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UBC Theses and Dissertations

Automatic model structure determination for adaptive control Kotzev, Anat 1992-12-31

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AUTOMATIC DETERMINATION  MODEL  STRUCTURE  FOR ADAPTIVE  CONTROL  By Anat  Kotzev  B . S c . C h e m i c a l E n g . , T e c h m o n (Israel I n s t i t u t e of T e c h n o l o g y ) , H a i f a , I s r a e l , 1979 M . S c . Chemical E n g . , Technion, Haifa, Israel,  A  THESIS T H E  SUBMITTED  IN  PARTIAL  REQUIREMENTS  FOR  F U L F I L L M E N T OF  T H E D E G R E E  O F  D O C T O R OF PHILOSOPHY  in T H E  FACULTY OF  G R A D U A T E  MECHANICAL  STUDIES  ENGINEERING  W e a c c e p t this thesis as c o n f o r m i n g to the r e q u i r e d  T H E  UNIVERSITY  OF  June  (c)  standard  BRITISH  COLUMBIA  1992  Anat Kotzev,  1982  1992  In  presenting this  degree at the  thesis  in  University of  partial  fulfilment  of  the  requirements  British Columbia, I agree that the  freely available for reference and study. I further  this thesis for scholarly purposes may be granted  department  or  his  or  her  representatives.  an advanced  Library shall make  it  agree that permission for extensive  copying of  by  for  It  is  by the  understood  that  head of copying  my or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  fl^f^  1^1 ^  n - j ^  Abstract  T h i s w o r k is a s t u d y of a d a p t i v e l y c o n t r o l l e d s y s t e m s w i t h p l a n t m o d e l s t r u c t u r e s t h a t m a y v a r y d u e t o c h a n g i n g o p e r a t i n g c o n d i t i o n s . M o s t closed l o o p a d a p t i v e c o n t r o l a l g o r i t h m s u s e i d e n t i f i c a t i o n m e t h o d s for d e t e r m i n a t i o n of t h e p a r a m e t e r s i n fixed s t r u c t u r e m o d e l s .  Those  p a r a m e t e r s , once e s t i m a t e d , are a s s u m e d t o be correct a n d u n c e r t a i n t i e s i n t h e v a l u e s a r e i g n o r e d . I f t h e s t r u c t u r e of t h e p l a n t d y n a m i c s changes o n - l i n e , t h e i n c o r r e c t m o d e l c a n l e a d to p o o r p e r f o r m a n c e a n d i n s t a b i l i t i e s . T h e a d a p t i v e a l g o r i t h m u s e d i n t h i s w o r k is t h e G e n e r a l i z e d P r e d i c t i v e C o n t r o l ( G P C ) algorithm.  It is r e p o r t e d t o be c a p a b l e of h a n d l i n g a n u m b e r of s i m u l t a n e o u s  a n d t h e r e f o r e was chosen.  problems  A l o n g w i t h h a n d l i n g on-Hne changes of p a r a m e t e r s , i t c l a i m s t o  overcome n o n m i n i m u m - p h a s e plants, open loop unstable plants, plants w i t h badly  damped  poles, p l a n t s w i t h v a r i a b l e or u n k n o w n t i m e delay, a n d p l a n t s w i t h u n k n o w n o r d e r . T h e g o a l of t h i s research is t o i n v e s t i g a t e a n d s t u d y G P C w i t h t h e on-Hne c h a n g e s i n t h e m o d e l s t r u c t u r e of t h e p l a n t , a n d c o r r e s p o n d i n g changes i n t h e o r d e r of t h e e s t i m a t e d m o d e l for G P C a n d t h e s t r u c t u r e of t h e c o n t r o l l e r a n d as well as t o p r o p o s e a m e t h o d t h a t o n - H n e , t h e n e e d for m o d e l o r d e r changes a n d d e t e r m i n e s t h e c o r r e c t  detects  one.  T h e r e are at least two m a j o r sources for s t r u c t u r e v a r i a t i o n s i n t h e e s t i m a t e d  model.  T h e first is t h e m o d e l a c t u a l l y b e i n g t i m e v a r i a n t a n d t h e second r e s u l t i n g f r o m t h e use o f i n h e r e n t l y n o n l i n e a r s y s t e m s a n d m i s - m o d e l i n g . T w o a p p l i c a t i o n s e x e m p l i f y i n g these v a r i a n t s were selected t o e x a m i n e t h e t e c h n i q u e s d e v e l o p e d i n t h e thesis. T h e first is a single  flexible  H n k m a n i p u l a t o r , whose changes i n m o d e l s t r u c t u r e are d u e t o n e w e x c i t e d v i b r a t i o n m o d e s . T h e s e c o n d is a two link r i g i d m a n i p u l a t o r w i t h h y d r a u l i c a c t u a t o r s c a u s i n g t h e s y s t e m t o b e h i g h l y n o n l i n e a r , whose m o d e l c o u l d c h a n g e d u e t o changes i n o p e r a t i n g p o i n t s . T h e effect  of m i s - m o d e l i n g on t h e t o t a l s y s t e m p e r f o r m a n c e a n d s t a b i l i t y was assessed. A cost f u n c t i o n was used as a m e a s u r e of t h e closed l o o p c o n t r o l l e d s y s t e m r e a c t i o n t o u n d e r , correct a n d o v e r - m o d e l i n g .  Its effectiveness  i n t e r m s of s t a b i l i t y a n d  performance  was m e a s u r e d i n c o n t e x t of t h e t w o a p p l i c a t i o n s . I n a d d i t i o n , e x p e r i m e n t a l d a t a f r o m  open  l o o p i d e n t i f i c a t i o n of t h e d y n a m i c m o d e l of a 2 1 5 B C a t e r p i l l a r , a n e x c a v a t o r t y p e m a c h i n e , c o n f i r m s t h e s t u d y of t h e b e h a v i o r of t h e cost f u n c t i o n for those c o n d i t i o n s . B a s e d o n t h e b e h a v i o r of t h e cost f u n c t i o n a n e w a l g o r i t h m was d e v e l o p e d .  The M O D  ( M o d e l O r d e r D e t e r m i n a t i o n ) a l g o r i t h m detects, determines a n d executes, on-line,  changes  t o t h e m o d e l o r d e r . It was i m p l e m e n t e d for b o t h a p p l i c a t i o n w h i c h were c o n t r o l l e d w i t h t h e G P C a l g o r i t h m . T h e results show t h a t g o o d p e r f o r m a n c e a n d s t a b i l i t y c a n be a c h i e v e d . T h e m a i n c o n t r i b u t i o n s of t h i s w o r k are: • T h e M O D a l g o r i t h m w h i c h b a s e d o n t h e b e h a v i o r of a cost f u n c t i o n , c o r r e c t s o n - l i n e m i s - m o d e l i n g of a d a p t i v e l y c o n t r o l l e d s y s t e m s w h i l e m a i n t a i n i n g g o o d  performance.  • G P C was successfuly i m p l e m e n t e d for h y d r a u l i c a l l y a c t u a t e d m a n i p u l a t o r s .  On-line  a u t o m a t i c change of t h e G P C o u t p u t h o r i z o n was i n t r o d u c e d t o a c h i e v e s u f f i c i e n t l y fast t r a n s i e n t response a n d a v o i d o v e r s h o o t s .  • E x p e r i m e n t a l d a t a f r o m a 2 1 5 B C a t e r p i l l a r m a n i p u l a t o r p r o v e d t h e n e e d for a c l o s e d loop approach.  Uo,  T a b l e of C o n t e n t s  Abstract  i  L i s t of T a b l e s  vii  list o f t a b l e s  vii  L i s t of Figures  viii  list o f figures  1  2  xii  INTRODUCTION  AND  STATEMENT OF OBJECTIVES  1  1.1  Introduction  1  1.2  Objectives and Motivation  2  1.3  T h e Thesis Outhne  3  1.4  Thesis Contribution  4  REVIEW  OF PREVIOUS WORK  6  2.1  Outhne  6  2.2  Adaptive Control - General Description  8  2.2.1  Introduction  8  2.2.2  Self T u n i n g R e g u l a t o r s - ( S T R )  9  2.2.3 2.3  Stabihty  10  Generahzed Predictive Control - G P C  11  2.3.1  11  Introduction  2.3.2  3  The G P C Algorithm  11  2.4  I n t r o d u c t i o n to M o d e l S t r u c t u r e a n d P a r a m e t e r D e t e r m i n a t i o n  16  2.5  R e v i e w of P r e v i o u s W o r k i n O r d e r D e t e r m i n a t i o n  19  SINGLE FLEXIBLE LINK MANIPULATOR  28  3.1  Outhne:  28  3.2  Single F l e x i b l e L i n k M a n i p u l a t o r  28  3.2.1  Introduction  28  3.2.2  E q u a t i o n s of M o t i o n for t h e Single F l e x i b l e L i n k  29  3.2.3  T h e S t a t e Space M o d e l  33  3.2.4  T h e Discrete T i m e M o d e l  35  3.2.5  O p e n L o o p D i s c r e t e T i m e M o d e l s for Different N u m b e r of M o d e s  3.2.6 3.3  . . .  C o n t r o l S t r a t e g y for t h e F l e x i b l e L i n k M a n i p u l a t o r  38  A n a l y s i s a n d R e s u l t s of S i m u l a t i o n a n d C o n t r o l W o r k P e r f o r m e d  38  3.3.1  38  3.3.2  Introduction Effects of U n d e r - M o d e h n g a n d O v e r - M o d e U n g o n t h e C o n t r o l l e d F l e x ible L i n k  3.3.3  U s e of a n E s t i m a t i o n C o s t F u n c t i o n as a C r i t e r i o n for C h a n g i n g  39 the  S t r u c t u r e of t h e P l a n t ' s M o d e l 3.3.4 3.3.5 3.4  4  49  Effects of O n L i n e C h a n g e s i n M o d e l o r d e r C o m p a r i s o n B e t w e e n t h e B e h a v i o r of T w o Different C o s t F u n c t i o n s  Conclusions  TWO 4.1  36  LINK MANIPULATOR WITH HYDRAULIC ACTUATORS  51 . .  55 62  65  Rigid Two Link Manipulator with Hydrauhc Actuators  68  4.1.1  68  4.1.2  Introduction E q u a t i o n s of M o t i o n for t h e R i g i d T w o L i n k M a n i p u l a t o r  68  4.1.3 4.2  4.4  5  74  4.2.1  74  Introduction C o n t r o l S t r a t e g y for t h e T w o h n k R i g i d M a n i p u l a t o r  74  A n a l y s i s a n d R e s u l t s of S i m u l a t i o n  76  4.3.1  System Parameters  76  4.3.2  Open Loop Analysis  77  4.3.3  Simulation Study and Results  79  4.3.4  Effects of O n - h n e C h a n g e s i n M o d e l O r d e r  93  Conclusions  93  MODEL ORDER DETERMINATION  95  5.1  Introduction  95  5.2  Cost Function - For Detection O f T h e M o d e l Structure  96  5.2.1  T h e C o s t F u n c t i o n for the F l e x i b l e L i n k M a n i p u l a t o r  5.2.2  T h e C o s t F u n c t i o n for t h e T w o L i n k M a n i p u l a t o r w i t h H y d r a u h c A c tuators  6  70  Control Strategy  4.2.2 4.3  E q u a t i o n s of M o t i o n of t h e H y d r a u h c A c t u a t o r  102  113  5.3  R e a s o n s for U n d e r a n d O v e r - M o d e l e d B e h a v i o r  127  5.4  M O D - Model Order Determination Algorithm  139  IMPLEMENTATION OF T H E MOD 6.1  ALGORITHM  I m p l e m e n t a t i o n of t h e O r d e r D e t e r m i n a t i o n A l g o r i t h m  145 145  6.1.1  T h e M e t h o d For The Flexible Link Manipulator  147  6.1.2  T h e Method For The Two Link Manipulator W i t h Hydrauhc Actuators  166  6.2  C o m p a r i s o n of m e t h o d ' s R e s u l t s w i t h O t h e r W o r k  176  6.3  Conclusions  179  7  CONCLUSIONS AND SUMMARY  181  7.1  M a i n R e s u l t s of t h e T h e s i s  181  7.2  S u g g e s t i o n s for F u t u r e W o r k  183  Bibliography  185  Appendices  191  A  192  B  E x p e r i m e n t a l R e s u l t s for t h e H y d r a u l i c A c t u a t e d M a n i p u l a t o r A.l  Introduction  192  A.2  D e s c r i p t i o n of t h e S y s t e m  192  A.3  Results  193  A.4  Conclusions  196  M o d a l A n a l y s i s for a C a n t i l e v e r B e a m  205  L i s t of T a b l e s  2.1  Différent final goals a n d s p e c i f i c a t i o n s for i d e n t i f i c a t i o n cases  16  3.1  N o . of f l e x i b l e m o d e s v s . o r d e r of s y s t e m  64  B.l  D a t a for t h e first five m o d e s of a c a n t i l e v e r b e a m  208  L i s t of F i g u r e s  2.1  B l o c k d i a g r a m for a self t u n i n g r e g u l a t o r , ( A s t r o m ^)  7  2.2  G e n e r a l p r o c e d u r e of process i d e n t i f i c a t i o n , ( I s e r m a n n  20  3.1  C o n f i g u r a t i o n of t h e single h n k flexible a r m  30  3.2  T w o m o d e flexible h n k w i t h t w o m o d e e s t i m a t o r m o d e l  40  3.3  T h e angle 6 a n d i t s d e r i v a t i v e s  41  3.4  T h e generaUzed g o o r d i n a t e  a n d its d e r i v a t i v e s  42  3.5  T h e g e n e r a h z e d c o o r d i n a t e q2 a n d i t s d e r i v a t i v e s  43  3.6  T h e torque input  44  3.7  T h e effect of u n d e r - m o d e l l i n g  46  3.8  T h e effect of o v e r - m o d e l l i n g  47  3.9  O v e r - m o d e l l e d 2 m o d e h n k w i t h 8''' o r d e r e s t i m a t o r  48  3.10 E s t i m a t o r cost f u n c t i o n for correct m o d e U i n g  50  3.11  O n h n e change of e s t i m a t o r m o d e l - correct to u n d e r - m o d e U n g  52  3.12  E s t i m a t o r cost f u n c t i o n for t h e o n h n e change i n m o d e l  53  3.13  O n Une change of e s t i m a t o r m o d e l - u n d e r m o d e U i n g t o c o r r e c t  54  3.14  E s t i m a t o r cost f u n c t i o n for t h e o n Une change i n m o d e l  56  3.15  O n l i n e change of e s t i m a t o r m o d e l - u n d e r m o d e l U n g t o c o r r e c t  57  3.16  E s t i m a t o r cost f u n c t i o n for t h e o n Une change i n m o d e l  58  3.17  G P C a n d e s t i m a t o r cost f u n c t i o n - onUne correct t o u n d e r - m o d e U n g  59  3.18  G P C a n d e s t i m a t o r cost f u n c t i o n - onUne u n d e r - m o d e U n g t o c o r r e c t (0.1 s e c o n d s )  60  3.19  G P C a n d e s t i m a t o r cost f u n c t i o n - onUne u n d e r - m o d e U n g t o c o r r e c t (0.2 s e c o n d s )  61  4.1  C o n f i g u r a t i o n of t h e t w o h n k m a n i p u l a t o r  69  4.2  Electrohydrauhc actuator  72  4.3  C o n t r o l s t r a t e g y for t h e t w o h n k m a n i p u l a t o r  75  4.4  ^1,^1,^1 for s q u a r e wave i n p u t  80  4.5  ^2,^2,^2 for square wave i n p u t  81  4.6  C o n t r o l a c t i o n a n d s p o o l d i s p l a c e m e n t for di a n d 62  82  4.7  P r e s s u r e s for Oi a n d 62  83  4.8  C o n t r o l a c t i o n a n d s p o o l d i s p l a c e m e n t for h y d r a u h c h n e a r i z e d m o d e l  85  4.9  P r e s s u r e s for 0i a n d 62 for h y d r a u h c h n e a r i z e d m o d e l  86  4.10  T h e effect of h i g h e r values of A^2 (lower case)  87  4.11 ^1 a n d ^2 for ATj^,^ = 50 a n d ATz^,^ = 20  89  4.12 61 a n d ^2 for iVj^,^ = 200 a n d iVa^,^ = 200  90  4.13  0i a n d 62 for N^^^ = 3 a n d iV„^^ = 3  91  4.14  9, a n d ^2 for  92  5.1  C o s t f u n c t i o n b e h a v i o r for o p e n l o o p f l e x i b l e h n k  98  5.2  C o s t f u n c t i o n b e h a v i o r for closed l o o p f l e x i b l e h n k  99  5.3  S c h e m a t i c d e s c r i p t i o n of t h e C . F . b e h a v i o r for t h e different a p p h c a t i o n s  5.4  C o s t f u n c t i o n b e h a v i o r for 2 m o d e L i n k a n d 2 m o d e e s t i m a t e d m o d e l  103  5.5  O u t p u t e r r o r b e h a v i o r for 2 m o d e h n k a n d 2 m o d e e s t i m a t e d m o d e l  104  5.6  C o s t f u n c t i o n b e h a v i o r for 2 m o d e h n k a n d 1 m o d e e s t i m a t e d m o d e l  = 10 a n d N^^^ = 1 0  . . . 101  with  l o g a r i t h m i c axis 5.7  C o s t f u n c t i o n b e h a v i o r for 2 m o d e h n k a n d 0 m o d e e s t i m a t e d m o d e l  105 with  l o g a r i t h m i c axis  106  5.8  C o s t f u n c t i o n b e h a v i o r for 2 m o d e h n k a n d 3 m o d e e s t i m a t e d m o d e l  107  5.9  C o s t f u n c t i o n b e h a v i o r for 2 m o d e h n k a n d 4 m o d e e s t i m a t e d m o d e l  108  5.10  C o s t f u n c t i o n b e h a v i o r for 1 m o d e H n k a n d 1 M o d e e s t i m a t e d m o d e l  110  5.11  O u t p u t b e h a v i o r for 1 m o d e Hnk a n d 0 m o d e e s t i m a t e d m o d e l  I l l  5.12  C o s t f u n c t i o n b e h a v i o r for 1 m o d e H n k a n d 0 m o d e e s t i m a t e d m o d e l  with  logarithmic axis  112  5.13  C o s t f u n c t i o n b e h a v i o r for 1 m o d e Hnk a n d 2 m o d e e s t i m a t e d m o d e l  114  5.14  C o s t f u n c t i o n b e h a v i o r for 1 m o d e H n k a n d 3 m o d e e s t i m a t e d m o d e l  115  5.15  C o s t f u n c t i o n b e h a v i o r for 1 m o d e Hnk a n d 4 m o d e e s t i m a t e d m o d e l  116  5.16  6i a n d 62 b e h a v i o r for 3 m o d e m o d e l a n d 3 m o d e e s t i m a t e d m o d e l  118  5.17  C o s t f u n c t i o n b e h a v i o r for 3 m o d e h y d r a u H c Hnks ( H n e a r i z e d p l a n t m o d e l ) a n d 3 mode estimated models  5.18  119  C o s t f u n c t i o n b e h a v i o r for 3 m o d e h y d r a u H c Hnks ( n o n H n e a r p l a n t m o d e l ) a n d 3 mode estimated models  5.19  120  C o s t f u n c t i o n b e h a v i o r for 3 m o d e h y d r a u H c Hnks a n d 2 m o d e e s t i m a t e d m o d e l 121  5.20 01 a n d O2 b e h a v i o r for 2 m o d e m o d e l a n d 3 m o d e e s t i m a t e d m o d e l  122  5.21  C o s t f u n c t i o n b e h a v i o r for 4 m o d e h y d r a u H c Hnks a n d 4 m o d e e s t i m a t e d m o d e l 123  5.22  C o s t f u n c t i o n b e h a v i o r for 5 m o d e h y d r a u H c Hnks a n d 3 m o d e e s t i m a t e d m o d e l 125  5.23 6i a n d 62 b e h a v i o r for 5 m o d e m o d e l a n d 3 m o d e e s t i m a t e d m o d e l  126  5.24  F l o w c h a r t of a n a d a p t i v e c o n t r o l l o o p w i t h m o d e l o r d e r d e t e r m i n a t i o n  5.25  F l o w c h a r t of t h e o r d e r d e t e r m i n a t i o n p r o c e d u r e  141  6.1  R e g i o n s for u n d e r , over a n d correct m o d e H n g  149  6.2  T h e o u t p u t b e h a v i o r of a two m o d e  flexible  . . . .  Hnk e s t i m a t e d i n i t i a U y w i t h a n  order 2 model 6.3  140  152  O r d e r changes of t h e e s t i m a t e d m o d e l for a t w o m o d e flexible H n k e s t i m a t e d initially w i t h order 2  153  6.4  T h e o u t p u t b e h a v i o r of a t w o m o d e f l e x i b l e h n k e s t i m a t e d i n i t i a l l y w i t h  an  order 4 model 6.5  155  O r d e r changes of t h e e s t i m a t e d m o d e l for a two m o d e flexible h n k e s t i m a t e d i n i t i a l l y w i t h order 4  6.6  156  T h e cost f u n c t i o n d e r i v a t i v e b e h a v i o r of a t w o m o d e  flexible  hnk estimated  initially with an order 4 m o d e l 6.7  157  T h e cost f u n c t i o n b e h a v i o r of a two m o d e flexible h n k e s t i m a t e d i n i t i a l l y w i t h an order 4 model  6.8  158  3 m o r e cases of o u t p u t b e h a v i o r of a two m o d e flexible h n k e s t i m a t e d i n i t i a l l y w i t h an order 4 m o d e l  6.9  159  3 m o r e cases of o r d e r changes of t h e e s t i m a t e d m o d e l for a t w o m o d e  flexible  hnk estimated initially with order 4 6.10  T h e o u t p u t b e h a v i o r of a one m o d e  160 flexible  hnk estimated initially with an  order 2 model 6.11  162  O r d e r changes of t h e e s t i m a t e d m o d e l for a one m o d e flexible h n k e s t i m a t e d i n i t i a l l y with order 2  6.12 T h e o u t p u t b e h a v i o r of a one m o d e flexible h n k i n i t i a l l y o v e r - m o d e l e d  163 . . . .  164  6.13 O r d e r changes of the e s t i m a t e d m o d e l for a one m o d e flexible h n k e s t i m a t e d i n i t i a l l y w i t h o r d e r 10 6.14  T h e o u t p u t b e h a v i o r of a h y d r a u h c a c t u a t e d t w o h n k m a n i p u l a t o r i n i t i a l l y under-modeled  6.15  6.16  165  169  O r d e r changes of t h e e s t i m a t e d m o d e l for h y d r a u h c a c t u a t e d t w o h n k m a n i p ulator initially under-modeled  170  T h e cost f u n c t i o n d e r i v a t i v e b e h a v i o r of h n k l i n i t i a l l y u n d e r - m o d e l e d  171  6.17 T h e cost f u n c t i o n b e h a v i o r o f h n k l i n i t i a l l y u n d e r - m o d e l e d  172  6.18  T h e o u t p u t b e h a v i o r of a h y d r a u h c a c t u a t e d t w o h n k m a n i p u l a t o r O v e r modeled  6.19  173  O r d e r changes of t h e e s t i m a t e d m o d e l for h y d r a u h c a c t u a t e d t w o l i n k m a n i p ulator over-modeled  174  6.20  P e r f o r m a n c e of a s y s t e m i n i t i a l l y m i s - m o d e l e d o n a l a r g e r t i m e s c a l e  175  A.l  Caterpillar 215B excavator  194  A.2  I n p u t o u t p u t b e h a v i o r for S l f  197  A.3  I n p u t o u t p u t b e h a v i o r for S 2 f  198  A.4  I n p u t o u t p u t b e h a v i o r for S 3 f  199  A.5  I n p u t o u t p u t b e h a v i o r for S 4 f  200  A.6  I n p u t o u t p u t b e h a v i o r for R l f  201  A.7  I n p u t o u t p u t b e h a v i o r for R 2 f  202  A.8  C o s t f u n c t i o n b e h a v i o r for t h e S i f cases  203  A. 9  C o s t f u n c t i o n b e h a v i o r for t h e R i f cases  204  B. l  M o d a l S h a p e s for a C a n t i l e v e r B e a m  209  Acknowledgments  I w o u l d l i k e t o t h a n k m y s u p e r v i s o r , P r o f e s s o r D a l e B . C h e r c h a s for t h e s u p p o r t ,  guidance  a n d e n c o u r a g e m e n t he p r o v i d e d i n t h e d e v e l o p m e n t of t h i s w o r k . I w o u l d h k e t o express m y g r a t i t u d e a n d a p p r e c i a t i o n t o P r o f e s s o r P e t e r D . L a w r e n c e , f o r f r u i t f u l w o r k a n d v a l u a b l e suggestions. I a m h i g h l y g r a t e f u l t o m y colleague D o u g L a t o r n e l l , for t h e f r i e n d s h i p , s u p p o r t a n d h e l p I r e c e i v e d d u r i n g a l l t h e years I w o r k e d o n t h i s thesis. A l s o , I w o u l d h k e t o t h a n k A l a n S t e e v e s for his g o o d a d v i c e , p a t i e n c e a n d s u p p o r t i n t h e use of t h e d e p a r t m e n t c o m p u t e r s y s t e m . A l l t h e staff i n t h e d e p a r t m e n t of M e c h a n i c a l E n g i n e e r i n g p r o v i d e d m e w i t h a l l t h e h e l p I n e e d e d , a n d always w i t h a s m i l e . T h e V A X S t a t i o n 3200 c o m p u t e r s y s t e m was p r o v i d e d t h r o u g h a n e q u i p m e n t g r a n t f r o m the B r i t i s h C o l u m b i a A d v a n c e d Systems Institute ( A S I ) . L a s t b u t not least I w o u l d Hke t o t h a n k t h e T e c h n i o n (Israel I n s t i t u t e of  Technology),  e s p e c i a l l y express m y w a r m e s t g r a t i t u d e to P r o f e s s o r R a m L a v i e , f r o m t h e f a c u l t y of C h e m i c a l E n g i n e e r i n g , w h o m a d e t h e last p e r i o d of w o r k i n g o n t h i s thesis s m o o t h a n d by g r a n t i n g m e t h e s t a t u s of a guest fellow at t h e T e c h n i o n .  comfortable  A k e y figure t o c o n t i n u i n g  m y r e s e a r c h w o r k i n I s r a e l was p r o v i d e d , v e r y p r o f e s s i o n a l l y , b y t h e c o m p u t i n g c e n t e r , b y M i r i a m B e n - H a i m and B e n Pashkoff - T h a n k Y o u .  To Milush Roi and R e u t  Chapter 1  INTRODUCTION AND STATEMENT OF  1.1  OBJECTIVES  Introduction  T h e r e s e a r c h i n t h i s thesis deals w i t h a d a p t i v e c o n t r o l s y s t e m s w h o s e p l a n t m o d e l p a r a m e t e r s a n d s t r u c t u r e m a y v a r y due t o c h a n g i n g o p e r a t i n g c o n d i t i o n s . T h e r e are n u m e r o u s  examples  of s y s t e m s t h a t n e e d a d a p t i v e c o n t r o l a l g o r i t h m s , ( A s t r o m et a l ^). O n e e x a m p l e is a r o b o t i c m a n i p u l a t o r w h o s e m o m e n t of i n e r t i a m a y v a r y w i t h i n a w o r k i n g c y c l e . A  flexible  robot m a y  h a v e u n e x p e c t e d m o d e s of v i b r a t i o n o c c u r r i n g a n d c h a n g i n g i t s m o d e l s t r u c t u r e .  Process  c o n t r o l also has changes i n d y n a m i c s , w h i c h d e p e n d o n o p e r a t i n g p a r a m e t e r s , s u c h as  flow  t h r o u g h t a n k s a n d pipes t h a t change w i t h p r o d u c t i o n r a t e . T h e s t r u c t u r e of t h e p l a n t ' s m o d e l is u s u a l l y d e t e r m i n e d b y i t s o r d e r a n d t h e n a t u r e o f i t s n o n h n e a r t e r m s . T h e m o d e l g e n e r a l l y u s e d for t h e a d a p t i v e a l g o r i t h m s is h n e a r a n d t h e r e f o r e its s t r u c t u r e is a c t u a l l y i t s o r d e r .  I n m o s t s t a b i h t y proofs for a d a p t i v e s y s t e m s , t h e b a s i c  a s s u m p t i o n is t h a t t h e m o d e l o r d e r is k n o w n , or at least t h e u p p e r b o u n d of t h e s y s t e m ' s o r d e r is k n o w n . I n t h e presence of a change i n t h e s y s t e m ' s o r d e r , s u c h as n e w s i g n i f i c a n t m o d e s i n a flexible m e c h a n i c a l s y s t e m , or i n t h e presence of a n y u n m o d e l e d d y n a m i c s , i n s t a b i h t i e s c a n o c c u r d u e t o i n c o r r e c t m o d e l s t r u c t u r e a n d therefore i n c o r r e c t p a r a m e t e r s . T h e o n - h n e changes i n t h e p l a n t ' s m o d e l s t r u c t u r e i f t h e y o c c u r , m a y result i n t h e n e e d t o i d e n t i f y those changes a c c o r d i n g l y , a n d adjust t h e o r d e r of t h e m o d e l for t h e a d a p t i v e a l g o r i t h m i n a d d i t i o n to the parameter identification. T h e i d e n t i f i c a t i o n m e t h o d s of a m o d e l off-hne c a n have t h e a d v a n t a g e of c h o o s i n g a m o d e l  o u t of a set of p r o p o s e d m o d e l s t r u c t u r e s . T h e m o d e l , i t s s t r u c t u r e a n d p a r a m e t e r s c a n l a t e r be v e r i f i e d w i t h m o d e l v a h d a t i o n m e t h o d s .  M a n y closed l o o p a d a p t i v e c o n t r o l a l g o r i t h m s  use i d e n t i f i c a t i o n m e t h o d s for o n - h n e d e t e r m i n a t i o n of t h e m o d e l p a r a m e t e r s ( A s t r o m et a l ^).  W e are not aware of a n y effective on-Hne s t r u c t u r e v a h d a t i o n m e t h o d s .  It is n o t  s u r p r i s i n g t h a t m o s t i f not aU i d e n t i f i c a t i o n m e t h o d s use a fixed m o d e l s t r u c t u r e t o e s t i m a t e t h e p a r a m e t e r s . I n some cases, t h e u n c e r t a i n t i e s i n t h e values are i g n o r e d , i.e. t h e i d e n t i f i e d p a r a m e t e r s are a s s u m e d to be c o r r e c t , a n d are used as i f t h e y were t h e t r u e ones.  T h i s is  caUed t h e c e r t a i n t y e q u i v a l e n c e p r i n c i p l e ( A s t r o m et a l ^ a n d M i d d l e t o n a n d G o o d w i n ^ ). A d a p t i v e s y s t e m s h a v e b e e n s a i d t o b e i n h e r e n t l y n o n H n e a r ( A s t r o m et a l ^) a n d r e H a n c e on that principle can lead to instabiHty.  1.2  Objectives and Motivation  O n e o b j e c t i v e of t h e research is t o i n v e s t i g a t e a n d s t u d y a c h o s e n a d a p t i v e c o n t r o l a l g o r i t h m w i t h a c h a n g i n g m o d e l s t r u c t u r e for t h e p l a n t , a n d w i t h i t , t h e c h a n g e i n e s t i m a t e d m o d e l o r d e r a n d o f t h e controUer s t r u c t u r e . A s e c o n d o b j e c t i v e is t o d e v e l o p a m e t h o d t o d e t e c t t h e n e e d for a m o d e l o r d e r c h a n g e , d e t e r m i n e t h e correct o r d e r , e x e c u t e i t o n - H n e a n d e n s u r e that the control system wiU m a i n t a i n its performance a n d stabiHty. Changes i n the order of t h e m o d e l a l o n g w i t h t h e a c c o m p a n y i n g p a r a m e t e r e s t i m a t i o n are t o be i n t e g r a t e d i n t o a c l o s e d l o o p a d a p t i v e s y s t e m . ( A U c a l c u l a t i o n s a n d e s t i m a t i o n s are d o n e on-Hne i n r e a l t i m e ) . T h e a p p r o a c h t a k e n t o t h e p r o b l e m is Hsted b e l o w :  1. A n a p p r o a c h to on-Hne o r d e r c h a n g e d e t e c t i o n a n d e s t i m a t i o n was d e v e l o p e d .  A n as-  s u m p t i o n c o n s i d e r e d is t h a t o r d e r changes are less f r e q u e n t a n d c o n v e r g e slower t h a n t h e changes i n t h e m o d e l p a r a m e t e r s . M o s t e x i s t i n g t e c h n i q u e s choose t h e o r d e r , e s t i m a t e t h e m o d e l p a r a m e t e r s , v a l i d a t e t h e o r d e r a n d i f i n c o r r e c t go t h r o u g h t h e w h o l e p r o c e d u r e a g a i n , off-Hne a n d i n o p e n l o o p , ( L j u n g ^).  2. E x p e r i m e n t a l e v i d e n c e f r o m a r e a l m a c h i n e , a 2 1 5 B C a t e r p i l l a r e x c a v a t o r was e x a m i n e d for o p e n l o o p o p e r a t i o n . T h e m o d e l o u t p u t a n d r e p r e s e n t i n g cost f u n c t i o n v a l u e s were c a l c u l a t e d a n d i n v e s t i g a t e d .  ( T h e r e s u l t s c o r r o b o r a t e t h e n u m e r i c a l r e s u l t s , see  A p p e n d i x A ) . It was shown t h a t a closed l o o p a p p r o a c h was n e e d e d .  3. T h e m o d e l d e t e r m i n a t i o n a l g o r i t h m was i m p l e m e n t e d w i t h G P C for t w o e x a m p h f y i n g a p p h c a t i o n s . T h e G e n e r a h z e d P r e d i c t i v e C o n t r o l ( G P C ) a l g o r i t h m was c h o s e n f o r t h e s t u d y , since i t c l a i m s t o effectively h a n d l e a n u m b e r of p r o b l e m a t i c s y s t e m c h a r a c t e r istics at t h e same t i m e ( C l a r k e 4, 5)  ^ h e r e are at least t w o reasons for p l a n t m o d e l  s t r u c t u r e v a r i a t i o n s . F i r s t , t h e m a c h i n e m o d e l i t s e l f changes s u c h as v i b r a t i o n m o d e s i n a f l e x i b l e l i n k . S e c o n d , use of a h n e a r r e p r e s e n t a t i o n of a h i g h l y n o n h n e a r  machine  m o d e l t r a c k i n g a n o p e r a t i n g p o i n t , s u c h as a t w o r i g i d h n k m a n i p u l a t o r w i t h h y d r a u h c a c t u a t o r s . T h e s t r u c t u r e f l e x i b i h t y of t h e single flexible h n k m a n i p u l a t o r m a y g i v e r i s e t o m o d e s of o s c i l l a t i o n s . D u r i n g a w o r k c y c l e of a r o b o t , n e w m o d e s c a n o c c u r d u e t o m o v e m e n t a n d c h a n g e i n t h e t i p ' s l o a d . T h i s resembles a n ' i n f i n i t e o r d e r ' s y s t e m . ( see C h a p t e r 3).  T h e two h n k r i g i d b o d y m a n i p u l a t o r w i t h h y d r a u h c a c t u a t o r s  a 'finite order' system.  resembles  T h e h y d r a u h c actuators can be m o d e l e d w i t h several m o d e l  s t r u c t u r e s a n d are h i g h l y n o n h n e a r .  C o u p h n g b e t w e e n different m o t i o n s of t h e a r m ' s  p a r t s generates n o n h n e a r t e r m s . ( see C h a p t e r 4).  1.3  T h e Thesis  Outline  C h a p t e r 2 presents a r e v i e w of p r e v i o u s w o r k done i n areas relevant t o t h e r e s e a r c h . C h a p t e r s 3,  4 and  flexible  5 present i n d e t a i l t h e w o r k d o n e t n t h i s r e s e a r c h . C h a p t e r 3 deals w i t h a s i n g l e  h n k m a n i p u l a t o r , i t s d y n a m i c e q u a t i o n s of m o t i o n , t h e c o n t r o l s t r a t e g y , r e s u l t s o f  c l o s e d l o o p s i m u l a t i o n a n d t h e effects of u n d e r a n d o v e r - m o d e h n g .  C h a p t e r 4 deals  the two h n k m a n i p u l a t o r actuated b y hydrauhc actuators and reports on the same  with topics  as C h a p t e r 3.  C h a p t e r 5 i n t r o d u c e s t h e chosen cost f u n c t i o n as a m e a s u r e of p l a n t m i s -  m o d e h n g . T h e b e h a v i o r of t h e cost f u n c t i o n a n d i t s d e r i v a t i v e is i n v e s t i g a t e d . T h e c o n c l u s i o n s d r a w n l e a d t o t h e p r o p o s a l of a m e t h o d t h a t d e t e c t s o n - h n e t h e need for a m o d e l  order  change, d e t e r m i n e s t h e c o r r e c t o r d e r a n d e x e c u t e s i t w h i l e d e v e l o p i n g or m a i n t a i n i n g g o o d system performance.  C h a p t e r 6 presents t h e i m p l e m e n t a t i o n of t h e M O D a l g o r i t h m o n  b o t h a p p h c a t i o n s a n d r e s u l t s are p r e s e n t e d . C h a p t e r 7 discusses t h e c o n c l u s i o n s d r a w n f r o m t h i s research.  A p p e n d i x A provides the experiment results f r o m operating the h e a v y d u t y  m a n i p u l a t o r , t h e 2 1 5 B C a t e r p i l l a r . A p p e n d i x B presents t h e m o d a l a n a l y s i s of a c a n t i l e v e r b e a m , t h e r e s u l t s of w h i c h were u s e d for t h e e q u a t i o n s of t h e single flexible h n k i n C h a p t e r 3.  1.4  Thesis Contribution  T h i s w o r k is a s t u d y of r o b o t i c s y s t e m s c o n t r o l l e d w i t h a n a d a p t i v e a l g o r i t h m ( G P C ) . T h e m a i n focus is o n t h e b e h a v i o r of those s y s t e m s w h e n t h e p l a n t is m i s - m o d e l e d a n d o n r e s t o r i n g its d e s i r e d p e r f o r m a n c e a n d s t a b i H t y i f n e e d e d . T h e m a i n research c o n t r i b u t i o n s are d e s c r i b e d here as:  1. W e d e v e l o p e d a m e t h o d caUed t h e M o d e l O r d e r D e t e r m i n a t i o n ( M O D ) a l g o r i t h m , t o d e t e c t m i s - m o d e H n g a n d i m p l e m e n t i t s c o r r e c t i o n on-Hne.  • A cost f u n c t i o n was s t u d i e d for c o r r e c t , u n d e r a n d o v e r - m o d e H n g , i t was f o u n d i t has s i g n i f i c a n t l y different b e h a v i o r for each case. • T h e cost f u n c t i o n b e h a v i o r for c o r r e c t , u n d e r a n d o v e r - m o d e H n g was s i m i l a r for two exemplifying appHcations. • W h e n M O D was i m p l e m e n t e d , t h e d e s i r e d p e r f o r m a n c e was r e s t o r e d for m i s m o d e H n g for b o t h a p p H c a t i o n s . • R u l e s for choosing the p a r a m e t e r s for t h e M O D a l g o r i t h m were d e f i n e d .  2. W e have f o u n d t h a t G P C c a n be successfuly i m p l e m e n t e d for h e a v y d u t y m a n i p u l a t o r s .  • T h i s w o r k e x a m i n e d some c o m p l e x c o n s i d e r a t i o n s , such as t h e effects o f n o n h n earities i n t h e a p p h c a t i o n of G P C t o a b r o a d c a t e g o r y of h y d r a u h c a U y a c t u a t e d manipulators. • T h e w o r k i n t r o d u c e d o n - h n e a u t o m a t i c change of t h e o u t p u t h o r i z o n ( f o r so t r a n s i e n t response c a n be sufficiently fast a n d u n d e s i r a b l e o v e r s h o o t  GPC)  avoided.  3. E x p e r i m e n t a l results f r o m a n o p e n l o o p e x p e r i m e n t o n a 2 1 5 B C a t e r p i U a r i n d i c a t e t h a t t h e cost f u n c t i o n b e h a v i o r i n o p e n l o o p does not v a r y s t r o n g l y t o b e r e h a b l y d e t e r m i n e d a n d a closed l o o p a p p r o a c h is r e q u i r e d . T h i s was also v e r i f i e d b y n u m e r i c a l s i m u l a t i o n s w i t h other apphcations.  Chapter 2  REVIEW OF PREVIOUS  2.1  WORK  Outline  T h i s c h a p t e r reviews some of t h e relevant w o r k p u b h s h e d i n t h e h t e r a t u r e c o n s i d e r i n g s o m e of t h e t o p i c s discussed i n t h i s thesis. A d a p t i v e c o n t r o l i n g e n e r a l a n d G P C i n p a r t i c u l a r , are d e s c r i b e d . S o m e r e v i e w on m o d e l o r d e r a n d p a r a m e t e r d e t e r m i n a t i o n is also g i v e n . A d a p t i v e c o n t r o l is t h e basic m o t i v a t i o n for t h e search for a m e t h o d t o c h a n g e t h e m o d e l structure on-hne.  S e c t i o n 2.2 describes a d a p t i v e c o n t r o l i n g e n e r a l t e r m s . A b l o c k d i a g r a m  ( F i g u r e 2.1) is presented to c l a r i f y h o w aU t h e c o m p o n e n t s ( i d e n t i f i c a t i o n , p l a n t m o d e l , c o n t r o l a l g o r i t h m , etc.) are i m p l e m e n t e d i n t h e c o m p l e t e c o n f i g u r a t i o n . I n t h e last two decades, m u c h research o n m o d e h n g , s y s t e m i d e n t i f i c a t i o n , m o d e l s t r u c t u r e d e t e r m i n a t i o n , a n d a d a p t i v e c o n t r o l a l g o r i t h m s has b e e n d o n e ( L j u n g ^, ^). M o s t o f t h e w o r k c o n c e r n i n g m o d e l s t r u c t u r e d e t e r m i n a t i o n a n d i t s v a h d a t i o n is d o n e off-line.  T h e advantage of  t h e off-hne m e t h o d s is i n t h e p o s s i b i l i t y of c h o o s i n g a m o d e l out of a set of h k e l y ones. O n t h e c o n t r a r y , o t h e r w o r k s w h i c h i n v o l v e on-line  (recursive) i d e n t i f i c a t i o n d e m a n d a n a s s u m p t i o n  o f a fixed m o d e l s t r u c t u r e , ( L j u n g ^, ^ a n d A s t r o m ^). A b r i e f r e v i e w of t h e a b o v e m o d e H n g a n d s t r u c t u r e d e t e r m i n a t i o n w i U be g i v e n i n S e c t i o n 2.4. I n S e c t i o n 2.3 we present t h e G P C ( G e n e r a l P r e d i c t i v e C o n t r o l ) a l g o r i t h m i n d e t a i l .  The  G P C has b e e n used i n t h e w o r k d o n e so f a r , f r o m w h i c h t h e r e s u l t s w i U b e p r e s e n t e d i n Chapters 3 and  5.  F l e x i b l e s t r u c t u r e m o d e l s are dealt w i t h i n t h i s w o r k . T h e n e e d for  flexibihty  arises w h e n  Process parameters  Estimation  Regulator parameters  Process U  F i g u r e 2.1: B l o c k d i a g r a m for a self t u n i n g r e g u l a t o r , ( A s t r o m '•)  t h e c e r t a i n t y e q u i v a l e n c e p r i n c i p l e is u s e d , or i n t h e presence of u n m o d e l e d d y n a m i c s ,  or  w h e n changes o c c u r i n t h e s t r u c t u r e . I n o r d e r t o m a i n t a i n s t a b i h t y of t h e a d a p t i v e c o n t r o l l e d s y s t e m , t h e flexible m o d e l s t r u c t u r e is p a r t i c u l a r l y i m p o r t a n t . S e c t i o n 2.5 i n c l u d e s a d i s c u s s i o n of t h e p r i o r w o r k d o n e i n t h e area of h n e a r m o d e l ' s  order  d e t e r m i n a t i o n i n t h e 1970's a n d some l a t e r w o r k o n h n e a r a n d n o n h n e a r m o d e l s t r u c t u r e determination.  2.2  Adaptive Control - General Description  2.2.1  Introduction  M u c h w o r k has b e e n d o n e a n d p u b h s h e d o n a d a p t i v e c o n t r o l (Astrom a n d W i t t e n m a r k ^, ^, Astrom a n d B o r r i s o n a n d L j u n g a n d W i t t e n m a r k ^. L a n d a u ^, E d g a r ^^).  M o s t of t h e  t e c h n i q u e s for t h e d e s i g n of c o n t r o l s y s t e m s a s s u m e t h a t t h e p l a n t a n d i t s e n v i r o n m e n t a r e k n o w n . T h i s is not o f t e n t h e case, since t h e p l a n t m i g h t b e t o o c o m p l e x , or b a s i c r e l a t i o n s h i p s m a y n o t b e f u l l y u n d e r s t o o d , or t h e process a n d t h e d i s t u r b a n c e s m a y c h a n g e w i t h o p e r a t i n g c o n d i t i o n s . A d a p t i v e c o n t r o l deals w i t h t h e a b o v e p r o b l e m s . T h e r e are f o u r m a i n c a t e g o r i e s of a d a p t i v e c o n t r o l :  1. S e l f T u n i n g R e g u l a t o r s - S T R 2. M o d e l R e f e r e n c e A d a p t i v e S y s t e m s - M R A S 3. A u t o - T u n i n g 4. G a i n S c h e d u h n g T h e S T R a n d t h e M R A S are t w o w i d e l y d i s c u s s e d a p p r o a c h e s for p l a n t s w i t h u n k n o w n p a r a m e t e r s . regulators.  T h e proposed  to solving the  problem  research c o n c e n t r a t e s o n self t u n i n g  2.2.2  Self T u n i n g Regulators -  (STR)  S T R are b a s e d on a f a i r l y n a t u r a l c o m b i n a t i o n of i d e n t i f i c a t i o n a n d c o n t r o l . I n F i g u r e 2.1 a b l o c k d i a g r a m of t h e s t r u c t u r e of a n S T R c o n t r o l l o o p is s h o w n . It has t w o f e e d b a c k l o o p s , i . e . a n i n n e r l o o p a n d an o u t e r l o o p . T h e i n n e r one is a n o r d i n a r y f e e d b a c k l o o p w i t h a p r o c e s s and a regulator.  T h e r e g u l a t o r has a d j u s t a b l e p a r a m e t e r s w h i c h are set b y t h e o u t e r  loop.  T h e a d j u s t m e n t s are based o n f e e d b a c k f r o m t h e process i n p u t s a n d o u t p u t s . T h e o u t e r l o o p is c o m p o s e d of a r e c u r s i v e p a r a m e t e r e s t i m a t o r a n d a design c a l c u l a t i o n . E s t i m a t i o n s c a n be d o n e o n t h e process p a r a m e t e r s or o n t h e r e g u l a t o r p a r a m e t e r s , d e p e n d i n g o n t h e c o n t r o l a l g o r i t h m . T h e s t a r t i n g p o i n t is a d e s i g n m e t h o d for k n o w n p l a n t s . S i n c e t h e p a r a m e t e r s are not k n o w n , t h e i r e s t i m a t e s are u s e d . T h e a s s u m p t i o n is t h a t t h e r e is a s e p a r a t i o n  between  i d e n t i f i c a t i o n a n d c o n t r o l , a n d t h e p a r a m e t e r s ' u n c e r t a i n t i e s are i n i t i a l l y not c o n s i d e r e d h e r e . A s a s i m p l e e x a m p l e , consider t h e p l a n t m o d e l e d by E q u a t i o n 2.1: y{t)  + ay{t  -  I) = buit -  (2.1)  1) + eit)  W h e r e u is t h e i n p u t , y is t h e o u t p u t a n d e(t) is a sequence of i n d e p e n d e n t , z e r o m e a n r a n d o m v a r i a b l e s . A c o n t r o l l a w t h a t w i l l give minimum v a r i a n c e c o n t r o l is : u{t)  (2.2)  ^ ^yit)  If a a n d b are u n k n o w n , t h e algorithm by Astrom a n d W i t t e n m a r k ^ c a n b e a p p h e d .  It  consists of t w o steps, each r e p e a t e d every s a m p h n g p e r i o d : • E s t i m a t e the parameter a i n the model: y{t)  = ay{t  -  1) + f3ou{t ~ 1) + e{t)  (2.3)  W h e r e e is t h e error. T h e r e s u l t i n g e s t i m a t e is à. • U s e t h e control l a w : nit)  = ^yit)  (2.4)  T h e e s t i m a t i o n of a c a n be d o n e r e c u r s i v e l y a n d on-hne. T h e a b o v e a l g o r i t h m was also g e n e r a h z e d i n Astrom &: W i t t e n m a r k ^. A s t r o m &; W i t t e n m a r k 11  10 a n d C l a r k e &: G a w t h r o p  p r o p o s e d a g e n e r a h z a t i o n of t h e a b o v e b a s i c a l g o r i t h m . S T R  are not c o n f i n e d to m i n i m u m v a r i a n c e c o n t r o l .  Edmunds  Astrom et a l .  ^ proposed  a l g o r i t h m s based o n pole p l a c e m e n t . M u l t i v a r i a b l e f o r m u l a t i o n s were g i v e n b y B o r r i s s o n 2.2.3  Stability  S t a b i h t y is a key r e q u i r e m e n t for a c o n t r o l s y s t e m ; however s t a b i h t y a n a l y s i s of a d a p t i v e s y s t e m s is difficult because t h e b e h a v i o r of s u c h systems is c o m p l e x  as a r e s u l t of t h e i r  n o n h n e a r c h a r a c t e r . T h e s t a b i h t y p r o b l e m c a n be a p p r o a c h e d i n s e v e r a l d i f f e r e n t w a y s . l o c a l s t a b i h t y t e c h n i q u e is of h m i t e d v a l u e since i t reveals l i t t l e a b o u t g l o b a l p r o p e r t i e s .  A  The  f u n d a m e n t a l s t a b i h t y c o n c e p t for n o n h n e a r s y s t e m s refers t o t h e s t a b i h t y o f a p a r t i c u l a r s o l u t i o n . O n e p o s s i b i l i t y is t o a p p l y L y a p u n o v ' s t h e o r y ( E d g a r  a n d A s t r o m ^).  it is o f t e n d i f f i c u l t to find a s u i t a b l e L y a p u n o v f u n c t i o n . C l o s e d l o o p s y s t e m s w i t h  However, bounded  i n p u t / o u t p u t signals a n d d e s i r e d a s y m p t o t i c p r o p e r t i e s c a n be a c h i e v e d , p r o v i d e d t h a t c e r t a i n a s s u m p t i o n s are m a d e as i n Astrom et a l . ^, E d g a r ^ ^ , as n o t e d b e l o w : G i v e n a p l a n t m o d e l of t h e t y p e :  A{q-')y{t)  = boq~^'+'^Biq'')u{t)  + e(f)  (2.5)  W h e r e A(q~^) & B(q~^) are p o l y n o m i a l s of degree n & m of t h e o u t p u t y ( t ) a n d t h e i n p u t u ( t ) r e s p e c t i v e l y , d is t h e t i m e delay a n d e(t) is a d i s t u r b a n c e t h a t c a n n o t b e m e a s u r e d , a n d q~^ is t h e b a c k w a r d shift o p e r a t o r . A l s o g i v e n are t h e f o l l o w i n g : T h e t i m e d e l a y d is k n o w n . T h e u p p e r b o u n d s o n t h e degrees of t h e p o l y n o m i a l s A & B a r e k n o w n , i.e. t h e o r d e r of t h e s y s t e m is k n o w n .  T h e p l a n t is a m i n i m u m p h a s e p r o c e s s .  T h e s i g n of bo is k n o w n . C o n s i d e r i n g those a s s u m p t i o n s , i n A s t r o m a n d W i t t e n m a r k ^ ^ , i t is s h o w n t h a t t h e c l o s e d  l o o p s y s t e m is stable i f b o u n d e d d i s t u r b a n c e a n d c o m m a n d s i g n a l (uc) gives b o u n d e d i n p u t (u) a n d o u t p u t  2.3  (y).  Generalized Predictive Control - G P C  2.3.1  Introduction  E q u a t i o n s of m o t i o n of a r o b o t i c m a n i p u l a t o r c o n t a i n n o n U n e a r i t i e s , i n e r t i a l c h a r a c t e r i s t i c s a n d d i s t u r b a n c e s t h a t v a r y d u r i n g a w o r k i n g c y c l e a n d m a y n o t a l w a y s be p r e d i c t a b l e ( F u ^^). I n m a n y cases, p e r f o r m a n c e o b t a i n e d w i t h f i x e d t i m e i n v a r i a n t c o n t r o l l e r s m a y n o t satisfactory.  be  L a t e l y , self t u n i n g p r e d i c t i v e a l g o r i t h m s have b e e n u s e d , since t h e r e s u l t s a r e  m o r e r o b u s t c o m p a r e d w i t h o t h e r self t u n i n g c o n t r o l a l g o r i t h m s , s u c h as P o l e P l a c e m e n t a n d M i n i m u m V a r i a n c e . T h e r o b u s t n e s s of p r e d i c t i v e a l g o r i t h m s is due t o t h e m i n i m i z a t i o n o f a m u l t i - s t e p cost f u n c t i o n , C l a r k e et a l  T h e basic p r e d i c t i v e m e t h o d c o n t a i n s t h e f o l l o w i n g  steps: 1. P r e d i c t i o n of t h e o u t p u t . 2. C h o i c e of t h e f u t u r e set p o i n t s , a n d m i n i m i z a t i o n of a cost f u n c t i o n c a l c u l a t e d f r o m t h e f u t u r e e r r o r s , b e t w e e n t h e f u t u r e o u t p u t s a n d f u t u r e set p o i n t s , w h i c h y i e l d s a set o f f u t u r e c o n t r o l signals. 3. T h e first t i m e step of t h e c o n t r o l signals is t h a t a c t u a l l y u s e d , a n d t h e w h o l e  procedure  is r e p e a t e d . T h i s is a r e c e d i n g - h o r i z o n c o n t r o l l e r .  2.3.2  The G P C Algorithm  T h e t y p e of c o n t r o l l e r s m e n t i o n e d a b o v e consider t h e o u t p u t at one p o i n t i n t i m e i n t h e f u t u r e . T h e G e n e r a h z e d P r e d i c t i v e C o n t r o l ( G P C ) , C l a r k e et a l  ^, a l g o r i t h m m i n i m i z e s a  cost f u n c t i o n t h a t considers t h e f u t u r e p r e d i c t e d o u t p u t s j steps a h e a d , t h e f u t u r e set p o i n t s  a n d f u t u r e c o n t r o l signals. T h e G P C is r o b u s t a n d deals w i t h o v e r p a r a m e t r i z a t i o n b e c a u s e of i t s p r e d i c t i v e c a p a b i h t i e s , a n d w i t h d e a d t i m e since i t uses a n e x p h c i t p l a n t m o d e l . r o b o t m a n i p u l a t o r c a n be n o n m i n i m u m phase a n d i n c o r r e c t l y p a r a m e t r i z e d ( e s p e c i a l l y t h e r e is s o m e  flexibility  The when  i n t h e h n k s ), c a n have d e a d t i m e i n t h e h y d r a u h c s y s t e m , a n d i f  s a m p l e d fast, c a n have i n s t a b i h t i e s . M a n y discussions a b o u t a d a p t i v e c o n t r o l a n d G P C c a n be f o u n d i n t h e h t e r a t u r e , s u c h as Astrom ^ ^ , T o m i z u k a  Demircioglu  LatorneU  ,  etc. . T h e G P C uses a p l a n t m o d e l w h i c h is a C A R I M A t y p e ( C o n t r o l l e d A u t o  Regressive  I n t e g r a t i n g M o v i n g A v e r a g e ) : i.e.  A{q-')yit) <0 A =  W h e r e y{t)  =  1) + eit)  (2.6)  is t h e c o n t r o l i n p u t , e{t)  is t h e u n m e a s u r e d d i s t u r -  = B{q-')u{t  -  c ( ç - ) ^ 1-g-i  is t h e m e a s u r e d o u t p u t , u(t)  b a n c e t e r m , ^ is u n c o r r e l a t e d r a n d o m s e q u e n c e , q~^ is t h e b a c k w a r d shift o p e r a t o r , A is t h e d i f f e r e n c i n g o p e r a t o r a n d A(q~^),  B(q~^)  a n d C{q^^) are p o l y n o m i a l s of degrees Ua, rib a n d  Tic r e s p e c t i v e l y . T h e a l g o r i t h m m i n i m i z e s a cost f u n c t i o n of t h e f o r m ( C l a r k e et a l  Auit  ^):  + i -  1)  W h e r e Ni is t h e m i n i m u m o u t p u t h o r i z o n , N2 is t h e m a x i m u m o u t p u t h o r i z o n , c o n t r o l h o r i z o n X{j) w{t  (2.7)  is t h e  is a c o n t r o l w e i g h t i n g sequence, y{t-\- j) is t h e o u t p u t j steps a h e a d a n d  + j) is t h e f u t u r e set p o i n t .  T h e G P C a l g o r i t h m p r e d i c t s f u t u r e o u t p u t s a n d a i m s at g o o d p e r f o r m a n c e a few  steps  a h e a d by m i n i m i z i n g t h e a b o v e cost f u n c t i o n , w h i c h gives a s e q u e n c e of f u t u r e c o n t r o l s i g n a l s a n d avoids large i n p u t signals w i t h s a t u r a t i o n . T o d e r i v e a j step a h e a d p r e d i c t o r o{y{t),  E q u a t i o n 2.6 s h o u l d b e m u l t i p h e d b y  q'^Ej{q~^)A  a n d the following identity used:  1 = Eiiq-')Aiq~')A  (2.8)  + q-'F,{q-')  T h i s is t h e D i o p h a n t i n e e q u a t i o n , w h e r e Ej a n d Fj are p o l y n o m i a l s i n t h e b a c k w a r d s h i f t o p e r a t o r . I n o r d e r t o get a u n i q u e s o l u t i o n , t h e degree of t h e f o l l o w i n g p o l y n o m i a l s is c h o s e n as: àeg{EM-'))  <3-l  and deg(F,(g-^)) <  àeg{A[q-'))  So t h e j step a h e a d o u t p u t y ( t + j ) i s : y{t + j) = F,{q^')y{t)  + Ei{q-')B{q-')Au{t  + j -  1) + EM-')i{t  + j)  (2.9)  C{q~^) is chosen to b e 1. T h e d i s t u r b a n c e sequence consists o n l y of f u t u r e values w h i c h are u n k n o w n , so t h e o p t i m a l p r e d i c t o r is:  y{t + j) = Fjiq-')yit) GAq-')  =  + Gj{q-')Au{t  + j -  1)  Ej{q-')Biq'')  T h e o b j e c t i v e of t h e p r e d i c t i v e c o n t r o l l a w is t o d e r i v e f u t u r e p l a n t o u t p u t s y(t + j), a f u t u r e set p o i n t sequence w{t + j).  (2.10)  It is d o n e as follows:  given  1. T h e f u t u r e set p o i n t s e q u e n c e w(t + j) is d e t e r m i n e d . 2. A set o f p r e d i c t e d e r r o r s is c a l c u l a t e d :  eit + j ) y { t  + j) - wit + j)  3. T h e cost f u n c t i o n i n E q u a t i o n 2.7 is m i n i m i z e d t o p r o v i d e a suggested s e q u e n c e o f f u t u r e c o n t r o l i n c r e m e n t s , a s s u m i n g t h a t after some c o n t r o l h o r i z o n N^y f u t u r e i n c r e m e n t s i n t h e c o n t r o l are zero, a n d t h e c o n t r o l s i g n a l is k e p t c o n s t a n t . T h e w e i g h t i n g f a c t o r w a s i n i t i a l l y selected as r e c o m m e n d e d b y C l a r k e et a l  ^, followed b y a d j u s t i n g t h e values  according to the controlled system. T h e o p t i m a l p r e d i c t i o n of y(t + j ) c a n b e w r i t t e n as:  y = G u + f  (2.11)  W h e r e f i n c l u d e s t h e c o m p o n e n t s o f t h e p r e d i c t e d o u t p u t y , w h i c h a r e k n o w n at t i m e t , a n d G is a l o w e r t r i a n g u l a r m a t r i x o f d i m e n s i o n N2 X N2. M i n i m i z a t i o n of e q u a t i o n 2.7 y i e l d s t h e c o n t r o l i n c r e m e n t v e c t o r :  (u) = ( G ' ^ G + A I ) - ^ G ' ^ ( w - f )  (2.12)  T h e first e l e m e n t of u is A u ( t ) so t h a t t h e c u r r e n t c o n t r o l u{t) is g i v e n b y :  u(t) = u ( t - l ) + g V - f ) Where  (2.13)  is t h e first r o w of ( G ^ G + A I ) - i g T .  T h e d e s i g n p a r a m e t e r s for t h i s a l g o r i t h m a r e :  1. M i n i m u m O u t p u t H o r i z o n , Nii is set t o t h e t i m e d e l a y K, i f k n o w n , t o save c o m p u t a t i o n l o a d since t h e c o n t r o l s i g n a l s have n o effect e a r h e r . T h e t i m e delay, K, i n d i s c r e t e  s y s t e m s c o r r e s p o n d s to t h e degree of t h e p o l y n o m i a l B, so w h e n K is n o t k n o w n ,  Ni  is g i v e n t h e m i n i m a l v a l u e for t h e t i m e delay, A^i = 1.  2. M a x i m u m  Output  H o r i z o n , A^2: t h e a u t h o r s r e c o m m e n d t o set i t a p p r o x i m a t e l y  at the value of t h e rise t i m e of t h e s y s t e m .  B o t h p a r a m e t e r s Ni  a n d N2 a r e u s e d i n  E q u a t i o n 2.11 i n w h i c h t h e n u m b e r of p r e d i c t i o n steps j v a r y f r o m Ni  t o A^2 a n d a r e  t h e n used i n c a l c u l a t i n g t h e c o n t r o l l a w i n E q u a t i o n 2.13.  3. T h e C o n t r o l H o r i z o n , N^: A m a j o r a d v a n t a g e of t h e G P C is i n t h e a s s u m p t i o n a b o u t f u t u r e c o n t r o l signals. A f t e r a n i n t e r v a l of  < N2, t h e p r o j e c t e d c o n t r o l i n c r e m e n t s  are a s s u m e d t o be zero. T h i s reduces t h e c o m p u t a t i o n b u r d e n , since and  is a n Nu x  dim(u)  =  m a t r i x . Usually, the control horizon can  be  chosen as A^u = 1 (for s t a b l e p l a n t s w i t h d e l a y or w i t h n o n m i n i m u m p h a s e ) , b u t w h e n p o o r l y d a m p e d or u n s t a b l e poles are p r e s e n t ,  should equal the n u m b e r of those  poles.  4. T h e C o n t r o l W e i g h t i n g S e q u e n c e , A ( j ) : A c t s as d a m p i n g of t h e c o n t r o l a c t i o n w h e n g r e a t e r t h a n zero.  Final  goal of  model  Type of  application  process  Required  model  of  accuracy  Identification  model  method  v e r i f i c a t i o n of  hnear / continuous  medium/  off-hne step  theoretical  time nonparametric/  high  response,  models  parameters  frequency  response,  parameter estimation controller  hnear nonparametric,  l o w for  off-hne  parameter  continuous time  input/output  step r e s p o n s e  tuning  behavior  computer aided  hnear parametric  m e d i u m for  o n - h n e , off-hne  d e s i g n of  (nonparametric)  input/output  parameter estimation  digital control  discrete t i m e  behavior  self-adaptive  hnear parametric  m e d i u m for  on-hne  digital control  discrete t i m e  input/output  e s t i m a t i o n i n close  behavior  loop  tuning  parameter  process  hnear / nonlinear  h i g h for  on-hne  parameter  parametric continuous  parametric  process  monitoring and  time  continuous  parameters  time  estimation  failure detection  parameter  T a b l e 2.1: Different final goals a n d s p e c i f i c a t i o n s for i d e n t i f i c a t i o n cases. 2.4  I n t r o d u c t i o n to M o d e l S t r u c t u r e a n d P a r a m e t e r  Determination  S y s t e m i d e n t i f i c a t i o n is b a s i c a U y a f u n c t i o n of b u i l d i n g a m a t h e m a t i c a l m o d e l b y a n a l y z i n g t h e r e l a t i o n s b e t w e e n o b s e r v e d i n p u t a n d o u t p u t . A c o n s i d e r a b l e a m o u n t o f w o r k has b e e n d o n e a n d c a n be f o u n d i n references L j u n g a n d S ô d e r s t ô m ^, L j u n g ^ a n d I s e r m a n n ^ 0 . It is i m p o r t a n t to first consider t h e final goal for t h e a p p h c a t i o n of t h e process m o d e l , since t h i s d e t e r m i n e s t h e t y p e of m o d e l , its a c c u r a c y r e q u i r e m e n t s , a n d t h e i d e n t i f i c a t i o n m e t h o d .  T a b l e 2.1 ( I s e r m a n n ^ )  shows some e x a m p l e s for r e l a t i o n s h i p s b e t w e e n d i f f e r e n t  final  goals a n d s o m e specifications of process i d e n t i f i c a t i o n . T h e a p r i o r i k n o w l e d g e of t h e p r o cess i s , for e x a m p l e , b a s e d o n g e n e r a l process u n d e r s t a n d i n g , o n p r i n c i p a l l a w s a n d o n p r e measurements. T h e c h o i c e of a m o d e l s t r u c t u r e is e s s e n t i a l for successfully i d e n t i f y i n g a s y s t e m . L j u n g ^ i n his b o o k describes t e c h n i q u e s a n d p r o c e d u r e s a b o u t m o d e l s t r u c t u r e s e l e c t i o n a n d m o d e l validation.  T h e choice of a m o d e l s t r u c t u r e s h o u l d be b a s e d o n g o o d u n d e r s t a n d i n g o f t h e  i d e n t i f i c a t i o n m e t h o d a n d a p r i o r i k n o w l e d g e of t h e s y s t e m . T h e r e are t h r e e b a s i c s t e p s i n c h o o s i n g a m o d e l s t r u c t u r e i.e.  1. C h o i c e of t y p e of t h e m o d e l set: h n e a r or n o n h n e a r m o d e l , i n p u t - o u t p u t or b l a c k b o x m o d e l s , etc. . 2. C h o i c e of t h e m o d e l size: m o d e l o r d e r for h n e a r s y s t e m s . 3. C h o i c e of m o d e l p a r a m e t r i z a t i o n : i d e n t i f i c a t i o n of m o d e l p a r a m e t e r s .  T h e q u a h t y of t h e m o d e l is u s u a l l y m e a s u r e d b y m i n i m i z i n g a c r i t e r i o n . T h e r e is a t r a d e off b e t w e e n  flexibihty  a n d p a r s i m o n y . F l e x i b i h t y w i l l g i v e , for a l a r g e r n u m b e r of p a r a m e t e r s ,  a b e t t e r fit for t h e m i n i m i z a t i o n of t h e c r i t e r i o n since t h e d a t a set is l a r g e r . O n t h e o t h e r h a n d i t is i m p o r t a n t , i n p r a c t i c e , t o e m p l o y t h e s m a l l e s t n u m b e r of p a r a m e t e r s for a d e q u a t e r e p r e s e n t a t i o n of p h y s i c a l s y s t e m s . T h e t y p e of t h e chosen m o d e l is u s u a l l y b a s e d o n a p r i o r i k n o w l e d g e a b o u t t h e s y s t e m a n d i n t u i t i o n . G e n e r a U y it is a d v i s a b l e t o s t a r t w i t h t h e s i m p l e s t p o s s i b l e m o d e l . O r d e r e s t i m a t i o n of a h n e a r s y s t e m is u s u a l l y b a s e d o n p r e U m i n a r y d a t a a n a l y s i s . relevant m e t h o d s f a l l i n t o t h e n e x t categories:  • S p e c t r a l a n a l y s i s of t h e t r a n s f e r f u n c t i o n .  The  • T e s t i n g r a n k s of s a m p l e d c o v a r i a n c e m a t r i x . • C o r r e l a t i n g variables. • E x a m i n i n g the i n f o r m a t i o n m a t r i x .  M o d e l v a h d a t i o n m e t h o d s c a n be as follows:  • C o m p a r i n g hnear models w i t h a priori knowledge of the system. • C o m p a r i n g measured and simulated outputs. • T e s t i n g r e s i d u a l s for i n d e p e n d e n c e of past i n p u t s . • C o m p a r i n g c r i t e r i a fit o b t a i n e d i n different m o d e l s t r u c t u r e s .  O f f - l i n e m e t h o d s are c a p a b l e of c h o o s i n g a m o d e l s t r u c t u r e a n d i d e n t i f y i n g i t s p a r a m e t e r s . T h e r e are m a n y cases w h e r e o n - h n e i d e n t i f i c a t i o n of t h e m o d e l is n e e d e d .  T h e r e are t w o  d i s a d v a n t a g e s t o r e c u r s i v e (on-Hne) i d e n t i f i c a t i o n , i n c o n t r a s t to off-hne i d e n t i f i c a t i o n . T h e first is t h a t t h e d e c i s i o n o n w h i c h m o d e l s t r u c t u r e t o use has t o be m a d e a n d f i x e d a p r i o r i , b e f o r e s t a r t i n g t h e r e c u r s i v e i d e n t i f i c a t i o n p r o c e d u r e . It is w e l l k n o w n t h a t p a r a m e t r i c m o d e l s c a n give l a r g e errors w h e n t h e o r d e r of t h e m o d e l does n o t agree w i t h t h e o r d e r of t h e p r o c e s s (see Astrom ^^).  I n t h e off-hne s i t u a t i o n , different t y p e s of models c a n be e x a m i n e d .  The  s e c o n d d i s a d v a n t a g e is t h a t r e c u r s i v e methods g e n e r a l l y are n o t as a c c u r a t e as off-hne ones ( L j u n g 3). F o r off-hne i d e n t i f i c a t i o n , t h e basic t h r e e steps are:  - g o o d choice of t h e d a t a r e c o r d m a k e s i t m a x i m a l l y i n f o r m a t i v e .  •  data recording  •  a set of candidate  models - t h e m o s t i m p o r t a n t a n d difficult choice of t h e s y s t e m i d e n -  t i f i c a t i o n . T h i s is t h e a c t u a l d e t e r m i n a t i o n of t h e m o d e l s t r u c t u r e .  •  determining  the best model in the set - g u i d e d by t h e d a t a ( t h i s is t h e i d e n t i f i c a t i o n  method). A f t e r h a v i n g s e t t l e d on t h e p r e c e d i n g t h r e e choices, one m u s t v a l i d a t e t h e m o d e l ,  i.e.  e s t a b h s h a c r i t e r i o n to e x a m i n e i f t h e m o d e l a c c e p t e d is g o o d e n o u g h . If i t is n o t , t h e e n t i r e p r o c e d u r e is r e p e a t e d a g a i n as s h o w n i n F i g u r e 2.2 . A f t e r d e t e r m i n i n g t h e s y s t e m m o d e l ' s s t r u c t u r e , t h e user m a y choose f r o m t h e i d e n t i f i c a t i o n a n d a d a p t i v e c o n t r o l t e c h n i q u e s t h a t are a v a i l a b l e . C o n s i d e r a b l e w o r k i n these a r e a s h a s b e e n d o n e a n d w i U not be d i s c u s s e d at t h i s p o i n t , b u t some i m p o r t a n t references a r e L j u n g a n d S ô d e r s t ô m ^, L j u n g Strejc  2.5  Astrom a n d E y k h o f f  Goodwin and Payne  Âstrôm   s t r ô m a n d W i t t e n m a r k ^,  .  R e v i e w of P r e v i o u s W o r k i n O r d e r  Determination  T o e m p h a s i z e t h e effect of i n c o r r e c t m o d e h n g , we refer t o R o h r s et a l  which analyzed the  effect of u n m o d e l e d d y n a m i c s o n t h e r o b u s t n e s s a n d s t a b i h t y of c o n t i n u o u s t i m e a d a p t i v e c o n t r o l a l g o r i t h m s . T h e c o n c l u s i o n i n t h i s p a p e r is t h a t a d a p t i v e a l g o r i t h m s p u b h s h e d i n t h e h t e r a t u r e are h k e l y to p r o d u c e u n s t a b l e c o n t r o l s y s t e m s i f t h e y are i m p l e m e n t e d o n p h y s i c a l s y s t e m s d i r e c t l y as t h e y a p p e a r i n t h e h t e r a t u r e . U n f o r t u n a t e l y , s t a b i h t y p r o o f s of a l l t h o s e a l g o r i t h m s have i n c o m m o n a v e r y r e s t r i c t i v e a s s u m p t i o n t h a t t h e o r d e r of t h e s y s t e m is k n o w n . So i f t h e r e are errors i n t h e s t r u c t u r e a s s u m p t i o n s , i n s t a b i l i t i e s c a n o c c u r . It is also n o t e d t h a t t h i s p r o b l e m c a n be p a r t i a l l y a l l e v i a t e d by sufficient e x c i t a t i o n , b u t t h e a m o u n t o f m o d e h n g e r r o r or t h e a m o u n t of d i s t u r b a n c e for w h i c h t h e a d a p t i v e s y s t e m c a n m a i n t a i n s t a b i h t y m a y be e x t r e m e l y s m a l l . I n t h e 1970's, several w o r k s were p u b h s h e d , s u c h as A k a i k e 2^, S c h w a r z  and Akaike  , Isermann  , p r i m a r i l y e s t a b h s h i n g e s t i m a t e s of a m e a s u r e of fit o f t h e m o d e l .  p r o c e d u r e s are off-hne ones a n d are p a r t of t h e m o d e l v a h d a t i o n t e c h n i q u e s .  Those  r  TASK  FINAL  ECONOMY  COAL  paoccss PHYSICAL LAWS P«EHEASLIAEHENTS O n U A T I N C CONDITIONS  DESIGN  OF  exPERIMENTS  SIGNAL Oi•NERATION. M E A S U R E * ENT AND O A T * STC« A C E  APPLICATION OF IDENTIFICATION ,."£THOp  ASSUMPTION  H  __rwOC£SS»<OOEL NON. •PASAMETRIC PARAMETBC-f-  MODEL  Cf  MODEL  STRUCTURE  MODEL  STRuaUSE  DETERMINATION  MO  VERIFICATION  F i g u r e 2.2: G e n e r a l p r o c e d u r e o f process i d e n t i f i c a t i o n , ( I s e r m a n n ^*^)  A s m e n t i o n e d , s e v e r a l c r i t e r i a for o r d e r e s t i m a t i o n were p u b l i s h e d . R i s s a n e n ^9 p r o p o s e s t h e f o U o w i n g cost f u n c t i o n : U{x,n,m,()  = Nlnr  +  ^  'jô^lnf da  ' + ( n + m + l)\n{N J  + 2) + 21n(n + l ) ( m + 1) (2.14)  Where X is t h e m o d e l : x{t) = / , [ x ( f ^l),...,x{t-  - m)]  n), e{t),e(i  i V - n u m b e r of s a m p l e d o b s e r v a t i o n s , e - disturbance. ^ - consist o f real v a l u e d p a r a m e t e r s . AT l o g f - t h e m i n i m i z e d l o g h k e h h o o d f u n c t i o n . T h e first a n d t h i r d t e r m s t o g e t h e r v i r t u a l l y c o i n c i d e w i t h a c r i t e r i o n d e r i v e d b y S c h w a r t z .  T h e most commonly  used criterion, derived b y A k a i k e (  " ) , is t h e A I C ( A k a i k e ' s  Information Theory Criterion) which is: A I C = ATlogr^ + 2Jb  (2.15)  Where Nlogr  - the minimized log-hkehhood function.  k - t h e number of parameters i n the model. M o r e c r i t e r i a were i n t r o d u c e d b y A k a i k e i n 1977  a n d b y H a n n a n a n d Q u i n n i n 1979  ^ ^ . I n 1989, G o u ^2 i n t r o d u c e d a n e w c r i t e r i o n for o r d e r e s t i m a t i n g o f a C A R A M A  model.  T h e f i n a l c o n c l u s i o n is t h a t e s t i m a t e s p r e s e n t e d are n o n - r e c u r s i v e a n d r e q u i r e a v a i l a b i h t y o f u p p e r b o u n d s for u n k n o w n o r d e r s ; therefore f u r t h e r r e s e a r c h is r e q u i r e d . A p p r o a c h e s for different s t r u c t u r e s h a v e b e e n e x p l o r e d s u c h as c a n o n i c a l s t r u c t u r e s ( G u i d o v z i a n d ^ ^ ) . T h e s e t r y t o close t h e g a p b y e x p l o r i n g a class o f s t a t e s p a c e c a n o n i c a l m o d e l s w i t h p a r t i c u l a r l y s i m p l e r e l a t i o n s t o i n p u t / o u t p u t difference  descriptions that can be d i -  r e c t l y i d e n t i f i e d f r o m i n p u t / o u t p u t sequences. H o w e v e r , i n o r d e r t o a v o i d e r r o r , t h e p r e v i o u s  e s t i m a t e o f t h e o r d e r a n d s t r u c t u r e of t h e process, out of a class of m o d e l s , is r e q u i r e d . F u r t h e r off-hne m e t h o d s , i n c l u d i n g M I M O ( M u l t i I n p u t M u l t i O u t p u t ) s y s t e m s ,  have  b e e n p u b h s h e d a n d w i U not be discussed here. M o s t of t h e r e c u r s i v e i d e n t i f i c a t i o n m e t h o d s for M I M O also assume a m o d e l s t r u c t u r e a p r i o r i . S u c h a m e t h o d b y G a u t h i e r  extends the  use of i n p u t / o u t p u t d e s c r i p t i o n i n t e r m s of p o l y n o m i a l m a t r i c e s for r e c u r s i v e i d e n t i f i c a t i o n i n c a n o n i c a l state space f o r m . H o w e v e r , before i d e n t i f y i n g t h e p o l y n o m i a l s ' c o e f f i c i e n t s ,  one  m u s t define t h e s t r u c t u r e a n d t h e p o l y n o m i a l s ' degree or at least, t h e i r u p p e r b o u n d s . L j u n g and Sodestrom  discuss i n t h e i r b o o k t h e c o n c e p t of i d e n t i f y i n g o v e r p a r a m e t r i z a -  t i o n of a m o d e l set a n d t h e choice of a m o d e l o r d e r . T h e choice of a m o d e l o r d e r is a d e h c a t e t r a d e off b e t w e e n g o o d d e s c r i p t i o n of t h e d a t a a n d t h e m o d e l c o m p l e x i t y . M o s t m e t h o d s f o r m o d e l o r d e r s e l e c t i o n are d e v e l o p e d for off-hne t e c h n i q u e s . T h e basic a p p r o a c h is t o c o m p a r e p e r f o r m a n c e of m o d e l s w i t h different o r d e r s a n d test w h e t h e r a h i g h e r o r d e r m o d e l is w o r t h while.  Recursive algorithms i n on-hne apphcations require identification of several models  s i m u l t a n e o u s l y . A m o d e l set is s a i d t o be i d e n t i f i a b l e i f i t s p a r a m e t e r s c a n be i d e n t i f i e d i . e . p a r a m e t e r s c a n b e u n i q u e l y d e t e r m i n e d f r o m t h e d a t a . L a c k of i d e n t i f i a b i l i t y c a n b e c a u s e d by non-exciting inputs and overparametrization. M I M O s y s t e m s , c a n be p a r a m e t r i z e d d e p e n d i n g o n t h e choice of s t r u c t u r e of t h e s y s t e m . T h e p r o b l e m has b e e n a v o i d e d b y s o m e researchers, a s s u m i n g t h a t t h e d e s i g n e r has e n o u g h a p r i o r i k n o w l e d g e of t h e s t r u c t u r e of t h e s y s t e m t o select s t a b l e p a r a m e t r i z a t i o n . O v e r b e e k and  Ljung  suggested a p r o c e d u r e t h a t p r o v i d e s a m e a n s of o b t a i n i n g t h e b e s t  model  structure. T h e m o d e l S t r u c t u r e Selection ( M S S ) a l g o r i t h m , the structure dealt w i t h i n t h a t w o r k , is t h e p a r a m e t r i z a t i o n of s y s t e m s a n d c a n be p e r f o r m e d i n a n u m b e r of w a y s .  The  t e c h n i q u e does not d e a l w i t h h o w t o select a n a p p r o p r i a t e o r d e r . T h e a l g o r i t h m receives as i n p u t a g i v e n s y s t e m w i t h a g i v e n p a r a m e t r i z a t i o n . It tests w h e t h e r t h i s p a r a m e t r i z a t i o n is w e l l c o n d i t i o n e d for i d e n t i f i c a t i o n p u r p o s e s . is c o n s i d e r e d .  If i t is not w e l l c o n d i t i o n e d , a n o t h e r s t r u c t u r e  T h e best s t r u c t u r e of a possible set is d e c i d e d u p o n a p r i o r i .  Nagy and  Ljung  describe i n their paper t h e subject of c o m p u t e r - aided m o d e l s t r u c t u r e selection.  I n o r d e r t o use a software package for s y s t e m i d e n t i f i c a t i o n , a n a p p r o p r i a t e m o d e l s t r u c t u r e s h o u l d b e chosen. A c o m m o n f e a t u r e for these m e t h o d s is t h a t t h e y m i x e x t e n s i v e n u m e r i c a l c o m p u t a t i o n s , code g e n e r a t i o n a n d s y m b o h c a l g e b r a . Davison  i n his p a p e r describes a m e t h o d o f m o d e l size r e d u c t i o n . M a n y p h y s i c a l p l a n t s  can be represented by simultaneous hnear differential equations w i t h constant coefficients, o f the form:  X = A x -|- B u  W h e r e t h e o r d e r o f m a t r i x A c a n b e l a r g e , for e x a m p l e c h e m i c a l p l a n t s o r n u c l e a r r e a c t o r s , w h i c h c a n p o s e n u m e r i c a l p r o b l e m s , t h e m e t h o d suggests t h e r e d u c t i o n o f t h e r a n k o f s u c h m a t r i c e s b y c o n s t r u c t i n g a m a t r i x o f l o w e r o r d e r w i t h t h e s a m e d o m i n a n t eigenvalues a n d eigenvectors as t h e o r i g i n a l s y s t e m . T h e paper by N i u , X i a o and Fisher  presents a s i m u l t a n e o u s r e c u r s i v e e s t i m a t i o n o f  t h e m o d e l p a r a m e t e r s a n d loss f u n c t i o n s for aU p o s s i b l e m o d e l orders f r o m z e r o t h r o u g h n i s d o n e b y u s i n g a u g m e n t e d i n f o r m a t i o n m a t r i x ( A I M ) a n d a UDU^ f a c t o r i z a t i o n a l g o r i t h m . T h e A I M m a t r i x is: k (2.16) 3=1  Where  is t h e regression v e c t o r a n d C „ is t h e A I M m a t r i x .  cj>„ ^ [-y{t  - n),v{t  - n),u{t  ~ n) • • • ~ y{t - l),v{t  - l),u{t  - 1),  -y{t)  W h e r e y is t h e o u t p u t , v is t h e noise a n d u is t h e i n p u t . T h e a l g o r i t h m is r e p o r t e d t o be c o m p u t a t i o n a l l y efficient a n d h a v e g o o d n u m e r i c a l p r o p e r t i e s d u e t o t h e use of t h e UDU^ a l g o r i t h m . I n a second p a p e r N i u a n d F i s h e r '^^ r e p o r t o n a  M I M O system identification technique using augmented U D factorization. T h i s work extends t h e A U D a l g o r i t h m f r o m S I S O s y s t e m s to M I M O s y s t e m s . It is b a s e d o n t h e c a n o n i c a l s t a t e s p a c e r e p r e s e n t a t i o n be G u i d o r z i  the Bierman's U D factorization  . This algorithm  t o o , is r e p o r t e d to poses exceUent n u m e r i c a l p r o p e r t i e s . T h e w o r k r e p o r t e d n e x t , ( W u h c h a n d K a u f m a n '^•^) is a t r i a l a n d e r r o r m o d e l estimation procedure.  order  T h e e s t i m a t i o n is based o n s a m p h n g a s i g n a l a n d c a l c u l a t i n g i t s  autocorrelation function:  Rin) = j^'e'<JMJ k = l,2,---,Lk  W h e r e x{t  — j)  (217)  + <  are s a m p l e d values of t h e s i g n a l x(t)  L  a n d L is t h e n u m b e r of s a m p l e s .  A  s y s t e m of n h n e a r e q u a t i o n s is g e n e r a t e d :  W h e r e A is a n o n z e r o vector of coefficients a n d R „ is a m a t r i x . T h e o r d e r is e s t i m a t e d by e x a m i n i n g t h e d e t e r m i n a n t of t h e m a t r i x R „ : If det R „ = 0 t h e n n>  N  and If det R „ 7^ 0 t h e n n<  N  Where n = 1,2,3, a n d N is t h e c o r r e c t o r d e r . A m e t h o d for s i m u l t a n e o u s l y s e l e c t i n g t h e o r d e r a n d i d e n t i f y i n g t h e p a r a m e t e r o f a n A u t o r e g r e s s i o n ( A R ) m o d e l , has b e e n d e v e l o p e d ,  ( K a t s i k a s et a l '^^).  d e f i n e d as: y{k)  =  J2'^iik)yik-i)+v{k) «=1  T h e A R m o d e l is  W h e r e a ; are t h e coefficients, y{k — i))  are p r e v i o u s o u t p u t s a n d v{k)  is z e r o m e a n w h i t e  noise. T h e o r d e r of t h e s y s t e m N i n u n k n o w n b u t i t is i n t h e r a n g e of 1 < AT < M , w h e r e M is k n o w n . T h e t r u e m o d e l w i U be one of a f a m i l y of m o d e l s w i t h t h e a b o v e r a n g e o f o r d e r . A p a p e r by ( B i r c h , L a w r e n c e et a l ^'^) deals w i t h f i t t i n g a n d e s t i m a t i n g a m o d e l t o E E G ( E l e c t r o e n c e p h a l o g r a p h y ) signals. T h e E E G s i g n a l is m o d e l e d w i t h a n A R m o d e l t y p e a n d a s p e c t r a l e s t i m a t i o n p r o c e d u r e is p e r f o r m e d . T h e s e l e c t i o n of t h e p r o p e r m o d e l o r d e r is d o n e by s o m e a p r i o r i k n o w l e d g e of e x p e c t e d r e s u l t s . V e r y h t t l e w o r k has been p u b h s h e d o n s t r u c t u r e d e t e r m i n a t i o n for n o n H n e a r s y s t e m s . O n e w h i c h t r e a t s m a t h e m a t i c a l m o d e l s for r e p r e s e n t a t i o n of t h e d y n a m i c o f s h i p r u d d e r y a w a n d r o l l m o t i o n s is i n Z h o u et a l  T h e w o r k checks suggested m o d e l s g i v e n i n t h e  H t e r a t u r e for t h e p r o b l e m w i t h t h e R e c u r s i v e P r e d i c t i o n E r r o r ( R P E ) i d e n t i f i c a t i o n m e t h o d s . It is a n off-Hne m e t h o d t h a t finds t h e best n o n H n e a r t e r m s for t h e m o d e l . o n - H n e m e t h o d has been d e s c r i b e d b y Z e r v o s a n d D u m o n t  A non Hnear  T h e p l a n t is m o d e l e d b y a n  o r t h o g o n a l L a g u e r r e n e t w o r k p u t i n t o s t a t e space f o r m . T h e n u m b e r o f t h e L a g u e r r e f i l t e r s u s e d d e p e n d s of t h e presence of t i m e d e l a y a n d u n d a m p e d m o d e s . T h e a c t u a l p l a n t o r d e r does n o t i n f l u e n c e t h e n u m b e r of L a g u e r r e filters ( N ) u s e d . U s u a l l y t h e c h o i c e is 5 < AT < 10, a n d N c a n b e c h a n g e d o n Hne. H e m e r l y '^^ presents a m e t h o d for o r d e r a n d p a r a m e t e r i d e n t i f i c a t i o n of i n d u s t r i a l p r o cesses. T h e processes t o be i d e n t i f i e d are d e s c r i b e d b y A R X m o d e :  y(t)  = -aiy(t  W h e r e y(t) is w h i t e noise.  -  1)  a „ y ( i - n) + biu{t -  is t h e s y s t e m s o u t p u t , u{t)  is t h e i n p u t ,  1) -f • • • + fc„w(i ~ n) +  w(t)  a n d bi are t h e coefficients a n d  w{t)  T h e p a r a m e t e r s are i d e n t i f i e d b y R e c u r s i v e L e a s t S q u a r e s ( R L S ) a l g o r i t h m  a n d for t h e o r d e r e s t i m a t i o n t h e P r e d i c t i v e L e a s t S q u a r e s ( P L S ) c r i t e r i o n is u s e d , ( R i s s a n e n  PLS{n.t)=y'£e'{n,i  + l) i=0  T h e best e s t i m a t e of t h e o r d e r s h o u l d b e  n(t) = argmin  PLS(n,t)  with  e{n, t + l) = y{t + l ) - 0 ^ ( n , t)^n,  t)  where  = [yU),---,y{J  - n + l),u{j),---  ,u{j  - n + 1)  T h e P L S c r i t e r i o n is h i g h l y i n t u i t i v e a n d at t i m e t t h e o r d e r e s t i m a t e n(t)  is t h e o r d e r  of t h e m o d e l w h i c h has g i v e n t h e least m e a n square p r e d i c t i o n e r r o r u p t o t h a t t i m e . T h e process c a n b e i d e n t i f i e d for different o p e r a t i n g p o i n t s b y v a r y i n g t h e e x c i t a t i o n a m p h t u d e a n d t h e r e f o r e g e t t i n g several h n e a r m o d e l s . a n d c h a n g e d i n r e a l t i m e i f necessary.  A c o n t r o l l e r c a n b e d e s i g n e d for e a c h  Medeiros and Hemerly  model  i n t e g r a t e d l a t t i c e f o r m for  constructing a m i n i m u m variance adaptive controller w i t h parameter a n d order estimation. A s d e s c r i b e d i n t h e f o r m e r p a p e r t h e o r d e r is e s t i m a t e d w i t h t h e P L S c r i t e r i o n . T h e l a t t i c e f i l t e r is a w a y t o p a r a m e t r i z e as f o U o w i n g :  ên+l(0 = ê„(f) + ^^n(<-l)  Vn+lit)  = r-nit - 1) +  pèn{t)  where:  ê4t)  Kit  =  y{t)~Ut/®n  - 1) = y ( f - n - 1) + à\y{t - n ) + • • • + àly{t - 1)  W h e r e pn are coefficients,  © „ e s t i m a t e d p a r a m e t e r s for o r d e r n , y(t)  is t h e  measured  o u t p u t , y „ is t h e e s t i m a t e d o u t p u t for m o d e l o r d e r n , a n d ê „ is t h e e r r o r . A set o f p r e d i c t i o n errors is c a l c u l a t e d a n d t h e e s t i m a t e d o r d e r of t h e m o d e l is t h e one t h a t has a m i n i m a l least squares e r r o r . A s c a n be i n f e r r e d f r o m t h e p r e v i o u s discussions, m u c h w o r k has b e e n p u b h s h e d  on  i d e n t i f i c a t i o n a n d a d a p t i v e c o n t r o l a l g o r i t h m s , m o s t l y for f i x e d o r d e r a n d s t r u c t u r e m o d e l s . V e r y l i t t l e r e s e a r c h w o r k has b e e n d o n e for flexible m o d e l s t r u c t u r e s p a r t i c u l a r l y for methods  a n d t h e r e is d e f i n i t e l y a n e e d for r e s e a r c h i n t h e a r e a .  on-hne  This work constitutes  c o n t r i b u t i o n t o t h e r e s e a r c h for o n - h n e m o d e l s t r u c t u r e d e t e r m i n a t i o n .  a  Chapter 3  SINGLE FLEXIBLE LINK MANIPULATOR  3.1  Outline:  I n t h i s C h a p t e r , we present t h e w o r k d o n e w i t h t h e flexible h n k m a n i p u l a t o r . T h e m a t h e m a t i c a l m o d e l a n d e q u a t i o n s o f m o t i o n have b e e n d e v e l o p e d .  N u m e r i c a l r e s u l t s for t h e c o n t r o l  of t h e m a n i p u l a t o r are p r e s e n t e d , as w e l l as t h e effects of m i s - m o d e h n g o n t h e m a g n i t u d e o f a cost f u n c t i o n a n d a n o u t h n e for a n i t e r a t i v e m e t h o d for t h e o r d e r e s t i m a t o r . S e c t i o n 3.2 develops a l l m a t h e m a t i c a l m o d e h n g i n v o l v e d i n t h e n u m e r i c a l s i m u l a t i o n o f a s i n gle  flexible  h n k m a n i p u l a t o r , i n c l u d i n g t h e h n e a r e q u a t i o n s of m o t i o n a n d c o n t r o l s t r a t e g y .  S e c t i o n 3.3 presents t h e n u m e r i c a l r e s u l t s a n d a n a l y s i s for t h e flexible h n k a p p h c a t i o n .  3.2  3.2.1  Single Flexible L i n k M a n i p u l a t o r  Introduction  A r o b o t is a c o m p l e x s y s t e m t o c o n t r o l not o n l y b e c a u s e i t is a n o n h n e a r s y s t e m a n d h a s v a r i a t i o n s i n t h e m o m e n t of i n e r t i a , b u t also b e c a u s e flexible h n k s t r u c t u r e or n o n h n e a r i t i e s , s u c h as h y s t e r e s i s or b a c k l a s h . T h e r e are t w o a p p r o a c h e s for t h e d e s i g n of c o n t r o l l e r s for s u c h s y s t e m s : i.e. t o d e s i g n one t h a t w i U not e x c i t e t h e p o o r l y d a m p e d m o d e s , or one t h a t a c t i v e l y d a m p s o s c i l l a t o r y m o d e s . T h e second o p t i o n is not u s e d i n i n d u s t r i a l r o b o t s .  Such control  s y s t e m s are c o m p h c a t e d , since t h e frequencies of t h e o s c i l l a t o r y m o d e s v a r y w i t h o r i e n t a t i o n and load.  T h e v a r i a t i o n s i n the o s c i l l a t o r y m o d e s are t h e r e a s o n for c h o o s i n g t h e  flexible  h n k as a n a p p h c a t i o n . N e w v i b r a t i o n m o d e s t h a t arise m e a n t h a t t h e s t r u c t u r e o f t h e m o d e l  has c h a n g e d o n h n e , so t h e m o d e l a n d t h e c o n t r o l s y s t e m s h o u l d b e u p d a t e d . D a t a for t h e chosen f l e x i b l e h n k c a n be f o u n d m 50 a n d 19. T h e a r m is a 1 m e t e r l o n g , f l e x i b l e m e c h a n i c a l s t r u c t u r e w h i c h c a n b e n d freely i n t h e h o r i z o n t a l p l a n e b u t is stiff i n v e r t i c a l b e n d i n g a n d i n t o r s i o n . Its m o t i o n is o n l y i n t h e h o r i z o n t a l p l a n e i.e. g r a v i t y effects are n o t i m p o r t a n t .  3.2.2  E q u a t i o n s of M o t i o n for t h e S i n g l e F l e x i b l e L i n k  T h e f l e x i b l e a r m is c o m p a r a b l e t o a c a n t i l e v e r b e a m .  F i g u r e 3.1 describes t h e  flexible  hnk  configuration. Where: XQ - is t h e reference a x i s . is t h e p o s i t i o n of a r i g i d a r m at 0 [rad.] f r o m XQ.  X w{x,t) 1B -  -  is t h e d e f l e c t i o n f r o m t h e r i g i d b o d y . is t h e m o m e n t of i n e r t i a a b o u t t h e h u b [kg • m ^ '  IH -  is t h e m o t o r ' s m o m e n t of i n e r t i a [kg •  TH -  is t h e t o r q u e a p p h e d by t h e m o t o r [N • m].  E I -  is Y o u n g ' s m o d u l e  .  [N/m^].  is t h e b e a m cross s e c t i o n a l m o m e n t of i n e r t i a [m"*  T h e d i s p l a c e m e n t of a n y p o i n t P a l o n g t h e b e a m at a d i s t a n c e x f r o m t h e h u b is g i v e n b y 0{t)  a n d t h e d e f l e c t i o n w(x,t),  m e a s u r e d f r o m t h e h n e Ox  w h i c h w o u l d be t h e a r m , h a d i t  b e e n r i g i d . T h e a s s u m p t i o n s m a d e are:  • t h e d e f l e c t i o n is s m a U - w{L,t)  <^ L  • shear d e f o r m a t i o n a n d r o t a r y i n e r t i a effects are n e g l e c t e d . • g r a v i t a t i o n effects for d e f l e c t i o n a n d m o v e m e n t i n a h o r i z o n t a l p l a n e are n e g l e c t e d . T h e d i s p l a c e m e n t y(x,t)  of a p o i n t p a l o n g t h e a r m is d e f i n e d as:  F i g u r e 3.1: C o n f i g u r a t i o n of the single h n k  flexible  arm  y{x,t)  = w{x,t)  (3.1)  + xe{t)  Let:  wix,t)  (3.2)  = J2M^)<liit) i=l  W h e r e ^i{x) is t h e i " * m o d e shape a n d qi{t) is t h e i * ' ' m o d e g e n e r a h z e d c o o r d i n a t e , n = 1,2, 3 , 4 , . . . is t h e n u m b e r of v i b r a t i o n m o d e s . ( N o t e t h a t w h e n i = 0, (l>o{x) = x, "^"^"^ = 1, qo{t) = 0 a n d qa{i) = 9 a r e t h e p a r a m e t e r s for a r i g i d b o d y . ) A U d e r i v a t i v e s as 6 are w i t h respect t o t i m e , a l l d e r i v a t i v e as Q' are w i t h respect t o x. T h e k i n e t i c e n e r g y of t h e s y s t e m i s :  2T,  = I,0^ + m f \ ' ^ Jo at  +  xèYdx  U s i n g E q u a t i o n 3.2 t h e k i n e t i c e n e r g y i s :  2n  = hé'  + è hi qUt) + 2èJ2lu «=1 t=l  Qiit)  (3.3)  T h e i n e r t i a i n t e g r a l s In, hi, hi are d e s c r i b e d i n A p p e n d i x B a n d t h e i r values are p r e s e n t e d i n T a b l e B . l . ( T h e n u m b e r i n d i c a t e s i f i t is h, h or h as i n d i c a t e d i n A p p e n d i x B , a n d t h e i shows for w h i c h m o d e t h e i n t e g r a l is c a l c u l a t e d ). T h e p o t e n t i a l energy is s t r a i n e n e r g y d u e t o b e n d i n g d e f o r m a t i o n i s :  2^ = E E r i=i j=i  EIcl/:cl>';qiqjdx  U s i n g orthogonality relations:  EI 4>'l cl>'! dx  = hi u;^  f (j)] pdx  i =  j  =  (3.4) 0  i^j  the p o t e n t i a l energy becomes:  2P = E  ql  ^3.  (3.5)  i=l  I n t r o d u c i n g a d i s s i p a t i o n f u n c t i o n D t h a t m a y be defined as: 2 ^  =  9  n E  n E  ^ij  t h e d a m p i n g force w i U be:  Mi  (3.6) i=i  C o m b i n i n g all together and applying Lagrange's the equations of m o t i o n  equation,  are:  for 1 = 0 - r i g i d b o d y :  lT'é^-Y.hiqi{t)^rH~cJ  (3.7)  »=i for i = 1 , 2 , 3 , . . . :  lu 0 + h^ qi{t)  = - / a . g i ( i ) - Co qi{t)  (3.8)  W h e r e : IT = IB + IH is t h e t o t a l m o m e n t of i n e r t i a , a n d Cj w h e r e i = 1 , 2 , 3 , . . . , n is t h e damping  coefficient.  3.2.3  T h e State Space  E q u a t i o n s 3.7 a n d  Model  3.8 c a n be p u t i n t o a m a t r i x f o r m , s u c h as:  M x = K X + hTH  W h e r e x is t h e state space v e c t o r defined as:  x=  [0,  qi, 92, ••• gn, 0,  qi,  (3.9)  qz, ••• Çn  M , K a n d b are m a t r i c e s d e f i n e d as:  M  =  ' 1  0  0  •••  ...  0  0  0  0  ...  • •  0  0  1  0  •••  ...  0  0  0  0  ...  • .  0  0  0  1  ••• . . .  0  0  0  0  •-.  • .  0  0  0  0  •••  ...  1  0  0  0  .••  • .  0  0  0  0  •••  ...  0  IT  hi  /i2  ha  • •  hn  0  0  0  •••  ...  0  hi  hi  0  ...  • •  0  0  Û 0  •••  ...  0  1X2  0  h2  0  •••  ...  0  /in  0  0  u  0  • • • •.  0  ...  hn  •  •  (3.10)  K  0  0  0  •• .  ...  0  1  0  0  ...  . •  0  0  0  0  •• .  ...  0  0  1  0  ...  . .  0  0  0  0  •• .  ...  0  0  0  1  . . . . .  0  0  0  0  •• •  ...  0  0  0  0  ••• • .  1  0  0  0  •• .  ...  0  - C o  0  0  0  . .  0  0  •• .  ...  0  0  -- C l  0  ••.  . .  0  ...  0  0  0  • • •  •  •  0  0  0  •••  . •  0  -/31  0  0  0  •• •  0  0  0  ••  b =- 0 0 0 ••• 1 0  0 0  - C 2  0  > . .  (3.11)  -c„  (3.12)  T h e s t a t e space m o d e l i n i t s final f o r m is: X = A • X + B • r/i  (3.13)  y tip = C • X Where: A =  .K  B =  b  C = f l 1 1 ••• 0 0 0  (3.14)  T h e o r d e r of t h e s y s t e m d e p e n d s o n t h e n u m b e r of m o d e s t h a t are i n c l u d e d . F o r e x a m p l e , a 3 m o d e m o d e l w i U be a n 8"* o r d e r m o d e l . T a b l e 3.1 shows t h e o r d e r of t h e s y s t e m v e r s u s t h e n u m b e r of m o d e s c o n s i d e r e d .  3.2.4  T h e Discrete T i m e  Model  T h e s y s t e m was c o n v e r t e d w i t h a z e r o o r d e r h o l d s a m p h n g f r o m a c o n t i n u o u s s t a t e s p a c e f o r m i n t o a discrete t i m e f o r m . F o r s a m p h n g w i t h p e r i o d h, t h e t i m e is:  tk  —k •h  T h e state space d i s c r e t e t i m e m o d e l has t h e f o l l o w i n g s t r u c t u r e :  x{{k  + l)h)  = ^x{kh)  y{kh)  =  Cx{kh)  *  = exp^''  +  ru{kh)  (3.15)  Where:  (3.16)  J\xp^^  dSB  I n o r d e r t o s i m p h f y c a l c u l a t i o n s , we e x p a n d e d i t i n t o a series, i . e . :  * . ^ e x p - . 5  = M + ^  + ^  + ... + ^  + ...  (3.17)  N o w t h e m a t r i c e s are g i v e n b y :  *  = 7  + A *  r  = * B  (3.18)  3.2.5  O p e n L o o p D i s c r e t e T i m e M o d e l s for D i f f e r e n t N u m b e r o f  Modes  T h e d i s c r e t e t i m e m o d e l s for ytip v e r . Th were d e v e l o p e d for cases w i t h a different n u m b e r o f modes.  T h e s t r u c t u r e of t h e d i s c r e t e m o d e l is i n t h e f o r m of E q u a t i o n 2.6, w h e r e t h e n o i s e  sequence is e q u a l t o z e r o . T h e s a m p h n g p e r i o d is  = 0.01 [sec.]. T h e d a m p i n g f a c t o r s a r e ,  Ci = 0.05 . R e m a r k : A U t h e f o U o w i n g d i s c r e t e t i m e m o d e l s for t h e different o r d e r s were c h e c k e d w i t h c o n t i n u o u s t i m e s i m u l a t i o n for t h e flexible h n k w i t h t h e s a m e n u m b e r of v i b r a t i o n m o d e s ( i . e . t h e s a m e m o d e l o r d e r ) . T h e r e s u l t s show t h e s a m e b e h a v i o r for t h e c o n t i n u o u s t i m e a n d t h e discrete t i m e models. F o r r i g i d b o d y - i = 0, m o d e l o r d e r is 2 :  ytip(tk)/o =  -  +I.m2ytip{tk  1) - 0.99822/tip(<fe - 2)  + 0 . 1 7 7 7 4 • 1 0 - ' TH{tk -  1) + 0.1773 • rnitk  (3.19) 2)  -  F o r one v i b r a t i o n m o d e z = 1, m o d e l o r d e r is 4 :  ytip(tk)/i =  +3.14233/eip(ffc +2.S781 ytipitk  1) - WUyupitk  - 3) - 0M73yupitk  - 2)  (3.20)  4)  -  + 0 . 1 4 5 8 • 10-Vj,(ifc - 1) + 0.0173 • lO'^THitk  -  2)  - 0 . 0 4 0 5 • 10-^TH{tk  -  4)  - 3) + 0.1344 • lO'^rnitk  F o r t w o v i b r a t i o n m o d e s i = 2, m o d e l o r d e r is 6 :  ytip(tk)/2=  +1.55062/ap(ife-l)+0.5763yup{tk  - 3) -  0.48743/tip(ffc-2) 1.2837y,ip(«fc -  + 1.03112/t.p(ife - 5) - 0.3870ytip{tk  4)  ~ 6)  (3.21)  + 0 . 0 5 3 4 IQ-^THitk  -  1) + 0.4899 IQ-^Tsitk  -  2)  + 0 . 1 0 4 3 lO-^mitk  - 3) - 0.1471 IQ-^sitk  -  4)  + 0 . 1 7 6 4 lO-^Hitk  - 5) + 0.0187 1 0 - ^ ^ ( 4 * -  6)  F o r t h r e e v i b r a t i o n m o d e s i = 3, m o d e l o r d e r is 8 :  ytip(tk)/3 =  + 0 . 0 6 6 2 -  1) + 2.0013 ytip(<fc - 2)  +0.6734y,ip(ffe - 3) -  1.70432/tip(<fc -  4)  ~1.15Uytij,{tk  - 5) + 0.9074ytip(<ife -  6)  +0.6223 ytipitk  -  7) -  (3.22)  8)  0.4120 yn^it^ -  + 0 . 0 5 7 6 • lO'^THitk  -  1) + 0.5583 • 10-\Hitk  -  2)  + 0 . 8 1 9 6 • lO'^Tnitk  -  3) -  0.1301 • lO-^Tsitk  -  4)  - 0 . 6 1 1 3 • 10'\H{tk  -  5) + 0.0127 • lO-^rnitk  -  6)  + 0 . 1 9 7 1 • lO'^rnitk  -  7) + 0.0217 • lO'^ffitk  -  8)  F o r f o u r v i b r a t i o n m o d e s i — 4, m o d e l o r d e r is 10 :  ytip(tk)/4 =  +l.mOytip{tk  -  1) + 1.24332/iip(<fc - 2)  -l.m7yup{tk  - 3) - 1.10162/tip(<fe -  +0.95442/tip(^fc - 5) + 0.84Myup{tk  4)  ~ 6)  - 1 . 0 6 4 5 3/t.p(ifc - 7) - 0.3661 ytip(<fc -  8)  + 0 . 8 8 7 8 y,ip(i;fe - 9) - 0.3062 2/,ip(<fc -  10)  + 0 . 0 5 8 3 • \0~^rii{ik  -  (3.23)  1) + 0.4965 • lO'^Tuitk  -  2)  3) - 0.6346 • \0~^Tu{tk  -  4)  + 0 . 0 2 1 9 • \0^^rii{tk  - 5) + 0.4743 • \0~^rii{tk  -  6)  - 0 . 2 8 4 6 • \0-^rH[tk  -  -  8)  + 0 . 2 6 0 2 • \0~'^Tii{tk -  7) - 0.1749 • lO-^rnitk  + 0 . 1 2 2 4 • 10-VH(tjfe - 9) + 0.0172 • IQ-^Hitk  -  10)  T h e s t r u c t u r e of a l l m o d e l s is s u c h t h a t yup at t h e present v a l u e i n t i m e d e p e n d s o n a s e q u e n c e of p r e v i o u s m e a s u r e d o u t p u t s ytip{tk — i) a n d p r e v i o u s i n p u t s Th(tk — i).  3.2.6  C o n t r o l S t r a t e g y for t h e F l e x i b l e L i n k M a n i p u l a t o r  Self t u n i n g a d a p t i v e c o n t r o l a l g o r i t h m s are t h e c o n t r o l s t r a t e g y used i n t h i s w o r k . a l g o r i t h m s c a n be d i r e c t S T R ' s or i n d i r e c t ones.  In the direct algorithms, the  These  controller  p a r a m e t e r s are e s t i m a t e d d i r e c t l y whereas t h e e s t i m a t i o n for t h e i n d i r e c t ones is d o n e o n t h e p l a n t ' s m o d e l p a r a m e t e r s r a t h e r t h a n t h e r e g u l a t o r ' s p a r a m e t e r s , w h i c h are c a l c u l a t e d later.  T h e first a l g o r i t h m , w h i c h has a l r e a d y b e e n i m p l e m e n t e d , is t h e G e n e r a l P r e d i c t i v e  C o n t r o l a l g o r i t h m ( G P C ) , a n d i t s results are p r e s e n t e d i n S e c t i o n 2.3. A s was p r e s e n t e d i n t h e p r e v i o u s s e c t i o n , t h e basic s t r u c t u r e of t h e h n e a r m o d e l for t h e flexible h n k is ( w i t h o u t noise):  A{q-')yit)  =  (3.24)  biq-')u{t)  F i g u r e 2.1 presents a b l o c k d i a g r a m of t h e c o n t r o l w h i c h was d e s i g n e d t o d e a l w i t h s t r u c t u r e of t h e p l a n t ' s m o d e l at a t i m e . structure.  one  W e w i l l e x a m i n e t h e effects of changes i n t h a t  A c t u a l changes s h o u l d b e d o n e i n t h e p a r a m e t e r e s t i m a t i o n b l o c k o n c e t h e i r  n u m b e r changes a n d i n t h e c o n t r o l l e r c a l c u l a t i o n s , since t h e d i m e n s i o n s of a l l p o l y n o m i a l s and matrices will  3.3  3.3.1  change.  A n a l y s i s a n d Results of S i m u l a t i o n a n d C o n t r o l W o r k  Performed  Introduction  T h e r e s u l t s p r e s e n t e d i n t h i s s e c t i o n are for n u m e r i c a l s i m u l a t i o n s of t h e single f l e x i b l e h n k c o n t r o l l e d w i t h t h e G P C a l g o r i t h m . T h e " m e a s u r e d " o u t p u t s are p r o d u c e d b y a s i m u l a t i o n  s o l v i n g t h e d y n a m i c e q u a t i o n s for t h e flexible h n k as p r e s e n t e d i n S e c t i o n 3.2 . T h o s e e q u a tions are r e f e r r e d to as the " r e a l p l a n t " w h o s e m o d e l is to be e s t i m a t e d . T h e u n k n o w n m o d e l is chosen i n t h e f o r m of E q u a t i o n 2.6 ( w i t h o u t t h e noise t e r m ) .  D i f f e r e n t m o d e l s for  the  different n u m b e r of m o d e s are p r e s e n t e d i n S u b s e c t i o n 3.2.5. I n a d d i t i o n , t h e p a r a m e t e r s o f those m o d e l s are p r e s e n t e d , even t h o u g h i n a c t u a l s i t u a t i o n s t h e y are u n k n o w n s . f o u n d by a n e s t i m a t i o n t e c h n i q u e  fitting  T h e y are  for t h e p l a n t t o b e e s t i m a t e d . T h e m e t h o d u s e d i n  t h i s w o r k so far has b e e n R e c u r s i v e L e a s t S q u a r e s ( R L S ) .  3.3.2  EflFects o f U n d e r - M o d e l i n g a n d O v e r - M o d e l i n g o n t h e C o n t r o l l e d F l e x i b l e Link  T h i s s e c t i o n w i l l present t h e s i m u l a t i o n results of i n v e s t i g a t i n g u n d e r a n d o v e r m o d e h n g t h e a c t u a l p l a n t , w i t h r e g a r d t o the s t r u c t u r e .  It m e a n s t h a t for e a c h  figure,  of  the a c t u a l  n u m b e r of m o d e s are s h o w n , i.e. m o d e s t h a t are u s e d i n t h e d y n a m i c e q u a t i o n s ( S e c t i o n 3.2 ), as w e l l as t h e n u m b e r of m o d e s t a k e n i n t o a c c o u n t i n t h e e s t i m a t o r m o d e l .  With  every  c h o i c e of e s t i m a t e d s t r u c t u r e for t h e p l a n t ' s m o d e l , t h e n u m b e r of p a r a m e t e r s t o b e e s t i m a t e d changes. I n F i g u r e 3.2, t h e flexible h n k was m o d e l e d w i t h t w o v i b r a t i o n m o d e s , w h i c h m e a n s t h a t t h e " r e a l " m o d e l of t h e fink is of 6"' o r d e r (see T a b l e 3.1).  T h e e s t i m a t e d s t r u c t u r e for t h e  d i s c r e t e t i m e h n e a r m o d e l was also of o r d e r 6. A s m e n t i o n e d b e f o r e , t h e t i p p o s i t i o n o f t h e flexible  h n k is c o n t r o l l e d w i t h t h e G P C a l g o r i t h m . T h e set p o i n t is a s q u a r e w a v e  t h e values o f ± 1 .  T h e o u t p u t , t h e t i p p o s i t i o n , follows i t v e r y w e l l .  between  F i g u r e 3.3 s h o w s  the  b e h a v i o r of t h e angle 9 a n d i t s d e r i v a t i v e s for t h e s a m e c o n d i t i o n s as i n F i g u r e 3.2. S i n c e i t is a h n e a r m o d e l a n d t h e l e n g t h of t h e a r m is 1 m . , a n d t h e effect of t h e v i b r a t i o n m o d e s is s m a l l , ytip a n d 9 a p p e a r to be t h e same.  F i g u r e s 3.4 a n d  3.5 present t h e v i b r a t i o n m o d e s  g e n e r a h z e d c o o r d i n a t e s qi a n d q2 a n d t h e i r d e r i v a t i v e s , a n d F i g u r e 3.6 shows t h e t o r q u e i n p u t  flex 2 mode link -  2 mode estimotor -  GPC control  F i g u r e 3.2: T w o m o d e flexible l i n k w i t h t w o m o d e e s t i m a t o r m o d e l  2 mode link - 2 mode estimotor h 0.4 H  GPC control 1  1  L  flex 2 mode link 1  ^  2 mode estimotor L 0.3-  GPC control  15  flex 2 mode link — 2 mode estimator — GPC control -J 10.04 -I 1 ^  10-  0.02-  5-  0.00-  'ë- 0-  -0.02 -  o  -5-10  -  4 6 TIME [sec]  CM  -10.-  TIME [sec]  10  -0.04 -0.06  —I  1—  4 6 TIME [sec]  10  flex 2 mode link -  2 mode estimator -  GPC control  F i g u r e 3.6: T h e t o r q u e i n p u t  T h e effect of under-modeling hnk  c a n be seen i n F i g u r e 3.7, w h e r e a " r e a l " f l e x i b l e t w o m o d e  (ô"* o r d e r p l a n t ) is e s t i m a t e d for a m o d e l w i t h a s t r u c t u r e of one v i b r a t i o n m o d e  order estimated model).  T h e r e s u l t is u n s t a b l e .  T h i s u n s t a b l e r e s u l t was e x p e c t e d  (4"* since  s t a b i h t y a n a l y s e s u s u a U y a s s u m e t h a t t h e e s t i m a t o r m o d e l s h o u l d be at least as c o m p l e x as t h e p l a n t itself. F i g u r e 3.8 presents t h e o p p o s i t e effect of over-modeling  w h e r e a f l e x i b l e one m o d e l i n k ( 4*''  o r d e r ) is m o d e l e d w i t h a f o u r m o d e e s t i m a t o r ( 10*'' o r d e r ) . T h e o u t p u t develops o s c i l l a t i o n s w h i c h d i v e r g e a n d ends w i t h i n s t a b i h t y . T h e f r e q u e n c y of o s c i l l a t i o n s is a b o u t 14 [ r a d . / s e c ] w h i c h is t h e f r e q u e n c y of t h e first m o d e s . W h e n t h e o v e r - m o d e h n g is closer, as i n F i g u r e 3.9, w h e r e a 2 m o d e f l e x i b l e h n k ( ô " * o r d e r ) is m o d e l e d w i t h a n e s t i m a t o r m o d e l of 3 m o d e s ( 8 " ' o r d e r ) , t h e response is n o t u n s t a b l e . H e r e t h e o u t p u t t r a c k s t h e s q u a r e w a v e of t h e set p o i n t as s h o w n i n F i g u r e 3.2, w h e r e t h e s i m u l a t e d m o d e l a n d t h e e s t i m a t o r ' s m o d e l m a t c h i n s t r u c t u r e . T h e s e r e s u l t s are also c h e c k e d i n C h a p t e r 5 for t h e two h n k r i g i d m a n i p u l a t o r w i t h t h e h y d r a u h c a c t u a t o r s . T h e conclusions  so far are t h a t u n d e r - m o d e h n g of a p l a n t w i t h a n a d a p t i v e c o n t r o l s y s t e m  w i l l m o s t p r o b a b l y result i n a n u n s t a b l e s y s t e m . T h i s is e x p e c t e d , i f one o b s e r v e s a s s u m p t i o n s m a d e i n s t a b i h t y proofs i n t h e h t e r a t u r e ( E d g a r e  and Astrom  w h i c h say t h a t t h e  e s t i m a t e d m o d e l s h o u l d be at least as c o m p l e x as t h e r e a l p l a n t m o d e l .  T h e r e is  more  f r e e d o m i n t h e choice of s t r u c t u r e for o v e r - m o d e h n g a p l a n t . If t h e e s t i m a t o r is close t o t h e a c t u a l m o d e l as i n F i g u r e 3.9, t h e n t h e s y s t e m behaves v e r y w e l l , b u t w h e n t h e  difference  g r o w s , as i n F i g u r e 3.8, i n s t a b i l i t y c a n o c c u r . It seems t h a t t h e i n s t a b i l i t i e s i n t h e o v e r - m o d e h n g  case are d u e t o d y n a m i c s i n t r o d u c e d i n  t h e c o n t r o l a l g o r i t h m t h r o u g h t h e e s t i m a t o r m o d e l - d y n a m i c s w h i c h do n o t a c t u a l l y e x i s t i n t h e s y s t e m b u t w h i c h are present for t h e c o n t r o l a l g o r i t h m , since t h e e s t i m a t o r w i l l n o n - z e r o values t o over m o d e l e d m o d e l p a r a m e t e r s .  give  T h e r e is a n effort m a d e t o c o n t r o l a n  e n t i r e l y different p l a n t t h a n t h e a c t u a l o n e , w h i c h is p r o j e c t e d t h r o u g h t h e " m e a s u r e d " v a l u e s  flex 2 mode link -  1 mode estimator -  GPC control  F i g u r e 3.7: T h e effect o f u n d e r - m o d e U i n g  flex 1 mode link -  4 mode estimotor -  GPC control  F i g u r e 3.8: T h e effect of o v e r - m o d e l h n g  F i g u r e 3.9: O v e r - m o d e l l e d 2 m o d e h n k w i t h 8 " ' o r d e r e s t i m a t o r  of ytip.  T h e n e x t s u b s e c t i o n describes t h e b e h a v i o r of a cost f u n c t i o n a n d a p o s s i b l e  method  of d e t e c t i n g t h e need to change t h e s t r u c t u r e a n d i n t e g r a t e t h e n e w one, o n h n e , i n t o t h e control system.  3.3.3  U s e o f a n E s t i m a t i o n C o s t F u n c t i o n as a C r i t e r i o n for C h a n g i n g t h e S t r u c ture of the Plant's  Model  T h e cost f u n c t i o n is a t o o l for a t t a i n i n g a n o p t i m a l b e h a v i o r of a p h y s i c a l p r o p e r t y o f t h e s y s t e m . O n e m a y w a n t t o o p t i m i z e a t r a j e c t o r y f o r a r o b o t a r m or t o o p t i m i z e t i m e o r o u t p u t e r r o r , as m a y b e l o g i c a l for t h i s case.  T h e cost f u n c t i o n chosen, E q u a t i o n 3.25, m i n i m i z e s  t h e o u t p u t e r r o r b e t w e e n t h e t i p p o s i t i o n ytip, t h e " m e a s u r e d " o u t p u t as c a l c u l a t e d f r o m t h e e q u a t i o n s of m o t i o n , a n d t h e one f r o m t h e e s t i m a t o r m o d e l ye*t-  t J{ytip,ye.t)  = ^[ytip  -  (3.25)  yest?  k=0 F i g u r e 3.10 presents t h e b e h a v i o r of t h e cost f u n c t i o n i n E q u a t i o n 3.25.  I t s v a l u e rises  i n i t i a l l y w h e n t h e r e is a difference b e t w e e n t h e m o d e l a n d p l a n t d y n a m i c s , a n d t h e n w h e n t h e e r r o r goes t o z e r o , i t settles t o a c o n s t a n t v a l u e . T h e r e a r e , of course, a d d i t i o n a l p o s s i b i h t i e s for t h e c h o i c e of a cost f u n c t i o n w h i c h a r e n o t h m i t e d t o t h e one m e n t i o n e d . A n o t h e r i n t e r e s t i n g cost f u n c t i o n , d e s c r i b e d i n S e c t i o n 2.3, is t h e one u s e d for t h e G P C a l g o r i t h m ( E q u a t i o n 2.7). ones p r e d i c t e d .  T h e r e a n e r r o r is also m i n i m i z e d ; h o w e v e r , t h e o u t p u t s are t h e  N o t o n l y s h o u l d t h e present o u t p u t t r a c k a set p o i n t , b u t t h e values  to  be m i n i m i z e d are p r e d i c t e d ones, so t h e cost f u n c t i o n ensures t h a t t h e f u t u r e e r r o r w i U be m i n i m a l .  I n o r d e r t o have r e a s o n a b l e c o n t r o l i n p u t s a n d not t o d e m a n d , for  example,  e x t r e m e l y h i g h i n p u t signals t h a t m a y d r i v e t h e s y s t e m t o s a t u r a t i o n , t h e t o t a l s u m of c o n t r o l i n c r e m e n t s is also m i n i m i z e d . T h e r e s u l t is m i n i m a l o u t p u t e r r o r , w i t h m i n i m a l c o n t r o l effort.  flex 2 mode link -  2 mode estimator -  GPC control  4 6 TIME Fsec.'  8  10  T h e b e h a v i o r of the two cost f u n c t i o n s for t h e flexible h n k a p p h c a t i o n w i U b e c o m p a r e d l a t e r i n this work.  3.3.4  E f f e c t s of O n L i n e C h a n g e s i n M o d e l o r d e r  C h a p t e r 5 w i l l present a f u U d i s c u s s i o n o n t h e effects of u n d e r a n d o v e r - m o d e h n g o n t h e c o s t f u n c t i o n for b o t h a p p h c a t i o n s of t h i s w o r k , t h e flexible h n k m a n i p u l a t o r a n d t h e h y d r a u h c a U y a c t u a t e d t w o h n k m a n i p u l a t o r . It w i U also present a m e t h o d t o d e t e c t s t r u c t u r e m o d e h n g e r r o r s a n d correct t h e m . I n t h i s s e c t i o n , p r e h m i n a r y d i s c u s s i o n o n t h e effects of t h e c h a n g e o f t h e e s t i m a t o r m o d e l s t r u c t u r e o n t h e c o n t r o U e d t i p p o s i t i o n of t h e flexible h n k is p r e s e n t e d . T h e h n k i t s e l f , t h e " r e a l " m o d e l , was chosen t o h a v e 2 v i b r a t i o n m o d e s ( i . e . 6*^ o r d e r ). T w o cases are p r e s e n t e d ; i n t h e first, t h e e s t i m a t o r m o d e l is i n i t i a l l y a 2 m o d e m o d e l (i.e. 6*'' o r d e r ) a n d is t h e n c h a n g e d i n t o a 1 m o d e m o d e l ( i.e. 4*'' o r d e r ) .  I n the second  case,  t h e e s t i m a t o r m o d e l is i n i t i a l l y a 1 m o d e m o d e l ( 4*'' o r d e r ) a n d is t h e n c h a n g e d i n t o a 2 m o d e m o d e l ( 6"* o r d e r ) to m a t c h t h e " r e a l " m o d e l m e n t i o n e d a b o v e . T h e n e x t figures w i U show n u m e r i c a l s i m u l a t i o n results for o n h n e m o d e l changes. F o r e a c h case, t h e t i p p o s i t i o n , t h e e s t i m a t o r cost f u n c t i o n , a n d t h e e s t i m a t o r o u t p u t e r r o r b e h a v i o r w i U b e p r e s e n t e d .  It  s h o u l d be n o t e d t h a t t h e c r i t e r i o n u s e d t o change t h e s t r u c t u r e of t h e e s t i m a t o r i n t h e cases p r e s e n t e d was t i m e , w h i c h is not t h e final one ( C h a p t e r 6 presents t h e f u U c r i t e r i a ) .  The  a c t u a l c r i t e r i a are t h e changes i n t h e values of t h e chosen cost f u n c t i o n a n d i t s d e r i v a t i v e s . I n F i g u r e s 3.11 a n d  3.12 t h e 6 " ' o r d e r m o d e l converges t o t h e set p o i n t , a n d after 5 s e c ,  t h e e s t i m a t o r m o d e l has b e e n c h a n g e d t o a n u n d e r - m o d e l e d s i t u a t i o n (order  = 4) w h e n t h e  w h o l e s y s t e m goes u n s t a b l e . I n F i g u r e 3.11 yup reaches i n s t a b i h t y after t h e c h a n g e of t h e e s t i m a t e d m o d e l s t r u c t u r e . F i g u r e 3.12 shows t h e changes i n t h e cost f u n c t i o n a n d o u t p u t error. I n F i g u r e 3.13, F i g u r e 3.14, F i g u r e 3.15, F i g u r e 3.16 t h e process s t a r t s w i t h t h e w r o n g e s t i m a t o r m o d e l a n d is c h a n g e d o n h n e t o t h e c o r r e c t one.  flex 2 mode link -  0  ^ & 2 mode estimator - GPC  2  3 TIME [sec.;  control  I n t h e e x a m p l e s p r e s e n t e d , t h e o u t p u t converges a n d t r a c k s t h e set p o i n t . I n F i g u r e s 3.13 t h e change i n m o d e l s is d o n e at a n e a r l y stage (0.1 sec w h i c h are 100 s a m p h n g s t e p s ) , so y^p converges weU. C o m p a r e d w i t h F i g u r e 3.2 i t is slower, b u t t h e r e s u l t s are s t i U s a t i s f a c t o r y . T h e e s t i m a t o r cost f u n c t i o n ( F i g u r e 3.14) converges t o a h i g h e r v a l u e ( o r d e r of m a g n i t u d e of 10""*) t h a n t h e one ( o r d e r of m a g n i t u d e 10"^ ) w h i c h exists w h e n b o t h m o d e l s m a t c h . I n F i g u r e s 3.15 a n d  3.16 t h e c o r r e c t i o n of the m o d e l is d o n e l a t e r , so t h e s y s t e m g a i n s  error from the wrong estimator m o d e l .  more  T h e convergence t a k e s l o n g e r t h a n i n F i g u r e 3.13,  a n d t h e e s t i m a t o r cost f u n c t i o n has m u c h l a r g e r values.  3.3.5  C o m p a r i s o n B e t w e e n t h e B e h a v i o r of T w o D i f f e r e n t C o s t  Functions  It is of i n t e r e s t t o c o m p a r e t h e e s t i m a t o r cost f u n c t i o n b e h a v i o r as p r e s e n t e d i n E q u a t i o n 3.25 a n d t h e G P C c o n t r o l a l g o r i t h m cost f u n c t i o n as p r e s e n t e d i n E q u a t i o n 2.7. F i g u r e s 3.17 3.19 present s u c h c o m p a r i s o n s for different m i s - m o d e h n g  3.18  cases.  W h e n w r i t i n g G P C e r r o r or e s t i m a t o r o u t p u t e r r o r , t h e c a l c u l a t i o n s are of t h e t e r m s w i t h i n t h e s u m s y m b o l s i n b o t h E q u a t i o n 2.7 a n d  3.25, r e s p e c t i v e l y .  T h o s e t e r m s are c a l c u l a t e d  at each t i m e s t e p . W h e n w r i t i n g t h e G P C cost f u n c t i o n or t h e e s t i m a t o r cost f u n c t i o n , t h e c a l c u l a t i o n s are a c c u m u l a t e d w i t h t i m e .  F i g u r e 3.17 is t h e cost f u n c t i o n for t h e c h a n g i n g  e s t i m a t o r m o d e l . It s t a r t s w i t h t h e c o r r e c t 6"" o r d e r m o d e l t h a t changes t o a 4"' o r d e r o n e after 5 s e c o n d s , as i n F i g u r e 3.11.  Figures  3.18 a n d  3.19 present t h e o p p o s i t e case, w h e r e  t h e difference b e t w e e n t h e t w o is t h e t i m e of s w i t c h i n g m o d e l s (as i n F i g u r e s 3.13 a n d ).  T h e t w o cost f u n c t i o n s are f u n c t i o n s of different v a r i a b l e s .  3.15  T h e e s t i m a t o r cost f u n c t i o n  ( E q u a t i o n 3.25 ) is t h e s u m of t h e s q u a r e e s t i m a t o r e r r o r , w h i c h is t h e difference  between  ytip, t h e m e a s u r e d v a l u e , a n d y^.t , t h e e s t i m a t o r o u t p u t . T h e G P C cost f u n c t i o n is t h e s u m o f t h e s q u a r e e r r o r b e t w e e n t h e p r e d i c t e d o u t p u t a n d f u t u r e set p o i n t s a n d t h e s u m o f t h e w e i g h t e d s q u a r e f u t u r e c o n t r o l effort. Y e t even t h o u g h t h e b e h a v i o r o f b o t h cost f u n c t i o n s is v e r y s i m i l a r , t h e i r values are different. T h i s m a y p r o m o t e t h e use of different p o s s i b l e cost  flex 2 mode link -  1 à 2 mode estimator -  GPC control  .0007 c  .0006  c .0005 •a^ -0004 o o ^ .0003 . J .0002 (0  .0001.0000 0  4 6 TIME [ s e c .  8  flex 2 mode link -  ^ ic 2 mode estimator -  GPC control  flex 2 mode link -  10 '=1 0  2 & 1 mode estimator -  GPC control  1  1  \  1  1  1  2  3  4  5  TIME [ s e c ]  F 6  flex 2 mode link -  1  2 mode estimator -  4 6 TIME [ s e c ]  4 6 TIME [ s e c ]  GPC control  flex 2 mode link — 1 & 2 mode estimator — GPC control  10000 * 10  Z 1500 d " 1000 o "  500 -1  r  4 6 TIME [ s e c ]  f u n c t i o n s for t h e m o d e l s t r u c t u r e changes a l g o r i t h m .  3.4  Conclusions  A n i n t e r e s t i n g p o i n t of t h i s research is t h e response  of adaptively controlled systems  o n - h n e changes i n t h e m o d e l s t r u c t u r e d u e t o v a r i a t i o n s i n o p e r a t i n g c o n d i t i o n s .  to  Adaptive  a l g o r i t h m s u s u a l l y use a n e s t i m a t i o n p r o c e d u r e for t h e p l a n t o r controUer p a r a m e t e r s i n w h i c h t h e s t r u c t u r e of t h e p l a n t ' s m o d e l is a s s u m e d t o be fixed. E s t i m a t e d values are c o n s i d e r e d  to  be c o r r e c t , a n d u n c e r t a i n t i e s i n those values are i g n o r e d ( t h e c e r t a i n t y e q u i v a l e n c e p r i n c i p l e ) . Reliance on that principle can lead to i n s t a b i h t y i n the s y s t e m . A g o o d e x a m p l e is a flexible h n k m a n i p u l a t o r , w h e r e changes i n l o a d d u r i n g a w o r k i n g c y c l e c a n r e s u l t i n t h e rise of v i b r a t i o n m o d e s w h i c h were not there b e f o r e . presents t h e e q u a t i o n s of m o t i o n for a single  flexible  This chapter  h n k m a n i p u l a t o r w h i c h is c o n t r o l l e d  with a General Predictive C o n t r o l algorithm ( G P C ) and the parameters estimated w i t h the R e c u r s i v e L e a s t S q u a r e s ( R L S ) a l g o r i t h m . S i m u l a t i o n r e s u l t s of t h e c o n t r o U e d s y s t e m a r e p r e s e n t e d . U n d e r - m o d e h n g of t h e p l a n t ' s d y n a m i c s (i.e. t h e o r d e r of t h e e s t i m a t o r is s m a l l e r t h a n t h e " r e a l " o r d e r ) leads to i n s t a b i h t y . O v e r - m o d e h n g c o u l d also l e a d t o i n s t a b i l i t y w h e n t h e gap b e t w e e n  t h e e s t i m a t e d m o d e l o r d e r a n d t h e a c t u a l s y s t e m ' s m o d e l is t o o  H o w e v e r , t h e r e are c o n d i t i o n s u n d e r w h i c h t h e s y s t e m b e h a v e s weU w i t h o v e r - m o d e h n g .  large. I t is  also s h o w n t h a t a s y s t e m w h i c h begins w i t h a n u n d e r - m o d e l e d e s t i m a t o r p l a n t , a n d is t h e n c h a n g e d t o t h e c o r r e c t one, w i U not b e c o m e u n s t a b l e u n d e r t h e right c o n d i t i o n s , as i t w o u l d h a v e i f t h e c h a n g e i n t h e e s t i m a t o r h a d not b e e n d o n e . It is suggested t h a t t h e cost f u n c t i o n p r e s e n t e d i n E q u a t i o n 3.25 m a y b e a c r i t e r i o n t o d e t e c t t h e n e e d for t h e e s t i m a t o r ' s o r d e r change.  A change of t h e e s t i m a t o r ' s m o d e l s t r u c t u r e o n  Une r e q u i r e s a change i n t h e c o n t r o U e r ' s s t r u c t u r e as weU. T h e work i n C h a p t e r 5 puts together the results i n C h a p t e r 3 that detect the need i n  m o d e l s t r u c t u r e change a n d e x e c u t e i t w h e n t h e s y s t e m is c o n t r o l l e d w i t h t h e G P C a d a p t i v e a l g o r i t h m . T h i s c h a p t e r shows t h e results of c o r r e c t i n g m i s - m o d e h n g b y u s i n g a t i m e c r i t e r i o n , i . e . , a s y s t e m t h a t c o u l d be u n s t a b l e , b u t w i t h t h e c o r r e c t i o n has a n a c c e p t a b l e  performance.  No. Order  of  Modes  of the  system  0  1  2  3  2  4  6  8  4 10  5 12  T a b l e 3.1: N o . of flexible m o d e s v s . o r d e r of s y s t e m  Chapter 4  TWO  LINK MANIPULATOR WITH HYDRAULIC A C T U A T O R S  Outline  R o b o t i c m a n i p u l a t o r s consist of U n k s ( r i g i d or  flexible),  connected by joints that c o n t r o l the  r e l a t i v e m o t i o n of n e i g h b o r i n g h n k s . T h e j o i n t s u s u a l l y h a v e p o s i t i o n sensors w h i c h m e a s u r e t h e r e l a t i v e m o t i o n a n d are a c t u a t e d b y e l e c t r i c , p n e u m a t i c or h y d r a u h c d r i v e s  These  s y s t e m s are s u b j e c t to n o n h n e a r i t i e s s u c h as c o u p h n g , c o u l o m b f r i c t i o n a n d b a c k l a s h . T h e i r i n e r t i a l c h a r a c t e r i s t i c s a n d loads v a r y d u r i n g o p e r a t i o n a n d are n o t always p r e d i c t a b l e H y d r a u h c a J l y a c t u a t e d m a n i p u l a t o r s are w i d e l y u s e d i n i n d u s t r y t o d a y .  Hydrauhc systems  have r e l a t i v e l y large t o r q u e t o weight r a t i o s , h i g h e r l o o p gains a n d w i d e r b a n d w i d t h s pared w i t h electrical motors  .  com-  H y d r a u h c r o b o t s are u s e d for h e a v y d u t y t a s k s r e q u i r i n g  p o s i t i o n a c c u r a c y , r a p i d d y n a m i c s a n d r a p i d s t a r t a n d s t o p . H o w e v e r , h y d r a u h c s y s t e m s are c o m p l e x , n o n h n e a r a n d d i f f i c u l t t o a n a l y z e for c o n t r o l p u r p o s e s  ,  I n t h e d e s i g n of a m a n i p u l a t o r c o n t r o l s t r a t e g y t w o k i n d s of p h y s i c a l q u a n t i t y s h o u l d be c o n s i d e r e d , those t h a t c a n be d e t e r m i n e d a c c u r a t e l y w i t h t h e values r e m a i n i n g r e l a t i v e l y c o n s t a n t , a n d those t h a t v a r y w i t h i n a r a n g e of values d u r i n g a w o r k i n g c y c l e . T h e s e c o n d t y p e of q u a n t i t i e s c a n n o t a l w a y s be a v o i d e d i n a c o n t r o l s y s t e m a n d m a y r e q u i r e a n o n - h n e c h a n g e of t h e c o n t r o U e r p a r a m e t e r s . E x a m p l e s i n h y d r a u h c s y s t e m s w o u l d b e e x t e r n a l a n d i n t e r n a l leakages, size of orifices, t e m p e r a t u r e changes, a c c u m u l a t i o n of o i l c o n t a m i n a t i o n , v i s c o s i t y changes of t h e h y d r a u h c fluid, d a m p i n g coefficient etc. I n t h e m o d e h n g of t h e h n k s , one c a n find changes i n t h e m o m e n t s o f i n e r t i a d u r i n g a w o r k i n g c y c l e w h e n a n e x t e r n a l l o a d  ( i n s o m e cases a n u n k n o w n e x t e r n a l l o a d ) is b e i n g p i c k e d u p a n d p u t d o w n . C o m p l i a n c e i n t h e h n k s m a y give o s c i l l a t o r y d y n a m i c s w i t h l o w d a m p i n g , i.e. e x c i t e v i b r a t i o n m o d e s t h a t change t h e m o d e l o f t h e c o n t r o l l e d s y s t e m . M o s t o f t h e t e c h n i q u e s for c o n t r o l s y s t e m d e s i g n a s s u m e t h e p l a n t a n d i t s e n v i r o n m e n t are k n o w n .  I n m a n y cases h o w e v e r , t h i s is n o t so, since t h e p l a n t m i g h t b e t o o  complex,  t h e m o d e l n o t f u l l y u n d e r s t o o d , or t h e process a n d t h e d i s t u r b a n c e s c h a n g i n g w i t h o p e r a t i n g conditions.  W h e n a s y s t e m ' s d y n a m i c m o d e l is u n c e r t a i n or has t h e p o s s i b i h t y of c h a n g i n g  its p a r a m e t e r s o n - h n e , a d a p t i v e c o n t r o l m a y be c o n s i d e r e d . C o n t r o l of r o b o t i c al  A n et a l  s y s t e m s has b e e n w i d e l y d i s c u s s e d i n t h e h t e r a t u r e b e f o r e , ( F u et  Asada  , C r a i g ^1 a n d o t h e r s . )  T h e dynamics of the actuators  usually ignored a n d the hnk m o t i o n provides second order equations w i t h c o u p h n g 1^. S e p e h r i et a l  are  effects  show a c o n t r o l s t r a t e g y i n w h i c h t h e h n k m o t i o n is c o n t r o l l e d b y a self  t u n i n g a l g o r i t h m ( m i n i m u m v a r i a n c e c o n t r o l ) , a n d t h e h y d r a u h c s is c o n t r o l l e d b y a c l a s s i c a l control algorithm. A s t u d y o n a h y d r a u h c m a n i p u l a t o r c o n t r o l l e d b y a n a d a p t i v e a l g o r i t h m was by V a h a i n 1988  'j'^g c o n t r o l a l g o r i t h m was b a s e d o n a o n e - s t e p - a h e a d  t r o U e r p r o p o s e d b y C l a r k e et a l i n 1975  presented  self-tuning con-  A n i n t e g r a l t e r m was i n t r o d u c e d t o a q u a d r a t i c  p e r f o r m a n c e c r i t e r i o n w h i c h was m i n i m i z e d t o find a c o n t r o l l a w t h a t was a p p h e d t o a h e a v y d u t y m a n i p u l a t o r i n two ways.  A n e x p e r i m e n t a l s t u d y was p e r f o r m e d i n o r d e r t o e v a l u a t e  t h e a p p h c a b i l i t y of t h e a d a p t i v e a l g o r i t h m t o c o n t r o l t h e m o v e m e n t  of t h e m a n i p u l a t o r ' s  l i n k s . T h e a u t o r e g r e s s i v e m o d e l chosen for t h e a d a p t i v e c o n t r o l process e x p e r i e n c e d d i f f i c u l ties c a u s e d b y m e c h a n i c a l a n d p h y s i c a l c h a r a c t e r i s t i c s a n d m e a s u r e m e n t noise. A s w e U , t h e s t u d y c o n s i d e r e d s i m u l a t i o n e v a l u a t i o n of t h e p r o b l e m .  T h e m o d e l chosen t o s i m u l a t e t h e  a c t u a l m a n i p u l a t o r was a h n e a r i z e d s e c o n d o r d e r m o d e l . A s p r e v i o u s l y m e n t i o n e d , i n 1987 C l a r k e et a l  ^ developed the Generalized P r e d i c t i v e  C o n t r o l t e c h n i q u e w h i c h m a y h a v e advantages for t h e c o n t r o l of c o m p l e x s y s t e m s s u c h as  heavy duty manipulators. T h e present w o r k apphes t h e G P C a l g o r i t h m t o a n e x t e n s i v e m e c h a n i c a l a n d h y d r a u h c s y s t e m m o d e l of a n i n d u s t r i a l h y d r a u h c m a n i p u l a t o r , t o assess t h e c o n t r o l c a p a b i h t y of t h i s m o r e recent a l g o r i t h m o n s u c h a n o n h n e a r s y s t e m , a n d t o s t u d y t h e effect o f t h e  design  p a r a m e t e r s , K o t z e v et a l ^'^ . F i r s t t h e d y n a m i c m o d e l o f t h e m a n i p u l a t o r is p r e s e n t e d as w e l l as t h e e q u a t i o n s m o t i o n of t h e h y d r a u h c a c t u a t o r i n c l u d i n g c o m p h a n c e ,  of  dead time a n d full d y n a m i c s of t h e  s e r v o v a l v e , r e s u l t i n g i n a r a t h e r c o m p l e x n o n h n e a r s y s t e m i n w h i c h t h e o r d e r of t h e e s t i m a t e d h n e a r m o d e l for t h e G P C m a y v a r y f r o m 6 t o 10 . T h e G P C a l g o r i t h m is also p r e s e n t e d i n d e t a i l . It uses for c o n t r o l p u r p o s e s a h n e a r i z e d m o d e l of t h e s y s t e m . T h e c o n t r o l l a w d e r i v e d d e p e n d s o n values of t h e m e a s u r e d o u t p u t f r o m t h e n o n h n e a r s y s t e m , a n d uses a s s u m e d a n d e s t i m a t e d p a r a m e t e r s for t h e h n e a r m o d e l . T h e c o n t r o l s t r a t e g y i n t h i s w o r k , consists of t w o a d a p t i v e l o o p s , i n w h i c h t h e p r o c e s s m o d e l c o n t a i n s t h e m a n i p u l a t o r h n k w i t h t h e h y d r a u h c a c t u a t o r . T h e r e is a n a d v a n t a g e i n c o m b i n i n g aU t h e s y s t e m states i n t o one c o n t r o l l o o p , w h e r e t h e s y s t e m is r e p r e s e n t e d b y a n i n p u t / o u t p u t m o d e l i n t h e G P C , since t h e e s t i m a t e d p a r a m e t e r s c a n reflect a l l c h a n g e s i n t h e s y s t e m as w e l l as t h e u n c e r t a i n t i e s , d i s t u r b a n c e s , n o n h n e a r i t i e s a n d c o u p h n g ,  provided  t h a t safety l i m i t s o n t h e r e q u i r e d s y s t e m v a r i a b l e s e x i s t . T h i s a p p r o a c h c a n b e i m p l e m e n t e d o n a n y h y d r a u h c m a n i p u l a t o r w i t h as m a n y h n k s a n d a c t u a t o r s as r e q u i r e d . It c a n also b e i m p l e m e n t e d o n m a n i p u l a t o r s w i t h o t h e r a c t u a t o r s s u c h as e l e c t r i c m o t o r s . T h e r e s u l t s s h o w t h e effects of t h e different c o n t r o l t u n i n g p a r a m e t e r s o n t h e c o n t r o l l e d s y s t e m  performance.  T h e G P C has a n i n h e r e n t i n t e g r a t o r w h i c h helps o v e r c o m e offsets b u t r e s u l t s i n u n d e s i r a b l e overshoot when operating robot manipulators. I n a d v a n c i n g t h e s t a t e of t h e art of p r e d i c t i v e c o n t r o l , i n t h i s w o r k s p e c i a l a t t e n t i o n is g i v e n t o t h e m a x i m u m o u t p u t h o r i z o n , w h i c h for l a r g e r values (i.e. l a r g e r p r e d i c t i o n h o r i z o n ) , has s t a b i h z i n g effects a n d d a m p s t h e o u t p u t b e h a v i o r b u t slows t h e t r a n s i e n t r e s p o n s e .  The  w o r k also i n t r o d u c e s a n o n - l i n e a u t o m a t i c c h a n g e of t h e m a x i m u m o u t p u t h o r i z o n so t h a t t h e t r a n s i e n t response c a n be sufficiently fast a n d u n d e s i r a b l e o v e r s h o o t s a v o i d e d .  Further  advances are also m a d e i n t h e selection of o t h e r G P C d e s i g n p a r a m e t e r s . T h e d y n a m i c e q u a t i o n s of t h e m a n i p u l a t o r a n d i t s a c t u a t o r s h a v e b e e n s i m u l a t e d i n a F O R T R A N program along w i t h the control a l g o r i t h m and the n u m e r i c a l simulations were p e r f o r m e d o n a V A X 3200 c o m p u t e r .  4.1  R i g i d T w o Link Manipulator with Hydraulic Actuators  4.1.1  Introduction  A t w o h n k r i g i d m a n i p u l a t o r is a c o m p l e x a n d n o n h n e a r s y s t e m . T h e c o u p h n g b e t w e e n  the  m o t i o n of t h e a r m ' s c o m p o n e n t s i n t r o d u c e s n o n h n e a r i t i e s . T h e h y d r a u h c a c t u a t o r s c o n s i s t o f servovalves a n d c y h n d e r s a n d m a y b e d e s c r i b e d as a t h i r d or a fifth o r d e r s y s t e m . T h e n e x t t w o sections w i U present t h e e q u a t i o n s of m o t i o n of t h e d y n a m i c s of t h e m a n i p u l a t o r h n k s , a n d the equations of m o t i o n of the h y d r a u h c actuators, w h i c h w i l l be expressed i n equations for s o l v i n g Pii,,,Piaut  T h e s t a t e space v e c t o r , of o r d e r 8, for t h i s s p e c i f i c s y s t e m i s :  X =  4.1.2  Ql  01  P2.„  P2.  (4.1)  E q u a t i o n s o f M o t i o n for t h e R i g i d T w o L i n k M a n i p u l a t o r  F i g u r e 4.1 shows t h e c o n f i g u r a t i o n of t h e t w o h n k m a n i p u l a t o r .  Using a general formulation, approach are:  , t h e d y n a m i c e q u a t i o n s of m o t i o n d e r i v e d v i a L a g r a n g e ' is  Figure 4.1: Configuration of the two link manipulator  F i g u r e 4.1: C o n f i g u r a t i o n of t h e two h n k m a n i p u l a t o r  W h e r e n is t h e n u m b e r o f degrees o f f r e e d o m , Dij t e r m s for effective a n d c o u p h n g i n e r t i a at j o i n t i d u e t o h n k j m o t i o n , Dijk t e r m s for t h e C o r i o h s a n d c e n t r i p e t a l forces at j o i n t i as a r e s u l t o f m o t i o n i n h n k s j a n d k, a n d Di are t e r m s for g r a v i t y l o a d i n g at j o i n t i. T h e k i n e t i c energy of t h e s y s t e m i s :  (4.3) +7712/1/2 cos  ^2(^1+^1^2)  T h e p o t e n t i a l energy o f t h e s y s t e m i s : P = —migli  cos 61  — 7712/1  cos di  — 77125/2  cos (di + O2)  (4-4)  C o m b i n i n g a n d applying Lagrange's equation, the nonhnear equations of m o t i o n are: Ti =  (77ii + 7712) ll  +  + 7712/2 + 27712/1/2 cos 62  77X2/0 + "^2/1/2 cos 02  -277x2/1/2 s i n 02^1^2 -  =  (4.5)  9,  «12/1/2  s i n ^2^2 + ("'•1 + " ^ 2 ) gh  T2  di  s i n 9i + 77125/2^171(^1 + Ô2)  7712/2 + «12/1/2 C0SÔ2  9i  (4.6)  + m 2 / 2 ^ 2 + 7712/1/2 s i n ^2^1  ^1^ + 77125/2  sin(ei +  ^2)  W h e r e r i a n d r2 are i n p u t t o r q u e s t o t h e j o i n t s .  4.1.3  E q u a t i o n s of M o t i o n of the H y d r a u l i c A c t u a t o r  T h e h n k s o f t h e m a n i p u l a t o r a r e a c t u a t e d b y h y d r a u h c a c t u a t o r s . E a c h h n k is a c t i v a t e d b y a h y d r a u h c m o t o r w h i c h is c o n n e c t e d t o a s e r v o v a l v e t h r o u g h e x p a n d a b l e hoses. T h e s e r v o v a l v e  m o n i t o r s t h e flow of t h e h y d r a u h c h q u i d .  F i g u r e 4.2 d e s c r i b e s a c r i t i c a l c e n t e r s y m m e t r i c  valve. T h e s u p p l y p r e s s u r e is k e p t c o n s t a n t w h i c h allows e a c h s e r v o v a l v e t o f u n c t i o n i n d e p e n dently.  T h e r e t u r n pressure is t h e a t m o s p h e r i c p r e s s u r e , since i t is c o n n e c t e d t o a s t o r a g e  t a n k . C o m p o n e n t s s u c h as check valves a n d rehef v a l v e s a r e for m a c h i n e safety. T h e s e r v o valves c o n t r o l t h e f l u i d p o w e r . T h e m o s t w i d e l y u s e d v a l v e has a s p o o l v a l v e t y p e c o n s t r u c t i o n , a n d is classified b y t h e w a y flow goes t h r o u g h t h e v a l v e . T h e v a l v e v a r i a b l e s are t h e s p o o l d i s p l a c e m e n t ( A V J , t h e flows i n a n d out of t h e v a l v e qi-^ a n d g^^, , t h e s u p p l y p r e s s u r e (P,up)) t h e r e t u r n p r e s s u r e (Près),  a n d t h e h n e pressures (P,-.^ a n d  T h e equations describing  t h e e q u a t i o n s of m o t i o n for t h e valves are n o n h n e a r . T h e flow e q u a t i o n s are  > 0,  g..„ = K.ai.e  (positive  Xv, v^P,up - Pi,„  qia^t = K^cilve Xv, \JPi^t  < 0,  = K,al.e  g.„„. = K,al.e  direction)  (negative  ~ Pre.  (4.8)  direction)  Xv, ^JPi,^ - Pre.  Xv, ^P.up  (4.7)  - Pi^,  (4.9)  (4.10)  Hydraulic Control Elerr>ents  F i g u r e 4.2: E l e c t r o h y d r a u l i c a c t u a t o r  L i n e a r i z a t i o n o f tliese e q u a t i o n s w i t h a T a y l o r series e x p a n s i o n a b o u t z e r o s p o o l d i s p l a c e m e n t , for i n i t i a l design p u r p o s e s o n l y , gives:  = i f . . Xv, - i^p. P,.„  (4.11)  =  (4.12)  ^F. + ^P.  W h e r e K^i a n d Kp- are t h e flow g a i n a n d t h e flow p r e s s u r e coefficients, r e s p e c t i v e l y . A first o r d e r m o d e l ,  describing the equations of t h e pipes m o d e l are:  (4.13)  K.-^{<li..-D^éi)  ^w  =  (4.14)  ^ ( ^ m ^ i - 9 w )  W h e r e Dm is t h e v o l u m e t r i c d i s p l a c e m e n t o f h y d r a u h c m o t o r , a n d ^ i s t h e h y d r a u h c comphance. T h e m o t o r a n d h n k d y n a m i c m o d e l is :  Ti  = (Pw„ - P w )  Dm  =  jrni ëi + bméi  + Ti  (4.15)  W h e r e t h e first t e r m expresses t h e m o v e m e n t of t h e h y d r a u h c m o t o r , t h e s e c o n d is a d a m p i n g t e r m , Ti expresses e x t e r n a l l o a d b y t h e finks m o v e m e n t a n d Ti is t h e a p p h e d t o r q u e t o t h e h n k i.  4.2  Control Strategy  4.2.1  Introduction  A s m e n t i o n e d before, t h e e q u a t i o n s of m o t i o n of a r o b o t i c m a n i p u l a t o r c o n t a i n n o n h n e a r i t i e s , inertial characteristics and disturbances that vary d u r i n g a working cycle a n d m a y not always be p r e d i c t a b l e . L a t e l y self t u n i n g p r e d i c t i v e a l g o r i t h m s have b e e n used since t h e r e s u l t s h a v e b e t t e r r o b u s t n e s s c o m p a r e d w i t h o t h e r self t u n i n g c o n t r o l a l g o r i t h m s s u c h as P o l e P l a c e m e n t and Generahzed M i n i m u m Variance (Astrom  T h e robustness of p r e d i c t i v e a l g o r i t h m s is  d u e t o t h e m i n i m i z a t i o n of a m u l t i - s t e p cost f u n c t i o n ^. T h e basic p r e d i c t i v e m e t h o d h a s t h e f o l l o w i n g steps:  1. P r e d i c t i o n of t h e o u t p u t i n t h e f u t u r e . 2. C h o i c e of t h e f u t u r e set p o i n t s , a n d m i n i m i z a t i o n of a cost f u n c t i o n c a l c u l a t e d f r o m t h e f u t u r e e r r o r s , b e t w e e n t h e f u t u r e o u t p u t s a n d f u t u r e set p o i n t s , y i e l d s a set o f f u t u r e c o n t r o l signals. 3. T h e first element of t h e c o n t r o l signals is a c t u a l l y u s e d a n d t h e w h o l e p r o c e d u r e is r e p e a t e d . T h i s is a r e c e d i n g - h o r i z o n  controUer.  T h e t y p e o f controUers m e n t i o n e d a b o v e c o n s i d e r t h e o u t p u t at one p o i n t o f t i m e i n t h e f u t u r e . T h e G e n e r a h z e d P r e d i c t i v e C o n t r o l ( G P C ) ^, ^, a l g o r i t h m m i n i m i z e s a cost f u n c t i o n t h a t c o n s i d e r s t h e f u t u r e p r e d i c t e d o u t p u t s j steps a h e a d , t h e f u t u r e set p o i n t s a n d f u t u r e control signals.  4.2.2  C o n t r o l S t r a t e g y for t h e T w o l i n k R i g i d M a n i p u l a t o r  F i g u r e 4.3 presents a b l o c k d i a g r a m for c o n t r o U i n g t h e t i p p o s i t i o n of t h e t w o h n k m a n i p u l a t o r w i t h the hydrauhc actuators.  Process parameters  Design  Estimation  Regulator  Process  Regulator parameters  F i g u r e 4.3: C o n t r o l s t r a t e g y for t h e t w o h n k m a n i p u l a t o r  T h e t i p l o c a t i o n error is t r a n s l a t e d t o angle changes i n t h e j o i n t s . T h e c o n t r o l c o n s i s t s of t w o a d a p t i v e loops, i n w h i c h t h e p r o c e s s m o d e l c o n t a i n s t h e m a n i p u l a t o r h n k w i t h t h e h y d r a u h c a c t u a t o r . E a c h of t h e j o i n t h n k s is c o n t r o U e d s e p a r a t e l y w i t h t h e g e n e r a l p r e d i c t i v e c o n t r o l a l g o r i t h m . T h e m o d e l for each l o o p w i U be a n I n p u t / O u t p u t t y p e o f m o d e l i n t h e form of ; Aiq-')y{t)=^b{q-')u{t)  + c{q-')e{t)  Where: y(t) is t h e o u t p u t - j o i n t angle. u{t) is t h e i n p u t t o t h e process - s p o o l d i s p l a c e m e n t e{t) is t h e noise sequence.  4.3  A n a l y s i s a n d R e s u l t s of S i m u l a t i o n  4.3.1  System  Parameters  T h e link parameters are:  h  50 c m .  =  I2 =  50  cm.  mi  =  1 kg.  m2  =  1 kg.  T h e h y d r a u h c actuator parameters are:  (4.16)  -  K.al.e  243.  Dm  =  8.2  ^ V  ^  o  =  65.  kg  cm^  t i l cm^  '  kgf Psup  -  250.  cm'  =  P ^ res  K^,  0  M  —  o  =  1387.  -  4.65  Kp.  cm  ^ cm?  T h e r a n g e of t h e s p o o l d i s p l a c e m e n t i s :  -0.5  cm.  <  Xvi  >  0.5  cm.  W h e r e i = 1,2 for t h e n u m b e r of h n k s .  4.3.2  O p e n Loop Analysis  T h e h n e a r i z e d e q u a t i o n s of m o t i o n , i.e. E q u a t i o n 4.11, E q u a t i o n 4.12, E q u a t i o n 4.13, E q u a t i o n 4.14, E q u a t i o n 4.15 a n d h n e a r i z e d E q u a t i o n 4.5, E q u a t i o n 4.6 p r o d u c e s t h e f o l l o w i n g s t a t e space f o r m , for a single h n k a n d are u s e d o n l y for p r e h m i n a r y s t u d y a n d d e s i g n :  = A  X  BXv,  X +  (4.17)  w h e r e t h e s t a t e space v e c t o r for one l o o p i s :  A U o w P L , = Pi„^ -  P.„.., a n d :  - 2 f A  =  0 \  Jii  0  1  0  0  (4.18)  W h e r e t h e eigenvalues are:  0.9997413  1.072178  (2^K B  =  0.9996837  )  (4.19)  0 0  /  T h e t r a n s f e r f u n c t i o n b e t w e e n t h e angle a n d s p o o l d i s p l a c e m e n t i s :  Oi{s) Xv,  2 f Dm s {Jli  .2 +  K,,  JU l,Kp,  (4.20)  3 + 2 f D^)  T h e s y s t e m was c o n v e r t e d w i t h a z e r o o r d e r h o l d s a m p h n g i n t o a d i s c r e t e t i m e f o r m , t h e i n p u t / o u t p u t model is:  Oiitk)^  1) - a2 Oi{tk - 2) -  -ai  Oiitk -  +h  Xv,{tk - l )  + b2 Xv,{tk  (4.21)  Biitk - 3)  - 2) + fci Xv,{tk -  1) + ei{tk)  F o r s a m p l i n g p e r i o d oî h = are:  al =  63 =  -2.9326,  0.0000005348.  components,  a2 =  0.005  [sec]  2.8652, a 3 =  W h e r e e{t)  t h e p a r a m e t e r s of t h e A a n d B  -0.9326,  61 =  0.000000554, 62 =  polynomials 0.00O002177,  is t h e u n m e a s u r e d d i s t u r b a n c e t e r m w h i c h i n c l u d e s  two  t h e first, d y n a m i c c o u p h n g b e t w e e n t h e h n k s a n d g r a v i t a t i o n effects, a n d t h e  s e c o n d a n u n c o r r e l a t e d r a n d o m noise sequence i f e x i s t s .  4.3.3  Simulation Study and  Results  S u b s e c t i o n 4.3.2 d e v e l o p e d t h e o p e n l o o p a n a l y s i s w h i c h was u s e d t o d e t e r m i n e t h e o r d e r o f t h e i n p u t / o u t p u t m o d e l for t h e G P C a l g o r i t h m , a n d t h e i n i t i a l values for i d e n t i f i c a t i o n o f i t s parameters.  T h e a c t u a l s y s t e m is a n o n h n e a r s y s t e m r e s u l t i n g f r o m c o u p h n g t e r m s d u e t o  r e l a t i v e m o t i o n of t h e h n k s , t h e g r a v i t y t e r m , s a t u r a t i o n h m i t s o n v a r i a b l e s , t h e h y d r a u h c s y s t e m etc. T h e n o n h n e a r i t i e s were i n c o r p o r a t e d i n t o t h e s i m u l a t i o n m o d e l w h i l e t h e m o d e l e s t i m a t e d a n d u s e d by G P C is h n e a r b y i t s n a t u r e . F i g u r e 4.4, F i g u r e 4.5, F i g u r e 4 . 6 , a n d F i g u r e 4.7 present t h e c o n t r o l l e d n o n h n e a r s y s t e m , ( t h e t w o h n k m a n i p u l a t o r a c t u a t e d hydrauhc actuators) which exhibit good performance points.  by  i n o u t p u t t r a c k i n g of t h e g i v e n set  T h e n o n h n e a r i t i e s are t r e a t e d as u n k n o w n d e t e r m i n i s t i c d i s t u r b a n c e s i n t h a t t h e  G P C assumes a h n e a r m o d e l for t h e a c t u a l s y s t e m . F i g u r e 4.4 a n d F i g u r e 4.5 show t h e b e h a v i o r of t h e o u t p u t s 6i, a n d 02 a n d t h e i r d e r i v a t i v e s t o s q u a r e w a v e s e t p o i n t s . F i g u r e 4.6 presents t h e s p o o l valves d i s p l a c e m e n t s a n d t h e c o n t r o l a c t i o n Au(t)  for b o t h h n k s a n d a c t u a t o r s .  F i g u r e 4.7 presents t h e hose pressures for b o t h  actuators. T h e d e s i g n p a r a m e t e r s u s e d for t u n i n g t h e G P C are n o t e d at t h i s p o i n t . N2, t h e m a x i m u m o u t p u t h o r i z o n , is c h a n g e d o n h n e . Ni, t h e m i n i m u m o u t p u t h o r i z o n , N^, t h e c o n t r o l h o r i z o n a n d A , t h e w e i g h t i n g f a c t o r are a d d i t i o n a l design p a r a m e t e r s . {N2,.  A t first a l o w e r v a l u e o f N2  )was u s e d , t o achieve a faster t r a n s i e n t response a n d l a t e r i t was i n c r e a s e d t o iV2,_  avoid overshoots.  to  I n t h e case p r e s e n t e d i n F i g u r e 4.4, F i g u r e 4.5, F i g u r e 4.6, a n d F i g u r e 4.7  F i g u r e 4.4:  ^1,^1,^1  for s q u a r e wave i n p u t  TIME [sec]  F i g u r e 4.5:  ^2,^2,^2  for s q u a r e wave i n p u t  F i g u r e 4.6: C o n t r o l a c t i o n a n d s p o o l d i s p l a c e m e n t for 9i a n d 82  GPC control — 2 link monipulotor with hydraulic octuotors 25-  1  r  10 TIME [sec]  15  10 TIME [sec]  15  10 TIME [sec]  15  10 TIME [sec]  15  F i g u r e 4.7: P r e s s u r e s for Oi a n d 62  20  t h e s y s t e m ' s p e r f o r m a n c e was a c h i e v e d w i t h t h e f o l l o w i n g design p a r a m e t e r s : 1. for 6x : Nu,  = 1, iV2,^^ = 70, N2,^^ = 100, iV„,^ = 1, A„, = 0.05  2. for 02 : Ni,^  =  1, N2,^^ = 40, N2,^^ = 60, N^,^ = 1,  = 0.05  N o t e t h a t i n s t e a d y s t a t e t h e r e is a c h a t t e r i n some of t h e p a r a m e t e r s , s u c h as t h e s p o o l v a l v e d i s p l a c e m e n t , t h e c o n t r o l i n c r e m e n t s i g n a l , t h e hose pressures a n d t h e a c c e l e r a t i o n s . I n s t e a d y s t a t e the s p o o l v a l v e c h a t t e r s a r o u n d t h e zero value w h i c h i t c a n n o t  maintain  due to nonhnearities i n the system a n d the m i s - m a t c h between the nonhnear m o d e l t h a t s i m u l a t e s t h e a c t u a l s y s t e m a n d t h e r e p r e s e n t i n g h n e a r m o d e l used b y t h e a d a p t i v e a l g o r i t h m . F i g u r e 4.8 a n d F i g u r e 4.9 show t h e r e s u l t s for t h e s a m e design p a r a m e t e r s as F i g u r e 4.6 a n d F i g u r e 4.7, o n l y i n t h i s case t h e m o d e l of t h e h y d r a u h c a c t u a t o r s y s t e m was h n e a r i z e d . T h e n o n h n e a r i t i e s d u e to c o u p l i n g b e t w e e n t h e m o v e m e n t of t h e h n k s or d u e t o s a t u r a t i o n i n t h e d i s p l a c e m e n t o f t h e s p o o l v a l v e r e m a i n i n t h e s i m u l a t e d m o d e l of t h e s y s t e m . T h e c h a t t e r i n g has b e e n r e d u c e d s i g n i f i c a n t l y . C h a n g e s i n t h e values of t h e d e s i g n p a r a m e t e r s a b o v e w i l l c h a n g e t h e b e h a v i o r of t h e system.  T h e effects of t h e o u t p u t h o r i z o n N2 were c h e c k e d i n b o t h l o o p s .  T h e larger the  v a l u e of N2, t h e slower t h e response. L a r g e r values have t h e t e n d e n c y t o s t a b i h z e t h e s y s t e m s i n c e i t uses p r e d i c t e d e r r o r s over a l a r g e r p e r i o d of t i m e . O n t h e o t h e r h a n d fewer p r e d i c t i o n steps w i U r e s u l t i n a m o r e r a p i d c o n t r o l a c t i o n , a n d t h e i n h e r e n t i n t e g r a t i o n t e r m o f t h e m o d e l u s e d for G P C causes t h e response t o h a v e m o r e overshoot a n d m o r e o s c i l l a t i o n s . 0i i n t h e u p p e r p a r t of F i g u r e 4.10 is t h e s a m e as t h e one i n F i g u r e 4.4 b u t t h e s e t p o i n t is c o n s t a n t i n s t e a d of a s q u a r e wave. T h e l o w e r p a r t shows 61 i n response t o t h e s a m e s e t p o i n t , b u t t h e o u t p u t h o r i z o n is c o n s t a n t a n d i t s values are t h e h i g h e r values for iVz,^ = 100 a n d N2,^ u s e d i n F i g u r e 4.4. T h e response for t h e l a r g e r values is slower.  = 70  GPC control - 2 link naonipulotor with hydroulic octuotofs 1.0  3*  0.5-  10 TIME [sec] E o .  10 TIME [sec]  15  20  10 TIME [sec]  15  20  1.0  CM  0.5-  T -  i2  0.0  a u u _ç  o 10 TIME [sec]  15  c o u  -T-  5  F i g u r e 4.8: C o n t r o l a c t i o n a n d s p o o l d i s p l a c e m e n t f o r h y d r a u l i c h n e a r i z e d m o d e l .  GPC control — 2 link manipulator with hydroulic octuotors 25-  10 TIME [sec]  r 10 TIME [sec]  10 TIME [sec]  10 TIME [sec]  F i g u r e 4.9; P r e s s u r e s for di a n d 62 for h y d r a u K c l i n e a r i z e d m o d e l .  15  F i g u r e 4.10: T h e effect of h i g h e r values of ATj (lower case)  PAGINATION  TEXT  ERROR.  ERREUR DE  COMPLETE.  NATIONAL  LIBRARY  CANADIAN  THESES  LE  OF  CANADA.  SERVICE.  TEXTE  PAGINATION.  EST  BIBLIOTHEQUE  SERVICE  DES  COMPLET.  NATIONALE  THESES  DU  CANADA.  CANADIENNES.  F i g u r e 4.12: Oi a n d 62 for iVj^^ ^ 200 a n d Niy^  = 200  F i g u r e 4.13:  a n d 62 for N^^^ ^ 3 a n d N^^^ = 3  GPC control — 2 link monipulotor with hydraulic actuators 0.6-1 ' 1 h  '  0  5  TIME  10 [sec]  15  F i g u r e 4.14: 9^ a n d ^2 for N^^^ = 10 a n d  20  = 10  4.3.4  E f f e c t s of O n - l i n e C h a n g e s i n M o d e l O r d e r  A n a n a l y s i s of t h e effect of u n d e r a n d over m o d e h n g for t h e t w o h n k h y d r a u h c m a n i p u l a t o r is g i v e n i n C h a p t e r 5. T h e a n a l y s i s is b a s e d o n t h e b e h a v i o r of a cost f u n c t i o n for t h e cases of c o r r e c t , u n d e r a n d o v e r - m o d e h n g .  4.4  Conclusions  In this chapter a rigid two hnk manipulator w i t h h y d r a u h c actuators controlled by a G P C a l g o r i t h m is p r e s e n t e d . T h e s y s t e m is h i g h l y n o n h n e a r , c o n t r o U e d w i t h a G P C a l g o r i t h m t h a t assumes a h n e a r s t r u c t u r e d i n p u t / o u t p u t m o d e l , w h o s e p a r a m e t e r s are e s t i m a t e d o n h n e .  The  c o n t r o l s t r a t e g y t r e a t s t h e c o n t r o l of e a c h h n k a n d i t s a c t u a t o r as one m o d e l . T h e c h a n g e s i n t h e s y s t e m p a r a m e t e r s are h a n d l e d w i t h o u t t h e n e e d t o i d e n t i f y t h e e x a c t cause of t h e c h a n g e . S u c h a n a p p r o a c h m a y need safety m e a s u r e s a n d b o u n d e d variables.  values o n s o m e o f t h e  T h e s y s t e m is weU b e h a v e d w h e n c o n t r o U e d w i t h t h e set of d e s i g n  system  parameters  s h o w n i n F i g u r e 4.4, F i g u r e 4.5, F i g u r e 4.6, a n d F i g u r e 4.7. G P C o v e r c o m e s t h e effects o f t h e n o n h n e a r i t i e s i n t h e a c t u a l m o d e l of t h e s y s t e m , ( c o u p h n g b e t w e e n h n k s , s a t u r a t i o n s e t c . ) I n t e r m s of a d v a n c i n g t h e s t a t e o f t h e art of p r e d i c t i v e c o n t r o l , the i n f l u e n c e of t h e m a i n G P C d e s i g n p a r a m e t e r s , N2 a n d N^, was s t u d i e d . It was r e c o m m e n d e d , b y C l a r k e et a l , t h a t t h e c o n t r o l h o r i z o n N^, s h o u l d b e chosen as h i g h as t h e n u m b e r of p o o r l y d a m p e d poles o f the system.  I n t h i s s t u d y t h e r e was no significant effect o n t h e o u t p u t b y c h o o s i n g i t so,  ( F i g u r e 4.13). W h e n A^^ is chosen as h i g h as 10 t h e o u t p u t s h a v e a m o r e o s c i U a t o r y n a t u r e . A s w e U , t h e m a x i m u m o u t p u t h o r i z o n N2 is s h o w n t o h a v e a s t a b i h z i n g effect i f t h e p r e d i c t i o n m a r g i n is w i d e e n o u g h . T h e larger t h e v a l u e N2, t h e slower a n d m o r e d a m p e d t h e r e s p o n s e . T h e o u t p u t h o r i z o n was f o u n d t o p l a y a role i n r e d u c i n g t h e i n h e r e n t o v e r s h o o t of t h e G P C a l g o r i t h m ( F i g u r e 4.10 c o m p a r e d t o F i g u r e 4.11 ). I n t h i s w o r k , i t s v a l u e was c h a n g e d o n hne a n d r e s u l t e d i n a r e l a t i v e l y q u i c k e r r e s p o n s e  at first a n d s i g n i f i c a n t l y r e d u c e d overshoot (if at a l l ) as s h o w n i n F i g u r e 4.4 a n d F i g u r e 4.5 c o m p a r e d t o F i g u r e 4.10.  T h e sluggish response t o h i g h e r values of N2 a n d t h e o s c i l l a t o r y  n a t u r e o f i t for lower values was s h o w n i n F i g u r e 4.12 c o m p a r e d t o F i g u r e 4.11. M a n y different i n d u s t r i a l m a n i p u l a t o r s i n use t o d a y are r e p r e s e n t e d b y 2 h n k h y d r a u h c a U y a c t u a t e d m e c h a n i s m s of t h e t y p e i n t h i s w o r k . T h i s w o r k s t u d i e s t h e b e h a v i o r of t h i s c a t e g o r y of m a n i p u l a t o r w h e n c o n t r o l l e d b y a n a d a p t i v e c o n t r o l a l g o r i t h m . T h e s t u d y c a n be e x p a n d e d t o a m a n i p u l a t o r w i t h a d d i t i o n a l h n k s . H y d r a u h c a U y a c t u a t e d m a c h i n e s are h i g h l y n o n h n e a r systems a n d their parameters may vary onhne d u r i n g a w o r k i n g cycle. W h e n controUed b y G P C a g o o d p e r f o r m a n c e of t h e o u t p u t i n t r a c k i n g a sequence of set p o i n t s was a c h i e v e d . T h u s i n a d d i t i o n t o a d v a n c i n g t h e s t a t e of t h e a r t i n c e r t a i n areas of p r e d i c t i v e c o n t r o l r e l a t e d t o d e s i g n p a r a m e t e r s , t h e w o r k d e s c r i b e d i n t h i s c h a p t e r has also e x a m i n e d  some  c o m p l e x c o n s i d e r a t i o n s s u c h as t h e effect o f h a r d n o n h n e a r i t i e s i n t h e a p p h c a t i o n of G P C t o a b r o a d c a t e g o r y of h y d r a u h c a U y a c t u a t e d m a n i p u l a t o r s .  Chapter 5  MODEL ORDER DETERMINATION  5.1  Introduction  A d a p t i v e c o n t r o l a l g o r i t h m s are designed a s s u m i n g t h a t t h e p l a n t m o d e l is d e f i n e d b y a  fixed  s t r u c t u r e . A q u e s t i o n a s k e d i n t h i s w o r k is h o w w i l l a n a d a p t i v e c o n t r o l a l g o r i t h m ( t h e G P C i n t h i s case) b e h a v e w h e n t h e t r u e p l a n t is n o t p e r f e c t l y d e s c r i b e d by a n y m o d e l o f a g i v e n class. T h e b e h a v i o r of a specific a l g o r i t h m is u n d e r s t o o d t h r o u g h a n a l y z i n g s t a b i h t y a n d p e r formance.  S t a b i h t y proofs usually require restrictive assumptions,for example, a s s u m p t i o n  o n t h e s t r u c t u r e of t h e m o d e l ( n u m b e r of poles, n u m b e r of zeros, t i m e delay, e t c . ) . I n m a n y r e s e a r c h s t u d i e s , s t a b i h t y proofs for a d a p t i v e c o n t r o l l e d s y s t e m s are d o n e for e x a m p l e s w h i c h d e a l p r i m a r i l y w i t h h n e a r s y s t e m s , a n d t h e signals are b o u n d e d w i t h s m a l l p e r t u r b a t i o n s (for e x a m p l e i n A s t r o m et a l ^). I f m o d e h n g errors are s u f f i c i e n t l y s m a l l , r o b u s t s t a b i h t y o f a d a p t i v e systems can be achieved ( B a h n a s a w i a n d M a h m o u d as u n m o d e l e d or o v e r - m o d e l e d d y n a m i c s , n o n h n e a r i t i e s , etc.)  ).  M o d e h n g errors  (such  a p p e a r as d i s t u r b a n c e s i n t h e  adaptive process. T h e present r e s e a r c h deals w i t h changes i n t h e s t r u c t u r e of t h e p l a n t , a n d t h e r e f o r e c h a n g e s i n t h e m o d e l o r d e r . W h e n t h e m o d e l o r d e r is not a c c u r a t e , t h e m o d e h n g e r r o r c a n b e l a r g e a n d i n s t a b i h t i e s c a n a p p e a r . B o t h a p p h c a t i o n s u s e d i n t h i s w o r k are a n a l y z e d w i t h m o d e h n g errors. T h e G P C uses a h n e a r i z e d m o d e l of t h e s y s t e m for c o n t r o l p u r p o s e s . A n y d i v e r s i o n f r o m  t h e h n e a r f o r m is as a d i s t u r b a n c e t o t h e a d a p t a t i o n m e c h a n i s m . T h e h y d r a u h c m a n i p u l a t o r e x a m p l e is a h i g h l y n o n h n e a r s y s t e m w h i c h is r e p r e s e n t e d b y a h n e a r m o d e l for t h e G P C . T h e m o d e h n g e r r o r s act as d i s t u r b a n c e s c a u s e d p a r t l y b y t h e n o n h n e a r i t i e s . C h a p t e r 4 a n a l y z e d t h i s s y s t e m a n d g o o d p e r f o r m a n c e is a c h i e v e d . W h e n t h e o r d e r of t h e s y s t e m is c h a n g e d , f o r e x a m p l e w h e n t i m e d e l a y or v a l v e d y n a m i c s are i n t r o d u c e d , a n d t h e c o n t r o l a l g o r i t h m h a s n o t b e e n u p d a t e d w i t h t h e c h a n g e s , t h e m o d e h n g e r r o r is l a r g e a n d i n s t a b i l i t i e s o c c u r nonhnear terms remain unchanged).  T h e single  flexible  (the  h n k m a n i p u l a t o r , is m o d e l e d b y a  h n e a r m o d e l t h a t m a t c h e s t h e o r d e r of t h e one for t h e a d a p t i v e a l g o r i t h m . G o o d  performance  is s h o w n i n C h a p t e r 3. W h e n t h e o r d e r of t h e s y s t e m does not m a t c h t h e o n e r e p r e s e n t i n g it for t h e a d a p t i v e a l g o r i t h m d u e t o v i b r a t i o n m o d e s , t h e m o d e h n g  e r r o r s are l a r g e a n d  instabihties occur. T h i s c h a p t e r presents a m e t h o d for d e t e c t i n g a n d c o r r e c t i n g t h e m o d e l o r d e r a n d h e n c e m i n i m i z i n g t h e m o d e h n g e r r o r . It e v a l u a t e s a cost f u n c t i o n a n d i t s d e r i v a t i v e . I f n e c e s s a r y , t h e r e p r e s e n t e d m o d e l o r d e r for t h e a d a p t i v e a l g o r i t h m is c h a n g e d o n - h n e t o r e d u c e m o d e h n g e r r o r s a n d t h e u n c e r t a i n p a r a m e t e r s i n t h e m o d e l e s t i m a t e d as is n o r m a l l y d o n e i n a n a d a p t i v e algorithm.  5.2  Cost Function - For Detection O f T h e M o d e l Structure  T h e g o a l , i n o r d e r t o achieve g o o d p e r f o r m a n c e , is t o r e d u c e m o d e h n g e r r o r s . A s a m e a s u r e o f t h e m o d e h n g e r r o r , we choose a cost f u n c t i o n w h i c h is t h e s q u a r e o f t h e difference  between  t h e o u t p u t m e a s u r e d f r o m t h e a c t u a l s y s t e m a n d t h e one o f t h e h n e a r m o d e l as u s e d b y t h e a d a p t i v e c o n t r o l a l g o r i t h m i.e.  t  J{ymea.,yc»t)  = Y^iymea.  -  Y^t]^  A cost f u n c t i o n is u s u a l l y chosen as a c r i t e r i o n t o be m i n i m i z e d a c c o r d i n g t o t h e  (5-1)  final  target.  F o r t h e e s t i m a t i o n p r o b l e m , for e x a m p l e , t h e process o u t p u t a n d t h e e s t i m a t e d  m o d e l o u t p u t are c o m p a r e d a n d some o p t i m a l a d j u s t m e n t between t h e t w o s h o u l d b e f o u n d . T h e o p t i m u m is d e f i n e d u s i n g a c r i t e r i o n w i t h respect t o o u t p u t signals or t o t h e  expected  e r r o r of t h e e s t i m a t e d p a r a m e t e r s values. F o r c o n t r o l p u r p o s e s t h e c r i t e r i o n is o p t i m i z e d i n o r d e r to achieve a d e s i r e d c o n t r o l l a w . T h e c r i t e r i o n c a n be a q u a d r a t i c f o r m w h i c h , for e x a m p l e , c o u l d be a f u n c t i o n o f t h e s t a t e v e c t o r a n d t h e i n p u t s i g n a l (see A s t r o m a n d W i t t e n m a r k  L j u n g ^ a n d E y k h o f f ^^).  In  C h a p t e r 3 t h e b e h a v i o r of t h e cost f u n c t i o n of t h e G P C a l g o r i t h m ( E q u a t i o n 2.7) is c o m p a r e d w i t h t h e one p r e s e n t e d i n E q u a t i o n 5.1 a n d was f o u n d t o be s i m i l a r . T h e b e h a v i o r of t h e cost f u n c t i o n i n E q u a t i o n 5.1 was s t u d i e d w i t h o p e n l o o p a n d c l o s e d l o o p c o n t r o l i n o r d e r t o d e t e r m i n e a m e t h o d of d e t e c t i n g a n o n - h n e c h a n g e i n t h e a c t u a l s y s t e m ' s m o d e l a n d of c h a n g i n g t h e m o d e l for t h e c o n t r o l a l g o r i t h m a c c o r d i n g l y .  T h e cost  f u n c t i o n b e h a v i o r was s t u d i e d i n t w o w a y s , first as a f u n c t i o n of o r d e r changes ( F i g u r e s and  5.2 ) i.e.  5.1  t h e v a l u e of t h e cost f u n c t i o n was r e c o r d e d at a c e r t a i n t i m e as a f u n c t i o n  of different e s t i m a t e d m o d e l o r d e r s ( t h e p l a n t ' s m o d e l r e m a i n e d u n c h a n g e d ) a n d s e c o n d , as a f u n c t i o n of t i m e .  I n each r u n t h e p l a n t m o d e l a n d o r d e r of t h e e s t i m a t e d one r e m a i n  u n c h a n g e d for t h e d e s i r e d p e r i o d of t i m e . T o e x a m i n e t h e o p e n l o o p b e h a v i o r of t h e cost f u n c t i o n as a f u n c t i o n o f t h e o r d e r , t h e e s t i m a t e d m o d e l o r d e r was c h a n g e d w h i l e t h e a c t u a l s y s t e m s t r u c t u r e r e m a i n e d u n c h a n g e d . F o r e x a m p l e , for t h e f l e x i b l e h n k (as d e s c r i b e d i n C h a p t e r 3 ), t h e a c t u a l s y s t e m h a d t w o v i b r a t i o n m o d e s ( p l a n t m o d e l o r d e r of 6), b u t t h e e s t i m a t e d m o d e l o r d e r was c h a n g e d ( f r o m 2 w h i c h is a r i g i d b o d y , t o 10 w h i c h is four v i b r a t i o n m o d e s ) . F i g u r e 5.1 shows t h e cost f u n c t i o n b e h a v i o r as a f u n c t i o n of t h e m o d e l o r d e r . T h e values at e a c h o r d e r are r e l a t i v e l y s m a l l , a n d i t is h a r d t o d i f f e r e n t i a t e b e t w e e n t h e m . O n t h e o t h e r h a n d . F i g u r e 5.2 shows t h e b e h a v i o r o f t h e s a m e cost f u n c t i o n w i t h t h e s a m e m o d e l o r d e r changes t o a closed l o o p s i t u a t i o n , w i t h t h e G P C a l g o r i t h m . T h i s t i m e , t h e r e s u l t s for each chosen o r d e r of t h e e s t i m a t e d m o d e l differ  F i g u r e 5.1: C o s t f u n c t i o n b e h a v i o r for o p e n l o o p  flexible  hnk  F i g u r e 5.2: C o s t f u n c t i o n b e h a v i o r for closed l o o p flexible h n k  e x t e n s i v e l y , a n d t h e cost f u n c t i o n c a n i n d i c a t e t h e size o f t h e m o d e h n g e r r o r . A p p e n d i x A shows the results of e x p e r i m e n t a l d a t a o b t a i n e d for t h e i d e n t i f i c a t i o n o f t h e d y n a m i c m o d e l of a C a t e r p i l l a r 2 1 5 B e x c a v a t o r , w h i c h is a t w o h n k m a n i p u l a t o r a c t u a t e d b y h y d r a u h c a c t u a t o r s . T h e i d e n t i f i c a t i o n d o n e was a n o p e n l o o p one i n w h i c h n o a d a p t i v e 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 . T h e r e s u l t s i n A p p e n d i x A show t h a t after t h e cost f u n c t i o n b e c o m e s flat at h i g h orders i t is easy t o m i s - c h o o s e t h e o r d e r for t h e s y s t e m ' s m o d e l , Figures A . 8 and  (see  A.9 )  O b s e r v i n g t h e t i m e b e h a v i o r of t h e cost f u n c t i o n reveals t h r e e p a r a m e t e r s t h a t c a n b e u s e d for d e t e c t i n g m i s - m o d e h n g . T h e p a r a m e t e r s are: t h e R i s e S l o p e J r , t h e Z e r o S l o p e Jz,  and  C h a n g e T i m e T^. F i g u r e 5.3 also shows a b e h a v i o r f o r m a t of t h e cost f u n c t i o n ( C . F . ) ( w h i c h are b a c k e d u p , l a t e r i n t h e w o r k , w i t h figures s h o w i n g t h e a c t u a l b e h a v i o r o f t h e C . F . ) for s e v e r a l c o n f i g u r a t i o n s , for t h e t w o a p p h c a t i o n s , w h e r e t h e t h r e e p a r a m e t e r s are s h o w n . I n a l l of t h e c o n f i g u r a t i o n s s h o w n i n F i g u r e 5.3 t h e m o d e l of t h e a c t u a l s y s t e m m a t c h e s t h e o r d e r of t h e e s t i m a t e d m o d e l b y t h e G P C . O n e a n d t w o m o d e flexible h n k s , a n d a h n e a r a n d n o n h n e a r m o d e l of t h e h y d r a u h c a U y a c t u a t e d m a n i p u l a t o r a l l have t h e s a m e n a t u r e o f b e h a v i o r , i . e . a l l t h r e e p a r a m e t e r s , for h n e a r i z e d m o d e l s of t h e s y s t e m s , have a s i m i l a r b e h a v i o r . T h e R i s e S l o p e JR has t h e o r d e r of m a g n i t u d e of 1 0 " ^ [deg/sec or c m / s e c d e p e n d i n g o n t h e s y s t e m ) , t h e C h a n g e T i m e Tc, f r o m 1.5 [sec]  to 5.4 [sec]  depending on the system's nature a n d the  c o n t r o l p a r a m e t e r s . T h e Z e r o S l o p e s t a b i h z e s o n different values w i t h a n o r d e r o f m a g n i t u d e o f 1 0 " ^ [deg. or c m . . T h e v a l u e s of t h e p a r a m e t e r s do not n e c e s s a r i l y h a v e t h e same o r d e r of m a g n i t u d e , as i t aU d e p e n d s o n t h e n a t u r e of t h e s y s t e m b e i n g c o n t r o U e d . H o w e v e r t h e n a t u r e o f t h e b e h a v i o r is t h e s a m e . T h i s is i m p o r t a n t , since regions c a n be d e f i n e d for e a c h of t h e p a r a m e t e r s for a s p e c i f i c s y s t e m so t h a t d e t e c t i o n of m i s - m o d e h n g c a n be a c h i e v e d . It w i U be s h o w n b e l o w t h a t t h e b e h a v i o r o f t h e C . F . is different w h e n a m i s - m o d e h n g o c c u r s . F i g u r e 5.3 also shows t h a t t h e b e h a v i o r o f t h e C . F . for a n o n h n e a r i z e d m o d e l of t h e h y d r a u h c a c t u a t e d m a n i p u l a t o r .  Flexible Link  F i g u r e 5.3: S c h e m a t i c d e s c r i p t i o n o f the C F . b e h a v i o r for t h e different a p p h c a t i o n s  W h e n t h e s y s t e m is u n d e r or o v e r - m o d e l e d ,  t h e b e h a v i o r of t h e cost f u n c t i o n  e x t e n s i v e l y i n some cases, a n d m o d e r a t e l y i n o t h e r s .  S i n c e those changes c a n b e  changes detected,  t h e a s s u m p t i o n s a b o u t t h e s t r u c t u r e of t h e e s t i m a t e d m o d e l w i l l be u p d a t e d a n d t h e r e s u l t s improved.  5.2.1  T h e C o s t F u n c t i o n for t h e F l e x i b l e L i n k M a n i p u l a t o r  C o s t F u n c t i o n of the F l e x i b l e T w o M o d e  Link  W h e n a f l e x i b l e two m o d e h n k ( o r d e r 6 ) is e s t i m a t e d b y a t w o m o d e h n k , t h e e r r o r is s m a l l a n d g o o d c o n t r o l is a c h i e v e d , as m e n t i o n e d i n C h a p t e r  3.  modehng  F i g u r e 5.4  shows  t h e b e h a v i o r o f t h e cost f u n c t i o n , w h i c h has l o w values i n t h e o r d e r of m a g n i t u d e of 10~^°, t h e C h a n g e T i m e is T^ = l.5[sec.]  t o a z e r o slope, w h i c h m e a n s t h a t for t > Tc, t h e e r r o r h a s  v e r y s m a l l values ( F i g u r e 5.5). M i s - m o d e h n g c a n be classified i n t o t w o categories:  under-modehng and  over-modehng.  W h e n t h i s s y s t e m is u n d e r - m o d e l e d w i t h a one m o d e e s t i m a t e d m o d e l ( o r d e r 4 ) , i n s t a b i h t i e s o c c u r , since t h e c o n t r o l a l g o r i t h m does not a c c o u n t for t h e u n m o d e l e d d y n a m i c s a n d c a n n o t o v e r c o m e i t as a d i s t u r b a n c e .  F i g u r e 5.6 presents t h e b e h a v i o r of t h e cost f u n c t i o n i n t h i s  case, s h o w i n g t h a t i t s values rise v e r y h i g h e v e n before t h e C h a n g e T i m e (1.5 s e c ) ; t h u s t h e m i s - m o d e h n g c a n be d e t e c t e d a n d c h a n g e d o n - h n e . F i g u r e 5.7 p r o d u c e s v e r y s i m i l a r r e s u l t s for u n d e r - m o d e h n g o f t h e e s t i m a t e d m o d e l w i t h t h e o r d e r of 2. W h e n t h e s y s t e m is o v e r - m o d e l e d , t h e r e a c t i o n t o t h e m i s - m o d e h n g is m o r e  moderate.  W h e n t h e e s t i m a t e d m o d e l is a 3 m o d e one (order 8), t h e cost f u n c t i o n a n d t h e o t h e r p a r a m e t e r s b e h a v e as i f t h e r e is no m i s - m o d e H n g ( F i g u r e 5.8 ). W h e n a 4 m o d e m o d e l  (order  10) is i n t r o d u c e d , t h e m o d e h n g e r r o r is l a r g e r , a n d t h e cost f u n c t i o n v a l u e rises b e y o n d  the  d e s i r e d v a l u e ( o r d e r of m a g n i t u d e goes t o 1 0 " * i n s t e a d of 1 0 ' ^ ° ); after Tc i t keeps o n r i s i n g a n d does not achieve t h e z e r o slope. ( F i g u r e 5.9).  *  10^  flex 2 mode link -  2 mode estimator -  4 6 TIME [ s e c ]  GPC control  8  10  *  "10*10  flex 2 mode link -  2 mode estimator -  GPC control  2.0  o  Q. -4-'  1.5  H  1.0 H  o o D  E  0.5 H  '•4-'  CO  0.0  0  4 6 TIME f s e c .  8  10  F i g u r e 5.6: C o s t f u n c t i o n b e h a v i o r for 2 m o d e h n k a n d 1 m o d e e s t i m a t e d m o d e l w i t h l o g a r i t h m i c axis  F i g u r e 5.7: C o s t f u n c t i o n b e h a v i o r for 2 m o d e h n k a n d 0 m o d e e s t i m a t e d m o d e l w i t h r i t h m i c axis  * 10"'*'  flex 2 mode link -  3 mode estimator -  GPC control  T h e c o n c l u s i o n so far is t h a t u n m o d e l e d d y n a m i c s affect t h e p e r f o r m a n c e a n d s t a b i h t y o f t h e s y s t e m faster a n d i n a m o r e i n t e n s i v e m a n n e r t h a n t h e over- m o d e h n g does. T h i s f a c t w i U h e l p t h e on-Hne m o d e l s t r u c t u r e d e t e c t i o n differentiate b e t w e e n u n d e r a n d o v e r - m o d e H n g . T h e s e c o n c l u s i o n s w i U be b a c k e d u p b y f u r t h e r r e s u l t s .  Flexible One M o d e  Link  T h e one m o d e Hnk (order 4 ) , w h e n e s t i m a t e d w i t h a one m o d e e s t i m a t e d m o d e l , p r o d u c e s  a  c o n t r o U e d s y s t e m w i t h g o o d p e r f o r m a n c e . F i g u r e 5.10 shows t h e b e h a v i o r of t h e cost f u n c t i o n for these c o n d i t i o n s .  T h e C . F . ( Hke t h e one i n F i g u r e 5.4) has a z e r o s l o p e a n d s t a b i h z e s  at t h e o r d e r of m a g n i t u d e of 10~® a n d  = 2sec. . T h e b e h a v i o r of t h e t w o cost f u n c t i o n s  is t h e s a m e b u t t h e values are different. H e r e t o o , the m i s - m o d e H n g is a d d r e s s e d b y t h e t w o categories: u n d e r - m o d e l i n g a n d o v e r - m o d e H n g . U n d e r - m o d e h n g , as i n F i g u r e 5.11, causes t h e u n s t a b l e response i f a n e s t i m a t e d m o d e l of a r i g i d b o d y is chosen. It p r o d u c e s a n o s c i U a t i o n f r e q u e n c y o f a b o u t li.bHz.,  t h e f r e q u e n c y of t h e first m o d e n o t a c c o u n t e d for b y t h e c o n t r o l  a l g o r i t h m (see T a b l e B . l ) . F i g u r e 5.12 shows t h a t t h e cost f u n c t i o n grows, a n d at Tc = 2, i t s v a l u e is a p p r o x i m a t e l y 50, w h e r e a s i n F i g u r e 5.10 i t was at t h e o r d e r of m a g n i t u d e of 1 0 " ^ . T h i s is a c l e a r i n d i c a t i o n t h a t t h e m o d e l chosen was n o t t h e r i g h t one. T h e s e r e s u l t s m a t c h t h e b e h a v i o r of t h e 2 m o d e s y s t e m for u n d e r - m o d e h n g as p r e s e n t e d i n F i g u r e 5.6 a n d F i g u r e 5.7. O v e r - m o d e H n g has a m o r e m o d e r a t e response. A t w o m o d e e s t i m a t e d m o d e l ( F i g u r e 5.13) behaves l i k e t h e 1 m o d e m o d e l , b u t a 3 m o d e e s t i m a t e d m o d e l ( F i g u r e 5.14) at Tc has a v a l u e o f t h e o r d e r of m a g n i t u d e of 10"^ w h e r e t h e 1 m o d e e s t i m a t e d m o d e l h a d t h e v a l u e o f 1 0 " ^ . T h i s case shows t h a t t h e C . F . c o n t i n u e s i n c r e a s i n g , a n d t h e w h o l e process b e c o m e s u n s t a b l e . T h e r a t e of a p p r o a c h i n g b a d p e r f o r m a n c e or even i n s t a b i H t y is m u c h slower t h a n t h e one for under-modehng.  A 4 m o d e e s t i m a t o r ( F i g u r e 5.15) has a s i m i l a r b e h a v i o r t o t h e p r e v i o u s  c a s e , b u t t h e cost f u n c t i o n v a l u e rises q u i c k e r . A t Tc t h e C . F . has t h e o r d e r of m a g n i t u d e  flex 1 mode link -  0.0  0.5  0 mode estimator -  1.0 TIME f s e c .  GPC control  1.5  2.0  F i g u r e 5.12: C o s t f u n c t i o n b e h a v i o r for 1 m o d e h n k a n d 0 m o d e e s t i m a t e d m o d e l l o g a r i t h m i c axis  with  o f 10 ^ a n d at < = 5sec. t h e o r d e r of m a g n i t u d e of 10 w h e r e t h e o r d e r of m a g n i t u d e f o r t h e p r e v i o u s case is 1 0 ' ^ . T h e c o n c l u s i o n t h a t c a n be d r a w n so far f r o m a n a l y z i n g t h e  flexible  h n k , is t h a t  u n d e r , over or c o r r e c t m o d e h n g , t h e cost f u n c t i o n b e h a v i o r is s i g n i f i c a n t l y d i f f e r e n t .  for  So a  m o d e l p l a n t m i s - m a t c h c o u l d be d e t e c t e d b y e x a m i n i n g t h e b e h a v i o r o f t h e cost f u n c t i o n a n d its derivative.  5.2.2  T h e C o s t F u n c t i o n for t h e T w o L i n k M a n i p u l a t o r w i t h H y d r a u l i c A c t u ators  T h e t w o h n k m a n i p u l a t o r is h i g h l y n o n H n e a r , w i d e l y u s e d i n t h e i n d u s t r y , a n d is t h e r e f o r e of i n t e r e s t i n t h i s i n v e s t i g a t i o n . C h a p t e r 4 presents a t h o r o u g h d i s c u s s i o n o f s u c h a s y s t e m c o n t r o U e d w i t h t h e G P C a l g o r i t h m w h i c h achieves g o o d p e r f o r m a n c e .  In this chapter, the  b e h a v i o r o f a cost f u n c t i o n for s u c h a s y s t e m w i U b e s t u d i e d . T h e h y d r a u h c a U y m a n i p u l a t e d r o b o t i c H n k is b a s i c a U y a t h i r d o r d e r s y s t e m , o r d e r o f 2 for t h e d y n a m i c s of t h e H n k s , a n d o r d e r of 1 for t h e h y d r a u h c s y s t e m s . T h e o r d e r of s u c h a s y s t e m c a n c h a n g e i f t h e H n k is n o t rigid, but  flexible  w i t h a n u n k n o w n n u m b e r of v i b r a t i o n m o d e s , or i f t h e h y d r a u H c s y s t e m  c o n t a i n s a t i m e d e l a y ( w h i c h w i U a d d a n o r d e r of one t o t h e basic s y s t e m ) , o r i f t h e s p o o l v a l v e d y n a m i c s i n f l u e n c e t h e process ( t h e n a n o r d e r of 2 is a d d e d t o t h e basic process b r i n g i n g i t t o o r d e r 5). I n t h i s d i s c u s s i o n t h e o r d e r change w i U b e i n t h e h y d r a u h c p a r t , w h e r e t h e r e s u l t s w i U be d i v i d e d i n t o two categories. F i r s t a h n e a r i z e d m o d e l of t h e m a c h i n e is i n t r o d u c e d a n d t h e b e h a v i o r of t h e cost f u n c t i o n s t u d i e d , a n d t h e n t h e f u U n o n H n e a r m o d e l for t h e s y s t e m is u s e d i n t h e s i m u l a t i o n a n d t h e b e h a v i o r of i t s C . F . s t u d i e d as weU. F i g u r e 5.16 shows t h e b e h a v i o r of 6i a n d 82 t o a step f u n c t i o n w h e n t h e h n e a r i z e d m o d e l of t h e s y s t e m a n d t h e m o d e l a s s u m e d for G P C m a t c h a n d are b o t h of o r d e r 3. F i g u r e 5.17 shows t h e b e h a v i o r of t h e cost f u n c t i o n a n d i t s d e r i v a t i v e for b o t h H n k s T h e s a m e p a t t e r n o f b e h a v i o r c a n be o b s e r v e d (as i n t h e flexible h n k ) . T h e R i s e S l o p e , t h e T i m e C h a n g e  Tc,  a n d t h e Z e r o S l o p e are f o u n d i n t h i s case t o o ( s c h e m a t i c d e s c r i p t i o n of these p a r a m e t e r s a n d t h e i r values are d e s c r i b e d i n F i g u r e 5.3). W h e n t h e a c t u a l n o n h n e a r s y s t e m is i n t r o d u c e d ( t h e o r d e r r e m a i n s 3), t h e b e h a v i o r o f t h e cost f u n c t i o n changes a n d so does t h e b e h a v i o r of t h e s y s t e m .  In order to m a i n t a i n  s t e a d y s t a t e values for s o m e of t h e h y d r a u h c p a r a m e t e r s , t h e s p o o l v a l v e c h a t t e r s a r o u n d i t s z e r o v a l u e . A s a r e s u l t , t h e o u t p u t e r r o r is c o n s t a n t (not z e r o ) , a n d t h e cost f u n c t i o n rises c o n s t a n t l y . F i g u r e 5.18 shows t h a t t h e C . F . behaves s i m i l a r l y t o t h e b e h a v i o r i n F i g u r e 5.17 u p t o the t i m e w h e n t h e c h a t t e r i n g b e g i n s . It c a n b e d e t e c t e d c l e a r l y o n t h e C . F . d e r i v a t i v e plot . S i n c e g o o d p e r f o r m a n c e is a c h i e v e d i n c o n t r o U i n g t h e n o n h n e a r s y s t e m w i t h G P C C h a p t e r 4 ), t h e cost f u n c t i o n i n d i c a t e s t h a t t h e o r d e r of t h e e s t i m a t e d ( h n e a r i z e d )  (see  model  m a t c h e s t h e o r d e r o f t h e a c t u a l n o n h n e a r m a c h i n e m o d e l a n d c a n be u s e d t o d e t e c t m o d e h n g e r r o r s of t h e s y s t e m . W h e n t h e o r d e r of t h e e s t i m a t e d m o d e l does not agree w i t h t h a t of t h e a c t u a l s y s t e m , i t is also e v i d e n t i n t h e cost f u n c t i o n b e h a v i o r . F i g u r e 5.19 shows a n u n d e r m o d e l e d first h n k , i n w h i c h i t s e s t i m a t e d m o d e l was o f o r d e r 2, whereas h n k 2 h a d a m a t c h i n g e s t i m a t e d m o d e l of o r d e r 3. T h e cost f u n c t i o n of L i n k 2 has a v e r y s i m i l a r b e h a v i o r t o t h e o n e seen i n F i g u r e 5.18 a n d i t s o u t p u t ( F i g u r e 5.20 ) s t a b i h z e s o n i t s set p o i n t . H o w e v e r , i t h a s a l a r g e r o v e r s h o o t d u e t o t h e c o u p h n g w i t h h n k 1, w h i c h is u n d e r - m o d e l e d b y i t s e s t i m a t e d model. O v e r - m o d e h n g , as i n t h e f l e x i b l e h n k case, reacts i n a m o r e m o d e r a t e w a y t h a n t h e u n d e r modehng.  I n F i g u r e 5.21 b o t h h n k s were over m o d e l e d w i t h a n e s t i m a t e d m o d e l of o r d e r 4.  T h e cost f u n c t i o n for b o t h h n k s g r o w s , i n d i c a t i n g t h e m i s - m a t c h b e t w e e n t h e m o d e l s .  The  d e r i v a t i v e h o w e v e r , decreases e v e n t u a l l y . T h e cost f u n c t i o n a n d i t s d e r i v a t i v e c h a n g e i n a m o d e r a t e m a n n e r c o m p a r e d w i t h t h e u n d e r - m o d e h n g case. I n o r d e r t o get a r e a h s t i c f i f t h o r d e r m o d e l for t h e s y s t e m , d y n a m i c s s h o u l d be i n t r o d u c e d t o t h e s e r v o v a l v e of t h e s y s t e m . U s u a l l y , for m o s t p r a c t i c a l p u r p o s e s , t h e s e r v o v a l v e d y n a m i c s  GPC control -  .6 H  2 link manipulator with hydraulic octuotors L_ 1 1 1 .  TIME [ s e c ]  . TO"  2 link monipulotor -  35  lineor plont -  linki -  order=3  Iink2- order«3  •10«  4 6 TIME [sec]  4 6 TIME [sec]  • XT' 40-  1  1  1  1  100-  80-  30-  /  >  <s 60>  J 2 0 -  1  j  c  3 | 1 0 -  u 0-  :5 40-  i 4 6 TIME [sec]  10  0 -  F i g u r e 5.17: C o s t f u n c t i o n b e h a v i o r for 3 m o d e h y d r a u h c h n k s ( h n e a r i z e d p l a n t m o d e l ) a n d 3 m o d e estimated models  2 link monipulotor — nonlineor plant — linki — order—3  Iink2— order=3  F i g u r e 5.18: C o s t f u n c t i o n b e h a v i o r for 3 m o d e h y d r a u h c h n k s ( n o n h n e a r p l a n t m o d e l ) a n d 3 mode estimated  models  2 link manipulator —linki— order—3 , Rnk2— order»3 10=1^  'ET  10" —I  1—  4 6 TIME [sec]  10"  10  « :  10' _,  10^:  10"°:  10-° Ê r "  10-i  —I  r—  10  4 6 TIME [sec]  i r r  10-»: 10"  b  I  :  ^10-'  \  ^  10-^ 10-^  4 6 TIME [sec]  10  10-^  —I  1—  4 6 TIME [sec]  10  GPC control — 2 link manipulator with hydroulic octuotors  —8 H 0  2  .8 H  '  I  0  I  2  1  1  1  1  6 TIME [sec]  8  '  '  <•  I  I  4  6 TIME [sec]  4  I  8  10  I  10  2 link monipulotor — linki  200-1  >  1  1  •  1  1  order=4  Iink2- order=4  1.  150-  c o  1  100-  8  •X 0  \  1  "^-i  2  ^  4 6 TIME [sec]  8  1  1  f-  10  25-1 -  140120 ^ 100-  1 1 §60o  / 80-  y  20-  0-J 0  2  /  /  1 4 6 TIME [sec]  B  10  1  ^  1  L  are fast e n o u g h t o be i g n o r e d . I n t h i s case, a n e q u i v a l e n t s e c o n d o r d e r s y s t e m w a s a d d e d t o t h e s t r u c t u r e of t h e e s t i m a t e d m o d e l . T h e i n i t i a l values for t h e e s t i m a t e d p a r a m e t e r s i n c l u d e d t h e d a t a for t h e servovalve d y n a m i c s w i t h n a t u r a l f r e q u e n c y of 20Hz  and a damping ratio  of 0.6 (see C a t a l o g , M o o g I n c . ^^). I n F i g u r e 5.22 h n k 1 has a n e s t i m a t e d m o d e l o f o r d e r 5, w h i c h grows c o n s t a n t l y d u e t o t h e e r r o r b e t w e e n t h e m o d e l s . F o r h n k 2, t h e o r d e r is 3 a n d t h e cost f u n c t i o n behaves h k e t h e one i n F i g u r e 5.18. F i g u r e 5.23 shows t h e b e h a v i o r o f t h e o u t p u t s . 9i c a n not achieve t h e g o a l of i t s set p o i n t d u e t o t h e m i s - m a t c h o f t h e m o d e l s , a n d $2 behaves w e l l since t h e m o d e l s m a t c h e a c h o t h e r . T h e c o n c l u s i o n s d r a w n f r o m t h i s s e c t i o n are s i m i l a r t o t h e ones f r o m t h e  flexible  hnk.  It is p o s s i b l e t o i d e n t i f y , t h r o u g h t h e cost f u n c t i o n a n d i t s d e r i v a t i v e , t h e case i n w h i c h the estimated model matches the actual system's model. under-modehng and over-modehng.  T h e cost f u n c t i o n also i n d i c a t e s  T h i s i n f o r m a t i o n f o r m s t h e b a s i c d a t a for t h e m e t h o d  for d e t e c t i n g o n - l i n e t h e o r d e r of a s y s t e m m o d e l a n d i t s changes.  5.3  R e a s o n s for U n d e r a n d O v e r - M o d e l e d  Behavior  M o s t a d a p t i v e c o n t r o l a l g o r i t h m s assume t h a t p l a n t d y n a m i c s c a n b e m o d e l e d b y one m e m b e r of a s p e c i f i e d class of m o d e l s .  Usually, there are uncertainties i n the estimated m o d e l  due  t o u n k n o w n b u t e s t i m a t e d p a r a m e t e r s or d i s t u r b a n c e s . T h e s e c a n b e f r o m e x t e r n a l s o u r c e s or i n t e r n a l ones s u c h as n o n h n e a r i t i e s i n t h e p l a n t d y n a m i c s w h i c h are not i n c l u d e d i n t h e estimated model.  I f t h e d i s t u r b a n c e s are b o u n d e d a n d t h e r e is sufficient e x c i t a t i o n b y t h e  i n p u t s i g n a l t o e s t i m a t e t h e m o d e l p a r a m e t e r s , t h e n t h e s y s t e m c a n be c o n t r o l l e d a n d s t a b i h t y r e t a i n e d . T h i s was d e m o n s t r a t e d i n C h a p t e r 3 b y c o n t r o U i n g a single flexible h n k m a n i p u l a t o r , a n d i n C h a p t e r 4 by c o n t r o U i n g a h y d r a u h c a U y a c t u a t e d t w o h n k m a n i p u l a t o r w h i c h is a h i g h l y n o n h n e a r s y s t e m . B o t h s y s t e m s were m o d e l e d b y a h n e a r m o d e l for c o n t r o l p u r p o s e s , w i t h p a r a m e t e r s e s t i m a t e d o n - h n e , a n d t h e n o n h n e a r i t i e s are c o n s i d e r e d t o be d i s t u r b a n c e s . B y t u n i n g the G P C parameters, acceptable F i g u r e s 4.4,  4.5,  4.6,  a n d good performance  c a n be a c h i e v e d ( see  4.7).  T h i s w o r k deals w i t h m o d e l / p l a n t m i s - m a t c h i n w h i c h t h e e s t i m a t e d m o d e l for t h e G P C a l g o r i t h m has a different s t r u c t u r e (i.e. h n e a r a n d different o r d e r ) f r o m t h a t of t h e r e a l p l a n t . S u c h m i s - m a t c h is a n o t h e r f o r m of u n c e r t a i n t y i n t h e a d a p t i v e c o n t r o U e r . T h e cost f u n c t i o n ( E q u a t i o n 5.1) a n d i t s t i m e v a r i a t i o n s were chosen as a m e a s u r e of t h a t p h e n o m e n o n .  When  t h e p l a n t a n d m o d e l m a t c h , t h e cost f u n c t i o n rises i n i t i a l l y for t h e t i m e p e r i o d t h a t i t t a k e s for t h e e s t i m a t e d m o d e l to a d j u s t , a n d t h e n s t a b i h z e s o n a close t o c o n s t a n t v a l u e , since t h e error between the models becomes very smaU. T h e b e h a v i o r of t h e cost f u n c t i o n J as a f u n c t i o n of t h e e s t i m a t e d m o d e l o r d e r ( F i g u r e 5.1, for o p e n l o o p i n v e s t i g a t i o n ) shows t h a t u n d e r - m o d e h n g , a n d o v e r - m o d e h n g are h a r d t o d e t e c t b y c o m p a r i s o n w i t h t h e correct s t r u c t u r e . O n t h e o t h e r h a n d , F i g u r e 5.2 (closed l o o p c a l c u l a t i o n s ) shows t h a t m o d e l m i s - m a t c h i n m o s t f o r m s is significant a n d c a n be d e t e c t e d .  T h e r e a l processes h a v e c o m p l e x  nonhnear  d y n a m i c s ( as i n t h e two h n k h y d r a u h c a U y a c t u a t e d m a n i p u l a t o r ) , a n d t h e a d a p t i v e c o n t r o U e r attempts to control the dynamics by a simple hnear model. T h e p a r a m e t e r s of t h e h n e a r e s t i m a t e d m o d e l d e p e n d s t r o n g l y o n t h e p r o p e r t i e s o f t h e i n p u t s i g n a l a n d i t s f r e q u e n c y c o n t e n t . P r o p e r e x c i t a t i o n is n e e d e d for g o o d e s t i m a t i o n r e s u l t s . T h e r e is self e x c i t a t i o n w h e n t h e e s t i m a t i o n is d o n e i n closed l o o p (as w h e n t h e a d a p t i v e c o n t r o U e r is u s e d ) , since t h e e s t i m a t i o n process is e x c i t e d b y t h e s i g n a l f r o m t h e f e e d b a c k .  The  f e e d b a c k c o u l d cause d e p e n d e n c i e s b e t w e e n t h e e l e m e n t s of t h e regression v e c t o r ( E q u a t i o n 5.4 w h i c h m e a n s t h a t t h e p a r a m e t e r s c a n n o t b e d e t e r m i n e d u n i q u e l y ( A s t r o m ^). E r r o r s d u e t o m o d e h n g e r r o r s arise w h e n t h e chosen m o d e l does n o t d e s c r i b e t h e s y s t e m c o m p l e t e l y , i t c a n cause p o o r p e r f o r m a n c e d e p e n d i n g o n t h e v a l u e of t h e m o d e h n g e r r o r a n d i t s n a t u r e . I n t h e f o U o w i n g m a t e r i a l , a d i s c u s s i o n o n t h e e s t i m a t i o n process R L S ( R e c u r s i v e L e a s t S q u a r e s ) , u s e d w i t h G P C (see L j u n g ^ a n d A s t r o m a n d W i t t e n m a r k ^^), a n d t h e effect o f u n d e r , a n d o v e r - m o d e l i n g is g i v e n . W h e n t h e p l a n t is h n e a r a n d i t s o r d e r is k n o w n , i t c a n be d e s c r i b e d b y t h e m a t h e m a t i c a l model:  y{t)  = ~aiy{t  -  1) - a2y{t - 2)  + bou{t - 1) + biu{t - 2) + • • •  (5.2)  or:  y{t)  = *^(t)0  (5.3)  w h e r e $ is t h e regression v e c t o r :  $^ = [ - 3 / ( f - l ) ,  -y{t-2),  a n d 0 is a vector of u n k n o w n p a r a m e t e r s :  u{t-l),  uit-2)-  (5.4)  0  =  [ai,  02,  • • •,  bo, bi.  (5.5)  T h e estimated m o d e l is:  m  = $^(0©  w h e r e 0 is a v e c t o r o f t h e e s t i m a t e d p a r a m e t e r s , a n d err(t)  err{t)  (5.6) is t h e e r r o r .  = yit)-y{t)  (5.7)  T h u s the R L S algorithm is:  0m  Pit) Pit)  è(t  u  Pit ^) - Pit - 1 ) -  a P j t - m t M t ) - m )  . . . .  - a p j t - m m ^ i m t - ^ ) ^ + AT^^t)Pit - i ) m  . . . . (^-^^  w h e r e : a ( i ) Ê [0,1] is a g a i n , 7 ( f ) > 0 is a n o r m a h z a t i o n t e r m , AT is t h e s a m p h n g p e r i o d , yit)  is t h e m e a s u r e d o u t p u t , a n d y ( i ) is t h e e s t i m a t e d o u t p u t .  Under-Modeling  W h e n u n d e r - m o d e h n g is c o n s i d e r e d t h e m e a s u r e d o u t p u t c a n be e x p r e s s e d as:  yit)  = ^''{t)e  (5.10)  + r,{t)  w h e r e T]{t) c o n t a i n s t h e u n m o d e l e d t e r m s .  r e a r r a n g i n g E q u a t i o n 5.8, E q u a t i o n 5.10 a n d E q u a t i o n 5.6 y i e l d s :  Where: 0 =  0 - 0  t h e regression v e c t o r , c o n t a i n s i n f o r m a t i o n o n p r e v i o u s o u t p u t s a n d i n p u t s t o t h e p l a n t a n d therefore i n f o r m a t i o n a b o u t t h e f e e d b a c k t o t h e c o n t r o l l e r . T h e u n m o d e l e d  dynamics,  a l t h o u g h u n k n o w n , are p a r t of t h e p l a n t ' s o u t p u t a n d of t h e f e e d b a c k s i g n a l . It c a n t h u s b e c o n c l u d e d t h a t $ a n d rj are d e p e n d e n t . E q u a t i o n 5.11 shows t h a t e v e n w h e n t h e e s t i m a t e d p a r a m e t e r s m a t c h t h e ones i n 0  the  t e r m w i t h ^i} i n i t c a n cause 0 t o d r i f t . T h i s effect c a n also be seen i n t h e e q u a t i o n s d e s c r i b i n g t h e cost f u n c t i o n . B a s i c a l l y w h e n t h e e r r o r grows t h e cost f u n c t i o n grows i n v a l u e i . e . : B a s e d on equations  5.10  5.5  5.6 a n d  5.7, t h e e r r o r i n t e r m s of t h e regression v e c t o r  the  difference b e t w e e n e s t i m a t e d a n d t r u e p a r a m e t e r s a n d t h e u n m o d e l e d d y n a m i c s i s : err{t)  = $^(f ) 0 + 7/(0  Where: 0 = [ôi,  • • •,  â„^,  6i,  • • •,  (5.12)  a n d t h e regression vector for t h i s case i s :  -y(t-n„),  = [~y{t~l), i t s d i m e n s i o n i s : dim{i^)  = 2nu, w h e r e  u(t-l),  u(t-n„)]  (5.13)  is t h e o r d e r of t h e m o d e l e d d y n a m i c s f o r t h e  u n d e r - m o d e h n g case, a n d t h e t e r m s y{t — i) a r e :  y{t-i)=.^'^(t~i)e  (5.14)  + ri(t-i) i = 1 , 2 , •••,7i„  T h e t e r m ^^(t — i)Q is d e r i v e d u s i n g e q u a t i o n 5.5:  #^(f - z ) 0 =  -a^y[t - j) + bAt  - j) = ny,n)  (5.15)  T h e cost f u n c t i o n is defined as: t (5.16)  J^Y^err' B a s e d o n E q u a t i o n 5.12 i t foUows t h a t : evr\t) = ( # ^ ( i ) 0 ) ' + 2 * ^ ( f ) 0 ï / ( i ) +  (5-17)  T h e s e c o n d t e r m o n t h e right h a n d side of E q u a t i o n 5.19 c o n t a i n s t h e r e g r e s s i o n v e c t o r $ , a n d t h e u n m o d e l e d d y n a m i c s 77, w h i c h as c a n b e c o n c l u d e d f r o m E q u a t i o n s are d e p e n d e n t .  5.14 a n d 5 . 1 5 ,  A s c a n be seen f r o m E q u a t i o n 5.17 t h e r e are t w o c o n t r i b u t i o n s t o t h e e r r o r ,  t h e m o d e h n g reflected i n t h e u n m o d e l e d d y n a m i c s t e r m , a n d t h e e s t i m a t i o n w h i c h is r e f l e c t e d i n t h e p a r a m e t e r s . B y e x p r e s s i n g t h e regression v e c t o r $ t e r m s , w i t h E q u a t i o n 5.15 t h e t e r m $ ^ ( i ) 0 is c a l c u l a t e d as:  * ' ' ( 0 ê = T^l-àMt  - j) + *(2/. « ) ) ] + h<t  - j)  (5.18)  S u b s t i t u t i n g E q u a t i o n s 5.17 a n d 5.18 i n t o E q u a t i o n  5.16 t h e cost f u n c t i o n is d e s c r i b e d  by: J=è(*''(fc)0f k=i + 2 Ehàj j=i  The unmodeled plant's output.  - j) - àj è r,{k)ny, k=i  X: vikHk fc=i  dynamics  ^) + h É n{k)u{k k=i  j)]  is a p h y s i c a l s i g n a l a n d t h o u g h u n k n o w n i t is p a r t o f t h e  T h e s e c o n d t e r m i n t h e r i g h t h a n d side o f E q u a t i o n  s t r o n g for u n d e r - m o d e h n g  (5.19)  5.17 ^^{k)Q,  will be  b e c a u s e of 0 a n d because o f t h e c o r r e l a t i o n b e t w e e n t h e r e g r e s s i o n  vector $ a n d the unmodeled  d y n a m i c s rj. W h e n e v a l u a t i n g t h e c o r r e l a t i o n b e t w e e n m e a -  s u r e m e n t s o f p a i r s of v a r i a b l e s , t h e c o r r e l a t i o n is d e t e r m i n i n g w h e t h e r t h e r e e x i s t s a p h y s i c a l r e l a t i o n s h i p b e t w e e n t h e t w o , or w h e t h e r t h e v a r i a t i o n s i n t h e o b s e r v e d values o f o n e q u a n t i t y are c o r r e l a t e d w i t h t h e v a r i a t i o n s i n t h e m e a s u r e d values o f t h e o t h e r . I n P r e s s ^2 t h e d i s c r e t e c o r r e l a t i o n o f t w o s a m p l e d f a n c t i o n s is defined b y :  Corrig,h)^  = J29U  + mk)  (5.20)  W h e n g a n d h are t h e s a m e f u n c t i o n t h e above is t h e a u t o c o r r e l a t i o n o f t h e s i g n a l . T h e c o r r e l a t i o n w i l l b e large at s o m e value of k i f t h e first f u n c t i o n g is a close c o p y o f t h e s e c o n d h b u t lags i t b y k . I n E q u a t i o n  5.19 s e v e r a l t e r m s are s u m m e d w i t h r e s p e c t t o t i m e .  r), y a n d u a r e r e a l p h y s i c a l signals t h e r e are t e r m s of a u t o c o r r e l a t i o n a n d c o r r e l a t i o n . t e r m s d o n o t exist i n t h e over m o d e h n g modehng. in time.  Since These  case as w i U b e s h o w n i n t h e d i s c u s s i o n o n o v e r -  T h e t h i r d t e r m i n t h e r i g h t h a n d side is a n a u t o c o r r e l a t i o n o f t w o rj signals s h i f t e d T h e fourth a n d the fifth terms i n E q u a t i o n  5.19, are c o r r e l a t i o n t e r m s b e t w e e n  rj a n d t h e o u t p u t y or 7/ a n d t h e i n p u t u . I n t h e u n d e r - m o d e h n g  case 7/ is a p a r t o f t h e  o u t p u t y a n d since there is feedback o f y i t is c o r r e l a t e d w i t h t h e i n p u t u a n d t h e r e f o r t h e  u n m o d e l e d d y n a m i c s 7/ as w e l l .  I n t h e u n d e r - m o d e h n g case, t h e c o r r e l a t i o n b e t w e e n  the  different v a r i a b l e s i n E q u a t i o n  5.19, is t h e reason for t h e r a p i d rise i n t h e cost f u n c t i o n ' s  values as was s h o w n i n S e c t i o n  5.2 for b o t h a p p h c a t i o n s .  Over-Modeling  W h e n o v e r - m o d e h n g is c o n c e r n e d y a n d y are:  y{t)  (5.21)  = $^(00  and:  m  (5.22)  = ^''m+7,(1)  T h e regression v e c t o r for t h e over m o d e h n g case i s :  = [-2/(^-1),  its d i m e n s i o n is: dimirjit))  -yit-2),  -yit-n),  (/im($-^(i)) =  uit-1),  uit-2),  uit-n)]  (5.23)  2 n , w h e r e n is t h e c o r r e c t o r d e r of t h e s y s t e m , a n d  = Uo ^ n, w h e r e Uo is t h e o v e r - m o d e l e d m o d e l o r d e r . T h e e r r o r is t h e n :  errit)  = $^(^0  (5-24)  - vit)  and :  err\t)  = ( ^ ' ^ ( i ) © ) ' " 2$^(O0t,(O + v'it)  *^(O0  = E  -àjyit - 0 + bAt  - j)  (5.25)  (5-26)  S u b s t i t u t i n g E q u a t i o n 5.26 i n t o E q u a t i o n 5.25 a n d i n t o E q u a t i o n 5.16 r e s u l t s i n t h e c o s t f u n c t i o n for  over-modehng:  J = E(*'^(^)êr  - 2 E [ - â , y: j=l  v{k)yik  - j)+bj  fc=l  vikHk  y:  -  (5.27)  j)]  *! = 1  +  i:vik)r,ik) k=l  I n t h i s case, r) c o n t a i n s a l l t h e e x t r a t e r m s of t h e e s t i m a t e d m o d e l . T h e s e d y n a m i c s a r e j u s t i n t h e e s t i m a t e d m o d e l a n d not i n t h e r e a l s y s t e m . T h i s m e a n s t h a t t h e r e is n o c o r r e l a t i o n b e t w e e n t h e regression v e c t o r $ a n d t h e e x t r a t e r m s T/, i.e. b o t h v a r i a b l e s are i n d e p e n d e n t . T h e correlation and autocorrelation terms i n E q u a t i o n  5.19 d o not exist i n e q u a t i o n  5.27,  a n d r]{k) i n t h e t h i r d a n d f o u r t h t e r m of t h e r i g h t h a n d side acts as a t i m e v a r y i n g coefficient. 7) t h e r e f o r e influences t h e c o n t r o l p a r a m e t e r s w h i c h i n f l u e n c e t h e i n p u t t o t h e p r o c e s s  (u),  b u t n o t t h e feedback of t h e c o n t r o l l e d s y s t e m . C l o s e d L o o p P o l e s for U n d e r , O v e r a n d  Correct-Modeling  T h e change i n t h e c o n t r o l l e r p a r a m e t e r s is a change i n t h e c o n t r o l l e r d y n a m i c s w h i c h d e t e r m i n e t h e l o c a t i o n of t h e closed l o o p poles.  T h e m i n i m i z a t i o n of t h e cost f u n c t i o n for  t h e e s t i m a t i o n w i l l d e t e r m i n e t h e p a r a m e t e r s u n i q u e l y , o n l y w h e n t h e m o d e l o r d e r is c o r r e c t . W h e n t h e m o d e l is over p a r a m e t r i z e d i t c a n r e s u l t i n a n y one of s e v e r a l s o l u t i o n s , a n d t h e c o r r e c t p a r a m e t e r s c a n n o t be d e t e r m i n e d . T h e closed l o o p poles show i n s o m e cases u n s t a b l e m o d e s i n d i c a t i n g t h a t t h e excess m o d e l d y n a m i c s a d d poles w h i c h are close t o t h e d o m i n a n t p o l e s of t h e s y s t e m d r i v i n g t h e e r r o r i n t o t h e h i g h e r values.  T h i s can result eventually i n  p o o r p e r f o r m a n c e or i n i n s t a b i h t y ( e s p e c i a l l y i f t h e r e is not e n o u g h e x c i t a t i o n i n t h e p r o c e s s ) . T h e r e a c t i o n for t h e o v e r - m o d e l e d d y n a m i c s is not as e x t r e m e as t o t h e u n d e r - m o d e l e d o n e s . N e x t we discuss some e x a m p l e s , f r o m t h e flexible h n k a p p h c a t i o n , for t h e b e h a v i o r o f t h e  closed l o o p poles w h i c h s h o w t h e u n s t a b l e m o d e s of t h e c o n t r o l l e d s y s t e m w h e n m i s - m o d e h n g poses a p r o b l e m . LatorneU  T h e c a l c u l a t i o n s of t h e closed l o o p p o l e d are b a s e d o n C l a r k e ^ a n d o n  . F i r s t t h e closed l o o p poles for c o r r e c t m o d e h n g :  c o r r e c t - m o d e l i n g - o r d e r 6 for p l a n t a n d m o d e l  pl  = +0.9871 + 0.0140J  p2 = + 0 . 3 9 5 5 + 0.4982J p3 = - 0 . 6 1 9 2 + 0.7582J p4 = +0.0360 + O.OOOOj p5 — - 0 . 6 1 9 2 -  0.7582J  p6 = + 0 . 3 9 5 5 -  0.4982J  p7 = + 0 . 9 8 7 1 -  0.0140i  A U poles for t h e correct m o d e h n g of a t w o m o d e flexible h n k ( o r d e r 6 ) are w i t h i n t h e u n i t c i r c l e i n d i c a t i n g a s t a b l e s y s t e m . T h e s y s t e m b e h a v i o r is p r e s e n t e d i n F i g u r e 3.2 a n d t h e cost f u n c t i o n i n F i g u r e 5.4. c o r r e c t - m o d e l i n g - o r d e r 4 for p l a n t a n d m o d e l  pi = +0.9869 + 0.0132J p2 = + 0 . 5 7 2 0 + 0.7359J p3 = + 0 . 0 3 3 7 + O.OOOOj p4 -  +0.5720 -  0.7359J  p5 = +0.9869 -  0.0132J  W h e n one m o d e o c c u r s for t h e flexible h n k a n d t h e s y s t e m is c o r r e c t l y m o d e l e d t h e G P C c o n r t o l achieves g o o d r e s u l t s a n d t h e cost f u n c t i o n behaves as p r e s e n t e d i n F i g u r e 5.10.  The  closed l o o p poles as s h o w n a b o v e are a l l i n t h e s t a b l e r e g i o n w i t h i n t h e u n i t c i r c l e . o v e r - m o d e l i n g - o r d e r 6 for p l a n t a n d o r d e r 8 for m o d e l  pl  = +0.9987 + 0.0299J  p2 = + 0 . 6 2 9 5 + O.OOOOj p3 = + 0 . 5 0 4 5 + 0.6561J pi  = - 0 . 1 5 6 7 + 0.8209i  p5 — + 0 . 3 9 1 2 + 0.5150J p6 = - 0 . 6 1 9 2 + 0 . 7 5 8 2 i p7 = - 0 . 6 9 1 0 + 0 . 2 5 4 9 i p8 = - 0 . 6 9 1 0 -  0.2549J  p9 = - 0 . 6 1 9 2 -  0.7582i  p l O = +0.3912 -  0.5150i  pll  = -0.1567 -  0.8209i  p l 2 = +0.5045 -  0.6561J  p l 3 = +0.9987 + 0.0299J A s p r e v i o u s l y m e n t i o n e d w h e n a s y s t e m is over p a r a m e t r i z e d t h e r e is no u n i q u e s o l u t i o n to the identification process.  If t h e excess d y n a m i c s a d d closed l o o p poles t h a t are close t o  t h e d o m i n a n t ones it c o u l d d r i v e t h e c o n t r o l l e d s y s t e m i n t o i n s t a b i l i t i e s . I n t h i s case a l l c l o s e d l o o p poles are s t a b l e a n d i t was s h o w n t h a t g o o d p e r f o r m a n c e was a c h i e v e d . H o w e v e r p6 a n d  p9 are close to t h e c i r c l e at a r a d i u s of 0.97889 a n d pi a n d p l 3 are at t h e r a d i u s o f 0 . 9 9 1 5 . A n y s m a l l change i n t h e s y s t e m s p a r a m e t e r s o r e v e n i n t h e c o n t r o l p a r a m e t e r s ( w h i c h a r e chosen for correct m o d e h n g of t h e best s y s t e m k n o w n t o t h e designer) c o u l d d r i v e t h e s y s t e m to instabihty. o v e r - m o d e l i n g - o r d e r 4 for p l a n t a n d o r d e r 10 for m o d e l  pl  = +1.2989 + 0 . 3 9 5 4 ;  p2 = +0.8569 + 0 . 7 8 1 5 ; p3 = +0.2784 + 0 . 9 5 7 3 ; p4 = - 1 . 0 2 0 4 + 0 . 7 5 9 2 ; p5 = - 0 . 0 3 4 8 + 0 . 0 0 0 0 ; p6 = - 1 . 0 2 0 4 -  0.7592;  p7 = +0.2784 -  0.9573;  p8 = +0.8569 -  0.7815;  p9 = + 1.2989 -  0.3954;  T h i s o v e r - m o d e h g case is one w i t h s i x u n s t a b l e m o d e s : pi a n d p9 at a r a d i u s o f 1.3577, p2 a n d p8 at 1.1597 a n d p6 a n d p4 at 1.2718, t h e s y s t e m is u n s t a b l e s as t h e cost f u n c t i o n i n d i c a t e s F i g u r e 5.15. u n d e r - m o d e l i n g - o r d e r 6 for p l a n t a n d o r d e r 4 for m o d e l  p i = +1.2795 + 0 . 0 0 0 0 ;  p2 = +0.7764 + 0 . 4 1 9 1 ;  p3 = +0.2420 + 0.6053J p4 = - 0 . 5 9 5 6 + 1.3359i p5 = +0.3840 + 0.5073J pQ = - 0 . 7 7 4 0 + O.OOOOi p7 = - 0 . 3 5 1 1 + O.OOOOj p8 = - 0 . 5 9 5 6 -  1.3359i  p9 = +0.2420 -  0.6053J  p l O = +0.7764 -  0.4191J  pU  = + 1 . 2 7 9 5 - O.OOOOj  W h e n u n d e r - m o d e h m g o c c u r s t h e r e a c t i o n o f t h e cost f u n c t i o n was m o r e r a p i d a n d t h e s y s t e m b e c a m e u n s t a b l e faster t h a n t h e u n d e r - m o d e h n g case.  T h e unstable modes are: p i  at r a d i u s of 1.2795, a n d p4,p8 at 1.3002. T h e o v e r - m o d e h n g of t h e t w o m o d e m a n i p u l a t o r p r e s e n t e d a b o v e was s t a b l e , i n t h i s case t h e u n s t a b l e poles are q u i t e f a r i n t h e u n s t a b l e r e g i o n . u n d e r - m o d e l i n g - o r d e r 4 for p l a n t a n d o r d e r 2 for m o d e l  p i = +1.8581 + O.OOOOj  p2 = +0.6089 + 1.3839J p 3 = +0.7904 + 0.8734J p4 = - 1 . 1 6 4 0 + 0.0ÛÛÛJ p 5 = - 0.2321 + O.OOOOj p6 = +0.6089 -  1.3839J  p7 = + 0 . 7 9 0 4 - Û.8734J I n t h i s u n d e r - m o d e l e d case t h e u n s t a b l e m o d e s a r e : pi at r a d i u s of 1.8581, p2 a n d p6 a t 1.5119, p4 at 1.164, a n d p3 a n d p7 at r a d i u s of 1.1779. i n t h i s case t o o , t h e u n s t a b l e poles are f u r t h e r i n t h e u n s t a b l e z o n e t h a t t h e ones for t h e o v e r - m o d e h n g case (for p l a n t o r d e r 4 a n d m o d e l o r d e r 10).  5.4  M O D - Model Order Determination Algorithm  S e c t i o n 5.2 presents t h e b e h a v i o r of t h e cost f u n c t i o n J a n d i t s t i m e v a r i a t i o n s for apphcations, the  flexible  both  h n k m a n i p u l a t o r a n d t h e h y d r a u h c a U y a c t u a t e d m a n i p u l a t o r , for  u n d e r , over a n d correct m o d e h n g .  T h e cost f u n c t i o n is a m e a s u r e of t h e a c c u m u l a t e d e r r o r  b e t w e e n t h e p l a n t a n d t h e m o d e l d y n a m i c s . T h e difference b e t w e e n u n d e r a n d o v e r - m o d e h n g is c l e a r i n t h e b e h a v i o r of J i t s t i m e d e r i v a t i v e s as d i s c u s s e d i n S e c t i o n 5.2.  T h i s section  p r e s e n t s a n a l g o r i t h m t o detect m i s - m a t c h b e t w e e n p l a n t a n d m o d e l , b a s e d o n t h e r e s u l t s a b o v e , a n d to correct t h e o r d e r .  It s h o u l d b e n o t e d t h a t c o r r e c t i n g m i s - m o d e h n g is n o t a  t a r g e t i n i t s e l f , b u t r a t h e r , is t o detect a p o s s i b l e r o u t e t o i n s t a b i h t y a n d p o o r p e r f o r m a n c e . T h u s , i f a n o v e r - m o d e l e d s y s t e m is weU c o n t r o U e d , t h e r e is n o r e a s o n t o i n t e r f e r e . T h e g o a l o f t h e m e t h o d p r e s e n t e d is t o detect p r o b l e m a t i c m i s - m a t c h cases, t o i d e n t i f y t h e i r n a t u r e , a n d t o c o r r e c t t h e m regardless of t h e i r cause. A s m e n t i o n e d i n C h a p t e r 2 ( F i g u r e 2.1), t h e a d a p t i v e s y s t e m c o n t a i n s t w o l o o p s ; one is a n o r d i n a r y feedback l o o p , a n d t h e s e c o n d l o o p identifies t h e e s t i m a t e d m o d e l p a r a m e t e r s a n d u p d a t e s t h e p a r a m e t e r s of t h e c o n t r o U e r . F i g u r e 5.24 shows a n a d a p t i v e s y s t e m b l o c k d i a g r a m w i t h a m o d e l d e t e r m i n a t i o n b l o c k w h i c h is a n a d d i t i o n t o t h e t w o l o o p s m e n t i o n e d I n t h e p r o c e d u r e a feedback l o o p is a d d e d t o t h e i d e n t i f i c a t i o n l o o p .  above.  T h i s loop calculates  the error between the measured and estimated outputs and m i n i m i z e s it w i t h the a l g o r i t h m g i v e n b e l o w . F i g u r e 5.25 shows t h e b l o c k d i a g r a m of t h e o r d e r d e t e r m i n a t i o n m e t h o d .  1  1 A 1  RLS  y  N ,  (.order)  Calculations  1 1  MOD Algorithm y  Design Contoller parameters  Adaptive controller  u  F i g u r e 5.24: F l o w c h a r t of a n a d a p t i v e c o n t r o l l o o p w i t h m o d e l o r d e r d e t e r m i n a t i o n  Initialization: 1. G u e s s i n i t i a l s t r u c t u r e (order). 2 . D e f i n e r e g i o n s of v a l u e s for c o s t f u n c t i o n b e h a v i o r : for correct, under-and over-modeling.  Adaptive Loop C h e c k v a l u e s of C . F . t tc J correct structure  Parameter Estimation  mis-modeling  C h e c k U n d e r - or Over- modeling  I i  C h a n g e structure of m o d e l  i  F i g u r e 5.25: F l o w c h a r t of t h e o r d e r d e t e r m i n a t i o n p r o c e d u r e  T h e o r d e r change is c a l c u l a t e d w i t h t h e M O D a l g o r i t h m w h i c h is p r e s e n t e d n e x t :  = i V ( f - 1) + A i V ( J ,  j,  t„a.r.  Tu,  Tc,  NOT)  (5.28)  W h e r e J is t h e cost f u n c t i o n w h i c h is d e s c r i b e d i n E q u a t i o n 5.1. Its d e r i v a t i v e J is t h e following:  Where:  • torder is t h e M O D ' s t i m e scale. I n t h e event of s e v e r a l o r d e r changes d u r i n g a w o r k i n g c y c l e , i n e v e r y c h a n g e tarder is set t o z e r o .  T h i s m o v e s t h e o r i g i n of t h e t i m e s c a l e  r e l a t i v e t o t h e a b s o l u t e t i m e t, a n d enables t h e t i m e p a r a m e t e r s ( t h a t w i l l b e s t a t e d n e x t ) for each o r d e r c h a n g e t o be c o n s i d e r e d .  • Tu is t h e t i m e for u n d e r m o d e h n g d e t e c t i o n .  • Tc is t h e t i m e w h e n t h e cost f u n c t i o n changes t o Z e r o S l o p e for c o r r e c t m o d e h n g . v a l u e w i l l be w i t h i n t h e r e g i o n  • NDT  7c,„i„  Its  ^ Tc > Tc^^^ .  is t h e n u m b e r of t i m e steps t o w a i t for c o n v e r g e n c e after a n o r d e r c h a n g e .  • No is t h e i n i t i a l guess for t h e o r d e r .  •  K^ait  is t h e n u m b e r of t i m e steps t o w a i t b e t w e e n i n d i c a t i o n of p o s s i b l e m i s - m o d e h n g  and its acceptance.  T h e o r d e r change f u n c t i o n , A.N b a s e d o n t h e b e h a v i o r of J i n E q u a t i o n 5.19 for u n d e r m o d e h n g a n d E q u a t i o n 5.27 for o v e r - m o d e h n g a n d for c o r r e c t m o d e h n g , is as f o l l o w s :  <AT•NDT 0  tarder  <Tc  J<  JR  >Tc AN  = <  (5.30) DU M  tarder [  < Tv  J > JR^  J >  JR^  <Tc DOM  tarder  >Tc where :  • jRy  a n d JR^ are m i n i m u m values for t h e cost f u n c t i o n a n d i t s d e r i v a t i v e , for i d e n t i f y i n g  under-modehng. • JR^ a n d JR^ are m i n i m u m values for t h e cost f u n c t i o n a n d i t s d e r i v a t i v e , to i d e n t i f y o v e r - m o d e h n g , for t < • Jz^ a n d Jz„  are m i n i m u m values for t h e cost f u n c t i o n a n d i t s d e r i v a t i v e , to i d e n t i f y  o v e r - m o d e h n g , for t > • JR^^^ a n d  Tc-  Tc-  are m a x i m u m values for t h e cost f u n c t i o n a n d i t s d e r i v a t i v e , to i d e n t i f y  correct-modehng.  • Jz,„  a n d Jz^^ are m i n i m u m values for t h e cost f u n c t i o n a n d i t s d e r i v a t i v e , t o i d e n t i f y  correct-modehng. • D U M is t h e u n d e r - m o d e h n g  a d d i t i o n to t h e o r d e r at each o r d e r c h a n g e s t e p .  • D O M is t h e o v e r - m o d e h n g s u b t r a c t i o n f r o m t h e o r d e r at each o r d e r c h a n g e step.  T h i s is a g r a d i e n t a l g o r i t h m designed t o m i n i m i z e t h e n u m b e r of steps t o a c h i e v e t h e correct o r d e r . B a s e d o n t h e r e s u l t s f r o m t h e i n v e s t i g a t i o n of t h e r o b o t i c a p p h c a t i o n s p r e s e n t e d i n t h i s w o r k p a r a m e t e r s i n i t i a l values were d e t e r m i n e d . T h e M O D a l g o r i t h m was i m p l e m e n t e d o n t h e f l e x i b l e a n d t h e h y d r a u h c m a n i p u l a t o r a n d was f o u n d t o be s t a b l e i n b e h a v i o r d u e t o several f a c t o r s :  •  K^ait  is t h e n u m b e r of t i m e steps t o w a i t a n d v e r i f y t h e need for o r d e r c h a n g e .  This  p r e v e n t s a r a n d o m i n c r e a s e i n t h e values o f t h e p a r a m e t e r s a n d o n u n n e c e s s a r y o r d e r change.  • tarder a r e l a t i v e t i m e o r i g i n is used a n d reset after a n o r d e r c h a n g e t o w h a t is b e h e v e d is t h e c o r r e c t v a l u e .  Chapter 6  I M P L E M E N T A T I O N O F T H EM O D A L G O R I T H M  6.1  Implementation  of the O r d e r D e t e r m i n a t i o n A l g o r i t h m  T h e a l g o r i t h m i m p l e m e n t a t i o n is d e s c r i b e d as follows:  1. I n i t i a h z a t i o n : d e f i n i t i o n s b y t h e user  (a) No : i n i t i a l value for t h e e s t i m a t e d m o d e l o r d e r . (b)  ?c,„i„  {c)  Tu : Tu < Tc^i^ : t i m e for u n d e r - m o d e h n g d e t e c t i o n  <Tc>  Tc,^^^ : t i m e r e g i o n for t h e t i m e change.  ( d ) Kuiait' n u m b e r of t i m e steps t o w a i t b e t w e e n i n d i c a t i o n o f p o s s i b l e m i s - m o d e h n g and its acceptance. (e) NUDT:  t h e n u m b e r of t i m e steps t o w a i t for c o n v e r g e n c e , w h e n a n o r d e r c h a n g e  was d o n e d u e t o u n d e r - m o d e h n g . (f) NODT:  t h e n u m b e r of t i m e steps t o w a i t for c o n v e r g e n c e , w h e n a n o r d e r c h a n g e  was d o n e d u e t o o v e r - m o d e h n g . (g) d a t a for c o r r e c t - m o d e h n g : i . at - i < T c • J H „ : m a x i m u m v a l u e for r i s i n g cost f u n c t i o n : J < JR,^ • JR„^ '• m a x i m u m value for r i s i n g slope J < J/t,„ i i . at - i > T c  • Jz,^ • m a x i m u m value for zero slope cost f u n c t i o n : J < Jz^^ • Jz,„ : m a x i m u m value for zero s l o p e J < Jz^, • Jz„i '• i n f l u e n c e of n o n l i n e a r i t i e s J ^ ^ , > Jz^ • Jzr,l '• i n f l u e n c e o f n o n l i n e a r i t i e s J ^ ^ , > Jz^ (h) d a t a for u n d e r - m o d e l i n g : i.  ai-t<Tu • J f l y : m i n i m u m value for cost f u n c t i o n ( t o i d e n t i f y u n d e r - m o d e l i n g ) :  J >  JRU  • JR^: m i n i m u m v a l u e for cost f u n c t i o n s l o p e ( t o i d e n t i f y  under-modeling):  (i) d a t a for o v e r - m o d e l i n g : i . at  -t<Tc  • JR^: m i n i m u m value for cost f u n c t i o n ( t o i d e n t i f y o v e r - m o d e H n g ) :  J  >  • JR^/. m i n i m u m v a l u e for cost f u n c t i o n , slope ( t o i d e n t i f y u n d e r - m o d e l i n g ) : J>JRo  i i . at -  t>Tc  • JZQ' m i n i m u m v a l u e for cost f u n c t i o n ( t o i d e n t i f y o v e r - m o d e l i n g ) :  • Jzfj'- m i n i m u m v a l u e for cost f u n c t i o n , s l o p e ( t o i d e n t i f y j  >  under-modeling):  JZO  2. T h e M O D A l g o r i t h m  (a)  J >  C h e c k t i m e r e l a t i v e t o Tc a n d values o f t h e cost f u n c t i o n J , a n d i t s s l o p e J .  (b)  D e t e r m i n e i f m i s - m o d e H n g is i n d i c a t e d , t h e n : • D e t e r m i n e u n d e r or o v e r - m o d e H n g • W a i t for Kuiait t i m e steps • T h e a l g o r i t h m has logic to h a n d l e m o d e l o r d e r changes a c c o r d i n g t o t h e t y p e of m i s - m o d e h n g • C o n v e r g e n c e t o a c c e p t a b l e s t r u c t u r e w h e n at i > T c : J < Jz„^ a n d J <  (c)  W h e n m i s - m o d e H n g is not i n d i c a t e d : • c o n v e r g e n c e for v e r i f i c a t i o n of t h e m o d e l m o d e l o c c u r s : J < Jz,^ a n d j <  (d)  6.1.1  Jz^  w h e n at t >  Tc:  Jz^  A t e a c h t i m e s t e p , t h e c o n t r o l a l g o r i t h m is a c t i v a t e d w i t h t h e present m o d e l o r d e r .  T h e M e t h o d For T h e Flexible L i n k Manipulator  D a t a for t h e F l e x i b l e L i n k T h e values for t h e p a r a m e t e r s p r e s e n t e d i n t h i s s e c t i o n are b a s e d o n t h e i n v e s t i g a t i o n d o n e i n S e c t i o n 5.2 for t h e b e h a v i o r of t h e cost f u n c t i o n of t h e flexible H n k . I n S e c t i o n 5.2 t h e M O D p a r a m e t e r s were d e t e r m i n e d f r o m s i m u l a t i o n r e s u l t s . I n o t h e r a p p h c a t i o n s of t h e a l g o r i t h m , s u c h s i m u l a t i o n s w o u l d first be r u n o n - h n e w i t h M O D t u r n e d off, t h e o r d e r of t h e w i U be c h a n g e d a n d b a s e d o n the cost f u n c t i o n b e h a v i o r p a r a m e t e r s w i l l b e T h e values for  TCmin,  Tc„,^,,  JR^,  JR^,  JZ,^,  model  determined.  J Z ^ , ^ZnU Jznl were d e t e r m i n e d f r o m  correct  m o d e H n g results s h o w n i n F i g u r e 5.4 a n d F i g u r e 5.10. V a l u e s for JRy, JR^J a n d Tu are f r o m d a t a based o n u n d e r - m o d e h n g modehng  F i g u r e 5.6, F i g u r e 5.7 a n d F i g u r e 5.12.  D a t a for t h e over-  case, JR^, JR^, JZ^, JZOI was o b t a i n e d f r o m F i g u r e 5.8, F i g u r e 5.9, F i g u r e 5.13,  F i g u r e 5.14, F i g u r e 5.15.  F i g u r e 6.1 shows t h e regions i n w h i c h J a n d J i n d i c a t e d t h e m i s - m o d e h n g . T h e s h a d e d areas s h o w for t h e flexible h n k at w h a t values u n d e r , over o r c o r r e c t m o d e h n g o c c u r .  1. T i m e d a t a  (a) r c , „ . „ =  lA[sec.]  ( b ) T - c ^ = 2.5[.ec.] (c) Tu =  0.5[sec.  2. D a t a for c o r r e c t m o d e h n g  (a) Jfi„, = 5 . 1 0 - « (b) j « „ . = 5 • 1 0 - « (c) J ^ , „ = 1 0 - ^ (d)  Jz„, =  10-^=^  (e) Jzr.i = Jz^ (f) Jz.1 =  Jz..  3. D a t a for u n d e r - m o d e h n g  (a)  JR^  =  10-"  (b)  JR,  =  10-^  4. D a t a for o v e r - m o d e h n g  (a) J H „ = 1 0 ^ « (b)  JR,  -  10-^  10"  KT  K)"' 10-  K'' 10^  10^  / / /  10-" H  10-"=10-»10"  '  /  -  !  /  10-"  /  1  /  /  ^  7  / /  10^ H  -  10  TIME [sec]  10"  -  /  o e r r M t fne«  4  6  10  TIME [see]  J  io^'^o"^o'=io'"io'"id-id- id-' id- lo"' lo" J  10^'10'  '*io'-"ioSo'-"io'-id-io- '10-'10^10 J  '10  F i g u r e 6.1: R e g i o n s f o r u n d e r , over a n d c o r r e c t m o d e l i n g  (c)  Jzo  -  10-«  (d)  Jzo  =  10-«  5. D a t a for o t h e r p a r a m e t e r s  (a)  DUM  = 2 , (4),  (b)  DOM  = 1 , (2)  (6)  R e s u l t s for t h e F l e x i b l e L i n k T h e d a t a for b o t h t h e t w o m o d e a n d t h e one m o d e e x a m p l e s w h e n m o d e l e d c o r r e c t l y , s h o w t h a t w h e n t < Tc, t h e values of J are a p p r o x i m a t e l y 0.5 10"® a n d at i > Tg J is of t h e o r d e r of m a g n i t u d e of 10"®.  W h e n u n d e r - m o d e l e d at s m a l l t, t h e values of J rise t o t h e o r d e r o f  m a g n i t u d e of 1 0 " ^ a n d h i g h e r , a n d so do t h e values of J. are m o r e m o d e r a t e .  For over-modehng the  changes  I n t h e case of s m a U o v e r - m o d e h n g (by o r d e r of 2 ), t h e m i s - m o d e h n g  c a n n o t be d e t e c t e d since t h e c o n t r o l a l g o r i t h m w o r k s w e l l a n d t h e C . F . d e r i v a t i v e ' s v a l u e s are s m a l l . W h e n o v e r - m o d e H n g is l a r g e r , t h e values of J c h a n g e i n a m o r e m o d e r a t e  slope.  T h e o b s e r v a t i o n of t h e values of J a n d J were m a d e o n t h e basis of r e s u l t s i n S e c t i o n 5.2.  J  a c c u m u l a t e s i t s values w i t h t i m e , a n d o r d e r changes on-Hne. Its values m a y b e h i g h e r , b u t i t s s h a p e r e m a i n s t h e s a m e a n d i m p o r t a n t . J values do not change a n d are v e r y i m p o r t a n t . T h e r e s u l t s t o be s h o w n d e s c r i b e t h e b e h a v i o r of t h e s y s t e m s dealt w i t h i n t h i s w o r k i n u n d e r a n d o v e r - m o d e h n g for different values for t h e d a t a n e e d e d b y t h e o r d e r d e t e r m i n a t i o n a l g o r i t h m . (Note:  e a c h t i m e a n o r d e r c h a n g e is d o n e , t h e a l g o r i t h m sets i t s i n t e r n a l t i m e t o z e r o , so  a l l p a r a m e t e r s for t h e v a l u e regions c a n be t r e a t e d i n t h e p r o p e r t i m e f r a m e a n d n o t i n t h e a b s o l u t e one). Figures  6.2,  6.3, show t h e b e h a v i o r of t h e 2 m o d e [order  = 6) f l e x i b l e h n k w h e n u n d e r -  m o d e l e d w i t h a s e c o n d o r d e r e s t i m a t e d m o d e l . C a s e A is t h e b e h a v i o r of t h e s y s t e m w h e n  n o o r d e r c o r r e c t i o n is done.  Y t i p goes u n s t a b l e .  d e s i r e d set p o i n t , b u t has a v e r y h i g h o v e r s h o o t .  I n case B , t h e s y s t e m converges t o  T h e a l g o r i t h m detects t h e u n d e r - m o d e h n g  s o o n e n o u g h , b u t does not h a v e t h e n e e d e d t i m e t o settle o n order as o v e r - m o d e h n g a g a i n a n d goes t o order  the  = 6. It u n d e r s t a n d s i t  = 4, w h i c h d r i v e s t h e s y s t e m i n t o i n s t a b i h t y . T h e  a l g o r i t h m t h e n changes t h e o r d e r t o t h e v a l u e o f order  = 10, w h i c h stops t h e r a p i d  change  i n t h e values of J. A f t e r Tc,„ax t h e o v e r - m o d e h n g has b e e n d e t e c t e d a n d t h e o r d e r is r e d u c e d to 8, J is r e d u c e d t o t h e r e g i o n a c c e p t e d as t h e correct m o d e h n g a n d t h e s y s t e m goes t o t h e v a l u e of t h e set p o i n t . I n case C some of t h e p a r a m e t e r s have b e e n a d j u s t e d . F i r s t , m o r e t i m e has b e e n g i v e n for t h e s y s t e m t o settle after t h e u n d e r - m o d e h n g was d e t e c t e d a n d c h a n g e d . I n a d d i t i o n , t h e values for  a n d Jza have b o t h b e e n i n c r e a s e d t o 10~^. A U t h e c h a n g e s  m a d e i n t h e p a r a m e t e r s of t h i s case were m a d e d o n e o r d e r t o i n c r e a s e t h e t i m e o f b e t w e e n o r d e r changes.  A s a r e s u l t , Y T I P has a l m o s t n o o v e r s h o o t .  convergence  It t a k e s a h t t l e m o r e  t i m e t o converge t o t h e set p o i n t ( a b o u t 4 seconds i n s t e a d of a b o u t 1.5 seconds); y e t , i t is f a r b e t t e r t h a n t h e u n d e r - m o d e l e d response p r e s e n t e d i n case B a n d ofcourse case A . T h e n e x t set of r e s u l t s c o m b i n e s u n d e r a n d o v e r - m o d e h n g i n t h e process o f c o r r e c t i o n a n u n d e r - m o d e h n g case of a t w o m o d e {order e s t i m a t e d m o d e l of order cases.  = 6)  flexible  hnk initiaUy under-modeled  with  = 4. F i g u r e 6.4, F i g u r e 6.5, F i g u r e 6.6, F i g u r e 6.7 s h o w t h r e e s u c h  C a s e A shows t h e u n s t a b l e b e h a v i o r of a t w o m o d e  flexible  h n k , controUed w i t h an  e s t i m a t e d m o d e l of o r d e r 4. I n case B , t h e o u t p u t converges t o t h e set p o i n t slower  (about  5 seconds i n c o n t r a s t t o 1.5 seconds) a n d has a n overshoot o f 27 p e r c e n t , b u t i t does n o t go u n s t a b l e h k e case A . T h e p a r a m e t e r s for t h e o r d e r d e t e r m i n a t i o n a l g o r i t h m are DOM Jza  —  = 2,  a n d Jz^ — 10~^. T h e a l g o r i t h m detects t h e u n d e r - m o d e h n g s o o n e n o u g h so as  n o t t o have a v e r y large o v e r s h o o t , a n d t h e o r d e r is set t o t h e c o r r e c t v a l u e o f 6.  Therefore,  t h e cost f u n c t i o n d e r i v a t i v e rises at first, b u t after t h e o r d e r c h a n g e , i t d r o p s a n d settles o n a n o r d e r of m a g n i t u d e of 10"^^ w h i c h is a n i n d i c a t i o n for convergence. DUM  I n case C , DOM  = 2,  — 6, JR^^ — 1 0 ® a n d JR^^ = 10~®. T h e o u t p u t has a l a r g e r o v e r s h o o t ( F i g u r e 6.4) s i n c e  flex 2 mode link — under—modelled estimator — GPC control 10-1  1  1  ^  8V Y 6 o  4-  4 6 TIME [sec]  1.0 1.5 TIME [sec]  3-  1—  4 6 TIME [sec]  —r 8  10  10  t h e o r d e r of t h e e s t i m a t e d m o d e l changes f r o m u n d e r - m o d e l e d (order [order  — 10) to (order  = 4) t o o v e r - m o d e l e d  = 8) a n d is i n a c c u r a t e for a l o n g e r t i m e . B u t t h e o u t p u t c o n v e r g e s at  t h e s a m e t i m e as case B . F i g u r e 6.8, F i g u r e 6.9, a g a i n show t h e b e h a v i o r of t h e 2 m o d e flexible h n k w h e n i n i t i a l l y u n d e r - m o d e l e d (order l a t e r over- m o d e l e d .  — 4) a n d d e p e n d i n g o n t h e o r d e r change of a l g o r i t h m p a r a m e t e r s , C a s e A has a l a r g e r NUDTu,  DUM  = 6, DOM  ^ 2. NUDTu  is t h e  p a r a m e t e r d e f i n i n g t h e n u m b e r of t i m e steps, after u n d e r - m o d e h n g is d e t e c t e d . W h e n i t h a s a l a r g e r v a l u e , t h e s y s t e m is i n t h e u n s t a b l e m o d e l o n g e r ; t h u s , t h e r e is a l a r g e o v e r s h o o t ( F i g u r e 6.8 case A ) . DUM  = 6 changes t h e o r d e r t o t h e v a l u e of 10, o v e r - m o d e h n g t h a t y i e l d s  h i g h e r values for t h e cost f u n c t i o n a n d i t s d e r i v a t i v e ; w i t h t i m e , t h e o r d e r is r e d u c e d t o 6. T h e w h o l e p r o c e d u r e r e s u l t e d , as m e n t i o n e d , i n a h i g h overshoot a n d i n a slower c o n v e r g e n c e . I n case B t h e t i m e t o change t h e o r d e r after u n d e r - m o d e h n g is d e t e c t e d was r e d u c e d , w h i l e aU o t h e r p a r a m e t e r s r e m a i n e d u n c h a n g e d .  T h e r e s u l t is a m u c h s m a l l e r o v e r s h o o t  and a  faster c o n v e r g e n c e . C a s e C presents t h e best result of t h e t h r e e , w i t h a n o v e r s h o o t o f a b o u t 15 p e r c e n t a n d convergence t o t h e s e t p o i n t w i t h i n less t h a t 6 seconds. T h e one m o d e flexible h n k (order t o u n d e r - m o d e h n g . F i g u r e s 6.10, flexible  = 4) w h e n u n d e r - m o d e l e d , reacts h k e t h e t w o m o d e h n k 6.11, present 3 cases. C a s e A is a one m o d e (order  — 4)  h n k u n d e r - m o d e l e d w i t h a n e s t i m a t e d m o d e l of o r d e r 2. T h e o u t p u t is u n s t a b l e U k e  t h e c o r r e s p o n d i n g i n F i g u r e 5.11. T h e cost f u n c t i o n a n d i t s d e r i v a t i v e rise t o h i g h values w h i c h i n d i c a t e t h e r e s p o n d i n g i n s t a b i h t y . I n case B , t h e o r d e r is c h a n g e d o n h n e t o s t a b i h z e t h e r e s p o n s e . W h e n u n d e r - m o d e h n g is d e t e c t e d , t h e o r d e r is c h a n g e d , b y t h e p a r a m e t e r s g i v e n a p r i o r i , t o a n over- m o d e h n g v a l u e of 8, a n d t h e n goes d o w n g r a d u a l l y t o t h e c o r r e c t v a l u e o f 4. T h e response has a n o v e r s h o o t of 83 p e r c e n t w h i c h is m a i n l y d u e t o t h e i n i t i a l i n s t a b i h t y a n d t h e l a t e r o v e r - m o d e h n g . B u t t h e o v e r a l l response, i n s t e a d of g o i n g u n s t a b l e , converges t o t h e set p o i n t after a n a c c e p t a b l e t i m e ( a b o u t 4 seconds).  Case C presents a better behavior  of t h e s y s t e m , i n w h i c h t h e o v e r s h o o t is s m a l l e r . T h e o r d e r c h a n g e of a l g o r i t h m p a r a m e t e r s  flex 2 mode link — under—modelled estimotor — GPC control 6.0-4, ^  0.0  0.2  4 6 TIME [sec]  0.4 (X6 TIME [sec]  98o  65-1 C: 4-  4 6 TIME [sec]  10  flex 2 mode link — under-modelled estimator — GPC control  TIME [sec]  flex 2 mode link - under-modelled estimator - GPC control  0) o 6-  5B: 4 6 TIME [sec]  4 6 TIME [sec]  4  4 6 TIME [sec]  10  was c h a n g e d , w i t h D U M t h a t has a s m a l l e r v a l u e of 4 i n s t e a d of 6. T h u s after d e t e c t i n g t h e u n d e r - m o d e h n g , t h e o r d e r rises t o t h e v a l u e of 6 ( t h e correct one is 4); yet t h e o v e r s h o o t  is  s t i l l q u i t e h i g h due t o t h e i n i t i a l i n s t a b i l i t y . F i g u r e s 6.12, 6.13, show 3 cases for i n i t i a l o v e r - m o d e h n g  (for a one m o d e h n k )  where  t h e different cases present different p a r a m e t e r s of t h e o r d e r d e t e r m i n a t i o n a l g o r i t h m . I n a l l of t h e 3 cases, t h e e s t i m a t e d m o d e l o r d e r for t h e c o n t r o l a l g o r i t h m is i n i t i a l l y 10, a n d t h e o u t p u t s v a r y s h g h t l y i n t h e overshoot  a n d t i m e of c o n v e r g e n c e . I n case A , DOM  = 4, a n d  t h e a l g o r i t h m , after d e t e c t i n g o v e r - m o d e h n g , sets t h e o r d e r o n 6. I n case B , t h e p a r a m e t e r s c h a n g e d a r e . D O M = 2, Tc„,,^  = 2.5 s e c , JR^ =  a l g o r i t h m detects t h e o v e r - m o d e h n g  1 0 " ^ , JR^ =  10"®.  T h e order  detection  a n d changes i t t o t h e v a l u e of 8, t h e n after 4.3 sec.  to  t h e o r d e r of 6. C a s e C has îbma» = 1-5 s e c , JR^ = 1 0 " ^ , JR^ = 10~^, w h i c h b r i n g s a q u i c k e r c h a n g e of t h e o r d e r f r o m 8 t o 6. T h e differences b e t w e e n t h e t h r e e r e s u l t s is s m a l l . C a s e s A a n d C h a v e a l m o s t n o difference because of t h e q u i c k change of order = 8 or 10 t o order  = 6;  C a s e B takes m o r e t i m e a n d t h e cost f u n c t i o n a n d i t s d e r i v a t i v e h m i t s are h i g h e r , so t h e r e s u l t has a s h g h t l y h i g h e r o v e r s h o o t a n d t a k e s a h t t l e l o n g e r t o  converge.  T h e c o n c l u s i o n s d r a w n so f a r f r o m t h e results of t h e f l e x i b l e h n k (one o r t w o m o d e s ) is t h a t u n d e r - m o d e h n g creates i n s t a b i l i t y , w h i c h c a n be c o n t r o l l e d b y d e t e c t i n g t h e u n d e r - m o d e h n g a n d c h a n g i n g i t t o t h e correct one or one close t o i t . T h e r e s u l t s m a y t a k e l o n g e r t o c o n v e r g e a n d have a n u n d e s i r e d l a r g e r overshoot  ( t h a n t h e correct m o d e h n g ) ,  b u t t h e o u t p u t is n o t  u n s t a b l e . T h e o v e r - m o d e h n g has a m u c h m o r e m o d e r a t e response, w h i c h is easier t o c o n t r o l after d e t e c t i n g i t a n d c h a n g i n g t h e o r d e r of t h e e s t i m a t e d m o d e l .  flex 1 mode link - mis-modelled estimotor -  0.5  1.0 TIME [sec]  1^  GPC control  2.0  4 6 TIME [sec]  F i g u r e 6.10: T h e o u t p u t b e h a v i o r o f a one m o d e flexible h n k e s t i m a t e d i n i t i a U y w i t h a n o r d e r 2 model  flex 1 mode link — mis—modelled estimator — GPC control 6  0.0  0.5  -|  1.0 TIME [sec]  r  1.5  4 6 TIME [sec]  flex 1 mode link -  mis-modelled estimotor — GPC control  4 6 TIME [sec]  4 6 TIME [sec]  F i g u r e 6.12: T h e o u t p u t b e h a v i o r o f a one m o d e flexible h n k i n i t i a l l y o v e r - m o d e l e d .  flex 1 mode link -J  T  r  4 6 TIME [sec]  mis-modelled estimator 110  GPC control  -r'  r  4 6 TIME [sec]  6.1.2  The M e t h o d For T h e Two Link Manipulator W i t h Hydraulic Actuators  D a t a for t h e T w o L i n k M a n i p u l a t o r T h e values for t h e p a r a m e t e r s p r e s e n t e d i n t h i s s e c t i o n are b a s e d o n t h e i n v e s t i g a t i o n d o n e i n S e c t i o n 5.2 for t h e b e h a v i o r of t h e cost f u n c t i o n of t h e T w o L i n k M a n i p u l a t o r . T h e v a l u e s for ^c,„i„,  2c,„„,,  JR„„  JR„,,  JZ„,,  Jz^,  Jz.,1, Jz„t were d e t e r m i n e d f r o m correct m o d e h n g r e s u l t s  shown in Figure 5.17 and Figure 5.18.  V a l u e s for JR^^, JR^ a n d Tu are f r o m d a t a b a s e d  o n u n d e r - m o d e h n g , F i g u r e 5 . 1 9 . D a t a for t h e o v e r - m o d e h n g obtained from Figure 5.21 and Figure 5.22.  1. T i m e d a t a  (a) (b)  =  2.0[.ec.  T c , „ „ . = 3 . 5 sec.  (c) Tu =  1.0 sec.  2. D a t a for c o r r e c t m o d e h n g  (a)  J«,„ = 5 . 10-®  (b)  ^ „  (c)  Jz,,„ = 5 . 1 0 - ®  (d)  J^,„ =  10-^^  (e)  Jz.i  =  10-«  (f)  Jz,j -  10-«  = 5 • 10-®  3. D a t a for u n d e r - m o d e h n g  (a)  JR,  =  10^  case, J r „ , J r „ , JZ„, JZO, w a s  (b)  Jn, =  10-==  4. D a t a for o v e r - m o d e l i n g  (a)  =  10-« 10-"  (b)  Jna  =  (c)  Jzo  = 10°  (d)  Jz„ =  10°  5. D a t a for o t h e r p a r a m e t e r s  (a)  DUM  = 1 , (2) , (3)  (b)  DOM  = 1 , (2)  R e s u l t s for t h e T w o L i n k M a n i p u l a t o r T h e d a t a for t h e h y d r a u h c m a n i p u l a t o r w h e n m o d e l e d c o r r e c t l y , show, as i n t h e p r e v i o u s a p p h c a t i o n , a t y p i c a l b e h a v i o r of t h e cost f u n c t i o n a n d i t s slope w h e r e t h e values are r e l a t i v e l y s m a l l . H e r e t o o , t h e changes are m o r e r a p i d for t h e u n d e r - m o d e h n g case t h a n for t h e o v e r m o d e h n g case.  F i g u r e 6.14, F i g u r e 6.15, F i g u r e 6.16, F i g u r e 6.17 present 3 cases.  A is t h e one w h e r e t h e o r d e r d e t e r m i n a t i o n a l g o r i t h m is not a c t i v a t e d .  Case  L i n k 2 is i n i t i a l l y  e s t i m a t e d w i t h a correct o r d e r 3 m o d e l a n d is weU c o n t r o U e d t o foUow a set p o i n t . L i n k 1 is i n i t i a l l y u n d e r - m o d e l e d w i t h a second o r d e r m o d e l , so t h a t t h e o u t p u t does n o t detect t h e set p o i n t . I n cases B a n d C , t h e o r d e r d e t e r m i n a t i o n a l g o r i t h m is a c t i v a t e d t o d e t e c t t h e c o r r e c t m o d e l and control the system.  T h e difference b e t w e e n t h e t w o cases is i n t h e p a r a m e t e r s  u s e d for t h e o r d e r d e t e r m i n a t i o n a l g o r i t h m . I n C a s e B , DUM  = 1 a n d DOM  = 1, so t h e  u n d e r - m o d e h n g w h e n d e t e c t e d , is c o r r e c t e d t o t h e correct o r d e r i n t h e first t r y . T h e r e s u l t is t h a t Oi converges t o t h e set p o i n t i n , a b o u t 7 seconds i n c o m p a r i s o n w i t h 5.5 seconds for  t h e case w h e r e b o t h h n k s are s t r u c t u r a l l y c o r r e c t l y m o d e l e d , as d e s c r i b e d s c h e m a t i c a l l y i n F i g u r e 5.3 . I n case C , D O M has c h a n g e d t o t h e v a l u e of 2. W h e n u n d e r - m o d e h n g is d e t e c t e d t h e a l g o r i t h m changes to t h e o r d e r 4, o v e r - m o d e h n g at first a n d t h e n r e d u c i n g i t . T h e r e s u l t is even slower t h a n i n case B ( a b o u t 9 seconds t o converge), b u t t h e set p o i n t is t r a c k e d , i n c o n t r a s t t o case A . T h e n e x t t h r e e cases present t h e b e h a v i o r o f t h e h y d r a u h c s y s t e m w h e n  over-modeled.  F i g u r e 6.18, F i g u r e 6.19, present t h e r e s u l t s . C a s e A shows t h e b e h a v i o r of t h e s y s t e m w h e r e h n k 1 is c o r r e c t l y m o d e l e d w i t h a t h i r d o r d e r s y s t e m a n d h n k 2 is over m o d e l e d (order 6i does n o t t r a c k t h e set p o i n t . I n case B , h n k 2 is i n i t i a l l y u n d e r - m o d e l e d (order t h e o r d e r d e t e c t i o n a l g o r i t h m is a c t i v a t e d , DOM (order  = 3 brings the system to  = 5 ) , w h i c h g r a d u a l l y is b r o u g h t d o w n t o t h e correct v a l u e (order  =  5).  = 2). W h e n over-modehng  = 3). A s a r e s u l t ,  t h e o u t p u t i n t h e first 3 seconds goes i n t h e u n s t a b l e d i r e c t i o n a n d t h e n s t a b i h z e s o n t h e set p o i n t .  I n case C , l i n k 2 i n i n i t i a l l y o v e r - m o d e l e d (order  — 5), a n d g r a d u a l l y t h e o r d e r  is c h a n g e d t o t h e correct one. T h e o u t p u t s t a b i h z e s faster (6.5 seconds, as c o m p a r e d t o 9.5 seconds). T h e hydrauhc actuated manipulator, hke the  flexible  hnk, when mis-modeled can  be  brought to the desired results w i t h the order d e t e r m i n a t i o n a l g o r i t h m . A g a i n , u n d e r - m o d e h n g affects t h e response of t h e o u t p u t m o r e t h a n o v e r - m o d e h n g a n d is m o r e d i f f i c u l t t o c o n t r o l , b u t b o t h are s o l v e d w i t h t h e o r d e r d e t e r m i n a t i o n a l g o r i t h m .  F i g u r e 6.20 p r e s e n t s r e s u l t s  s i m i l a r t o t h o s e p r e s e n t e d i n case B of F i g u r e 6.18, b u t o n a l a r g e r t i m e scale.  Once the  m i s - m o d e h n g is d e t e c t e d a n d c o r r e c t e d , t h e s y s t e m w i l l persist w i t h t h e s u i t a b l e e s t i m a t e d m o d e l s t r u c t u r e a n d w i U y i e l d t h e d e s i r e d response.  2 link monipulotor -  0.6  linki - order=2 0.6 0.5  0.3 o ^  Iink2- order=  P  0.4 H  0.0 . . .  - Votve 1  0.2 H  *''-0.3 Volv» 2  m  *'-0.6 -  O.H -  0.0  -0.9 -1.2  Valve 1  . -  4 6 TIME [sec]  10  -0.1-  —I  r—  4 6 TIME [sec]  Volve 2  8  06„  0.4-  O  "  0.2-.  0.0--;  6 . .  -  -0.2-  Volve 2 -T  -0.4-  Volve 1  1—  4 6 TIME [sec]  8  10  F i g u r e 6.14: T h e o u t p u t b e h a v i o r o f a h y d r a u h c a c t u a t e d t w o h n k m a n i p u l a t o r i n i t i a l l y under-modeled  10  2 link manipulator - linki — order=2  —1  r-  Ilnk2- order=3  4 6 TIME [sec]  4 6 TIME [sec]  5.J •S4.o  « -  Vot™ 1 -  r— 4 6 TIME [sec]  —1  Vi*«« 2  10  F i g u r e 6.15: O r d e r changes o f t h e e s t i m a t e d m o d e l for h y d r a u h c a c t u a t e d t w o h n k m a n i p u lator initially under-modeled  F i g u r e 6.16: T h e cost f u n c t i o n d e r i v a t i v e b e h a v i o r of h n k l i n i t i a l l y u n d e r - m o d e l e d  2 link monipulotor -I L  linki  6 TIME [sec]  4  4  5  TIME [sec]  cost function - valve 1  —1  1—  6 TIME [sec ]  4  10  F i g u r e 6.17: T h e cost f u n c t i o n b e h a v i o r o f h n k l i n i t i a l l y u n d e r - m o d e l e d  2 link manipulator — linki — order = 5 0.6  0.6 0.5  0.5 /  r> 0.4  r :  0.2  /  . . .  /  0.1  -  Votve 1  -  0.2-  .- • •  / /  0.4-  0.3-  / /  0.0  T)  /  •' 0.3 ^ '  -  /  o  ft  Iink2— order=3  0.1-  VoPve 2  . . -•  -  m  0.0  -0.1  -0.1-  -0.2  -0.2  _ voiv« 2  6 TIME [sec]  10  4  6 TIME [sec]  4  10  0.6 0.5 V o  0.4-  "  0.3-  t.  0.2<->  0.10.0--;  .  -0.1-0.2-  0  - VoPve 1  volv« 2  6 TIME [sec]  4  10  F i g u r e 6.18: T h e o u t p u t b e h a v i o r of a h y d r a u h c a c t u a t e d t w o h n k m a n i p u l a t o r O v e r - m o d e l e d  2 link monipulotor -  linki -  order = 5  Iink2- order = 3  6.  5. H ^4. H  ^3.4-  •  1-  •D O  -  œ  Volv« 1  2.-I. . .  1. -  - Valvt 1  Volve 2  4 6 TIME [sec]  10  -T  1—  4 6 TIME [sec]  8  10  Votv« 1  -  -  4 6 TIME [sec]  0.  —r 8  Votvt 2  10  F i g u r e 6.19: O r d e r changes of t h e e s t i m a t e d m o d e l for h y d r a u h c a c t u a t e d t w o h n k m a n i p u lator over-modeled  2 link manipulator - linki - order-2  Iink2- order=3 1  \  / /  1'  ••  \  / / / •  1  / /  •. \ '•. \ •. \ '•, \ •  ,-' .•'  / .•' / / /,•' /.• /.''  . . .  -4  -  Votn 1 Vdv* 2  1  1  10  15 TIME [sec]  20  25  30  5H fe4H T3  i  o . . .  1-  0  -  Valvt 1 Vol» 2  10  15 TIME [sec]  —I—  20  25  30  F i g u r e 6.20: P e r f o r m a n c e o f a s y s t e m i n i t i a l l y m i s - m o d e l e d o n a l a r g e r t i m e scale  6.2  C o m p a r i s o n of m e t h o d ' s R e s u l t s w i t h O t h e r  Work  T h e m e t h o d d e v e l o p e d i n t h i s t h e s i s , d e t e c t s , d e t e r m i n e s a n d executes o n - h n e c h a n g e s i n m o d e l o r d e r . T h e M O D a l g o r i t h m is a g r a d i e n t a l g o r i t h m b a s e d o n t h e b e h a v i o r o f a c h o s e n cost f u n c t i o n a n d i t s t i m e d e r i v a t i v e . T h e cost f u n c t i o n b e h a v i o r enables t h e a l g o r i t h m t o d i s t i n g u i s h b e t w e e n t y p e s of m i s - m o d e h n g of t h e s y s t e m ( u n d e r a n d o v e r - m o d e h n g ) .  When  m i s - m o d e h n g creates p r o b l e m s i n c o n t r o U i n g t h e s y s t e m i t changes t h e m o d e l ' s o r d e r . I n i t i a h z a t i o n of t h e M O D a l g o r i t h m is based e i t h e r o n a p r i o r i k n o w l e d g e of t h e s y s t e m a n d s i m u l a t i o n r e s u l t s or o n p r e h m i n a r y tests of t h e s y s t e m . T h e c o n t r o l s t r a t e g y for a s y s t e m is designed b a s e d o n t h e b e s t k n o w l e d g e of t h e s y s t e m available. A m o d e l order change w i U occur if a n operating point change on-hne a n d therefore t h e c o n d i t i o n s change or i f t h e i n i t i a l i d e n t i f i c a t i o n of t h e s y s t e m was n o t a c c u r a t e .  The  M O D a l g o r i t h m a c t i v a t e s a m o d e l s t r u c t u r e change o n l y w h e n t h e cost f u n c t i o n i n d i c a t e s t h a t m i s - m o d e h n g is a p r o b l e m .  T h e c o m p u t a t i o n a l b u r d e n o f t h i s a l g o r i t h m is r e l a t i v e l y  s m a U , a n d b y u s i n g i t t h e d e s i r e d b e h a v i o r of t h e s y s t e m is a c h i e v e d . A d i s c u s s i o n of t h e advantages a n d d i s a d v a n t a g e s of e x i s t i n g t e c h n i q u e s w i U n o w b e p r e s e n t e d . A m o r e d e t a i l e d d i s c u s s i o n of t h e m e t h o d s , has b e e n r e v i e w e d i n S e c t i o n s  2.4 a n d  2.5. Identifying a system depends  s t r o n g l y o n t h e choice of t h e m o d e l s t r u c t u r e .  Off-hne  m e t h o d s have t h e a d v a n t a g e of c h o o s i n g a m o d e l s t r u c t u r e , i d e n t i f y i n g i t s p a r a m e t e r s a n d t h e n vahdating the m o d e l .  If t h e r e s u l t s are not s a t i s f a c t o r y , different t y p e s of m o d e l s c a n  be e x a m i n e d t o f i n d t h e best m o d e l for t h e s y s t e m a n d t h e o p e r a t i n g c o n d i t i o n s . m o d e l v a l i d a t i o n t e c h n i q u e s were p u b h s h e d by A k a i k e  97  , Isermann  90  Schwarz  Off-hne 9ft  and  o t h e r s . T h e s e w o r k s p r o p o s e different c r i t e r i a as a m e a s u r e of t h e fit of t h e m o d e l . A s w e U , different m e t h o d s for s t a t e space r e p r e s e n t a t i o n have b e e n d e v e l o p e d . were p r o p o s e d by G u i d o v z i  . Davison  Canonical structures  presented a m e t h o d for m o d e l o r d e r r e d u c t i o n  by s e l e c t i n g t h e d o m i n a n t eigenvalues a n d eigenvectors of t h e s y s t e m . A m i n i m a l d e s c r i p t i o n of a s y s t e m w i t h a l l s i g n i f i c a n t d y n a m i c s is a basis for d e s i g n i n g a c o n t r o l l e r , b u t t h e m e t h o d does n o t address t h e q u e s t i o n of c h a n g i n g c o n d i t i o n s a n d therefore changes i n t h e n u m b e r o f d o m i n a n t eigenvalues. T h e r e are also several r e c u r s i v e m e t h o d s .  Overbeek and Ljung  suggested t h e m o d e l  s t r u c t u r e s e l e c t i o n ( M S S ) a l g o r i t h m i n w h i c h t h e s t r u c t u r e s differ i n t h e p a r a m e t r i z a t i o n o f t h e m o d e l , b u t t h e r e is n o r e c u r s i v e o r d e r s e l e c t i o n t h a t i s , t h e o r d e r is chosen a p r i o r i . a l g o r i t h m c a l c u l a t e s at e a c h t i m e t h e e n t i r e set of s t r u c t u r e s a n d c o m p a r e s  The  t h e m on-Une  r e s u l t i n g i n a p o s s i b i l i t y of a h i g h c o m p u t a t i o n a l b u r d e n . N i u , X i a o a n d F i s h e r  present a  s i m u l t a n e o u s r e c u r s i v e e s t i m a t i o n of p a r a m e t e r a n d o r d e r . T h e o r d e r is f o u n d b y c a l c u l a t i n g a cost f u n c t i o n for a l l possible orders u p t o a k n o w n u p p e r b o u n d . T h e o r d e r w h i c h c o r r e s p o n d s t o t h e m i n i m a l v a l u e of t h e cost f u n c t i o n is t h e one t h a t is u s e d . F u r t h e r w o r k B y Fisher  i m p l e m e n t e d t h e a b o v e a l g o r i t h m for M I M O sestems.  Hemerly  N i u and  presented  a  m e t h o d for o n h n e o r d e r a n d p a r a m e t e r i d e n t i f i c a t i o n u s i n g t h e R L S a l g o r i t h m a n d t h e P L S criterion. Mereiros and Hemerly  integrated the above m e t h o d w i t h lattice f o r m filters for  a m i n i m u m v a r i a n c e c o n t r o l l e r . T h i s w o r k is t h e closest i n n a t u r e t o t h e w o r k p r e s e n t e d i n t h i s thesis.  H o w e v e r , i t r e q u i r e s c o m p u t a t i o n of a cost f u n c t i o n for aU p o s s i b l e o r d e r s  t o a n u p p e r b o u n d ) , at e a c h t i m e step. mis-modehng.  (up  T h e r e is n o reference or d i s c u s s i o n o f t h e t y p e o f  O t h e r r e c u r s i v e a n d off-hne o r d e r a n d p a r a m e t e r s i d e n t i f i c a t i o n m e t h o d s  r e p o r t e d ( o p e n l o o p m e t h o d s ) , s u c h as t h e one b y W u h c h a n d K a u f m a n  were  and Katsikas  T h e s e m e t h o d s are b a s e d o n a p r i o r i defined c r i t e r i o n , a n d c a l c u l a t i o n s of a l l p o s s i b l e o r d e r s a n d t h e choice of t h e one w h i c h gives t h e best p e r f o r m a n c e of t h e c r i t e r i o n . T h e r e are several a d v a n t a g e s to t h e M O D m e t h o d p r e s e n t e d i n t h i s t h e s i s : 1. T h e m e t h o d is a n o n - h n e m e t h o d t h a t detects t h e n e e d t o c h a n g e t h e m o d e l ' s  order  a n d i m p l e m e n t s i t w h i l e u s i n g w e l l k n o w n m e t h o d s for t h e i n d e n t i f i c a t i o n a n d a d a p t i v e  c o n t r o l processes.  M o s t of t h e p u b h s h e d w o r k s t h a t were p r e s e n t e d i n S e c t i o n  discuss m o d e l o r d e r d e t e r m i n a t i o n , e i t h e r off-hne or o n - h n e b u t o p e n l o o p , w i t h  2.5 no  control algorithm implemented.  • O f f - h n e m e t h o d s c a n e s t i m a t e a m o d e l t h e n check a n d v a l i d a t e i t a n d i f t h e r e s u l t s do not satisfy, a n o t h e r m o d e l s t r u c t u r e is c h o s e n , u n t i l a g o o d r e p r e s e n t i n g m o d e l is a c h i e v e d . If a m o d e l is chosen w i t h one of t h e off-hne m e t h o d s i t s s t r u c t u r e i s f i x e d w h e n u s e d o n - h n e , for c o n t r o l p u r p o s e s for e x a m p l e . R e p r e s e n t i n g w o r k s c a n be f o u n d i n A k a i k e  Isermann  , Schwartz  Rissanen  Guidovzi  and Davison • Recursive identification methods  a n d s t r u c t u r e s e l e c t i o n are m a i n l y p a r a m e t e r  s e l e c t i o n m e t h o d s w h e r e t h e o r d e r is f i x e d . T h e p o s s i b l e s t r u c t u r e s are s c a n n e d a n d t h e best c h o s e n , O v e r b e e k a n d L j u n g  or s i m u l t a n e o u s o r d e r a n d p a r a m e t e r  e s t i m a t i o n w i t h t h e s a m e p r i n c i p l e , a set of p o s s i b l e o r d e r s are c h o s e n a n d a c o s t f u n c t i o n is c a l c u l a t e d for a l l t h e set each t i m e s t e p . T h e chosen o r d e r is t h e o n e t h a t c o r r e s p o n d t o t h e m i n i m a l v a l u e of t h e cost f u n c t i o n . Fisher  , N i u and Fisher  , Wuhch and K a u f m a n  See N i u , X i a o a n d  ^ Katsikas  and Hemerly  47  2. C o s t f u n c t i o n b e h a v i o r i n d i c a t e s t h e best e s t i m a t e d m o d e l o r d e r t o t h e M O D a l g o r i t h m .  • It was f o u n d t h a t t h e cost f u n c t i o n has a different b e h a v i o r for u n d e r a n d for o v e r - m o d e h n g , b u t s i m i l a r b e h a v i o r for t h e t w o a p p h c a t i o n s . • T h e cost f u n c t i o n i n d i c a t e s t h e best p o s s i b l e o r d e r for t h e present o p e r a t i n g p o i n t a n d does not s e a r c h for t h e exact m o d e l . • A n i n i t i a l o r d e r is p r o v i d e d a n d there is no need t o a s s u m e o n u p p e r b o u n d t o t h e order.  • T h e cost f u n c t i o n b e h a v i o r i n d i c a t e s i n d i c a t e s t h e i n f l u e n c e of t h e c l o s e d  loop,  e s p e c i a l l y i n the m i s - m o d e h n g cases. • T h e r e is no need to e x a m i n e a n d s c a n a set of m o d e l s for e a c h t i m e s t e p as t h e cost f u n c t i o n is c a l c u l a t e d for t h e present m o d e l o r d e r .  T h e w o r k done by H e m e r l y  , c o m b i n e s a r e c u r s i v e i d e n t i f i c a t i o n process a n d r e c u r s i v e  o r d e r e s t i m a t i o n for a m o d e l r e p r e s e n t e d by l a t t i c e filter f o r m w i t h a m i n i m u m v a r i a n c e control algorithm.  T h e u p p e r b o u n d of t h e o r d e r is a s s u m e d t o b e k n o w n a n d t h e  o r d e r is e s t i m a t e d by s c a n n i n g a l l p o s s i b l e P L S f u n c t i o n s a n d c h o o s i n g t h e o r d e r t h a t c o r r e s p o n d s t o t h e m i n i m a l value.  T o t h e best of o u r k n o w l e d g e , t h e r e is n o o t h e r m e t h o d h k e t h e one p r e s e n t e d i n t h i s thesis.  B a s e d o n t h e t i m e b e h a v i o r of a cost f u n c t i o n t h e a l g o r i t h m d e t e c t s a n d e x e c u t e s  o n - h n e changes of m o d e l o r d e r . It was e s t a b U s h e d t h a t b o t h , u n d e r a n d o v e r - m o d e H n g c a n cause p o o r p e r f o r m a n c e a n d i n s t a b i h t y a n d for b o t h , o r d e r c o r r e c t i o n is d o n e i f r e q u i r e d . T h e a l g o r i t h m detects changes i n t h e cost f u n c t i o n b e h a v i o r w h i c h is m o n i t o r e d o n h n e . T h e c o m p u t a t i o n b u r d e n is f a i r l y s m a l l a n d t h e a l g o r i t h m has s t a b l e c h a r a c t e r i s t i c s a n d is a b l e b a s e d o n s o m e a p r i o r i k n o w l e d g e w i t h i n a few i t e r a t i o n s t o m a i n t a i n d e s i r e d p e r f o r m a n c e o f the system.  6.3  Conclusions  T h i s c h a p t e r p r e s e n t e d t h e b e h a v i o r of t w o k i n d s of r o b o t i c m a n i p u l a t o r s c o n t r o l l e d w i t h t h e G P C a l g o r i t h m i n m i s - m o d e h n g of t h e e s t i m a t e d m o d e l for t h e c o n t r o l a l g o r i t h m . T h e b e h a v i o r o f t h e cost f u n c t i o n a n d i t s d e r i v a t i v e for t h e one a n d t w o m o d e flexible h n k a n d for t h e t w o l i n k h y d r a u h c a c t u a t e d m a n i p u l a t o r s h o w e d a p a t t e r n of b e h a v i o r for u n d e r , o v e r a n d c o r r e c t s t r u c t u r e for t h e e s t i m a t e d m o d e l .  B a s e d o n these r e s u l t s , a n o r d e r d e t e r m i n a t i o n  a l g o r i t h m was p r e s e n t e d .  D e p e n d i n g on its parameters, m i s - m o d e h n g can be corrected to  give a n a c c e p t a b l e response f r o m these s y s t e m s , e v e n w h e n t h e y are i n i t i a l l y u n s t a b l e or h a v e bad performance.  Chapter 7  CONCLUSIONS AND  7.1  M a i n Results of the  SUMMARY  Thesis  T h i s w o r k has c h a l l e n g e d t h e c o n c e p t of u s i n g a fixed s t r u c t u r e m o d e l for a p l a n t c o n t r o l l e d b y an adaptive control a l g o r i t h m ( G P C ) . Generally, i n order to implement an adaptive a l g o r i t h m for a s y s t e m , t h e p l a n t is m o d e l e d by a h n e a r m o d e l i n w h i c h p a r a m e t e r s are e s t i m a t e d o n h n e . T h i s c a n r e s u l t i n u n c e r t a i n t i e s i n p a r a m e t e r v a l u e s , e s p e c i a l l y w h e n t h e m o d e l o r d e r is i n c o r r e c t l y chosen. T w o r o b o t i c a p p h c a t i o n s chosen for t h e s t u d y were m o d e l e d , s i m u l a t e d a n d c o n t r o U e d . T h e single flexible h n k c a n give rise t o m o d e s of o s c i U a t i o n s o n - h n e d u r i n g a w o r k i n g c y c l e , a n d therefore have o n - h n e changes of t h e p l a n t d y n a m i c s .  T h e two h n k m a n i p u l a t o r w i t h  h y d r a u h c a c t u a t o r s c a n be m i s - m o d e l e d , b u t i t was also chosen b e c a u s e of i t s h i g h l y n o n h n e a r n a t u r e a n d i t s e x t e n s i v e use i n i n d u s t r y . Chapters 3 and  4 present t h e d y n a m i c m o d e h n g  of t h e s y s t e m s , i m p l e m e n t a t i o n of  t h e G P C a l g o r i t h m a n d t h e t u n i n g of t h e c o n t r o l p a r a m e t e r s t o achieve g o o d  performance  c o n t r o l . C h a p t e r 5 presents a m o d e l o r d e r d e t e r m i n a t i o n ( M O D ) a l g o r i t h m for d e t e c t i n g t h e need to c h a n g e t h e m o d e l s t r u c t u r e , c o r r e c t i n g t h e o r d e r a n d e x e c u t i n g i t o n - h n e . T h e s t u d y t h a t l e d t o t h e a b o v e m e t h o d b e g a n w i t h t h e e s t a b h s h m e n t of a cost f u n c t i o n as a m e a s u r e of t h e e r r o r between t h e p l a n t a n d m o d e l d y n a m i c s . T h e m a i n results of t h i s s t u d y are:  m o d e l i n g , t h e cost f u n c t i o n rises i n i t i a l l y ( w h e n t h e e r r o r goes t o z e r o ) , a n d t h e n s e t t l e s o n a c o n s t a n t v a l u e . W h e n u n d e r - m o d e l e d , t h e cost f u n c t i o n ' s i n i t i a l rise is s t e e p e r , g o i n g t o m u c h h i g h e r values a n d l e a d i n g t o i n s t a b i h t i e s .  W h e n over-modeled,  the  b e h a v i o r is m u c h m o r e m o d e r a t e , b u t t h e p e r f o r m a n c e d e t e r i o r a t e s a n d t h e s y s t e m c a n go u n s t a b l e . W h e n u n d e r - m o d e h n g is i n v o l v e d , t h e r e g r e s s i o n v e c t o r is c o r r e l a t e d t o t h e u n m o d e l e d d y n a m i c s . H o w e v e r , t h e o v e r - m o d e h n g does n o t i n c l u d e t h a t c o r r e l a t i o n a n d therefore t h e response is m o r e m o d e r a t e .  T h e excess of d y n a m i c s i n t h e m o d e l ( o v e r -  m o d e h n g ) causes t h e c o n t r o l a l g o r i t h m t o t r y a n d c o n t r o l d y n a m i c s t h a t are n o t t h e r e ; t h u s , t h e c o n t r o l p a r a m e t e r s are no longer w e l l t u n e d .  B o t h a p p h c a t i o n s s h o w s i m i l a r cost f u n c t i o n b e h a v i o r . B a s e d o n t h e a b o v e r e s u l t s , a n on-hne m o d e l o r d e r d e t e r m i n a t i o n ( M O D ) a l g o r i t h m is g i v e n to d e t e c t t h e n e e d t o change t h e m o d e l o r d e r a n d t o correct i t o n o n - h n e . R e s u l t s f r o m i m p l e m e n t i n g t h e m e t h o d o n b o t h a p p h c a t i o n s show t h a t for u n d e r a n d o v e r - m o d e h n g , i n s t a b i l i t i e s are a v o i d e d a n d d e s i r e d p e r f o r m a n c e is r e s t o r e d . G e n e r a h z e d P r e d i c t i v e C o n t r o l ( G P C ) c a n be a p p h e d t o h e a v y d u t y m a n i p u l a t o r s w h i c h are h i g h l y n o n h n e a r s y s t e m s . T h e h y d r a u h c a U y a c t u a t e d h e a v y d u t y m a n i p u l a t o r s a r e u s e d e x t e n s i v e l y i n l a r g e resource b a s e d i n d u s t r i e s , a n d a n y i m p r o v e m e n t i n e f f i c i e n c y m a y result i n m a j o r  financial  benefits. T h e r e f o r e , t h e r e s u l t s i n C h a p t e r 4 t h a t a d v a n c e  t h e s t a t e of t h e a r t w i U be s t a t e d n e x t ( K o t z e v et at ^^):  — T h i s w o r k e x a m i n e d t h e effect of n o n h n e a r i t i e s i n t h e a p p h c a t i o n o f G P C t o a w i d e r a n g e of h y d r a u h c a U y a c t u a t e d m a n i p u l a t o r s . -  S p e c i a l a t t e n t i o n is g i v e n to the m a x i m u m o u t p u t h o r i z o n . T h e w o r k i n t r o d u c e s a n o n - h n e a u t o m a t i c change of t h e m a x i m u m o u t p u t h o r i z o n so t h a t t h e t r a n s i e n t  response c a n be s u f f i c i e n t l y fast a n d u n d e s i r a b l e overshoots a v o i d e d . T h e s e l e c t i o n of o t h e r G P C d e s i g n p a r a m e t e r s is also a d d r e s s e d .  • E x p e r i m e n t a l results f r o m a n open loop experiment on a heavy d u t y m a n i p u l a t o r , a 2 1 5 B C a t e r p i l l a r , i n d i c a t e t h a t the cost f u n c t i o n b e h a v i o r i n o p e n l o o p does n o t v a r y s t r o n g l y e n o u g h for m i s - m o d e h n g t o be r e h a b l y d e t e r m i n e d . T h i s has also b e e n v e r i f i e d by n u m e r i c a l s i m u l a t i o n s w i t h o t h e r a p p l i c a t i o n s . A closed l o o p a p p r o a c h is n e e d e d .  7.2  S u g g e s t i o n s for F u t u r e W o r k  T h e g o a l of t h i s thesis was t o s t u d y t h e b e h a v i o r of a s y s t e m c o n t r o U e d w i t h a n a d a p t i v e c o n t r o l a l g o r i t h m w h e n p l a n t m i s - m o d e h n g o c c u r s , u n d e r s t a n d t h e b e h a v i o r , a n d suggest  a  m e t h o d t o o v e r c o m e p r o b l e m s t h a t arise, s u c h as p o o r p e r f o r m a n c e a n d i n s t a b i h t y . S u c h a s t u d y was c o n d u c t e d a n d a m e t h o d t h a t p r o v i d e s g o o d r e s u l t s is p r e s e n t e d . T h e s c o p e o f t h e i n v e s t i g a t i o n c a n be m a d e b r o a d e r for f u r t h e r g e n e r a l i z a t i o n of t h e r e s u l t s :  • M o r e adaptive algorithms, predictive a n d non-predictive, should be tried. • M o r e s y s t e m s s h o u l d b e c h e c k e d , not o n l y f r o m t h e r o b o t i c f a m i l y . T h i s c o u l d g e n e r a l i z e t h e c o n c l u s i o n s o n t h e b e h a v i o r of those s y s t e m s i n m i s - m o d e h n g a n d m a y c o m e u p w i t h p a r a m e t e r s t o c h a r a c t e r i z e i t . F o r e x a m p l e t h e t i m e c o n s t a n t of t h e s y s t e m c o u l d have a n i n f l u e n c e o n t h e r e s u l t s .  • A closed l o o p e x p e r i m e n t w i t h t h e o r d e r d e t e r m i n a t i o n m e t h o d i m p l e m e n t e d s h o u l d b e d o n e w i t h a 2 1 5 5 C a t e r p i U a r or a s i m i l a r s y s t e m to f i n d a d v a n t a g e s a n d d i s a d v a n t a g e s , since t h e a i m is to i m p l e m e n t i t for i n d u s t r i a l use.  • M o r e a t t e n t i o n s h o u l d b e g i v e n t o o t h e r possible o r d e r d e t e r m i n a t i o n m e t h o d s a n d t h e i r p e r f o r m a n c e s h o u l d be c o m p a r e d w i t h t h e present one.  • T h e s e n s i t i v i t y of t h e i d e n t i f i c a t i o n a l g o r i t h m a n d its i n f l u e n c e o n t h e cost f u n c t i o n s h o u l d be c h e c k e d .  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Vibrations  Theory  and  Applications,  Appendix  A  E x p e r i m e n t a l R e s u l t s for t h e H y d r a u l i c A c t u a t e d  A.l  Manipulator  Introduction  A t U B C ( 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 ) a n e x c a v a t o r , C a t e r p i l l a r 2 1 5 B , w h i c h is e n gaged i n a teleoperation project, Sepehri  ^ a s a v a i l a b l e t o us for s o m e e x p e r i m e n t s .  The  g o a l i n t h e e x p e r i m e n t was t o i d e n t i f y t h e d y n a m i c m o d e l of t h e m a n i p u l a t o r w i t h o p e n l o o p , check t h e b e h a v i o r of t h e cost f u n c t i o n ( E q u a t i o n 5.1), a n d c o m p a r e i t w i t h t h e r e s u l t s o f a s i m u l a t i o n of a s i m i l a r m a n i p u l a t o r c o n t r o l l e d w i t h G P C , as d e s c r i b e d i n C h a p t e r 4 a n d C h a p t e r 5.  T h e r e s u l t s show t h a t after t h e cost f u n c t i o n b e c o m e s flat at h i g h o r d e r s , i t is  easy t o choose a m i s - m a t c h e d o r d e r for the s y s t e m . B u t r e s u l t s i n C h a p t e r 5 for t h e t w o h n k m a n i p u l a t o r w i t h t h e h y d r a u h c a c t u a t o r show t h a t i f t h e o r d e r is w r o n g l y c h o s e n , t h e n t h e instabihties can occur.  A.2  D e s c r i p t i o n of t h e  System  T h e C a t e r p i l l a r 2 1 5 B e x c a v a t o r is a m o b i l e t h r e e degree of f r e e d o m m a n i p u l a t o r . T h e h n k s are t h e " S w i n g " , w h i c h is t h e base t h a t r o t a t e s . T h e " B o o m " a n d t h e " S t i c k " are t w o h n k s operated through hydrauhc cyhnders.  T h e e n d effector, t h e " b u c k e t " , w h i c h is u s e d t o d i g  a n d c a r r y h e a v y loads is also o p e r a t e d w i t h a h y d r a u h c a c t u a t o r . excavator's structure.  T h e m o t i o n s of t h e " B o o m "  F i g u r e A . l describes  a n d " S t i c k " are c o u p l e d b y  v a l v e s , w h i c h aUow for a faster m o v e m e n t of one h n k w h e n t h e o t h e r is slower.  the  cross-over  T h e e x p e r i m e n t for t h i s w o r k is t h e i d e n t i f i c a t i o n of t h e d y n a m i c s of t h e " B o o m " i t s a c t u a t o r t h a t were o p e r a t e d .  The couphng  b e t w e e n t h e valves was e h m i n a t e d .  and The  e s t i m a t i o n a l g o r i t h m u s e d was a R e c u r s i v e L e a s t S q u a r e s . T h e i n p u t is t h e s i g n a l f r o m t h e s p o o l v a l v e , a n d t h e o u t p u t , f r o m t h e a n g u l a r p o s i t i o n of t h e h n k .  A.3  Results  S i x r u n s were m a d e w h e r e i n e a c h , t h e n e x t i n p u t c h a r a c t e r i s t i c s were g i v e n :  • S l f - a m p h t u d e of 1.0 volt a n d f r e q u e n c y of 4 seconds. • S 2 f - a m p h t u d e of 1.5 volt a n d f r e q u e n c y of 3 seconds. • S 3 f - a m p h t u d e of 1.5 volt a n d f r e q u e n c y of 2 seconds. • S 4 f - a m p h t u d e of 1.3 volt a n d f r e q u e n c y of 6 seconds.  • R l f - r a n d o m i n p u t , m a x i m u m a m p h t u d e of 2 v o l t . • R 2 f - r a n d o m i n p u t , m a x i m u m a m p h t u d e of 3 v o l t .  Figures A . 2  A.3  A.4  A.5  A.6  A . 7 present t h e b e h a v i o r of t h e m e a s u r e d o u t p u t , t h e  e s t i m a t e d m o d e l o u t p u t , a n d t h e cost f u n c t i o n w h i c h is a n i n d i c a t i o n of t h e e r r o r b e t w e e n t h e t w o o u t p u t s . I n aU s i x cases, t h e d r a w i n g of t h e m e a s u r e d o u t p u t v s . t h e m o d e l o u t p u t show v e r y h t t l e difference b e t w e e n t h e t w o , as does t h e cost f u n c t i o n w h i c h grows fast t o a v a l u e a n d d r i f t s s l o w l y f r o m i t d u e t o t h e n o n h n e a r i t i e s w h i c h are not m o d e l e d i n t h e h n e a r m o d e l for t h e e s t i m a t i o n a l g o r i t h m . F i g u r e A . 8 presents t h e b e h a v i o r of t h e cost f u n c t i o n for t h e f o u r S i f r u n s . T h e values of t h e cost f u n c t i o n i n aU r u n s was t a k e n after 2000 s a m p h n g steps. T h e r e s u l t s c o n f i r m t h e d i s c u s s i o n i n C h a p t e r 2 t h a t t h e cost f u n c t i o n for o p e n l o o p i d e n t i f i c a t i o n w i l l h a v e h i g h values for u n d e r - m o d e h n g a n d w i l l r e a c h a p l a t e a u for h i g h e r  o r d e r s of t h e e s t i m a t e d m o d e l . T h e same goes for t h e R i f e x p e r i m e n t s , as s h o w n i n F i g u r e A . 9 , t h e m e a s u r e d a n d t h e m o d e l o u t p u t agree w i t h e a c h o t h e r q u i t e w e l l , a n d t h e cost f u n c t i o n , after 2000 s a m p h n g steps, a g a i n has h i g h values for u n d e r - m o d e h n g a n d reaches a p l a t e a u for h i g h e r o r d e r values. I n C h a p t e r 5 the cost f u n c t i o n for t h e closed l o o p a l g o r i t h m b e h a v e s differently. modehng.  T h e r e is a clear difference b e t w e e n t h e values for u n d e r - m o d e h n g a n d c o r r e c t F o r o v e r - m o d e h n g , t h e r e c o u l d also be a c l e a r difference f r o m c o r r e c t  r e s u l t i n g i n i n s t a b i h t y of t h e s y s t e m i f left u n a t t e n d e d .  modehng,  A.4  Conclusions  T h i s e x p e r i m e n t was d o n e w i t h a n e x c a v a t o r w h i c h is u s e d e x t e n s i v e l y i n t h e forest a n d c o n s t r u c t i o n i n d u s t r y . T h e s y s t e m is a h i g h l y n o n h n e a r one, a n d i t is therefore of i n t e r e s t t o check t h e b e h a v i o r of t h e cost f u n c t i o n w h e n o p e n l o o p i d e n t i f i c a t i o n is d o n e . T h i s c o n f i r m s t h e fact t h a t even w h e n a n i d e n t i f i c a t i o n is d o n e o p e n l o o p a n d a n over m o d e l e d m o d e l is chosen t h e closed l o o p c o n t r o l l e d s y s t e m m a y r u n i n t o p e r f o r m a n c e a n d s t a b i h t y p r o b l e m s . T h u s , t h e r e is a n e e d to detect o n - h n e a significant m o d e l m i s - m a t c h as m e n t i o n e d i n C h a p t e r 5.  T h e r e s u l t s show (see F i g u r e s  A . 8 and  A . 9 ) , t h a t w h e n i d e n t i f i c a t i o n is d o n e i n  o p e n l o o p t h e b e h a v i o r o f t h e cost f u n c t i o n is not s u c h t h a t a n e r r o r i n m o d e l s t r u c t u r e c a n b e e a s i l y d e t e r m i n e d . T h u s a closed l o o p a p p r o a c h is n e e d e d . simulations i n chapter  5.  T h i s has also b e e n s h o w n i n  Caterpillar 215B oxcavotor — one link identification — s1f_3 0.3 u ^ 0.2 H  I  1-0.1  I-  I  "  0.0-1  I 3  o. c  500  1000 1500 no. of time steps  2000  2500  500  1000 1500 no. of time steps  2000  2500  -0.2-0.3  500  1000 1500 no. of time steps  2000  2500  Coterpillor 215B axcovotor — one link identification — s2f_3 .50 o .45^.40-  c « 'J  .  ll  l' '  E 0 u  a m  .  0.2-f  O.H 0.0-1  Ô -0-H J30-  •» -0.2 H  .25 .20  1  ri  1  ^ -0.3 H  Vn  1000  2000 3000 no. of tinne steps  4000  1000  2000 3000 no. of time steps  4000  .200 .195.190-  8 -185 H o  .180.175  -0.4  1000  2000 3000 no. of time steps  4000  u J  I, 1''  ,  Caterpillar 215B axcovotor — one link identification — s3f_3 0.3' o ^ 0.2"c E 0.1u u "5. 0.0 tr,  I  '•B  0 I  «-0.2H 1  LU. 500  2000 1000 1500 no. of time steps  500  1000 1500 2000 no. of time steps  ^-0.3 H  a.  c 2500 " -0.4  2500  500  2000 1000 1500 no. of time steps  2500  Coterpillor 215B axcovotor — one link identification — s4f_3 r-' 0.3^  0.2 H  g  0.1  3a.  —I  f- -I  0.0-1  §.-0.1  CI)  I ^ -0.2  500  .222-  1  1000 1500 2000 no. of time steps 1  1  2500  c  "" -0.3 1  -  .220-  J  3000 3 a.  .218-  -  -216-  -  1 .214-  -  8 .210.208.206.2040  1  500  1  1  1  1000 1500 2000 no. of time steps  1  2500  3000  500  1000 1500 2000 no. of time steps  2500  3000  Caterpillar 215B oxcovator - one link identification — r1f_3  1000  2000 3000 4000 no. of time steps  5000  1000  2000 3000 no. of time steps  5000  ^.070 H o .065 - .• •  4000  1000  2000 3000 4000 no. of time steps  5000  Caterpillar 215B oxcavotor - one link identificotion I I rrr 0.4  r2f_3  E' u  0.2 H  0.0 4  -0.2 H  1000  2000 3000 no. of time steps  4000  1000  2000 3000 no. of time steps  4000  -0.4  1000  2000 3000 no. of time steps  4000  Appendix  B  M o d a l A n a l y s i s for a C a n t i l e v e r B e a m  T h e a n a l y s i s is b a s e d o n B i s p h n g h o f f et. a l E q u a t i o n ( P D E ) for t h e d e f l e c t i o n w(x,t)  a n d T s e et. a l  The Partial Differential  i n a cantilever b e a m , considering b e n d i n g a n d  s h e a r i n g s t r a i n s a n d n e g l e c t i n g shear d e f o r m a t i o n a n d r o t a r y i n e r t i a effects, is: (Pw  d^w  W h e r e E is Y o u n g ' s m o d u l u s , I is t h e cross s e c t i o n a l m o m e n t of i n e r t i a of t h e a r m , a n d m is t h e mass p e r u n i t l e n g t h . T h e a b o v e P D E is s e p a r a b l e , so l e t :  w{x,t)  dp  W h e r e 6" =  + u;\  (B.3)  = 0  G e n e r a l s o l u t i o n s for t h e a b o v e e q u a t i o n s are:  q{t)  (f>{x) = Csinh{bx)  = Asin{ujt)  <^(0) = 0,  ^  + Dcosh{bx)  = 0  (B.5)  + Bcos{ut)  T h e b o u n d a r y c o n d i t i o n s for c a n t i l e v e r b e a m are: at a; = 0 :  (B.2)  <l>{x)q{t)  + Esin{bx)  + Fcos{bx)  (B.6)  at X = L :  dx^  dx^  ~~ ^'  "  ^  S o l v i n g t h e a b o v e e q u a t i o n s w i t h the b o u n d a r y c o n d i t i o n s result w i t h : T h e n a t u r a l frequencies b y : = 1  cos{bL)cosh(bL) W h e r e bL = 0.59 697r,  1.49427r,  (B.7)  |7r,  T h e m o d e shapes b y :  ^{x)  = D[A{sinh{bx)  — sin[bx))  + cosh[bx)  — cos{bx)  (B.8)  Where: A  =  sin(bL)  —  sinh(bL)  cos(bL)  +  cosh{bL)  (B.9)  D is a n o r m a h z e d coefficient w h e r e (j>{L) = 1 so: ^  _ cosh{bL)  +  cos{bL)  2sinh{bL)cosh{bL)  (B.IO)  A d y n a m i c m o d e l has t h r e e i n t e g r a l s as a f u n c t i o n of ^(as), w h i c h are ( a s s u m i n g u n i f o r m mass d i s t r i b u t i o n m ) :  11 — m • I Jo  x^(x)dx  12 = m-  (t>'{x)dx =  73 =  For a r m w i t h the next d a t a : L = 1  [meter  E I = 574.024 [N It = 0.2817 m  [kg-m'  = 1  m = 0.8451  -m']  [kg/m  EI  I Jo  I  Jo  dx'  =  2mD ——P mD'L  (B.ll)  T a b l e B . l presents t h e d a t a for t h e first five m o d e s . F i g u r e B . l shows t h e m o d a l s h a p e s for the cantilever beam.  Mode  6/  Delta  D  1  1.8752  91.644  -0.7266  0.5  h 0.2404  h 0.2113  h 1774.4  2  4.6942  574.294  -0.9805  -0.5  -0.03835  0.2113  69681.4  3  7.854  1606  -0.9999  0.5  0.00138  0.2113  544928  4  10.9956  3151  -0.99998  -0.5  -0.00699  0.2113  2097707  5  14.137  5208.77  -0.9999  0.5  0.00423  0.2113  5732162  (rad)  w  (rad/dec)  T a b l e B . l : D a t a for t h e first five m o d e s of a c a n t i l e v e r b e a m  m o d e  » h o p «  «  m o d e  m o d e  [meters]  shape  >  1  no  no  3  [meters]  shope  »  m o d e  2  [meters]  shope  >  no  no  4  [meters]  

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