Estimated Parameters 49 F i g . 4.4b The Process Output and The P r e d i c r e d Output 49 F i g . 4.5a The Estimated Parameters. 50 F i g . 4.5b The Process Output and The P r e d i c r e d Output 50 F i g . 4.6a The Estimated Parameters. ...52 F i g . 4.6b The Process Output and The P r e d i c r e d Output 52 F i g . 4.7a The Estimated Parameters. ..53 F i g . 4.7b The Process Output and The P r e d i c r e d Output 53 F i g . 4.8a The Process Output f o r the F i r s t - O r d e r System. ...54 F i g . 4.8b The Reference S i g n a l . 54 F i g . 4.9a The Process Output 56 - v i -F i g . 4»9b The C o n t r o l S i g n a l . ..56 F i g . 4.10a The Process Output ....,57 F i g . 4.1 Ob The C o n t r o l S i g n a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 F i g . 4.11a The Estimated Parameters. ..........59 F i g . 4.11b The Process Output and The P r e d i c r e d Output 59 F i g . 4.12a The Estimated Parameters. 60 F i g . 4.12b The Process Output and The P r e d i c r e d Output 60 F i g . 4.13a The Estimated Parameters. ..61 F i g . 4.13b The Process Output and The P r e d i c r e d O u t p u t . . . . . . 61 F i g . 4.14a The Process Output. 63 F i g . 4.14b The C o n t r o l S i g n a l . ...63 F i g . 4.15a The Process Output f o r the Second-Order System. ..65 F i g . 4.15b The Reference S i g n a l 65 F i g . 4.16a The Process Output ........66 F i g . 4.16b The C o n t r o l S i g n a l ....66 - v i i -ACKNQWLEDGEMENT I wish to thank Dr. M.S. Davies for h i s patience, help, and encouragement during the l i f e of t h i s p r o j e c t . 1 I . ADAPTIVE CONTROL 1. I n t r o d u c t i o n Adaptive c o n t r o l has been a c h a l l e n g e f o r c o n t r o l e n g i n e e r s f o r a long .time, the f i r s t r e s e a r c h papers on the s u b j e c t being p u b l i s h e d i n the e a r l y 1960's, The i n t e r e s t i n a d a p t i v e c o n t r o l arose from the need f o r improving the performance of i n c r e a s i n g l y complex e n g i n e e r i n g systems i n v o l v i n g l a r g e u n c e r t a i n t i e s , of systems w i t h unknown and time v a r y i n g parameters. An a d a p t i v e system i s one t h a t c o n t i n u a l l y monitors changes i n t;ke p r o c e s s behaviour and a d j u s t s i t s control, s t r a t e g y a u t o m a t i c a l l y to maintain good, performance. T h e r e f o r e , these systems ar,e g e n e r a l l y n o n l i n e a r . In s p i t e of much i n t e r e s t i n a d a p t i v e c o n t r o l , only modest p r o g r e s s has been made in i n t r o d u c i n g these c o n t r o l l e r s i n t o i n d u s t r y . C o n t r o l l e r s a l l o w i n g p r o p o r t i o n a l , i n t e g r a l and d e r i v a t i v e a c t i o n (PID) are e x t e n s i v e l y used, with ..almost 90 p e r c e n t of e l e c t r o n i c analog' c o n t r o l l e r s used i n the i n d u s t r y being three term c o n t r o l l e r s of t h i s type [27]. As i s e v i d e n t , most i n d u s t r i a l p r ocesses can be c o n t r o l l e d s a t i s f a c t o r i l y with PI or PID c o n t r o l l e r s , and indeed many pr o c e s s e s have been designed to be c o n t r o l l e d by PID c o n t r o l l e r s [ 1 ] , In c o n t r o l loops that are not c r i t i c a l , these three term c o n t r o l l e r s w i l l undoubtedly c o n t i n u e to be used e x t e n s i v e l y i n the f u t u r e . With an i n c r e a s i n g - demand f o r e f f i c i e n c y i n the use of 2 energy and raw m a t e r i a l s , however, there i s a c o r r e s p o n d i n g i n c r e a s e i n the number of c o n t r o l problems which r e q u i r e c o n t r o l l e r s more s o p h i s t i c a t e d than the c o n v e n t i o n a l PID type. Altough i t i s f a i r l y easy to tune a PI c o n t r o l l e r which has only two terms, once o p e r a t i n g i n an i n s t a l l a t i o n where there may be s e v e r a l hundred of these c o n t r o l l e r s , i t i s a s u b s t a n t i a l amount of work to keep a l l of them w e l l - t u n e d i f f o r any reason the most d e s i r a b l e s e t t i n g s f l u c t u a t e . A PID c o n t r o l l e r which has three or four parameters, however, i s not very easy to tune, p a r t i c u l a r l y i f the process dynamics- are slow. T h e r e f o r e the d e r i v a t i v e a c t i o n i s o f t e n switched o f f i n i n d u s t r i a l c o n t r o l l e r s although i t might be used to advantage. More c o m p l i c a t e d c o n t r o l l e r s which i n c l u d e feedforward, feedback and observers o f t e n have a l a r g e number of parameters which can be a d j u s t e d , and t h i s can not be done without a s y s t e m a t i c adjustment procedure. Adaptive c o n t r o l l e r s attempt to address t h i s problem. The d i f f i c u l t y of implementing advanced c o n t r o l procedures i s a reason i n e x p l a i n i n g why modern c o n t r o l theory has not been used more e x t e n s i v e l y . Some of the a l g o r i t h m s proposed f o r example, are c o m p u t a t i o n a l l y too d i f f i c u l t to implement on sma l l computers. In d e s i g n i n g advanced c o n t r o l l e r s , i t i s o f t e n necessary to develop a mathematical model f o r the p r o c e s s and i t s d i s t u r b a n c e s , and to d e r i v e the c o n t r o l l e r parameters u s i n g these model and d i s t u r b a n c e c h a r a c t e r i s t i c s . The a p p r o p r i a t e mathematical model can be obtained from p h y s i c a l c o n s i d e r a t i o n or system i d e n t i f i c a t i o n . The drawback with such a procedure i s 3 that i t i s time consuming, and may need the conti n u o u s a t t e n t i o n of personnel with s k i l l s i n m o d e l l i n g , system i d e n t i f i c a t i o n and c o n t r o l d e s i g n . The ad a p t i v e c o n t r o l l e r can be regarded as a convenient combination of system i d e n t i f i c a t i o n and a c o n t r o l design techniques that can operate without s u p e r v i s i o n . In a d d i t i o n , i f the system parameters and i t s d i s t u r b a n c e s t a t i s t i c s are slowly v a r y i n g with time, a p r o p e r l y designed a d a p t i v e c o n t r o l l e r can c o n t i n u o u s l y tune the system f o r c l o s e to o p t i m a l performance. Recent i n c r e a s e d i n t e r e s t i n a d a p t i v e c o n t r o l has c o n c e n t r a t e d on procedures that s i m p l i f y the implementation of a d a p t i v e a l g o r i t h m s on minicomputers and microcomputers, thereby r e d u c i n g the c o s t , and p r o v i d i n g c o n t r o l engineers with a v i a b l e a l t e r n a t i v e to PID c o n t r o l l e r s . The success of these a d a p t i v e c o n t r o l l e r s has j u s t i f i e d the i n c r e a s e d i n t e r e s t i n them. Reference [14] d e s c r i b e s a s u c c e s s f u l a p p l i c a t i o n of an a d a p t i v e c o n t r o l l e r f o r a paper machine. The design of a d a p t i v e a u t o p i l o t s f o r s t e e r i n g l a r g e tankers seems l i k e l y to r e p l a c e c o n v e n t i o n a l a u t o p i l o t s , i n recent t e s t s [23], the a d a p t i v e a u t o p i l o t i n s t a l l e d on the tanker outperformed the c o n v e n t i o n a l PID a u t o p i l o t under almost a l l c o n d i t i o n s . T h i s d e s i g n i s based on r e c u r s i v e l e a s t squares e s t i m a t i o n with a minimum v a r i a n c e c o n t r o l l e r , the s o - c a l l e d " s e l f - t u n i n g r e g u l a t o r " . Other designs which can r e p l a c e the c o n v e n t i o n a l PID a u t o p i l o t are d i s c u s s e d i n [22], which d e s c r i b e s a design based on model r e f e r e n c e : t e c h n i q u e s . 4 2. Adaptive Parameter Estimation and Control The two primary approaches to parameter i d e n t i f i c a t i o n that have been developed are; i) Recursive Least Squares (RLS), and i i ) Model Reference Adaptive Systems (MRAS). Both methods have received great attention in the l i t e r a t u r e . The previous work done on these methods w i l l be b r i e f l y described, although i t i t must be realized that there have been hundreds of papers published on these methods during the past 20 years. The Recursive Least Squares method has been developed and successfully implemented in the industry, when combined with a suitable control algorithm. [2] i s a good survey of t h i s method. F i g . 1.1 shows the block diagram of a self-tuning regulator. In t h i s method the parameters of a difference equation model are determined by minimizing the cost: K V= E |y(kT)-y(kT)| 2 with respect to these parameters, here y(kT) i s the process output and y ( k T ) i s the prediction model output. The recursive algorithm can be found in [2] or [3]. The parameter estimates are updated after each sampling, and used in the tuning of the the control algorithm. In the work of Peterka [18], Astrom, and Wittenmark [4], and Wittenmark [19], the recursive least 5 u PLANT y + J u ADJUSTABLE y MODEL PARAMETER ESTIMATOR (a) r -u - c - CONTROLLER u PLANT y i. PARAMETER 1 1 ESTIMATOR 1 1 i j REGULATOR (b) F i g . 1.1 The Block Diagram of (a) RLS Adaptive I d e n t i f i e r (b) Self-Tuning Regulator (STR). 6 s q u a r e s pa rame te r e s t i m a t i o n i s combined v?ith a m i n i m a l v a r i a n c e c o n t r o l l e r „ In t h i s c a se the model s t r u c t u r e may be chosen so t h a t the e s t i m a t e d p a r a m e t e r s c o i n c i d e w i t h the c o n t r o l l e r p a r a m e t e r s , so t h a t the c o n t r o l l e r p a r a m e t e r s a t e c a l c u l a t e d d i r e c t l y . In the work of C e g r e l l and H e d q v i s t [ 1 4 ] , and B o r r i s o n and S y d i n g [15] s u c c e s s f u l a p p l i c a t i o n o f t h e s e ' s e l f - t u n i n g r e g u l a t o r s ' a r e r e p o r t e d . An i n t e r e s t i n g a s p e c t o f t h e second work. [15] was t h a t i t was p e r f o r m e d u s i n g t e l e c o m m u n i c a t i o n s between a p l a n t i n K i r u n a i n n o r t h e r n Sweden and a compute r i n Lund i n s o u t h e r n Sweden, 1800 km a p a r t . In [10 ] K u r z , I s e rmann , and Schumann e x p e r i m e n t a l l y compare and e v a l u a t e two r e c u r s i v e pa rame te r e s t i m a t i o n m e t h o d s , t he (RLS) e s t i m a t i o n and R e c u r s i v e Maximum L i k e l i h o o d (RML) e-3 t i mat i o n and 6 d i f f e r e n t c o n t r o l a l o g o r i t h m s , so c o n s i d e r i n g a t o t a l o f 12 a d a p t i v e c o n t r o l schemes. These a l g o r i t h m s were imp lemented on a p r o c e s s -.computer and t e s t e d on s i m u l a t e d a n a l o g p r o c e s s e s . I t was f ound t h a t the c o n t r o l a l g o r i t h m s p e r f o r m b e t t e r when used w i t h RLS e s t i m a t i o n , method , and t he c o n t r o l a l g o r i t h m c o u l d be a p p l i e d t o d i f f e r e n t t ype of p r o c e s s e s , e .g . . minimum phase and non-minimum p h a s e , s t a b l e and u n s t a b l e p r o c e s s e s , and p r o c e s s e s w i t h t i m e d e l a y . The a d a p t i v e r e g u l a t o r based on the (RLS) seems t o have a b r o a d f i e l d of a p p l i c a t i o n s , h o w e v e r a p r o o f o f the convegence o f a d a p t a t i o n a l g o r i t h m of t h i s t y p e i s n o t a v a i l a b l e i n s p i t e o f v e r y good e x p e r i m e n t a l r e s u l t s [ 3 ] , A n o t h e r p r o b l e m w i t h (RLS) e s t i m a t i o n i s t h a t not a l l the p a r a m e t e r s of the model can be d e t e r m i n e d f rom the i n p u t - o u t p u t o b s e r v a t i o n s when 7 feedback e x i s t s i n the p r o c e s s . Astrom and Wittenmark [4] e x p l a i n how to overcome t h i s problem and suggest some g u i d e l i n e s f o r unbiased parameter e s t i m a t i o n which a l l o w the esti m a t e d parameters to become very c l o s e t o t h e i r o p t i m a l v a l u e s . The Model Reference Adaptive method (MRAS), seems to have been i n t r o d u c e d by Whitaker, Yarmon,and Kezer [28] i n 1958. F i g u r e 1.2 shows the b a s i c c o n f i g u r a t i o n of standard MRAS. In t h i s method the input u i s fed both to the p r o c e s s and to the a d j u s t a b l e model with the d i f f e r e n c e between the p r o c e s s output and the model output being used to generate an e r r o r e ( t ) . T h i s e r r o r v e c t o r i s used by an adapt i v e mechanism t o a d j u s t the model parameters and f o r c e e-*-0 as t-*- °° . S i n c e i t s i n t r o d u c t i o n , there have been many papers and d o c t o r a l d i s s e r t a t i o n s d e d i c a t e d to the v a r i o u s types of MRAS. Landau [8] p r o v i d e s an e x c e l l e n t survey of MRAS from 1964 t o 1973, a recent book [9] d e s c r i b e s s e v e r a l d i f f e r e n t t echniques i n MRAS and p r o v i d e s good examples of a p p l i c a t i o n s . Landau uses a h y p e r s t a b i l i t y theorem proposed by Popov [16] f o r the proof of the convergence of MRAS. The a d a p t a t i o n laws have been form u l a t e d based on d i f f e r e n t methods, Pearson [13] developed an a l g o r i t h m based on the g r a d i e n t method, and t h i s method has been used by others such as Serdyukov [29], Here a g a i n there i s no a n a l y t i c a l proof of the convergence of the parameters e s t i m a t e s , although the experimental r e s u l t s a re good. A new method was proposed by Martin-Sanchez [ 2 0 ] , f o r an ad a p t i v e a l g o r i t h m which i s r e l a t e d to Popov's h y p e r t a b i l . i t y c r i t e r i o n DISTURBANCES PLANT y(t) u(t) DISTURBANCES e ( t ) ADJUSTABLE MODEL dft) ADAPTIVE MECHANISM Fig. 1.2 Basic Configuration of a Model Reference Adaptive System (MRAS) 9 [16] as a p p l i e d to d i s c r e t e systems [ 7 ] , but i s s i g n i f i c a n t l y d i f f e r e n t from the MRAS• f o r m u l a t i o n * In t h i s method the adjustment of the i d e n t i f i c a t i o n and c o n t r o l block parameters i s c a r r i e d out based the i n f o r m a t i o n a v a i l a b l e from the input and output of the p r o c e s s . T h i s method w i l l be d e s c r i b e d i n d e t a i l i n Chapter I I . [ 24 ] c o n t a i n s a l a r g e number of papers presented i n the 1979 Yal e U n i v e r s i t y Workshop on the A p p l i c a t i o n of Adaptive C o n t r o l . I t i n c l u d e s good t u t o r i a l s on both RLS e s t i m a t i o n and MRAS. A p p l i c a t i o n s d e s c r i b e d i n t h i s book are based mostly on these two methods. 3. Computer S i m u l a t i o n In order to assess t h e i r s u i t a b i l i t y f o r microcomputer implementation, the- r e c u r s i v e l e a s t squares e s t i m a t i o n method proposed by Astrom and Wittenmark [ 3 ] , and the Martin-Sanchez method were s i m u l a t e d u s i n g Amdahl/470 computer. The programmes were w r i t t e n i n F o r t r a n and the matrix o p e r a t i o n s were performed by s u b r o u t i n e s . The t r a n s f e r f u c t i o n s of the pro c e s s e s s i m u l a t e d were i d e n t i c a l f o r both methods, a f i r s t - o r d e r p r ocess and a second-order p r o c e s s . The r e s u l t s were very s a t i s f a c t o r y f o r both methods when the systems were no i s e f r e e and the estimated parameters reached t h e i r c o r r e c t v a l u e s . The performance of the two methods d e t e r i o r a t e d with the i n t r o d u c t i o n of nois e at the input and output of the two p r o c e s s e s . The (RLS) e s t i m a t i o n was found to be s l i g h t l y more s e n s i t i v e to the s e l e c t i o n of i n i t i a l parameters when system i d e n t i f i c a t i o n was performed. The computation time was a l i t t l e 10 h i g h e r f o r (RLS) e s t i m a t i o n . The number of a r i t h m e t i c o p e r a t i o n s , e s p e c i a l l y m u l t i p l y and d i v i d e , was much more i n the case of (RLS) e s t i m a t i o n as the order of the p r o c e s s e s , and so the number of parameters to be i d e n t i f i e d i n c r e a s e d . In the i d e n t i f i c a t i o n of the second-order p r o c e s s , the RLS method needed about 100 m u l t i p l y and d i v i d e o p e r a t i o n s compared with o n l y 25 o p e r a t i o n s f o r Martin-Sanchez method. Based on the r e s u l t s from the computer s i m u l a t i o n i t was d e c i d e d that the Martin-Sanchez method w i l l be c o n s i d e r e d i n our work mainly because i t i s c o m p u t a t i o n a l l y simpler than (RLS) f o r microcomputer implementation and a l s o because i t i s a method t h a t has not been implemented on a m i c r o p r o c e s s o r b e f o r e . 4. The O u t l i n e of T h i s T h e s i s The aim of t h i s t h e s i s i s to implement and e v a l u a t e the Martin-Sanchez method on a m i c r o p r o c e s s o r . T h i s system has been found to perform the i d e n t i f i c a t i o n and c o n t r o l on l i n e with f a i r l y good r e s u l t s , as w i l l be d e s c r i b e d . Chapter II w i l l cover the a n a l y s i s of the a l g o r i t h m , the f o r m u l a t i o n , the proof of h y p e r s t a b i 1 i t y and the proof of the convergence of the parameters. Chapter I I I w i l l d i s c u s s the implementation of the a l g o r i t h m on m i c r o p r o c e s s o r and the i n t e r f a c i n g of the d i f f e r e n t p a r t s of the system. I t w i l l a l s o a n a l y s e the software employed. Chapter IV w i l l cover the t e s t i n g of the i d e n t i f i e r and the c o n t r o l l e r f o r the f i r s t - o r d e r systems (with at most two parameters) and 11 s e c o n d - o r d e r systems ( 4 p a r a m e t e r s maximum). C h a p t e r V w i l l d i s c u s s t h e r e s u l t s and w i l l e v a l u a t e the i d e n t i f i e r and the c o n t r o l l e r . 12 I I . IDENTIFICATION AND CONTROL ALGORITHM 1 . Overview In t h i s c h a p t e r , the i d e n t i f i c a t i o n and c o n t r o l a l g o r i t h m w i l l be d e s c r i b e d . The m a t h e m a t i c a l f o u n d a t i o n of the a l g o r i t h m , the p r o o f of the h y p e r s t a b i l i t y of the parameter e s t i m a t e s , and the p r o o f of the convergence of the c o n t r o l b l o c k p a r a m e t e r s w i l l be o u t l i n e d . 2. D i s c r e t e - T i m e Systems A s i n g l e - i n p u t - s i n g l e - o u t p u t (SISO) l i n e a r t i m e - i n v a r i a n t c o n t i n u o u s time p r o c e s s i n a n o i s e f r e e environment can be r e p r e s e n t e d by the t r a n s f e r f u n c t i o n o s + a s +....+a. s +o. G(s)= (2.1) m-f / '° where m>n. To c o n t r o l a p r o c e s s such as (2.1) d i g i t a l l y i t i s c o n v e n i e n t t o d e v e l o p a d i s c r e t e e q u i v a l e n t o f G(s) p r e c e d e d by a sample and h o l d b l o c k r e p r e s e n t i n g the D/A c o n v e r s i o n a t t h e o u t p u t of the d i g i t a l c o n t r o l l e r . The d i s c r e t i z a t i o n o f G ( s ) w i l l r e s u l t i n a model of the p r o c e s s which can be r e p r e s e n t e d i n t he z - t r a n s f o r m domain o r , e q u i v a l e n t l y , by an i n p u t - o u t p u t d i f f e r e n c e e q u a t i o n of the form: y ( k ) = a ( y ( k - l ) + a _ 2 y ( k - 2 ) + + a n y ( k - n ) + b u ( k - 1 ) + b 0 u ( k - 2 ) + +bu(k-m) (2.2) 1 2 m 13 We w i l l study the i d e n t i f i c a t i o n and c o n t r o l of systems which can be re p r e s e n t e d by a d i f f e r e n c e equation such as (2.2) us i n g the a l g o r i t h m f i r s t proposed by Martin-Sanchez [ 2 0 ] . T h i s a l g o r i t h m w i l l be implemented on a Texas Instrument TM 990 microcomputer system to be d e s c r i b e d in d e t a i l i n Chapter I I I . T h i s chapter i s d e d i c a t e d to the a n a l y s i s of the ad a p t i v e a l g o r i t h m . 3. An A l g o r i t h m f o r Model Reference Adaptive Systems In t h i s a l g o r i t h m , an ad a p t i v e c o n t r o l b l o c k i s developed which uses only i n f o r m a t i o n from the input and output of the pr o c e s s . The c o n t r o l l e r attempts to behave as the exact i n v e r s e of the p l a n t . In t h i s way the c o n t r o l block output, equal to the p r o c e s s i n p u t , d r i v e s the pr o c e s s output to f o l l o w the c o n t r o l block input or r e f e r e n c e . The i d e n t i f i c a t i o n can be c a r r i e d out on l i n e , and u n l i k e the standard MRAS which needs some a ' p r i o r i knowledge of the parameters of the p l a n t , t h i s method can c a r r y out the i d e n t i f i c a t i o n f o r any i n i t i a l parameters. The two modes of the system, i d e n t i f i c a t i o n and c o n t r o l , are shown i n F i g u r e s 2.1a and 2.1b. F i g u r e 2.1a resembles an MRAS i d e n t i f i c a t i o n block diagram, but the f o r m u l a t i o n of the a l g o r i t h m i s q u i t e d i f f e r e n t . In the i d e n t i f i c a t i o n mode, F i g u r e 2.1a, an input u i s fed to both the process and the i d e n t i f i c a t i o n b l o c k . The comparison between the p l a n t output and the i d e n t i f i c a t i o n block output generates an e r r o r e , which w i l l be used by the a d a p t i v e mechanism to a d j u s t the i d e n t i f i c a t i o n block parameters so as to make e-*-0 14 PLANT ADJUSTABLE MODEL I t i ADAPTIVE MECHANISM (a) d(k+i; CONTROLLER -1 u(k) d(k) PLANT y(k) e(k) ADAPTIVE MECHANISM (b) Fig.. 2.1 Configuration of the Adaptive System i n : (a) I d e n t i f i c a t i o n Mode, (b) Control Mode 15 as t~ * o o . In the c o n t r o l mode, F i g u r e 2.2b, the d e s i r e d output i s the input to the c o n t r o l b l o c k , which uses the a d a p t i v e mechanism t o generate an input t o the p l a n t . The e r r o r e i s a measure of the d e v i a t i o n of the p l a n t output from the d e s i r e d output. T h i s e r r o r i s again used to update the c o n t r o l l e r parameters, so as to make e-*»0. I t i s assumed that the p l a n t i s output c o n t r o l l a b l e and can be represented by an inp u t - o u t p u t equation of the form (2.2). Let d(k) be the d e s i r e d output of the p l a n t a t sampling i n s t a n t k, and of the same dimension as y. An a l g o r i t h m i s r e q u i r e d which w i l l generate the c o n t r o l s i g n a l u(k) from d(k+l) and the e r r o r e(k)=y (k)-d(k)«, The a d a p t i v e mechanism used to a d j u s t the model parameters i s the same d u r i n g both the i d e n t i f i c a t i o n and the c o n t r o l modes of o p e r a t i o n . The parameter e s t i m a t i o n scheme w i l l be shown to be a s y m p t o t i c a l l y h y p e r s t a b l e by use of the Popov h y p e r s t a b i l i t y c r i t e r i o n [16], If the c o n t r o l l e r i s designed p r o p e r l y then the e r r o r e-*-0 as t-*- co, and t h e r e f o r e y(k)-*-d(k). The parameters of the c o n t r o l l e r w i l l remain unchanged when e=0, but i f the e r r o r i s non-zero due to process p e r t u r b a t i o n s or the v a r i a t i o n of the p l a n t ' s parameters, then proper c o n t r o l a c t i o n w i l l be taken t o ac h i e v e e q u i l i b r i u m . This, of course w i l l be done by a d j u s t i n g the the c o n t r o l l e r parameters. 16 3.1. Formulation Consider a process . c h a r a c t e r i z e d by the d i f f e r e n c e e q uation where y(k) and u(k) are the output and the input v e c t o r s of the p l a n t at sampling i n s t a n t k. Furthermore, i f the i n p u t v e c t o r u and the output v e c t o r y_ are assumed of the same dimension, say ( r x l ) , then A and B are r e a l m a t r i c e s of dimension ( r x r ) . In a d d i t i o n , i t i s assumed that the p l a n t i s output c o n t r o l l a b l e . I t i s d e s i r e d to f i n d an i d e n t i f i e r t h a t w i l l e s t i m a t e the m a t r i c e s A and B u s i n g only the input u and the output y_ . T h i s i d e n t i f i c a t i o n procedure w i l l i n c l u d e the r e f e r e n c e model: where A ( k - l ) and B(k-1) are the e s t i m a t e s of the m a t r i c e s A and B at the sampling i n s t a n t k-1; d(k) w i l l t h e r e f o r e be the p r e d i c t e d p l a n t output at sampling i n s t a n t k, from the i n f o r m a t i o n a v a i l a b l e at i n s t a n t k-1. Equation (2.4) a l s o d e s c r i b e s the c o n t r o l b l o c k , but now the output sequence i s the c o n t r o l l e r u(k) o b t a i n e d from the i n f o r m a t i o n a v a i l a b l e on d ( k + l ) . Thus equation (2.4) becomes y(k) = Ay(k-1)+Bu(k-i) (2 .3) d(k)= A(k-1)y(k-1)+ B(k-1)u(k-1) ( 2 . 4 ) u(k)= [B(k)]-'d(k+1)- [ B ( k > ] - , A ( k ) y ( k ) ( 2 . 5 ) In the above equation d(k+1) r e p r e s e n t s the d e s i r e d output of 17 the p l a n t at i n s t a n t k+1. u(k) w i l l be the input to the p l a n t and, s i n c e the i d e n t i f i c a t i o n block and the c o n t r o l block are governed by the same set of e q u a tions, t h i s input w i l l i d e a l l y ensure that the process ouput i s always equal to the d e s i r e d outputc The e r r o r f o r both the i d e n t i f i c a t i o n block and the c o n t r o l block w i l l be d e f i n e d as e(k)= y ( k ) - d(k) (2.6) To ensure t h a t the e r r o r converges to zero as t-**oo, an s t a b l e a d a p t i v e mechanism to a d j u s t A(k) and B(k) i s developed. T h i s uses the f o l l o w i n g parameter update: A(k)= AA(k)+ A(k-1) (2.7) B(k)= AB(k)+ B(k-1) (2.8) These new e s t i m a t e s of A and B can be used to p r e d i c t the p l a n t output at i n s t a n t k. Let g(k) be the p r e d i c t e d output at t h i s i n s t a n t g(k)= A(k)y(k-1)+ B(k)u(k-1) (2.9) T h i s i s the a ' p o s t e r i o r i model which w i l l generate a new a ' p o s t e r i o r i e r r o r d e f i n e d by: s(k)= y ( k ) - g(k) (2.10) The a d a p t i v e mechanism proposed by Martin-Sanchez uses the e r r o r s(k) to determine the incremental m a t r i c e s AA(k) and 18 AB(k) independent of the i n i t i a l c o n d i t i o n s ( A ( 0 ) , B ( 0 ) , y ( 0 ) , u ( 0 ) ) . ' The mechanism i s based on the f o l l o w i n g theorem: THEOREM 1: The system d e f i n e d by equations (2.3) to (2.10) and whose gen e r a l c o n f i g u r a t i o n i s shown i n " F i g u r e 2.1 w i l l be a s y m p t o t i c a l l y h y p e r s t a b l e i f the m a t r i c e s A(k) and. B(k) are generated by: A(k)= A(k-1)+ os(k)y(k-1) B(k)= B ( k - l ) + 0 S ( k ) u ( k - 1 ) where a and p are p o s i t i v e constant c o e f f i c i e n t s , t o be s e l e c t e d depending on the process c h a r a c t e r i s t i c s . The concept of h y p e r s t a b i l i t y i s a l e s s r e s t r i c t i v e form of a b s o l u t e s t a b i l i t y . I f the c o n t r o l system can be decomposed i n t o a l i n e a r t i m e - i n v a r i a n t p a r t and a n o n l i n e a r t i m e 7 v a r y i n g p a r t , as shown i n F i g u r e 2.2, the H y p e r s t a b i l i t y Theorem of Popov s p e c i f i e s s u f f i c i e n t c o n d i t i o n s which guarantee the s t a b i l i t y of the combined system. The main c r i t e r i a to be s a t i s f i e d are the i n e q u a l i t y k, I u , ( k ) y ( k ) > -«? (2.13a) /c=i where a* i s a p o s i t i v e c o n s t a n t , and a l s o the input/output r e l a t i o n s h i p i s : u ; ( k ) = F ( y ( i ) , i ) f o r i< k, and u ( k ) = - u ( ( k ) f o r k> 0. (2.13b) (2.11) (2.12) 19 \ u(k) LINEAR TIME-INVARIANT PLANT y(k) u x(k) NONLINEAR TIME-VARYING BLOCK Fig. 2.2 General Configuration of an Asymptotically Hyperstable Feedback Autonomous System. 20 The f i r s t c r i t e r i o n i s s i m i l a r t o the f a m i l i a r c o n s t r a i n t on the i n e q u a l i t y c o n d i t i o n i n the c o n v e n t i o n a l Popov c r i t e r i o n . The system i s a s y m p t o t i c a l l y h y p e r s t a b l e , i . e . y(k)-»~0 as k-*-°° i f the l i n e a r p l a n t i s s t r i c t l y p o s i t i v e r e a l d i s c r e t e . I t i s only h y p e r s t a b l e , i . e . y(k) i s bounded as k-#-o°, i f the l i n e a r p l a n t i s p o s i t i v e r e a l . As shown in F i g u r e 2.3, the system equations (2.4) to (2.12) form a n o n l i n e a r feedback autonomous system whose s t a t e can be d e f i n e d by the v e c t o r s ( k ) , e q u i v a l e n t to the system i n v e s t i g a t e d by Popov. The essence i s to convert the a d a p t i v e i d e n t i f i e r i n t o an e q u i v a l e n t feedback system with a l i n e a r block i n the forward path and a n o n l i n e a r t i m e - v a r y i n g block i n the feedback path. The l i n e a r p a r t i n t h i s case reduces to a u n i t y g a i n . The Popov h y p e r s t a b i l i t y c r i t e r i o n [16] f o r such an e q u i v a l e n t system i n the d i s c r e t e form [ 7 ] s t a t e s that asymptotic h y p e r s t a b i l i t y i s assumed i f the f o l l o w i n g i n e q u a l i t y i s s a t i s f i e d f o r a l l k. (2.13c) where s ( k ) ' i s the transpose of. v e c t o r s ( k ) . I f the paramter updating a l g o r i t h m can be shown to s a t i s f y the above i n e q u a l i t y , i t w i l l be a s y m p t o t i c a l l y h y p e r s t a b l e . 21 Fi g . 2.3a The nonlinear asymptotically hyperstable feedback autonomous system. s(k) b— -s(k) NON-LINEAR — TIME-VARYING BLOCK Fig . ",2.3b- The equivalent autonomous system. 22 3.2. H y p e r s t a b i l i t y of the C o n t r o l System The proof of h y p e r s t a b i l i t y f o r the c o n t r o l system with the a d a p t i v e mechanism g i v e n i n equations (2.11) and (2.12) i s from [20] by Martin-Sanchez. T h i s i s a c t u a l l y a p a r t i c u l a r case of Popov's theorem f o r d i s c r e t e systems i n v e s t i g a t e d by I.D. Landau [ 7 ] . The o u t l i n e proof i s as f o l l o w s : From (2.3),(2.9), and (2.10) s(k)= [A-A(k)]y(k-1)+ [B-B(k)]u(k-1) (2.14) I f (2.14) i s m u l t i p l i e d by - s ( k ) ' , - s ( k ) ' s ( k ) = s ( k ) ' [ A ( k ) - A ] y ( k - 1 ) + s ( k ) ' [ B ( k ) - B ] u ( k - 1 ) (2.15) - l ' s ( k ) ' s ( k ) = E' s(k) • [A(k)-A]y(k-1)+ s ( k ) ' [B(k)-B]u(k-1) k=o k=o (2.16) Equ a t i o n (2.16) can be decomposed to make the f o l l o w i n g two c o n d i t i o n s : I s ( k ) ' [ A ( k ) - A ] y ( k - 1 ) >-\* V K (2.17) l ' s(k) ' [B(k)-B]u(k-1 ) >-X 2\/k (2 . 18 ) k.-o b 1 where X and X. are c o n s t a n t s , a b C o n d i t i o n s (2.17) and (2.18) can be w r i t t e n i n s c a l a r form: I I Z s; (k)[a-.(k)-a..]y. (k-1) >-k2 \/ k (2.19) 23 1 E f s D ( k ) [ b J k ) - b J u q ( k - l ) M 2 k V (2.20) The above c o n d i t i o n s w i l l be s a t i s f i e d i f the parameters a (k) and b (k) are formed a c c o r d i n g to the a l g o r i t h m s k a •. (k)= I s ; ( h ) y ; ( h - T ) , i = l r n j = 1,n (2.21) KJ h z 0 J k b (k)= E s (h)u (h~1), p=1,n q=1,n (2.22) If (2.21) and (2.22) are s u b s t i t u t e d i n (2.19) and (2.20), the r e s u l t i n g equations are a s p e c i a l case of the more g e n e r a l e q u a t i o n k, k k, k, E x ( h ) [ E x(h)+c]=1/2[ E x(k)+c] 2+1/2 E x ( k ) 2 (2.23) k=o h=o k.o k'O where - ( c 2 / 2 ) > - X 2 , c=constant V k i X.=cpnstant The l e f t hand s i d e of t h i s equation i s always p o s i t i v e ; t h e r e f o r e , equations (2.21) and (2.22) can be expressed i n a new form. a;.(k)= s :(k)y ; (k-1 )+a.,(k-1 ) i = 1,n j = 1,n (2.24) b (k)= s p ( k ) u q ( k - l )+b p^(k-1) p=l,n q=1,n (2.25) If (2.24) and (2.25) are converted i n t o m a t r i x form, equations (2.11) and (2.12) are o b t a i n e d . The parameter updating of (2.21), (2.22) thus c a r r i e s with i t the important p r o p e r t y of h y p e r s t a b i 1 i t y . 24 3.3. R e l a t i o n Between e(k) and s(k) Since the a ' p o s t e r i o r i e r r o r s(k) cannot be obtained b e f o r e the parameters are updated and the parameters cannot be updated u n l e s s s(k) i s a v a i l a b l e ; a r e l a t i o n between s(k) and the a ' p r i o r i e r r o r e(k) must be e s t a b l i s h e d . T h i s can be done as f o l l o w s . From (2.11), (2.12), and (2.14), s(k)= [ A - A ( k - 1 ) - c s ( k ) y ( k - 1 ) ' ] y ( k - 1 ) + [ B - B ( k - 1 ) - * s ( k ) u ( k - l ) v ] u ( k - 1 ) (2.26) From eq u a t i o n s (2.3), (2.4), (2.6), and (2.26), s(k)= e(k ) [ a y(k-1 ) ' y(k-1)+pu(k-1)'u(k-1 ) ] (2.27) s o l v i n g (2.27) f o r s ( k ) , s(k)= e(k)/[1+ a y(k-1 ) ' y(k-1)+pu(k-1 )'u(k-1 )] (2.28) Let *(k) be d e f i n e d as * ( k ) = [ l + a y ( k-1 ) ' y ( k-1)+pu(k-1)'u ( k-1) ] - 1 (2.29) (2.28) can now be w r i t t e n as s(k)= *(k) e(k) (2.30) The a d a p t i v e a l g o r i t h m equations (2.11) and (2.12) can now be r e w r i t t e n i n a form, that avoids the need to c a l c u l a t e s(k) i n 2 5 o r d e r t o u p d a t e t h e p a r a m e t e r s . A ( k ) = c * ( k ) e ( k ) y ( k - 1 ) ' + A ( k - 1 ) ( 2 . 3 1 ) B ( k ) = **(k)e(k)u(k-1)»+ B ( k - 1 ) ( 2 . 3 2 ) 3.4. C o n v e r g e n c e o f t h e C o n t r o l B l o c k P a r a m e t e r s I t i s i m p o r t a n t t o r e a l i z e t h a t h y p e r s t a b i l i t y o f s ( k ) d o e s n o t a s s u r e t h e c o n v e r g e n c e o f t h e m o d e l p a r a m e t e r s t o t h e i r t r u e v a l u e s . I t i s e a s y t o g i v e e x a m p l e s o f s y s t e m f o r w h i c h t h e c o n t r o l l a w s b a s e d on i n c o r r e c t p a r a m e t e r s c a n g i v e g o o d o u t p u t e r r o r b e h a v i o u r . The h y p e r s t a b i l i t y o f t h e c o n t r o l s y s t e m i m p l i e s a t l e a s t a l o c a l c o n v e r g e n c e o f t h e c o n t r o l b l o c k p a r a m e t e r s . The p r o o f o f t h i s p a r a m e t e r c o n v e r g e n c e i s g i v e n by M a r t i n - S a n c h e z b a s e d on t h e work o f Ngumo a n d Noda [21 ] . The m e t h o d u s e s g r a d i e n t p a r a m e t e r s e s t i m a t i o n t e c h n i q u e a n d i s o u t l i n e d b e l o w : A c c o r d i n g t o t h e f o r m u l a t i o n i n S e c t i o n 3.3, t h e i t h component o f t h e a ' p o s t e r i o r i e r r o r s ( k ) i s g i v e n by s < ; ( k ) = t ( k ) e ; ( k ) ( 2 . 3 3 ) where * ( k ) i s d e f i n e d by ( 2 . 2 9 ) and e ( k ) by e-"(k)=y- ( k ) : d ; ( k ) ( 2 . 3 4 ) L e t e a n d e ( k - 1 ) be 9= [ a - ( , . . . , a t n , b u , . . . , b n ] ' ( 2 . 3 5 ) 6; (k-1 )= [ a : . ( k - 1 )•>,:.. , a : j k - 1 ) ,b-,(k-1 ) , ... , b L n ( k - 1 ) ]» ( 2 . 3 6 ) 26 Let x (k-1 ) be ( k - l ) = [ y , ( k - 1 ) , . . . , y n ( k - 1 ) , u , ( k - 1 ) , . . . , u n ( k - 1 ) ] ' (2.37) From (2.3),(2.35), and (2.37) y; (k)= x(k-1 ) '6t- (2.38) From (2.4),(2.36), and (2.37) d;(k)= x ( k - l ) ' G - ( k - 1 ) (2.39) The a l g o r i t h m p r e v i o u s l y developed f o r a d j u s t i n g the i d e n t i f i c a t i o n and c o n t r o l block parameters can now be w r i t t e n i n the f o l l o w i n g form: G;(k)= G-(k-1)+*(k)e;(k)x(k-1) (2.40) The a l g o r i t h m s are e q u i v a l e n t t o a g r a d i e n t parameter e s t i m a t i o n method which minimizes the f o l l o w i n g c o s t f u n c t i o n : J= [e- (k ) P/2 (2.41) T h i s i s a s p e c i a l case of the c o s t f u n c t i o n c o n s i d e r e d i n [11] or the c o s t f u n c t i o n used i n the u s u a l minimum v a r i a n c e a d a p t i v e c o n t r o l l e r [ 1 ] . From (2.40) and (2.34) G; ( k ) = e - ( k - 1 ) + * ( k ) x ( k - 1 ) [ y / ( k ) - d : ( k ) ] (2.42) 27 From (2.42),(2.38) , and (2.39) e[ ( k ) = e L ( k - i ) + * ( k ) x ( k - i ) x ( k - i ) ' [e- -©. (k-1) ] (2.4.3) Let G;(k) be the i d e n t i f i c a t i o n e r r o r : e t-(k)= eL~e-L (k) (2.44) From (2.43) and (2.44) e l ( k ) = [ l - * ( k ) x ( k - l ) x ( k - 1 ) ' ]G; (k-1) (2.45) T h i s i s the case c o n s i d e r e d in [ 2 1 ] , which proves A yy j j © ^  (k) | | 2 w i l l converge t o z e r o as t->oo, u n l e s s G^  (k.) and x(k) become o r t h o g o n a l . Such an o r t h o g o n a l i t y can be avoided by u s i n g an input s i g n a l with adequate frequency components. 3.5. C o n c l u s i o n s In t h i s Chapter an i d e n t i f i c a t i o n and c o n t r o l a l g o r i t h m was i n t r o d u c e d , the i d e n t i f i c a t i o n method i s s i m i l a r t o a model r e f e r e n c e approach but the f o r m u l a t i o n i s d i f f e r e n t and c o u l d be viewed e s s e n t i a l l y as a s i m p l i f i e d form of r e c u r s i v e l e a s t squares e s t i m a t i o n method. The a d a p t i v e c o n t r o l method was shown to e s t a b l i s h a h y p e r s t a b l e system a c c o r d i n g to Popov theorem, t h i s however does not imply that the c o n t r o l l e r w i l l reduce the n o i s e i n the c o n t r o l system d r a s t i c a l l y or even p r o v i d e a much b e t t e r performance than a PID c o n t r o l l e r . T h i s a d a p t i v e c o n t r o l l e r i s s i m i l a r t o a minimum v a r i a n c e c o n t r o l l e r , used i n the s e l f - t u n i n g r e g u l a t o r s , with a s i m p l i f i e d f o r m u l a t i o n f o r the parameter updates. 28 I I I . HARDWARE AND SOFTWARE 1. Overview The i d e n t i f i e r and the c o n t r o l l e r developed i n Chapter II are t e s t e d by implementing the i d e n t i f i c a t i o n and the c o n t r o l a l g o r i t h m on a microcomputer system. The system c o n s i s t s o f : (a) Texas Instrument TM 990/101M-1 microcomputer board. (b) Texas Instrument TM 990/510 card c h a s s i s . (c) Texas Instrument SILENT 700 (745 model) e l e c t r o n i c data t e r m i n a l . (d) Analog Devices RTI-1241-S (A/D), (D/A) c o n v e r s i o n board. (e) GSC GOF-2A-1T t r i p l e output power s u p p l y . The components were s e l e c t e d based on the system r e q u i r e m e n t s , c o m p a t i b i l i t y ( s i m p l i c i t y to i n t e r f a c e ) , ease of programming, and f i n a l l y , the t o t a l c o s t of the system. The f i r s t p a r t of t h i s chapter w i l l b r i e f l y d e s c r i b e the d i f f e r e n t p a r t s of the system, and the i n t e r f a c i n g . The second p a r t w i l l e x p l a i n the software p r o d u c t i o n and implementation. 2. The Hardware 2.1. TM 990/101M-1 Microcomputer The TMS 9900 microprocessor i s the heart of the TM990/101M-1. T h i s 16 b i t CPU f e a t u r e s a memory to memory a r c h i t e c t u r e and the e x t e n s i v e i n s t r u c t i o n set i n c l u d e s hardware m u l t i p l y and d i v i d e . There are a t o t a l of e i g h t a d d r e s s i n g modes. The c l o c k r a t e i s 3 MHz. The communication 29 w i t h the TI 745 t e r m i n a l i s c a r r i e d out through a m o d i f i e d EIA RS-232-C s e r i a l I/O i n t e r f a c e . There are three programmable i n t e r v a l t i m ers which are used to set up the i n t e r r u p t s e r v i c e r o u t i n e (ISR), and t h e r e f o r e the sampling r a t e . The debugging i s done by using the TIBUG debug monitor which p r o v i d e s an i n t e r a c t i v e i n t e r f a c e between the user and the TM990/101M-1 microcomputer. The TIBUG program i s l o c a t e d on two TI 2708 EPROM's and a l s o uses about 40 words ( i n t h i s m i c r o p r o c e s s o r every word i s two bytes) of RAM. The TIBUG p r o v i d e s seven software r o u t i n e s and s e t s the Baud Rate a u t o m a t i c a l l y everytime the microprocessor i s r e s e t . T h i s system i s set to operate at 300 Baud. Some of the other important f e a t u r e s of t h i s microcomputer board are l i s t e d i n Appendix A. 2.2. RTI-1241-S Board To i n t e r f a c e the d i g i t a l i d e n t i f i c a t i o n and c o n t r o l system to the continuous time process s i m u l a t i o n , analog to d i g i t a l and d i g i t a l to analog c o n v e r s i o n i s necessa r y . The analog d e v i c e s RTI-1241-S board p r o v i d e s both the analog input and output. The RTI-1241 s e r i e s has been designed f o r simple and e f f i c i e n t i n t e r f a c i n g of the analog s i g n a l s to TM 990 microcomputers. The standard board used i n the c o n t r o l system i s d e s c r i b e d in Appendix A. 30 3. The Software The software implementation f o r the i d e n t i f i c a t i o n and c o n t r o l of two SISO systems w i l l be d e s c r i b e d here. These two systems a r e : i ) a f i r s t - o r d e r system, and i i ) a second-order system. The continuous-time t r a n s f e r f u n c t i o n r e p r e s e n t i n g these two system a r e : Where u i s the undamped n a t u r a l frequency; and Q, the damping r a t i o of the system [26]. The software r e q u i r e d f o r i d e n t i f i c a t i o n and the c o n t r o l of G1 (s) w i l l be developed and from that the software f o r the i d e n t i f i c a t i o n and the c o n t r o l of G 2 ( s ) can be e a s i l y d e r i v e d . 3.1. The F i r s t - O r d e r Systems The t r a n s f e r f u n c t i o n of the f i r s t - o r d e r system G ( s ) , i n d i s c r e t e form can be r e p r e s e n t e d by the d i f f e r e n c e e quation G,(s) = K/(s+r) (3.1) G 2 ( s ) = K « 2 / ( s 2 + 2 £ o + u 2 ) . (3.2) y(k) = a ( y(k-l)+b,u(k-1) (3.3) where: a f= e x p ( - r T ) , and b, = ( K / r ) [ l - e x p ( - r T ) ] . To i d e n t i f y the parameters a, and b^ , the a d j u s t a b l e model d(k)= A(k-1)y(k-1)+B(k-1)u(k-1) (3.4) 3 1 w i l l be used; where A(k - 1 ) and B(k-1) are the e s t i m a t e s of a, and b f . From the d e f i n i t i o n s i n Chapter I I , the a ' p r i o r i e r r o r i s e(k)= y ( k ) - d ( k ) (3 .5) The a ' p o s t e r i o r i e r r o r i s s(k)= e ( k ) / [ 1 + a y ( k - 1 ) ' y ( k - 1 ) + £ u ( k - 1 ) ' u ( K - 1 ) ] . (3 .6a) For the s c a l a r systems y(k- 1)'=y(k-1), t h e r e f o r e s(k)= e ( k ) / [ l + c y ( k - 1 ) 2 + * u ( K - l ) 2 ] . (3 .6b) The parameters are a d j u s t e d a c c o r d i n g to the f o l l o w i n g a l g o r i t h m : A(k)= A ( k - 1 ) + a s ( k ) y ( k - 1 ) , (3 .7) B(k)= B ( k - 1 ) + * s ( k ) u ( k - 1 ) . (3 .8) To c o n t r o l the pro c e s s , the a ' p o s t e r i o r i e r r o r computation and the parameter adjustment w i l l remain the same, but the input to the p r o c e s s w i l l be the- c o n t r o l l e r u ( k ) , where: u(k)= [ B ( k ) ] - 1 [ d ( k + 1 ) - A ( k ) y ( k ) ] . (3 .9) The f l o w c h a r t f o r the two modes of o p e r a t i o n f o r the f i r s t order system are shown i n F i g u r e s 3.1 and 3 . 2 . 32 ( V START _ I read A/D channel: Input u(k) read A/D channel: Output y(k) compute the a d j u s t a b l e model output d(k)= A ( k - 1 ) y ( k - 1 ) + B ( k - I ) u ( k - 1 ) compute: the e r r o r e(k)= y ( k ) - d ( k ) s(k)= e ( k ) / ( 1 + C y ( k - 1 ) 2 + * u ( k - l ) 2 ) 4. update the model parameters A(k)= A(k - 1)+ c s ( k ) y ( k - 1 ) B(k)= B(k - 1)+ p s ( k ) u ( k - l ) update the input and the output y(k - 1 )-•—y(k) and u ( k- 1 ) u ( k ) I . * F i g , 3 .1 Flowchart f o r the i d e n t i f i c a t i o n of a f i r s t order system 33 ( START i read A/D channel: Input d(k) read A/D channel: Output y(k) compute: the e r r o r e(k) = y ( k ) - d ( k ) s(k)= e ( k ) / { 1 + c y ( k - 1 ) 2 + * u ( k - l ) 2 ) update the model parameters A(k)= A(k-1)+ os(k)y(k-1) B(k)= B(k-1)+ j s ( k ) u ( k - 1 ) compute the c o n t r o l l e r value u(k) u(k)= [ B ( k ) ] " 1 [ d ( k + 1 ) - A ( k ) y ( k ) ] !! update the input and the output y ( k-1 ) - * — y ( k) and u( k-1 ) u ( k ) t F i g . 3 . 2 Flowchart f o r the c o n t r o l of a f i r s t order system. 34 3.2. The Second-Order Systems In the case of a second-order process, the procedure f o r the i d e n t i f i c a t i o n and c o n t r o l of the f i r s t - o r d e r system can be extended t o o b t a i n the equations given below. The t r a n s f e r f u n c t i o n i n d i s c r e t e form i s r e p r e s e n t e d by the d i f f e r e n c e e quation y(k) = a, y(k-1 )+a 2y(k-2)+b f u(k-1 )+b 2u(k-2) . (3.10) where: a, = 2exp(-£uT)cos (oT) a 2= -exp(-2£oT) b, = K{l-exp(-£oT)[cos(oT) + U / 0 - £ 2 ) 2 ) s i n ( o T ) ]} b 2 = K{exp(-£«T) [exp(-£«T)-cos(uT) + U / ( 1-£ 2 ) 2 )sin(«T) ] } . The problem i s to i d e n t i f y the four parameters a, ,a a ,b^  ^b^. • I d e n t i f i c a t i o n : The a d j u s t a b l e model: d(k)= A1(k-1)y(k-1)+A2(k-1)y(k-2)+ B1(k-1)u(k-1)+B2(k-1)u(k-2) (3.11) The a ' p r i o r i e r r o r : e(k) = y ( k ) - d ( k ) (3.12) The a ' p o s t e r i o r i e r r o r : s(k)= e ( k ) / { 1 + o [ y ( k - l ) 2 + y ( k - 2 ) 2 ] + e [ u ( k - 1 ) 2 + u ( k - 2 ) 2 ] } (3.13) The parameters are updated by the a l g o r i t h m g i v e n i n equations (3.14a-d) 35 A1 (k)= A1 (k-1 ) + cs(k)y(k--1 ) (3. H a ) A2(k) = A2(k-1)+os(k)y(k~2) (3.14b) B1(k)= B1 (k-1 )+*s(k)u(k-1) (3,14c) B2(k)= B2(k-1)+£s(k)u(k-2) (3.1 4 d) • C o n t r o l : In the c o n t r o l mode the c o n t r o l l e r u(k) must be computed. The d e s i r e d output d(k+l) i s a v a i l a b l e . u ( k ) = [ B ( k ) ] " 1 [ d ( k + 1 ) - A 1 ( k ) y ( k ) - A 2 ( k ) y ( k ~ 1 ) - B 2 ( k ) u ( k - l ) ] (3.15) The computation sequence f o r the second-order system has a f l o w c h a r t s i m i l a r to the one f o r the f i r s t - o r d e r systems. The only d i f f e r e n c e i s the number of parameters being e s t i m a t e d , and t h e r e f o r e the amount of computation. 3.3. A r i t h m e t i c O p e r a t i o n s The implementation of the equations developed i n the p r e v i o u s s e c t i o n s on the micr o p r o c e s s o r r e q u i r e s the f o l l o w i n g a r i t h m e t i c o p e r a t i o n s : • ADD: The microprocessor performs the add o p e r a t i o n i n 2's complement form. • SUBTRACT: Same as ADD. » MULTIPLY: The TM 9 9 0 microcomputer has hardware m u l t i p l y ; but i t i s performed i n n a t u r a l b i n a r y . To perform the m u l t i p l i c a t i o n i n 2's complement, a simple s u b r o u t i n e would do the job. The flo w c h a r t i s shown i n F i g u r e 3.3 f o r two 16 b i t 36 START 4_ COMPUTE: C= A xor B M = /A| * |B| F i g . 3.3 Flowchart for two's Complement M u l t i p l i c a t i o n . 3 7 numbers, A, B. The r e s u l t i s a 32 b i t number, s t o r e d i n 2 c o n s e c u t i v e words. • DIVIDE: S i m i l a r to m u t i p l y o p e r a t i o n ; but here a 32 b i t number i s d i v i d e d by a 16 b i t number. The same technique can be used to perform signed d i v i s i o n . 3.4. The Sampling Rate A very important f u n c t i o n of the software i s t o . set the sampling r a t e . T h i s i s done by the implementation of an I n t e r r u p t S e r v i c e Routine (ISR), which takes advantage of the TMS 9901 programmable systems i n t e r f a c e as an i n t e r v a l timer* The timer has a r e s o l u t i o n of 21.33 microseconds. A 14 b i t number w i l l be counted down to zero (every decrement takes 21.33 microseconds), and the c l o c k w i l l be r e l o a d e d a u t o m a t i c a l l y . A maximum time of 349 m i l l i s e c o n d s can be a c h i e v e d . Table 3.1 l i s t s some of the sampling r a t e s and the c o r r e s p o n d i n g hexadecimal codes. Upon the i n i t i a l i z a t i o n of the system, the sampling r a t e i s s e l e c t e d by the o p e r a t o r l o a d i n g the proper hexadecimal code i n t o the d e s i g n a t e d memory l o c a t i o n TCONST. Any sampling r a t e can be c r e a t e d with a 21.33 microseconds r e s o l u t i o n so t h a t , f o r example, a sampling r a t e of 3 Hz or T = 1/3 seconds, i s obtained by t a k i n g K= 0.333/21.33X10" 6= 15625 The number K represented i n hex, i s 3D09. The counter uses b i t s 1-14. T h i s number i s t h e r e f o r e s h i f t e d to the l e f t and a '1' i s p l a c e d i n 15th b i t to enable the i n t e r r u p t . F i n a l l y the hexadecimal code to be used by the operator would thus be 7A13. SAMPLES PER 14 BIT 16 BIT SECONDS CLOCK REGISTER CLOCK REGISTER 3 3D09 7A1 3 5 24A1 4963 10 1250 24A1 20 0928 1251 30 061B 0C37 40 0494 0929 50 03A9 0753 60 030D 06 1B 70 029D 053B 80 024A 0495 90 0209 0413 100 01D5 02AB TABLE 3 . 1 The sampling rates and their correspondig hexadecimal coded data. 39 3.5. The Computer Programme The i d e n t i f i c a t i o n and c o n t r o l programmes were w r i t t e n i n assembly language, and then transformed i n t o machine codes. The code was t r a n s f e r r e d to the microcomputer on two EPROM's. The EPROM*s were loaded using a programme w r i t t e n i n F o r t r a n which c r e a t e s two f i l e s , the f i r s t f i l e c o n t a i n s b i t s 0-7 and the second f i l e c o n t a i n s b i t s 8-15 of every word. The programmes f o r both the f i r s t - o r d e r and the second-order systems c o n s i s t of f i v e s u b r o u t i n e s : 1) The I n t e r r u p t S e r v i c e Routine (ISR): T h i s r o u t i n e i s used to set the sampling r a t e . The ISR w i l l use the r a t e p r o v i d e d by the system o p e r a t o r . At t h i s time t h i s r a t e i s any number between 3 to 46875 i n t e r r u p t s / s e c o n d with a 21.33 microseconds r e s o l u t i o n . T h i s r a t e can be f u r t h e r slowed down by r e p e a t i n g the counter countdown be f o r e the i n t e r r u p t o c c u r s . 2) A s u b r o u t i n e to compute the a ' p o s t e r i o r i e r r o r s ( k ) . The r o u t i n e a c t u a l l y computes l / s ( k ) , because t h i s number w i l l always be l a r g e r than u n i t y . 3) A s u b r o u t i n e to update the parameters A(k) and B ( k ) . T h i s s u b r o u t i n e computes the increments AA(k) and AB(k) and adds them to the o l d parameters. A good c h o i c e of the parameters a and p, used i n the e q u a t i o n s (2.29), (2.31), and (2.32) i n Chapter I I , w i l l enhance the performance of the i d e n t i f i e r . Computer s i m u l a t i o n showed that small v a l u e s of o and p (0.1<[c,£3<1.0) w i l l slow down the convergence but a l s o that the i d e n t i f i e r performs b e t t e r f o r these v a l u e s than when used w i t h l a r g e r v a l u e s 40 ( [ G , £ ] > 1 . 0 ) . T h i s i s e s p e c i a l l y the case when the input changes are sudden, as i n a square wave r e f e r e n c e , f o r example. For s i m p l i c i t y i n the computation these parameters are s e l e c t e d as powers of two i . e . 2,4,... or 0.5,0.25,... so a l l o w i n g the m u l t i p l i c a t i o n to be performed by s h i f t i n g the r e g i s t e r s to the l e f t or r i g h t . 4) A s u b r o u t i n e which computes the p r e d i c t e d output d ( k ) . T h i s s u b r o u t i n e w i l l be a c t i v e only when the system i s o p e r a t i n g i n the i d e n t i f i c a t i o n mode. The f i r s t c a l l of the main programme i s to t h i s s u b r o u t i n e . 5) A s u b r o u t i n e to compute the c o n t r o l l e r u ( k ) . T h i s s u b r o u t i n e , c a l l e d only i n the c o n t r o l mode, i s the l a s t s u b r o u t i n e c a l l i n the main programme. At sampling i n s t a n t k, t h i s r o u t i n e . w i l l read the (A/D) channel to o b t a i n d ( k + l ) . The v a l u e of the c o n t r o l l e r output u(k) i s checked before being sent to the D/A c o n v e r t e r and consequently, t o the p r o c e s s . The f o l l o w i n g c o n s t r a i n t s are a p p l i e d : a) i f -2047<u(k)<2047 u(k)= u(k) b) i f u(k)>2047 u(k)= 2047 c) i f u(k)<-2047 u(k)= -2047 The above s a t u r a t i o n c o n s t r a i n t s are necessary because of the c o n v e r s i o n board 12 b i t r e s o l u t i o n . I f the sampling r a t e i s too h i g h , the parameter B l ( k ) becomes very small and t h i s leads to l a r g e c o n t r o l l e r v a l u e s , i n h i b i t i n g the convergence of the system. T h i s i s not a problem when i d e n t i f i c a t i o n i s being performed because the i n v e r s e of B1(k) i s not used. ;,. 41 The program to i d e n t i f y and c o n t r o l a f i r s t - o r d e r system i s l e s s than 400 words (800 b y t e s ) , and l e s s than 600 words (1200 bytes) f o r the second-order systems. Intermediate v a l u e s of the system i n p u t , output, p r e d i c t e d output, c o n t r o l l e r , and the parameters are s t o r e d , e n a b l i n g the operator to i n s p e c t the g r a d u a l change of these parameters. The e x e c u t i o n time f o r these programs a l l o w s a sampling r a t e of about 100 Hz f o r the f i r s t - o r d e r system and of about 50 Hz for the second-order systems. 42 IV. MICROCOMPUTER EVALUATION OF THE ADAPTIVE  IDENTIFIER AND CONTROLLER 1. OVERVIEW In t h i s Chapter the implementation of the i d e n t i f i e r and c o n t r o l l e r developed i n Chapter III w i l l be e v a l u a t e d on the microcomputer system. The processes to be i d e n t i f i e d and c o n t r o l l e d are s i m u l a t e d by using o p e r a t i o n a l a m p l i f i e r s as i n t e g r a t o r s and i n v e r t e r s . The e f f e c t of such f a c t o r s as the sampling r a t e , and the c h o i c e of i n i t i a l v a l u e s f o r the process parameters w i l l be i n v e s t i g a t e d . S ince the i d e n t i f i e r i s designed t o be used with the c o n t r o l l e r , the performance of the i d e n t i f i e r i s e v a l u a t e d f o r the c l o s e d l o o p p r o c e s s e s . The i d e n t i f i e d parameters may be b i a s e d because of the c o r r e l a t i o n between the input and the output, however, the process output i n a l l cases converges to the i n p u t . For f i r s t - o r d e r p rocesses t h i s was not a s i g n i f i c a n t problem and the estimated parameters are v e r y c l o s e to the t r u e parameters. The second-order processes are h a r d e r to i d e n t i f y and more s e n s i t i v e to the s e l e c t i o n of sampling r a t e and the i n i t i a l v a l u e s of the parameters. In both c a s e s the estimated parameters do not converge to the c o r r e c t v a l u e s but o s c i l l a t e c l o s e to the t r u e v a l u e s with a s m a l l b i a s . T h i s Chapter i s d i v i d e d i n t o two p a r t s . The f i r s t p a r t e v a l u a t e s the i d e n t i f i c a t i o n and c o n t r o l of f i r s t - o r d e r systems, and three d i f f e r e n t processes w i l l be i n v e s t i g a t e d . The second p a r t e v a l u a t e s the i d e n t i f i c a t i o n and c o n t r o l of 43 second-order systems. F i g u r e 4.1 shows the system i n the i d e n t i f i c a t i o n mode, and F i g u r e 4.2 shows the system i n i t s c o n t r o l mode. 2. F i r s t - O r d e r Processes A f i r s t - o r d e r process can be c h a r a c t e r i z e d by a d i f f e r e n c e equation y(k)= a,y(k-1) + b tu(k-1) (4.1) where y_ and u are the output and the input r e s p e c t i v e l y . To i d e n t i f y t h i s p r o c e s s i s to e s t i m a t e the parameters a, and b, . In the i d e n t i f i c a t i o n mode the operator must p r o v i d e the sampling r a t e and some reasonable i n i t i a l v a l u e s f o r a., and b 1 . The s e l e c t i o n of these v a r i a b l e s i s b r i e f l y d e s c r i b e d here. The sampling r a t e should be s e l e c t e d based on the i n f o r m a t i o n a v a i l a b l e from the process and the random i n p u t . I f the p r o c e s s i s unknown t h i s r a t e can be s e l e c t e d o n l y based on the input to get some knowledge about the p r o c e s s and then choose a new r a t e i f d e s i r e d . A good c h o i c e of sampling r a t e , a l t h o u g h important, i s not c r i t i c a l , any value 10 to 100 times the h i g h e s t frequency of the input s i g n a l w i l l produce s a t i s f a c t o r y r e s u l t s . A very f a s t sampling r a t e , however, i s not d e s i r a b l e because at very high r a t e s the parameters a and b w i l l reach t h e i r l i m i t s , a-^ -1 and b-»-0. The c h o i c e of i n i t i a l parameters i s a l s o important. I t should be noted.that the i d e n t i f i c a t i o n computer program w i l l r u(t) u(k) j ADAPTIVE IDENTIFIER I ADJUSTABLE MODEL ADAPTIVE MECHANISM Fig. 4.1 The Block Diagram of the Adaptive System i n I d e n t i f i c a t i o n Mode. INVERSE MODEL J * u(k) D/A u(t) ADAPTIVE MECHANISM PLANT y( t ) A/D d(k) y(k) 4-e(k) 4.2 The Block Diagram of the Adaptive System i n the Control Mode, 4 6 use a set value (hard l i m i t ) any time i t encounters an unreasonable v a l u e . T h e r e f o r e , at every sample the parameters are compared and a l t e r e d i f they exceed the l i m i t . For the f i r s t - o r d e r systems the estimated parameter A ( k ) has an upper l i m i t of 1.0 . T h i s i s a v a l i d assumption because i f A(k)>1.0 the system i s u n s t a b l e . The parameter B(k) does not have a hard l i m i t when the system i s i n i d e n t i f i c a t i o n mode, but has a lower l i m i t on i t s magnitude when the system i s i n the c o n t r o l mode. T h i s l i m i t i s a number g r e a t e r than zero, i . e . |B1|>0.05. Numerical a n a l y s i s and computer s i m u l a t i o n of d i f f e r e n t a l g o r i t h m s have shown that the input s i g n a l to the pro c e s s must have enough i n f o r m a t i o n a v a i l a b l e f o r the a l g o r i t h m to i d e n t i f y the process parameters. Such an input s i g n a l c o u l d be a: white n o i s e , a square wave or a s i n u s o i d a l i n p u t s can a l s o be used. To e v a l u a t e the i d e n t i f i c a t i o n and c o n t r o l a l g o r i t h m on the microcomputer system, three processes are si m u l a t e d , P1, P2, and P3 with the f o l l o w i n g t r a n s f e r f u n c t i o n s . P1 G(s)=l0.0/(s+5.0) P2 G(s)=0.5/(s+1.0) P2 G(s)=30.0/(s+20.0) These three processes w i l l be i d e n t i f i e d and the c o n t r o l a l g o r i t h m w i l l be e v a l u a t e d f o r the process P1. 47 2.1 I d e n t i f i c a t i o n The process P1 can be represented by a d i f f e r e n c e equation where the c o e f f i c i e n t s depend on the sampling r a t e and i t s parameters. Using the t r a n f o r m a t i o n formulas given i n Chapter I I I , process P1 can be represented by the d i f f e r e n c e e q u a t i o n : y(k)= 0.37y(k-l)+ 1.26u(k-l) at 5 samples/sec. y(k)= 0.61y(k~1)+ 0.78u(k-l) at 10 samples/sec. y(k)= 0.78y(k-l)+ 0.44u(k-l) at 20 samples/sec. To i d e n t i f y process P1 the sampling r a t e s i n d i c a t e d i n the above equations are used. The estimated parameters o b t a i n e d are shown i n F i g u r e s 4.3a, 4.4a, and 4.5a. These v a l u e s are very c l o s e t o the t r u e parameters of the p r o c e s s . As can be seen, the parameters change with sampling r a t e . The sampling r a t e s e l e c t e d should not be too h i g h . Since as the sampling r a t e i s i n c r e a s e d even f u r t h e r (>50), then the parameters converge to t h e i r l i m i t s i . e . A ( k ) — M . 0 , B(k)->0. F i g u r e s 4.3b, 4.4b, and 4.5b show the process output and the output p r e d i c t e d by the m i c r o p r o c e s s o r . The p r e d i c t e d output c l o s e l y f o l l o w s .the process output. The sampling rate s e l e c t e d f o r P2 was 5 s a m p l e s / s e c , a t t h i s r a t e P2 can be c h a r a c t e r i z e d by the d i f f e r e n c e equation y(k)= 0.82y(k-1)+ 0.09u(k-1) 48 > 50 LO F i g . 4.5a The Estimated Parameters. 20 Samples/Second. 51 The e s t i m a t e d parameters are shown i n F i g u r e 4.6a, once again they are very c l o s e to the t r u e v a l u e s . In t h i s case a very slow (low frequency) input s i g n a l was used. The p r e d i c t e d output shown i n F i g u r e 4.6b i s n o i s y but s t i l l v e ry c l o s e t o the p r o c e s s output. A slower sampling r a t e c o u l d perhaps improve the process i d e n t i f i c a t i o n . The p r o c e s s P3 i s a f a s t e r process than P1 and P2, t h e r e f o r e , the sampling r a t e s e l e c t e d was 100 samples/second. At t h i s r a t e the d i f f e r e n c e equation i s given by: y(k)= 0.82y(k-1)+ 0.27u(k-1) The i d e n t i f i c a t i o n r e s u l t s f o r the process P2 are shown i n F i g u r e 4.7. 2.2. C o n t r o l The performance of the a d a p t i v e c o n t r o l l e r i s e v a l u a t e d and p r o c e s s P1 was s e l e c t e d f o r t h i s t e s t . F i g u r e 4.8 shows the response of t h i s process to a s t e p i n p u t , i t i s d e s i r a b l e t h a t the t r a n s i e n t response be f a s t e r than the present response. T h i s can be done by i n t r o d u c i n g the a d a p t i v e c o n t r o l l e r developed i n Chapter III i n t o the system. The r e f e r e n c e i n p u t goes through t h i s c o n t r o l l e r which in turn generates a c o n t r o l s i g n a l to the p r o c ess, as shown i n F i g u r e 4.2. In the c o n t r o l mode hi g h e r sampling r a t e s c o u l d be used to a c h i e v e a very f a s t response, however, the c o n t r o l s i g n a l may be very l a r g e and t h e r e f o r e i m p r a c t i c a l . Since the l a r g e s t v a l u e handled by the LD Qo UJ >—i u_ I—1 Y—V a o a i 1 1 1 1 T 1 1 1 i I r 20 40 60 80 TIME (SAMPLES) i r 100 F i g . 4.6a The Estimated Parameters, 5 Samples/Second. 53 LO » I •—TM Q3_ -F i g . 4.7a The Estimated Parameters. 1 0 0 Samples/Second. 55 (A/D) or (D/A) c o n v e r s i o n board i s ±10 v o l t s , c a r e should be taken to prevent overflow. T h i s may be done by s c a l i n g the da t a . As mentioned i n Chapter I I I any time t h e r e i s an overflow, the h i g h e s t value i s used f o r the c o n t r o l l e r , ±10 v o l t s . In a d d i t i o n , i n t h i s mode of o p e r a t i o n the parameter B1 has a lower l i m i t to prevent overflow. For the process P1 t h i s l i m i t i s set at 0.05. In the c o n t r o l mode, the parameters of the a d a p t i v e c o n t r o l l e r remain constant o n l y i f the steady s t a t e has been reached. Since the time to reach steady s t a t e depends on the c o n t r o l s i g n a l , i f the c o n t r o l s i g n a l i s l i m i t e d then the system response w i l l be slower. The a d a p t i v e c o n t r o l of process P1 was c a r r i e d out a t d i f f e r e n t sampling r a t e s ; a l l r e s u l t s were s a t i s f a c t o r y a l t h o u g h slow r a t e s i n t r o d u c e d only s l i g h t improvements over the system with no c o n t r o l l e r . F i g u r e 4.9 shows the response of the p r o c e s s with the a d a p t i v e c o n t r o l l e r o p e r a t i n g a t 20 samples/second. As i s shown, the c o n t r o l s i g n a l changes d r a m a t i c a l l y when the r e f e r e n c e input goes through a sudden change. To s i m p l i f y the p l o t t i n g of the dat a , the p l o t of the c o n t r o l s i g n a l i s l i m i t e d to ±4 v o l t s , the l i m i t i s ±10 v o l t s on the a c t u a l system. F i g u r e 4.10 shows the c o n t r o l a c t i o n at 50 samples/second. The r e s u l t i s s t i l l very s a t i s f a c t o r y . Note that a l t h o u g h the c o n t r o l s i g n a l i s s a t u r a t e d f o r a g r e a t e r number of samples t h i s does not re p r e s e n t a longer time i n t e r v a l than that i n F i g u r e 4.9b. A f a s t e r r a t e can be ac h i e v e d on t h i s system but the s a t u r a t i o n of the c o n t r o l l e v e l w i l l e v e n t u a l l y l e a d to i n s t a b i l i t y . 5 6 F i g . 4.9b The Control Signal 20 Samples/Second. 5 7 F i g . 4.10a The Process Output. Fig. 4.10 The Control Signal. 50 Samples/Second 58 3. Second-Order Processes The c l o s e d loop t r a n s f e r f u n c t i o n of the second-order p r o c e s s s i m u l a t e d f o r t h i s p a r t i s : C(s) 39.5 R(s) s 2+ 6.5s +28.0 where: o= 5.28 rad/sec. and Q= 0.61. 3.1. I d e n t i f i c a t i o n U sing the f o r m u l a t i o n given i n Chapter I I I , the above p r o c e s s can be r e p r e s e n t e d by a d i f f e r e n c e equation y ( k ) = 0.7ly ( k-1) -0.28y (k-2)+ 0.49u(k-1)+ 0.31u(k-2) at 5 samples/sec. y ( k ) = 1.33y ( k - l ) -0.53y (k-2)+ 0.16u ( k - l ) + 0.12u(k-2) at 10 samples/sec. y ( k ) = 1.66y ( k - l ) -0.72y (k-2)+ 0.05u ( k - l ) + 0.04u(k-2) at 20 samples/sec. In the i d e n t i f i c a t i o n mode the parameters A1 and A2 have upper l i m i t s ; the l i m i t s a r e : |A1|<2.0 and |A2|<1.0. T h i s w i l l ensure t h a t a l l the roots are i n s i d e the u n i t c i r c l e . In a d d i t i o n , the c o n s t r a i n t B2 < B1 w i l l r e s u l t i n a minimum phase system. The i d e n t i f i c a t i o n was c a r r i e d out with the system o p e r a t i n g at the same sampling r a t e s i n d i c a t e d i n the above e q u a t i o n s . The e s t i m a t e d parameters are shown i n F i g u r e s 4.11a, 4.12a, and 59 o CM" in ] o F i g . 4.11a The Estimated Parameters. 5 Samples/Second. o F i g . 4.11b The Process Output and The Predicted Output. 60 Fig. 4.12b The Process Output and The Predicted Output. Fig. 4.13b The process Output and The Fredicted Output. 62 4.13a, along with the p r o c e s s output and the output of the p r e d i c t i o n model, F i g u r e s 4.11b, 4.12b, and 4.13b. As can be seen, the parameters B1 and B2 become very s m a l l as the sampling r a t e i n c r e a s e s . Again the b i a s i n e s t i m a t e s of B1 and B2 does not prevent the r e f e r e n c e model from t r a c k i n g the p r o c e s s output. 3.2. C o n t r o l The process simulated f o r the e v a l u a t i o n of the i d e n t i f i e r i s used now to e v a l u a t e the a d a p t i v e c o n t r o l l e r . The parameters A1 and A2 have upper l i m i t s h i g h e r than the l i m i t s s e t i n the i d e n t i f i c a t i o n mode, but the l i m i t s are low enough t o prevent o v e r f l o w i n the a r i t h m e t i c o p e r a t i o n s . The l i m i t f o r parameter B1 must be s e l e c t e d with c a r e , a very small value (0< B1 <0.1) c o u l d cause a very underdamped system with a l a r g e overshoot when a sudden change occurs i n the r e f e r e n c e i n p u t . However, a l a r g e value f o r the l i m i t (B1 >1.0) w i l l cause an overdamped response. For a second-order process the s e l e c t i o n of the sampling r a t e i s c r i t i c a l ; a f a s t r a t e can cause i n s t a b i l i t y and a n o i s y output. A slow r a t e , on the other hand, r e s u l t s i n a h i g h l y underdamped response when B1 has a v e r y s m a l l l i m i t , and an overdamped response when the l i m i t i s set a t a l a r g e v a l u e . The c o n t r o l a c t i o n was c a r r i e d out at 10 samples/second, the parameter B 1 was l i m i t e d to 0.5 from below and A1 and A2 had an upper l i m i t of 2.0, and the r e f e r e n c e input i s a square wave. The r e s u l t i s shown in F i g u r e 4.14. The a d a p t i v e c o n t r o l l e r has speeded up the response s i g n i f i c a n t l y when 63 o CQ' a I D a >-Tsl I—rv _J ID 1 o<=> U3-J I CD CO - I >— 121 a CJ ! ! O CO _J I I I 60 (SAMPLES) 3 00 F i g . 4.14a The Process Output. 10 Samples/Second ^ = 0 . 2 5 /3=0.50 i i i 1 r 20 40 TIM (SAMPLES) ]00 Fig.-4.14b The Control Signal. 6 4 compared with F i g u r e 4.13b. The simulated process parameters are changed so that i t becomes a h i g h l y underdamped system. The s t e p response of the process i s now shown i n F i g u r e 4.15. For t h i s p r o c e s s the a d a p t i v e c o n t r o l l e r can reduce the overshoot s i g n i f i c a n t l y . Parameter B1 has a lower l i m i t s i n c e the process has a gain l e s s than u n i t y . The r e s u l t i s shown i n F i g u r e 4.16. As can be seen, although there i s almost no overshoot, the process does not have a smooth t r a n s i e n t response. The n o i s e i n the response, and slow response are caused by c o n t r o l l e r l i m i t and q u a n t i z a t i o n n o i s e . In the c o n t r o l of the two p r o c e s s e s d e s c r i b e d the parameters a and i were s e l e c t e d based on the g a i n of the p r o c e s s . 65 F i g . 4.15b The Reference Signal. 66 CD oo" o ta a >-a ~1 1 60 (SAMPLES) i 1 ]00 I—CM _ ZD 1 I a OO _ ! — I — 40 TIME F i g . 4.16a The Process Output. <* = 0 . 5 -a co" a LD a ZDR C M cr: UJo 10 Samples/Second. • ° 3 O CO „ 1 r 20 40 T I M E 60 ( S A M P L E S ) Fig. 4.16b The Control Signal. 67 V. CONCLUSIONS 1. Summary Adaptive control techniques have been reviewed and. re s u l t s in the area have been examined. The sel f - t u n i n g regulators and model reference adaptive systems have received the greatest attention from researchers and control engineers. A method proposed by Martin-Sanchez was found to be computationally simple to implement on a microprocessor. The i d e n t i f i c a t i o n algorithm i s e s s e n t i a l l y a simple form of the recursive least squares estimation method, and the c o n t r o l l e r was found to be sim i l a r to the minimum '"'var'^lahce'" c o n t r o l l e r used 'in the self- t u n i n g regulators. The method has been . established as asymptotically hyperstable using the Popov h y p e r s t a b i l i t y theorem. The hardware necessary for the implementation of t h i s method on a TI TM 990 microprocessor was described; The software was written in assembly language and uses integer arithmetic. In Chapter IV the algorithm was implemented on the microcomputer system and evaluated. As was shown, the result for the system i d e n t i f i c a t i o n are good and the system performed reasonably well. The estimated parameters were very close to the true parameters but, at some sample points, the predicted parameters were noisy although convergence had been reached. This e f f e c t caused by the use of the integer arithmetic in the system, did not prevent the ove r a l l performance of the adaptive i d e n t i f i e r being s a t i s f a c t o r y . The performance of the adaptive c o n t r o l l e r , although 6 8 acceptable, was not as good as the adaptive i d e n t i f i e r , and to achieve a good performance, some a ' p r i o r i knowledge about the structure of the process was often needed. The most important shortcoming of t h i s adaptive c o n t r o l l e r i s that the c o n t r o l l e r always t r i e s to reach steady state, the desired output, in one sample. This action demands very large amplitude control signals which are unrealizable and can cause i n s t a b i l i t y . Another problem with t h i s algorithm l i e s in the d i f f i c u l t y of s e l e c t i n g the parameters c and p dicussed in Chapter I I . These parameters were found to be very important when the system i s in the control mode. The simplest choice of these parameters i s based on the gain of the process. In addition to problems associated with the algorithm, the integer arithmetic operations of the r e a l i z a t i o n also induce errors into the system. 2. Future Work The algorithm implemented on t h i s microcomputer can be used as a r e l i a b l e i d e n t i f i c a t i o n method, but the control algorithm i s not a robust method and needs modification to provide a r e l i a b l e c o n t r o l l e r . An a n a l y t i c a l procedure for choosing the a and £ parameters would be invaluable. However, the selection of these parameters must be made so as to maintain the economy of computation which i s a very important aspect of t h i s method for microcomputer implementation. The microcomputer system could be combined with a f l o a t i n g point arithmetic operation board such as the Am 9511 arithmetic processing unit to provide more convenient and r e l i a b l e 69 implementation. Another a l t e r n a t i v e would be to w r i t e r o u t i n e s which use the hardware m u l t i p l y and d i v i d e c a p a b i l i t y of the TM 990 m i c r o p r o c e s s o r to perform f l o a t i n g p o i n t a r i t h m e t i c . Such methods, however, may be very slow. The a d d i t i o n of a h i g h l e v e l language and bulk storage would add to the convenience of f u r t h e r expansion. S e l f - t u n i n g r e g u l a t o r s are c o m p u t a t i o n a l l y awkward to implement on microcomputers, but the minimum v a r i a n c e c o n t r o l l e r i s u s u a l l y very r e l i a b l e . I t i s t h e r e f o r e suggested t h a t t h i s microcomputer be used f o r the implementation of s e l f - t u n i n g r e g u l a t o r s to p r o v i d e a comparison w i t h the present work. 70 APPENDIX A The System Hardware A . I . TM 990/101M-1 Microcomputer Some of the i m p o r t a n t f e a t u r e s of t h i s microcomputer board a r e l i s t e d below: • The TMS 9901 programmable system i n t e r f a c e i s a m u l t i f u n c t i o n a l component, which p r o v i d e s i n t e r r u p t , I / O p o r t s , and an i n t e r v a l t i m e r . T h i s component w i l l be used t o s e t up the ISR t o p r o v i d e a v a r i a b l e s a m p l i n g r a t e . * The TMS 9902 Asynchronous Communications C o n t r o l l e r (ACC). T h i s component p r o v i d e s an i n t e r f a c e between the m i c r o p r o c e s s o r and a s e r i a l asynchronous communication c h a n n e l . The TMS 9902 ACC a c c e p t s EIA RS-232-C p r o t o c o l , i t t h e r e f o r e f a c i l i t a t e s t h e communication between t h e t e r m i n a l and the m i c r o p r o c e s s o r s e r i a l I / O p o r t s . The TMS 9902 ACC can a l s o be used as an i n t e r v a l t i m e r w i t h l o w e r r e s o l u t i o n , 64 m i c r o s e c o n d s compared w i t h 21.33 m i c r o s e c o n d s f o r the TMS 9901 PSI . The TMS 9901 PSI and TMS 9902 ACC p r o v i d e some o t h e r v e r y u s e f u l f u n c t i o n s which a r e d e s c r i b e d i n d e t a i l i n TI TMS 9900 M i c r o c o m p u t e r Manual. ® 17 i n t e r r u p t i n p u t a r e a v a i l a b l e on t h e TM 990/101M. Two i n t e r r u p t s a r e non-maskable and o t h e r s a r e maskable. There a r e t h r e e i n t e r r u p t s o u r c e s on b o a r d : The two s e r i a l I / O p o r t s , and the TMS 9901 PSI i n t e r v a l t i m e r . The 15 maskable i n t e r r u p t s a r e 71 a l s o a u t o m a t i c a l l y p r i o r i t i z e d , • Hardware m u l t i p l y and d i v i d e i s a v a i l a b l e on the TM 990 f a m i l y . T h i s f e a t u r e was one of the reasons t h a t t h i s microcomputer was s e l e c t e d . • The hardware r e g i s t e r s a v a i l a b l e on the TM 990 boards are the program counter (PC), s t a t u s r e g i s t e r (ST), and the workspace p o i n t e r (WP). WP always c o n t a i n s the address of the f i r s t word of a set of 16 contiguous words which can be used l i k e an accumulator. T h i s f e a t u r e i s very u s e f u l when branching to a s u b r o u t i n e , because i t a l l o w s the user t o use a new s e t of 16 word workspace, when branching to a s u b r o u t i n e , the l a s t t h r e e r e g i s t e r s (R13,R14,R15) c o n t a i n the o l d PC, WP, and ST r e g i s t e r . • MEMORY- The TM990-101M-1 microcomputer p r o v i d e s 2K words of RAM and 2K words of EPROM. I t should be mentioned t h a t the TIBUG monitor software o c c u p i e s 1K word of EPROM, t h e r e f o r e only 1K word of EPROM i s a v a i l a b l e f o r programming. A.2. RTI-1241-S Board The standard board used i n the c o n t r o l system has the f o l l o w i n g f e a t u r e s : » The r e s o l u t i o n i s 12 b i t s . The c o d i n g of the data i s j u m p e r - s e l e c t a b l e and i n three forms; i ) N a t u r a l B i n a r y , i i ) O f f s e t B i n a r y , and i i i ) 2's complement. • There are 16 s i n g l e ended (SE) or 8 d i f f e r e n t i a l (DIFF) input c h a n n e l s . These can be doubled on board or can be expanded . to 256 SE or 128 DIFF input channels u s i n g an 72 expansion board-® The software-programmable gain permits subranging over the g a i n s of 1, 2, 4, 8. • Memory-Mapped i n t e r f a c e : I t was mentioned e a r l i e r t h a t t h i s board has been designed to be used with a TM 990 microcomputer board. The host computer when i n t e r f a c e d with the RTI-1241 board, p e r c e i v e s the analog i n t e r f a c e board as a block of e i g h t words i n memory. The communication i s i n the same way i t s t o r e s and r e t r i e v e s data. The computer i s t h e r e f o r e f r e e t o do other t h i n g s while the c o n v e r s i o n i s " b e i n g performed. To perform the (A/D).conversion: (a) Set the GAIN, (b) Set the MULTIPLEXER channel, (c) Send the CONVERSION COMMAND ( s t a r t the c o n v e r s i o n ) . There are only three i n s t r u c t i o n s needed t o do the (A/D) c o n v e r s i o n . I t takes about 35 microseconds t o execute the above i n s t r u c t i o n s . To perform the (D/A) c o n v e r s i o n : (a) Send the data to one of the two DAC's. Only one i n s t r u c t i o n i s r e q u i r e d . The DAC output range i s j u m p e r - s e l e c t a b l e . In t h i s system i t i s s e t f o r ±10 v o l t s o p e r a t i o n with 2's complement c o d i n g . 73 A , 3 . TI 745 T e r m i n a l The c o m m u n i c a t i o n b e t w e e n t h e m i c r o c o m p u t e r a n d t h e s y s t e m o p e r a t o r i s done t h r o u g h t h e TI 745 e l e c t r o n i c d a t a t e r m i n a l . T h i s t e r m i n a l c a n be e a s i l y i n t e r f a c e d t o t h e TM 990 m i c r o c o m p u t e r b o a r d u s i n g t h e E I A RS-232-C c a b l i n g . The 745 t e r m i n a l comes w i t h a modem w h i c h c a n c o n n e c t t h e m i c r o c o m p u t e r b o a r d t o a n o t h e r d e v i c e , s u c h a s a l a r g e c o m p u t e r . The Baud R a t e f o r t h i s t e r m i n a l c a n be e i t h e r 110 o r 300 B a u d . The c o n t r o l s y s t e m i s s e t f o r t h e 300 Baud R a t e . A.4. TM 990/510 C a r d Cage The TM 990/510 c a r d c a r d c a g e h a s f o u r s l o t s , two o f t h e s e s l o t s c o n t a i n t h e TM 990 m i c r o c o m p u t e r b o a r d a n d t h e R T I - 1 2 4 1 c o n v e r s i o n b o a r d . The b a c k p a n n e l c o n t a i n s t h e a d d r e s s b u s , d a t a b u s , a n d c o n t r o l l i n e s t o p e r m i t memory, I/O, and DMA. e x p a n s i o n o f CPU m o d u l e s . A.5. The Power S u p p l y The DC v o l t a g e s n e e d e d f o r t h e s y s t e m a r e : T h e s e v o l t a g e s a r e p r o v i d e d by a GSC (GOF-2A-1T) t r i p l e o u t p u t power s u p p l y . I t p r o v i d e s : + 5 V ±12 V ±15 V + 5 V at 5 Amps ±15 V a t 1 .5 Amps the ±12 V i s o b t a i n e d from ±15 V using two r e g u l a t o r s . 75 REFERENCES [ I ] Astrom K.J. " S e l f - T u n i n g R e g u l a t o r s , Design P r i n c i p l e s and A p p l i c a t i o n s , " Department of Automatic C o n t r o l , Lund I n s t i t u t e of Technology, pp. 1-14. 1980. [2] Astrom K.J. and Eykhoff P., "System I d e n t i f i c a t i o n , A Survey," Automatica V o l . 7 No.2, pp. 123-162, March 1971. [3] Astrom K.J., B o r r i s o n V., Ljung L., and Wittenmark B., "Theory and A p p l i c a t i o n of S e l f - T u n i n g Regulators, 5' Automatica, V o l . 13, No. 1, pp. 457-476 Sept. 1977. [4] Astrom K.J. and Wittenmark B., "On S e l f - T u n i n g R e g u l a t o r s , " Automatica, V o l . 9, No. 2, pp. 185-199, March 1973. [5] D r e s s i e r R.M., "An Approach to Model Reference Adaptive c o n t r o l System," IEEE Trans, on AC V o l . AC-12 No.1, pp. 75-80, Feb. 1967. [6] Landau I.D., "A H y p e r s t a b i l i t y C r i t e r i o n f o r the Model Reference Adaptive C o n t r o l System," IEEE Trans, on AC, V o l AC-14 pp. 552-555, Oct. 1 9 6 9 . V . . : [7] Landau I.D., " S y n t h e s i s of D i s c r e t e Model Reference Adaptive Systems," IEEE Trans, on AC, pp. 507-509, Oct. 1971 . [8] Landau I.D., "A Survey of Model Reference Adaptive Techniques, Theory and A p p l i c a t i o n s , " Automatica Vol-10, pp. 353-379, 1974. [9] Landau I.D., "Adaptive C o n t r o l , The Model Reference  Approach," Dekker 1979 New York. [10] Kurz H., Isermann R., and Schumann R., "Experimental Comparison and A p p l i c a t i o n of V a r i o u s Parameter-Adaptive C o n t r o l A l g o r i t h m s , " Automatica, V o l . 16, pp. 117-133, 1980. [ I I ] Goodwin G.C., Johnson J r . C.R., and S i n K.S., " G l o b a l Convergence f o r Adaptive One-Step-Ahead Optimal C o n t r o l l e r s Based on Input Matching," IEEE Trans, on AC, V o l . AC-26 No.6 , pp. 1269-73, Dec. 1981. [12] Ljung L., " A n a l y s i s of R e c u r s i v e S t o c h a s t i c A l g o r i t h m s , " IEEE Trans, on AC, AC-22, pp. 539-551, 1977. 76 [13] Pearson A.E., "An Adaptive C o n t r o l A l g o r i t h m f o r L i n e a r Systems," IEEE Trans, on AC, AC-14, pp. 497-503, 1969. [14 ] C e g r e l l T., and Hedqvist T., " S u c c e s s f u l A d a p t i v e C o n t r o l of Paper Machines," Automatica V o l . 1 1 , pp. 53-59, 1975. [15] B o r r i s o n U„, and Syding R., " S e l f - T u n i n g C o n t r o l of An Ore Crusher," Automatica, V o l . 12, pp. 1-7, 1976. [16] Popov V.M., "The S o l u t i o n of a New S t a b i l i t y Problem f o r C o n t r o l l e d Systems," Automation and Remote C o n t r o l , V o l . 24, pp. 1-23, Jan. 1963. [ 17 ] Codbole S., and Smith C.F., "A New C o n t r o l Approach Using the Inverse System," IEEE Trans, on AC, pp. 698-720, 1972. [18] Peterka V., "Adaptive D i g i t a l R e g u l a t i o n of Noisy Systems," IFAC Symposium on I d e n t i f i c a t i o n and Process Parameter E s t i m a t i o n , 1970. [19] Wittenmark B., "A S e l f - T u n i n g R e g u l a t o r , " Department of AC Lund I n s t i t u t e of Technology, 1973. [20] Martin-Sanchez J.M., "A New S o l u t i o n t o A d a p t i v e C o n t r o l , " IEEE p r o c e e d i n g s , V o l . 64 No.8, pp, 1209-18, Aug. 1976. [21] Nagumo J . , and Noda A., "A L e a r n i n g method f o r System I d e n t i f i c a t i o n , " IEEE Trans, oh AC V o l . AC-12, pp. 282-287, 1967. [22] Van Amorongen J . , and Udink Ten Cate A . J . , "Model Reference A u t o p i l o t f o r S h i p s , " Automatica V o l . 11 p. 441, 1975. [23] K a l s t r o m C.G., Astrom K.J., T h o r e l l N.E., E r i k s s o n J . , and Sten L., "Adaptive A u t o p i l o t s f o r Takers," Automatica, V o l . 15, pp. 241-254, 1979. [24] Monopoli R.V., and Narendra K.S., " A p p l i c a t i o n s of  A d a p t i v e C o n t r o l , " Academic P r e s s , 1980. [25] Anderson B.D., "A S i m p l i f i e d Viewpoint of H y p e r s t a b i l i t y , " IEEE Trans, on AC, V o l . AC-13, No.3, pp. 292-294, June 1968. [26] Ogata K., "Modern C o n t r o l Theory," P r e n t i c e - H a l l , 1970. [27] Auslander D.M., Takahashi Y.,and Tomisuka, " D i r e c t D i g i t a l C o n t r o l p r a c t i c e and A l g o r i t h m f o r M i c r o p r o c e s s o r A p p l i c a t i o n s , " IEEE Proc. V o l 66, No.2, pp. 199-208, Feb. 1978. i 77 [28] Whitaker, Yarmon, and Kezer, "Design of a Model Reference Adaptive System f o r A i r c r a f t , " MIT Instrumentation Laboratory Rept. R-164, Sept. 1958. [29] Serdyukov Y. A., "Synthe s i s of a G e n e r a l i z e d Adaptation A l g o r i t h m f o r a Non-search S e l f A d j u s t i n g System with Reference Model," Auto, and Remote C o n t r o l 31, pp, 1078-84, 1970. 

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