UBC Theses and Dissertations

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UBC Theses and Dissertations

Automatic model structure determination for adaptive control 1992

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A U T O M A T I C 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 F O R A D A P T I V E C O N T R O L B y A n a t K o t z e v B . S c . C h e m i c a l E n g . , T e c h m o n (Israel Ins t i tu te of T e c h n o l o g y ) , H a i f a , Israe l , 1979 M . S c . C h e m i c a l E n g . , Techn ion , H a i f a , I s r a e l , 1982 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S M E C H A N I C A L E N G I N E E R I N G W e accept this thesis as c on forming to the required s tandard 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 J u n e 1992 (c) A n a t K o t z e v , 1992 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying 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 fl^f^ 1^1 ̂  n - j ^ DE-6 (2/88) A b s t r a c t T h i s work is a s t u d y of a d a p t i v e l y contro l l ed systems w i t h p lant m o d e l s t ruc tures t h a t m a y vary due to changing opera t ing cond i t i ons . M o s t closed loop adapt ive contro l a l g o r i t h m s use ident i f i ca t i on methods for d e t e r m i n a t i o n of the parameters i n fixed s t r u c t u r e m o d e l s . T h o s e parameters , once e s t i m a t e d , are assumed to be correct a n d uncerta int ies i n the va lues are ignored . I f the s t ruc ture of the p lant d y n a m i c s changes on- l ine , the incorrect m o d e l c a n l e a d to poor per formance and ins tab i l i t i e s . T h e adapt ive a l g o r i t h m used i n th is work is the 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 ) a l g o r i t h m . It is reported to be capable of h a n d l i n g a n u m b e r of s imul taneous p r o b l e m s a n d therefore was chosen. A l o n g w i t h h a n d l i n g on-Hne changes of parameters , 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 p lants , open loop unstab le p lants , p lants w i t h b a d l y d a m p e d poles, p lants w i t h variable or u n k n o w n t i m e delay, and plants w i t h u n k n o w n order . T h e goa l of this research is to invest igate a n d s t u d y G P C w i t h the on-Hne changes i n the m o d e l s t r u c t u r e of the p l a n t , a n d corresponding changes i n the order of the e s t i m a t e d m o d e l for G P C a n d the s t ruc ture of the contro l ler a n d as well as to propose a m e t h o d t h a t detects on-Hne, the need for m o d e l order changes a n d determines the correct one. T h e r e are at least two m a j o r sources for s t ruc ture var iat ions i n the e s t i m a t e d m o d e l . T h e first is the m o d e l ac tua l ly be ing t i m e var iant a n d the second resu l t ing f r o m the use of i n h e r e n t l y nonl inear systems and m i s - m o d e l i n g . T w o app l i ca t i ons e x e m p l i f y i n g these var iants were selected to examine the techniques developed i n the thesis . T h e first is a single flexible Hnk m a n i p u l a t o r , whose changes i n m o d e l s t ruc ture are due to new e x c i t e d v i b r a t i o n modes . T h e second is a two link r i g id m a n i p u l a t o r w i t h h y d r a u l i c ac tuators caus ing the s y s t e m to be h i g h l y non l inear , whose m o d e l cou ld change due to changes i n o p e r a t i n g po in ts . T h e effect of m i s - m o d e l i n g on the t o t a l system per formance and s tab i l i t y was assessed. A cost func t i on was used as a measure of the closed loop contro l led sys tem r e a c t i o n to u n d e r , correct and over -model ing . Its effectiveness i n terms of s t a b i l i t y a n d p e r f o r m a n c e was measured i n context of the two app l i ca t i ons . 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 o p e n loop ident i f i ca t i on of the d y n a m i c m o d e l of a 215B C a t e r p i l l a r , an excavator t y p e m a c h i n e , conf irms the s tudy of the behavior of the cost func t i on for those condi t ions . B a s e d on the behavior of the cost f u n c t i o n a new a l g o r i t h m was developed. T h e 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 - l ine , changes to the m o d e l order. 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 contro l l ed w i t h t h e G P C a l g o r i t h m . T h e results show that good per formance and s tab i l i t y can be ach ieved . T h e m a i n contr ibut ions of this work are: • T h e M O D a l g o r i t h m w h i c h based on the behavior of a cost f u n c t i o n , corrects o n - l i n e m i s - m o d e l i n g of adapt ive ly contro l led systems whi le m a i n t a i n i n g good per f o rmance . • 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 . O n - l i n e a u t o m a t i c change of the 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 to achieve suf f i c ient ly fast t rans ient response a n d avo id overshoots. • E x p e r i m e n t a l d a t a f r o m a 215B C a t e r p i l l a r m a n i p u l a t o r proved the need for a c losed loop approach . Uo, T a b l e of C o n t e n t s A b s t r a c t i L i s t of Tables v i i list of tables v i i L i s t of F i g u r e s v i i i list of figures x i i 1 I N T R O D U C T I O N A N D S T A T E M E N T O F O B J E C T I V E S 1 1.1 I n t r o d u c t i o n 1 1.2 Ob jec t ives and M o t i v a t i o n 2 1.3 T h e Thes is O u t h n e 3 1.4 Thes is C o n t r i b u t i o n 4 2 R E V I E W O F P R E V I O U S W O R K 6 2.1 O u t h n e 6 2.2 A d a p t i v e C o n t r o l - G e n e r a l D e s c r i p t i o n 8 2.2.1 I n t r o d u c t i o n 8 2.2.2 Self T u n i n g Regulators - ( S T R ) 9 2.2.3 S t a b i h t y 10 2.3 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 11 2.3.1 I n t r o d u c t i o n 11 2.3.2 T h e G P C A l g o r i t h m 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 and 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 Prev i ous 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 3 S I N G L E F L E X I B L E L I N K M A N I P U L A T O R 28 3.1 O u t h n e : 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 I n t r o d u c t i o n 28 3.2.2 E q u a t i o n s of M o t i o n for the Single F l e x i b l e L i n k 29 3.2.3 T h e State 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 Discrete T i m e M o d e l s for Different N u m b e r of M o d e s . . . 36 3.2.6 C o n t r o l S t rategy 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 38 3.3 A n a l y s i s and Resu l t s of S i m u l a t i o n and C o n t r o l W o r k P e r f o r m e d 38 3.3.1 I n t r o d u c t i o n 38 3.3.2 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 on the C o n t r o l l e d F l e x - ib le L i n k 39 3.3.3 Use of an E s t i m a t i o n Cos 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 the S t r u c t u r e of the P l a n t ' s M o d e l 49 3.3.4 Effects of O n L i n e Changes i n M o d e l order 51 3.3.5 C o m p a r i s o n Between the 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 . . 55 3.4 C o n c l u s i o n s 62 4 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 A T O R S 65 4.1 R i g i d 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 t u a t o r s 68 4.1.1 I n t r o d u c t i o n 68 4.1.2 E q u a t i o n s of M o t i o n for the 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 E q u a t i o n s of M o t i o n of the H y d r a u h c A c t u a t o r 70 4.2 C o n t r o l Strategy 74 4.2.1 I n t r o d u c t i o n 74 4.2.2 C o n t r o l S t rategy for the T w o h n k R i g i d M a n i p u l a t o r 74 4.3 A n a l y s i s and Resu l t s of S i m u l a t i o n 76 4.3.1 S y s t e m Parameters 76 4.3.2 O p e n L o o p A n a l y s i s 77 4.3.3 S i m u l a t i o n S t u d y and Resul ts 79 4.3.4 Effects of O n - h n e Changes i n M o d e l O r d e r 93 4.4 Conc lus i ons 93 5 M O D E L O R D E R D E T E R M I N A T I O N 95 5.1 I n t r o d u c t i o n 95 5.2 C o s t F u n c t i o n - For D e t e c t i o n O f T h e M o d e l S t r u c t u r e 96 5.2.1 T h e Cost 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 102 5.2.2 T h e Cost F u n c t i o n for the 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 113 5.3 Reasons 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 - 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 139 6 I M P L E M E N T A T I O N O F T H E M O D A L G O R I T H M 145 6.1 I m p l e m e n t a t i o n 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 145 6.1.1 T h e M e t h o d 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 147 6.1.2 T h e M e t h o d F o r 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 t u a t o r s 166 6.2 C o m p a r i s o n of method ' s Resul ts w i t h O t h e r W o r k 176 6.3 C o n c l u s i o n s 179 7 C O N C L U S I O N S A N D S U M M A R Y 181 7.1 M a i n Resu l t s of the Thes is 181 7.2 Suggestions for F u t u r e W o r k 183 B i b l i o g r a p h y 185 A p p e n d i c e s 191 A E x p e r i m e n t a l R e s u l t s for the 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 192 A . l I n t r o d u c t i o n 192 A . 2 D e s c r i p t i o n of the S y s t e m 192 A . 3 Resul ts 193 A . 4 Conc lus i ons 196 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 205 L i s t of Tables 2.1 Différent final goals and specif ications for ident i f i cat ion cases 16 3.1 N o . of f lexible modes vs. order of sys tem 64 B . l D a t a for the first five modes of a canti lever 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 regulator , ( A s t r o m ^) 7 2.2 G e n e r a l procedure of process ident i f i ca t i on , ( 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 the single h n k flexible a r m 30 3.2 T w o mode flexible h n k w i t h two mode es t imator m o d e l 40 3.3 T h e angle 6 a n d i ts derivatives 41 3.4 T h e generaUzed goordinate a n d its der ivat ives 42 3.5 T h e generahzed coordinate q2 a n d its der ivat ives 43 3.6 T h e torque i n p u t 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 over -mode l l ing 47 3.9 O v e r - m o d e l l e d 2 mode h n k w i t h 8''' order es t imator 48 3.10 E s t i m a t o r cost func t i on for correct modeUing 50 3.11 O n h n e change of es t imator m o d e l - correct to under - modeUng 52 3.12 E s t i m a t o r cost func t i on for the on hne change i n m o d e l 53 3.13 O n Une change of e s t imator m o d e l - under modeUing to correct 54 3.14 E s t i m a t o r cost f u n c t i o n for the on Une change i n m o d e l 56 3.15 O n l ine change of e s t imator m o d e l - under mode lUng to correct 57 3.16 E s t i m a t o r cost func t i on for the on Une change i n m o d e l 58 3.17 G P C and est imator cost func t i on - onUne correct to u n d e r - m o d e U n g 59 3.18 G P C and es t imator cost func t i on - onUne under -modeUng to correct (0.1 seconds) 60 3.19 G P C and es t imator cost f u n c t i o n - onUne under -modeUng to correct (0.2 seconds) 61 4.1 C o n f i g u r a t i o n of the two h n k m a n i p u l a t o r 69 4.2 E l e c t r o h y d r a u h c ac tuator 72 4.3 C o n t r o l strategy for the two h n k m a n i p u l a t o r 75 4.4 ^1,^1,^1 for square 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 ac t ion and spool d isplacement for di a n d 62 82 4.7 Pressures for Oi a n d 62 83 4.8 C o n t r o l ac t ion a n d spoo l d isplacement for h y d r a u h c hnear i zed m o d e l 85 4.9 Pressures for 0i a n d 62 for h y d r a u h c hnear ized m o d e l 86 4.10 T h e effect of higher values of A 2̂ (lower case) 87 4.11 ^1 a n d ^2 for ATĵ ,̂  = 50 a n d ATẑ ,̂  = 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 = 10 a n d N^^^ = 1 0 92 5.1 Cos t func t i on behavior for open loop f lexible h n k 98 5.2 Cos t func t i on behav ior for closed loop f lexible h n k 99 5.3 Schemat i c descr ipt ion of the C . F . behavior for the different apphcat ions . . . 101 5.4 Cos t func t i on behavior for 2 mode L i n k a n d 2 mode e s t i m a t e d m o d e l 103 5.5 O u t p u t error behavior for 2 m o d e hnk and 2 mode e s t i m a t e d m o d e l 104 5.6 Cos t funct i on behavior for 2 mode h n k and 1 mode 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 105 5.7 Cos t func t i on behavior for 2 mode hnk and 0 mode 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 106 5.8 Cos t funct i on behavior for 2 mode h n k a n d 3 mode e s t i m a t e d m o d e l 107 5.9 Cos t funct i on behavior for 2 mode h n k and 4 mode e s t i m a t e d m o d e l 108 5.10 Cost f u n c t i o n behavior for 1 mode Hnk 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 behavior for 1 mode Hnk a n d 0 mode e s t i m a t e d m o d e l I l l 5.12 Cos t func t i on behavior for 1 mode Hnk and 0 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 112 5.13 Cost f u n c t i o n behavior for 1 mode 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 Cost f u n c t i o n behav ior for 1 mode Hnk a n d 3 mode e s t i m a t e d m o d e l 115 5.15 Cost f u n c t i o n behavior for 1 mode Hnk a n d 4 mode e s t i m a t e d m o d e l 116 5.16 6i a n d 62 behavior for 3 mode m o d e l a n d 3 mode e s t i m a t e d m o d e l 118 5.17 Cost f u n c t i o n behav ior for 3 mode hydrauHc Hnks (Hnearized p lant m o d e l ) a n d 3 mode e s t i m a t e d models 119 5.18 Cost f u n c t i o n behavior for 3 mode hydrauHc Hnks (nonHnear p lant m o d e l ) a n d 3 mode es t imated models 120 5.19 Cost func t i on behav ior for 3 mode hydrauHc Hnks a n d 2 mode e s t i m a t e d m o d e l 121 5.20 01 a n d O2 behavior for 2 mode m o d e l a n d 3 mode e s t i m a t e d m o d e l 122 5.21 Cos t f u n c t i o n behavior for 4 mode hydrauHc Hnks a n d 4 mode e s t i m a t e d m o d e l 123 5.22 Cos t f u n c t i o n behavior for 5 mode hydrauHc Hnks a n d 3 mode e s t i m a t e d m o d e l 125 5.23 6i a n d 62 behav ior for 5 mode m o d e l and 3 mode e s t i m a t e d m o d e l 126 5.24 F l o w chart of an adapt ive contro l loop w i t h m o d e l order d e t e r m i n a t i o n . . . . 140 5.25 F l o w chart of the order d e t e r m i n a t i o n procedure 141 6.1 Regions for under , over a n d correct modeHng 149 6.2 T h e o u t p u t behavior of a two mode flexible Hnk e s t i m a t e d i n i t i a U y w i t h an order 2 m o d e l 152 6.3 O r d e r changes of the e s t imated m o d e l for a two mode flexible Hnk e s t i m a t e d i n i t i a l l y w i t h order 2 153 6.4 T h e output behavior of a two mode f lexible hnk es t imated i n i t i a l l y w i t h a n order 4 mode l 155 6.5 O r d e r changes of the e s t i m a t e d m o d e l for a two mode 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 4 156 6.6 T h e cost funct i on der ivat ive behavior of a two mode 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 157 6.7 T h e cost func t i on behav ior 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 a n order 4 m o d e l 158 6.8 3 more cases of ou tput behav ior of a two mode 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 a n order 4 m o d e l 159 6.9 3 more cases of order changes of the es t imated m o d e l for a two mode 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 4 160 6.10 T h e o u t p u t behavior of a one mode 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 a n order 2 m o d e l 162 6.11 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 mode 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 2 163 6.12 T h e o u t p u t behavior of a one m o d e flexible h n k i n i t i a l l y over -mode led . . . . 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 mode 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 10 165 6.14 T h e o u t p u t behavior of a h y d r a u h c a c t u a t e d two h n k m a n i p u l a t o r i n i t i a l l y under -mode led 169 6.15 O r d e r changes of the e s t imated m o d e l for hydrauhc a c t u a t e d two h n k m a n i p - u l a t o r i n i t i a l l y under -mode led 170 6.16 T h e cost func t i on der ivat ive behav ior 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 behavior 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 172 6.18 T h e output behavior of a h y d r a u h c ac tuated two h n k m a n i p u l a t o r O v e r - modeled 173 6.19 O r d e r changes of the e s t i m a t e d m o d e l for h y d r a u h c ac tuated two l i n k m a n i p - u la tor over-modeled 174 6.20 Per formance of a sys tem i n i t i a l l y mis -mode led on a larger t i m e scale 175 A . l C a t e r p i l l a r 215B excavator 194 A . 2 I n p u t output behavior for S l f 197 A . 3 I n p u t output behavior for S2f 198 A . 4 I n p u t o u t p u t behavior for S3f 199 A . 5 I n p u t output behavior for S4f 200 A . 6 I n p u t o u t p u t behavior for R l f 201 A . 7 I n p u t o u t p u t behavior for R 2 f 202 A . 8 C o s t func t i on behav ior for the S i f cases 203 A . 9 C o s t f u n c t i o n behavior for the R i f cases 204 B . l M o d a l Shapes for a C a n t i l e v e r B e a m 209 A c k n o w l e d g m e n t s I wou ld l ike to t h a n k m y superv isor , Professor D a l e B . Cherchas for the s u p p o r t , g u i d a n c e a n d encouragement he p r o v i d e d i n the development of this work . I w o u l d hke to 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 to Professor Pe t e r D . L a w r e n c e , for f r u i t f u l work and valuable suggestions. I a m h igh ly grate ful to m y colleague D o u g L a t o r n e l l , for the f r i endship , suppor t a n d h e l p I received d u r i n g a l l the years I worked on this thesis . A l s o , I w o u l d hke to t h a n k A l a n Steeves for his good advice , pat ience a n d support i n the use of the depar tment c o m p u t e r s y s t e m . A l l the staff i n the depar tment 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 me w i t h a l l t h e h e l p I needed, a n d always w i t h a smi le . T h e V A X S t a t i o n 3200 c o m p u t e r sys tem 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 grant 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 Ins t i tu te ( A S I ) . L a s t but not least I wou ld Hke to t h a n k the T e c h n i o n (Israel I n s t i t u t e of T e c h n o l o g y ) , espec ia l ly express m y warmest g ra t i tude to Professor R a m L a v i e , f r o m the fa cu l ty of C h e m i c a l E n g i n e e r i n g , who m a d e the last per i od of w o r k i n g on this thesis s m o o t h a n d c o m f o r t a b l e by g r a n t i n g me the status of a guest fellow at the T e c h n i o n . A key f igure to c o n t i n u i n g m y research work i n Israel was prov ided , very profess ional ly , by the c o m p u t i n g center , b y M i r i a m B e n - H a i m and B e n Pashkof f - T h a n k Y o u . To M i l u s h R o i a n d R e u t C h a p t e r 1 I N T R O D U C T I O N A N D S T A T E M E N T O F O B J E C T I V E S 1.1 I n t r o d u c t i o n T h e research i n this thesis deals w i t h adapt ive c o n t r o l systems whose p lant 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 vary due to changing opera t ing condi t ions . T h e r e are n u m e r o u s e x a m p l e s of systems t h a t need adapt ive contro l a lgor i thms , ( 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 whose moment of i n e r t i a m a y vary w i t h i n a w o r k i n g cyc le . A flexible robot m a y have u n e x p e c t e d modes of v i b r a t i o n o c c u r r i n g a n d changing its m o d e l s t r u c t u r e . Process contro l also has changes i n d y n a m i c s , w h i c h depend on opera t i ng parameters , such as flow t h r o u g h t a n k s and pipes that change w i t h p r o d u c t i o n rate . T h e s t r u c t u r e of the plant 's m o d e l is usua l ly d e t e r m i n e d by i ts order a n d the n a t u r e of i t s nonhnear t e rms . T h e m o d e l general ly used for the adapt ive a lgor i thms is h n e a r a n d therefore its s t r u c t u r e is a c tua l l y i ts order. I n most s tab ihty proofs for adapt ive sys tems , the bas ic a s s u m p t i o n is that the m o d e l order is k n o w n , or at least the upper b o u n d of the sys tem's order is k n o w n . I n the presence of a change i n the system's order , such as new s igni f icant modes i n a flexible mechan i ca l sys tem, or i n the presence of any 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 can occur due to incorrect m o d e l s t ruc ture a n d therefore incorrect parameters . T h e on-hne changes i n the p lant ' s m o d e l s t ructure i f they o c cur , m a y result i n the need t o i dent i f y those changes accord ing ly , a n d adjust the order of the m o d e l for the adapt ive a l g o r i t h m i n a d d i t i o n to the p a r a m e t e r ident i f i cat ion . T h e ident i f i ca t i on methods of a m o d e l off-hne can have the advantage of choos ing a m o d e l out of a set of proposed m o d e l s t ruc tures . T h e m o d e l , i ts s t ruc ture a n d parameters c a n l a t e r be veri f ied w i t h m o d e l v a h d a t i o n methods . M a n y closed loop adapt ive contro l a l g o r i t h m s use ident i f i cat ion methods for on-hne d e t e r m i n a t i o n of the m o d e l parameters ( A s t r o m et a l ^). W e are not aware of any effective on-Hne s t ruc ture vahdat i on methods . It is not s u r p r i s i n g that most i f not aU ident i f i ca t i on methods use a fixed m o d e l s t r u c t u r e to e s t i m a t e the parameters . In some cases, the uncertaint ies i n the values are ignored , i .e. the i d e n t i f i e d parameters are assumed to be correct , a n d are used as i f they were the t rue ones. T h i s is caUed the certainty equivalence pr inc ip le ( 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 systems have been said to be inherent ly nonHnear ( A s t r o m et a l ^) a n d reHance o n that p r inc ip l e can lead to ins tab iHty . 1.2 O b j e c t i v e s a n d M o t i v a t i o n O n e ob jec t ive of the research is to invest igate a n d s t u d y a chosen 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 changing m o d e l s t ruc ture for the p lant , a n d w i t h i t , the change i n e s t i m a t e d m o d e l order a n d of the controUer s t r u c t u r e . A second ob jec t ive is to develop a m e t h o d t o detect the need for a m o d e l order change, de termine the correct order , execute i t on-Hne a n d ensure t h a t the contro l sys tem wiU m a i n t a i n its per formance a n d stabiHty. Changes i n the order of the m o d e l a long w i t h the a c c o m p a n y i n g parameter e s t i m a t i o n are to be i n t e g r a t e d in to a c losed loop adapt ive system. ( A U ca lcu lat ions and est imat ions are done on-Hne i n rea l t i m e ) . T h e approach taken to the p r o b l e m is Hsted below: 1. A n approach to on-Hne order change detec t ion and e s t i m a t i o n was deve loped . A n as- s u m p t i o n considered is that order changes are less frequent a n d converge slower t h a n the changes i n the m o d e l parameters . M o s t ex i s t ing techniques choose the order , es t i - m a t e the m o d e l parameters , va l idate the order and i f incorrect go t h r o u g h the who le procedure again , off-Hne a n d i n open loop , ( L j u n g ^). 2. E x p e r i m e n t a l evidence f rom a real mach ine , a 2 1 5 B C a t e r p i l l a r excavator was e x a m - ined for open loop operat ion . T h e mode l o u t p u t a n d represent ing cost f u n c t i o n va lues were ca l cu lated and invest igated . ( T h e results corroborate the 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 that a closed l oop approach was needed. 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 two e x a m p h f y i n g apphcat ions . T h e Generahzed 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 chosen for t h e s tudy , since i t c la ims to effectively handle 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 - ist ics at the same t i m e ( C l a r k e 4, 5) ^ h e r e are at least two reasons for p l a n t m o d e l s t r u c t u r e var iat ions . F i r s t , the machine m o d e l i t se l f changes such as v i b r a t i o n m o d e s i n a f lexible l i n k . Second, use of a hnear representat ion of a h i g h l y nonhnear m a c h i n e m o d e l t r a c k i n g an operat ing po in t , such as a two 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 ac tuators . T h e s t ruc ture f l ex ib ihty of the single flexible h n k m a n i p u l a t o r m a y g ive r ise to modes of osc i l lat ions . D u r i n g a work cyc le of a robot , new modes can o c c u r d u e t o movement and change i n the t ip ' s l oad . T h i s resembles an ' in f in i te o rder ' 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 ac tuators resembles a ' f in i te order ' sys tem. 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 ruc tures a n d are h i g h l y nonhnear . C o u p h n g between different mot ions of the a r m ' s par t s generates nonhnear t e rms . ( see C h a p t e r 4). 1.3 T h e T h e s i s O u t l i n e C h a p t e r 2 presents a rev iew of previous work done i n areas relevant to the research. C h a p t e r s 3, 4 a n d 5 present i n de ta i l the work done t n th is research. C h a p t e r 3 deals w i t h a s ingle flexible h n k m a n i p u l a t o r , i ts d y n a m i c equations of m o t i o n , the c o n t r o l strategy, results of c losed loop s i m u l a t i o n and the effects of under a n d over -modehng. C h a p t e r 4 deals w i t h the two 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 actuators and reports on the same top ics as C h a p t e r 3. C h a p t e r 5 in troduces the chosen cost f u n c t i o n as a measure of p l a n t m i s - modehng . T h e behavior of the cost funct i on a n d its der ivat ive is invest igated . T h e c o n c l u s i o n s d r a w n lead to the proposa l of a m e t h o d that detects on-hne the need for a m o d e l o r d e r change, determines the correct order a n d executes i t wh i l e developing or m a i n t a i n i n g g o o d sys tem per formance . C h a p t e r 6 presents the i m p l e m e n t a t i o n of the M O D a l g o r i t h m o n b o t h apphcat ions and results are presented. C h a p t e r 7 discusses the conclusions d r a w n f r o m this research. A p p e n d i x A provides the exper iment results f r o m oper a t i ng the h e a v y d u t y m a n i p u l a t o r , the 215B C a t e r p i l l a r . A p p e n d i x B presents the m o d a l analysis of a c a n t i l e v e r b e a m , the results of w h i c h were used for the equat ions of the single flexible h n k i n C h a p t e r 3. 1.4 T h e s i s C o n t r i b u t i o n T h i s work is a s tudy of r obo t i c systems contro l led w i t h a n adapt ive a l g o r i t h m ( G P C ) . T h e m a i n focus is on the behavior of those systems w h e n the p lant 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 desired per formance and s tabiHty i f needed. T h e m a i n research contr ibut i ons are descr ibed here as: 1. W e developed a m e t h o d caUed the 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 , to detect mis -modeHng a n d i m p l e m e n t its correc t ion on-Hne. • A cost func t i on was s tud ied for correct , under and over -modeHng, i t was f o u n d i t has s igni f icant ly different behav ior for each case. • T h e cost funct i on behavior for correct , under a n d over -modeHng was s i m i l a r for two e x e m p l i f y ing appHcat ions . • W h e n M O D was i m p l e m e n t e d , the desired per formance was restored for m i s - modeHng for b o t h appHcat ions . • R u l e s for choosing the parameters for the M O D a l g o r i t h m were def ined. 2. W e have f ound that G P C can be successfuly i m p l e m e n t e d for heavy d u t y m a n i p u l a t o r s . • T h i s work e x a m i n e d some complex considerat ions , such as the effects o f n o n h n - earities i n the a p p h c a t i o n of G P C to a b road category 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 . • T h e work i n t r o d u c e d on-hne a u t o m a t i c change of the o u t p u t h o r i z o n ( for G P C ) so transient response can be suff iciently fast a n d undes irable overshoot a v o i d e d . 3. E x p e r i m e n t a l results f r o m an open loop exper iment on a 215B C a t e r p i U a r i n d i c a t e t h a t the cost func t i on behavior i n open loop does not vary strongly to be rehab ly d e t e r m i n e d a n d a closed loop approach is requ ired . T h i s was also veri f ied 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 other apphcat ions . C h a p t e r 2 R E V I E W O F P R E V I O U S W O R K 2.1 O u t l i n e T h i s chapter reviews some of the relevant work pubhshed i n the h t e r a t u r e cons ider ing some of the topics discussed i n this thesis . A d a p t i v e contro l i n general a n d G P C i n p a r t i c u l a r , are descr ibed . Some review on m o d e l order a n d parameter d e t e r m i n a t i o n is also g iven . A d a p t i v e contro l is the basic m o t i v a t i o n for the search for a m e t h o d to change the m o d e l s t r u c t u r e on-hne. Sect ion 2.2 describes adapt ive contro l i n general t e rms . A b lock d i a g r a m ( F i g u r e 2.1) is presented to c lar i fy how aU the components ( ident i f i ca t i on , p lant 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 the complete conf igurat ion . I n the last two decades, m u c h research on m o d e h n g , sys tem ident i f i ca t i on , 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 adapt ive contro l a lgor i thms has been done ( L j u n g ^, ^). M o s t of the w o r k concern ing m o d e l s t ruc ture d e t e r m i n a t i o n a n d its v a h d a t i o n is done off-line. T h e advantage of the off-hne methods is i n the poss ib i l i ty of choosing a m o d e l out of a set of h k e l y ones. O n the cont rary , o ther works w h i c h invo lve on-line (recursive) ident i f i ca t i on d e m a n d a n a s s u m p t i o n of a fixed m o d e l s t ruc ture , ( L j u n g ^, ^ and A s t r o m ^). A br ie f rev iew of the above 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 iven i n Sect ion 2.4. I n Sect ion 2.3 we present the 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 . T h e G P C has been used i n the work done so far , f rom w h i c h the results w i U be presented i n C h a p t e r s 3 a n d 5. F l e x i b l e s t r u c t u r e models are dealt w i t h i n this work . T h e need for flexibihty arises w h e n Process parameters Regulator parameters Estimation U Process 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 regulator , ( A s t r o m '•) the cer ta in ty equivalence pr inc ip le is used, or i n the 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 occur i n the s t ruc ture . I n order to m a i n t a i n s tab ihty of the a d a p t i v e c o n t r o l l e d sys tem, the flexible m o d e l s t ruc ture is p a r t i c u l a r l y i m p o r t a n t . Sec t i on 2.5 inc ludes a discussion of the pr ior w o r k done i n the area of hnear m o d e l ' s o r d e r d e t e r m i n a t i o n i n the 1970's and some later work o n h n e a r a n d nonhnear 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 . 2.2 A d a p t i v e C o n t r o l - G e n e r a l D e s c r i p t i o n 2.2.1 I n t r o d u c t i o n M u c h work has been done a n d pubhshed o n a d a p t i v e contro 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 and L j u n g and 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 techniques for the design of contro l systems assume t h a t the p lant and i ts e n v i r o n m e n t are k n o w n . T h i s is not often the case, since the p lant m i g h t be too complex , or bas ic r e l a t i o n s h i p s m a y not be fu l l y u n d e r s t o o d , or the process a n d the d is turbances m a y change w i t h o p e r a t i n g cond i t i ons . A d a p t i v e contro l deals w i t h the above prob lems . T h e r e are four m a i n categories of a d a p t i v e c ont ro l : 1. Sel f T u n i n g Regulators - S T R 2. M o d e l Reference A d a p t i v e Systems - 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 and the M R A S are two wide ly discussed approaches to so lv ing the p r o b l e m for p lants w i t h u n k n o w n parameters . T h e proposed research concentrates o n self t u n i n g regu la to rs . 2.2.2 Self T u n i n g R e g u l a t o r s - ( S T R ) S T R are based on a fa i r ly n a t u r a l c o m b i n a t i o n of ident i f i cat ion a n d contro l . I n F i g u r e 2.1 a b lock d i a g r a m of the s t ruc ture of a n S T R contro l loop is shown. It has two feedback l o o p s , i .e . an inner loop a n d an outer loop. T h e inner one is an o r d i n a r y feedback loop w i t h a process a n d a regulator . T h e regulator has ad justable parameters w h i c h are set by the o u t e r l o o p . T h e ad jus tments are based on feedback f r o m the process inputs and o u t p u t s . T h e o u t e r l o o p is composed of a recursive p a r a m e t e r e s t imator a n d a design ca l cu la t i on . E s t i m a t i o n s c a n be done on the process parameters or on the regulator parameters , depend ing on the 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 po int is a design m e t h o d for k n o w n p lants . S ince the p a r a m e t e r s are not k n o w n , the ir est imates are used . T h e a s s u m p t i o n is that there is a separat i on b e t w e e n ident i f i ca t i on a n d contro l , a n d the p a r a m e t e r s ' uncerta int ies are i n i t i a l l y not cons idered here. A s a s imple example , consider the p lant mode led by E q u a t i o n 2.1: y{t) + ay{t - I) = buit - 1) + eit) (2.1) W h e r e u is the i n p u t , y is the o u t p u t a n d e(t) is a sequence of independent , zero m e a n r a n d o m variables . A contro l law that w i l l give minimum variance contro l is : u{t) ^ ^yit) (2.2) If a a n d b are u n k n o w n , the algorithm by Astrom a n d W i t t e n m a r k ^ can be a p p h e d . It consists of two steps, each repeated every s a m p h n g per i od : • E s t i m a t e the parameter a i n the m o d e l : y{t) = ay{t - 1) + f3ou{t ~ 1) + e{t) (2.3) W h e r e e is the error. T h e resu l t ing est imate is à. • Use the control law: nit) = ^yit) (2.4) T h e es t imat i on of a can be done recurs ive ly and on-hne. T h e above a l g o r i t h m was also generahzed i n Astrom &: W i t t e n m a r k ^. Astrom &; 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 proposed a generahzation of the above bas ic a l g o r i t h m . S T R are not confined to m i n i m u m variance contro l . E d m u n d s Astrom et a l . ^ p r o p o s e d a lgor i thms based on pole p lacement . M u l t i v a r i a b l e formulat ions were g iven b y B o r r i s s o n 2.2.3 Stabi l i ty S t a b i h t y is a key requirement for a contro l system; however s t a b i h t y ana lys i s of a d a p t i v e systems is dif f icult because the behav ior of such systems is c omplex as a result of t h e i r nonhnear character . T h e s tab ih ty p r o b l e m can be approached i n several dif ferent ways . A l o c a l s t a b i h t y technique is of h m i t e d value since i t reveals l i t t l e about g l o b a l p roper t i e s . T h e f u n d a m e n t a l s tab ihty concept for nonhnear systems refers to the s t a b i h t y of a p a r t i c u l a r so lu t i on . O n e poss ib i l i ty is to a p p l y L y a p u n o v ' s theory ( E d g a r a n d A s t r o m ^). H o w e v e r , i t is often dif f icult to find a su i tab le L y a p u n o v func t i on . C l o s e d loop systems w i t h b o u n d e d i n p u t / o u t p u t signals a n d desired a s y m p t o t i c propert ies can be achieved , p r o v i d e d that c e r t a i n assumpt ions are made as i n Astrom et a l . ^, E d g a r ^ ^ , as noted below: G i v e n a p lant m o d e l of the type : 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 the o u t p u t y ( t ) a n d the i n p u t u ( t ) respect ive ly , d is the t i m e delay a n d e(t) is a d i s turbance t h a t can not be m e a s u r e d , a n d q~^ is the b a c k w a r d shift operator . A l s o g iven are the fo l lowing : T h e t i m e delay d is k n o w n . T h e upper bounds o n the degrees of the p o l y n o m i a l s A & B are k n o w n , i .e . the order of the sys tem is k n o w n . T h e p lant is a m i n i m u m phase process. T h e s ign of bo is k n o w n . C o n s i d e r i n g those assumpt ions , i n A s t r o m a n d W i t t e n m a r k ^ ^ , i t is shown t h a t the c losed loop sys tem is stable i f b o u n d e d d i s turbance a n d c o m m a n d s ignal (uc) gives b o u n d e d i n p u t (u) a n d o u t p u t (y) . 2.3 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 2.3.1 I n t r o d u c t i o n E q u a t i o n s of m o t i o n of a robot ic m a n i p u l a t o r c onta in nonUneari t ies , i n e r t i a l charac te r i s t i c s and d isturbances that vary d u r i n g a w o r k i n g cyc le a n d m a y not always be p r e d i c t a b l e ( F u ^^). I n m a n y cases, per formance ob ta ined w i t h f ixed t i m e invar iant control lers m a y not be sat is factory. Late ly , self t u n i n g pred ic t ive a lgor i thms have been used, since the resu l ts are more robust compared w i t h other self t u n i n g contro l a lgor i thms , such as P o l e P l a c e m e n t a n d M i n i m u m Var iance . T h e robustness of pred i c t ive a lgor i thms is due to the 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 , C l a r k e et a l T h e basic pred ic t ive m e t h o d contains the f o l l o w i n g steps: 1. P r e d i c t i o n of the o u t p u t . 2. C h o i c e of the future set po ints , 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 the fu ture errors , between the future outputs a n d future set po in ts , w h i c h y ie lds a set o f fu ture contro l signals. 3. T h e first t i m e step of the contro l signals is that a c tua l l y used , a n d the who le p r o c e d u r e is repeated . T h i s is a receding-hor izon control ler . 2.3.2 T h e G P C A l g o r i t h m T h e t y p e of control lers ment i oned above consider the o u t p u t at one po int i n t i m e i n the 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 that considers the future pred i c ted outputs j steps ahead , the f u t u r e set po ints a n d future contro l signals. T h e G P C is robust and deals w i t h o v e r p a r a m e t r i z a t i o n because of i ts pred ic t ive capabiht ies , and w i t h dead t ime since i t uses a n exphc i t p lant m o d e l . T h e robot m a n i p u l a t o r can be n o n m i n i m u m phase a n d incorrec t ly p a r a m e t r i z e d (espec ia l ly w h e n there is some flexibility i n the hnks ), can have dead t i m e i n the 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, can have instab iht ies . M a n y discussions about adapt ive contro l a n d G P C c a n be f ound i n the hterature , such as Astrom ^ ^ , T o m i z u k a D e m i r c i o g l u L a t o r n e U , etc. . T h e G P C uses a plant m o d e l w h i c h is a C A R I M A type ( C o n t r o l l e d A u t o Regress ive In tegra t ing M o v i n g Average ) : i .e. A{q-')yit) = B{q-')u{t - 1) + eit) (2.6) < 0 = c ( ç - ) ^ A = 1 - g - i W h e r e y{t) is the measured o u t p u t , u(t) is the contro l i n p u t , e{t) is the u n m e a s u r e d d i s t u r - bance t e r m , ^ is uncorre la ted r a n d o m sequence, q~^ is the b a c k w a r d shift operator , A is t h e di f ferencing operator and 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 respect ively . 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 the f o r m ( C l a r k e et a l ^): W h e r e Ni is the m i n i m u m output h o r i z o n , N2 is the m a x i m u m o u t p u t h o r i z o n , is the contro l h o r i z o n X{j) is a contro l we ight ing sequence, y{t-\- j) is the o u t p u t j steps ahead a n d w{t + j) is the fu ture set po int . Auit + i - 1) (2.7) T h e G P C a l g o r i t h m predicts fu ture o u t p u t s a n d aims at g o o d per formance a few steps ahead by m i n i m i z i n g the above cost f u n c t i o n , w h i c h gives a sequence of fu ture c o n t r o l s ignals 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 derive a j step ahead pred i c tor o{y{t), E q u a t i o n 2.6 shou ld be m u l t i p h e d b y q'^Ej{q~^)A a n d the fo l lowing ident i ty used: 1 = Eiiq-')Aiq~')A + q-'F,{q-') (2.8) T h i s is the D i o p h a n t i n e equat ion , where Ej a n d Fj are p o l y n o m i a l s i n the b a c k w a r d shift operator . I n order to get a unique so lu t i on , the degree of the f o l l owing p o l y n o m i a l s is chosen as: àeg{EM-')) < 3 - l a n d deg (F , (g -^ ) ) < àeg{A[q-')) So the j step ahead output y ( t + j ) is : 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 be 1. T h e d i s turbance sequence consists on ly of fu ture values w h i c h are u n k n o w n , so the o p t i m a l p r e d i c t o r is : y{t + j) = Fjiq-')yit) + Gj{q-')Au{t + j - 1) (2.10) GAq-') = Ej{q-')Biq'') T h e ob jec t ive of the pred i c t ive contro l law is to derive future p lant o u t p u t s y(t + j), g i v e n a f u t u r e set po int sequence w{t + j). It is done as follows: 1. T h e future set point sequence w(t + j) is de termined . 2. A set of predic ted errors is ca l cu la ted : eit + j ) y { t + j) - wit + j) 3. T h e cost funct i on i n E q u a t i o n 2.7 is m i n i m i z e d to provide a suggested sequence o f f u t u r e contro l increments , a s s u m i n g that after some contro l h o r i z o n N^y fu ture i n c r e m e n t s i n the contro l are zero, a n d the contro l s ignal is kept constant . T h e we ight ing fa c to r was i n i t i a l l y selected as r e c o m m e n d e d by C l a r k e et a l ^, fol lowed by a d j u s t i n g t h e values ac cord ing to the contro l led s y s t e m . T h e o p t i m a l pred i c t i on of y(t + j ) can be w r i t t e n as: y = G u + f (2.11) W h e r e f inc ludes the components of the pred i c ted output y , w h i c h are k n o w n at t i m e t , a n d G is a lower t r i a n g u l a r m a t r i x of 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 equat ion 2.7 y ie lds the contro l increment vector : (u) = ( G ' ^ G + A I ) - ^ G ' ^ ( w - f ) (2.12) T h e first element of u is A u ( t ) so t h a t the current contro l u{t) is g iven by : u(t) = u ( t - l ) + g V - f ) (2.13) W h e r e is the first row of ( G ^ G + A I ) - i g T . T h e design parameters for th is a l g o r i t h m are: 1. M i n i m u m O u t p u t H o r i z o n , Nii is set to the t i m e delay K, i f k n o w n , to save c o m p u - t a t i o n l oad since the contro l s ignals have no effect earher. T h e t i m e delay, K, i n d iscrete systems corresponds to the degree of the p o l y n o m i a l B, so w h e n K is not k n o w n , Ni is given the m i n i m a l value for the t i m e delay, A^i = 1. 2. M a x i m u m O u t p u t H o r i z o n , A^2: the authors r e c o m m e n d to set i t a p p r o x i m a t e l y at the value of the rise t i m e of the sys tem. B o t h parameters Ni a n d N2 are used i n E q u a t i o n 2.11 i n w h i c h the n u m b e r of p red i c t i on steps j vary f r o m Ni to A 2̂ a n d are t h e n used i n ca l cu la t ing the contro l law 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 advantage of the G P C is i n the a s s u m p t i o n a b o u t fu ture contro l signals. A f t e r a n in terva l of < N2, the pro jec ted c o n t r o l i n c r e m e n t s are assumed to be zero. T h i s reduces the c o m p u t a t i o n b u r d e n , since d im (u) = a n d is an Nu x m a t r i x . U s u a l l y , the contro l h o r i z o n c a n be chosen as A^u = 1 (for stable p lants w i t h delay or w i t h n o n m i n i m u m phase) , b u t w h e n p o o r l y d a m p e d or unstab le poles are present, shou ld equal the n u m b e r o f those poles. 4. T h e C o n t r o l W e i g h t i n g Sequence , A ( j ) : A c t s as d a m p i n g of the contro l a c t i o n w h e n greater t h a n zero. Final goal of Type of Required accuracy Identification model application process model of model method ver i f i cat ion of hnear / cont inuous m e d i u m / off-hne step theoret i ca l t i m e n o n p a r a m e t r i c / h i g h response, models parameters frequency response, parameter e s t i m a t i o n control ler hnear n o n p a r a m e t r i c , low for off-hne parameter cont inuous t i m e i n p u t / o u t p u t step response t u n i n g behav ior c o m p u t e r a ided hnear p a r a m e t r i c m e d i u m for on-hne, off-hne design of (nonparametr i c ) i n p u t / o u t p u t parameter e s t i m a t i o n d i g i t a l contro l discrete t i m e behav ior sel f -adaptive hnear p a r a m e t r i c m e d i u m for on-hne p a r a m e t e r d i g i t a l contro l discrete t i m e i n p u t / o u t p u t e s t i m a t i o n i n close t u n i n g behav ior loop process hnear / non l inear h i g h for on-hne p a r a m e t e r parameter p a r a m e t r i c continuous p a r a m e t r i c process m o n i t o r i n g a n d t i m e cont inuous parameters fa i lure detect ion t i m e e s t i m a t i o n Tab le 2.1: Different final goals and speci f ications for ident i f i ca t i on 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 D e t e r m i n a t i o n S y s t e m ident i f i ca t ion is basicaUy a func t i on 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 by a n a l y z i n g the re lat ions between observed i n p u t and o u t p u t . A considerable amount of w o r k has been done and can be found i n references L j u n g and Sôderstôm ^, 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 the final goal for the a p p h c a t i o n of the process m o d e l , since th i s de termines the type of m o d e l , its accuracy requ irements , and the ident i f i ca t i on m e t h o d . Tab le 2.1 ( Isermann ^ ) shows some examples for re lat ionships between di f ferent f i n a l goals a n d some specif ications of process ident i f i ca t i on . T h e a p r i o r i knowledge of t h e p r o - cess i s , for example , based on general process u n d e r s t a n d i n g , on p r i n c i p a l laws a n d o n p r e - measurements . T h e choice of a m o d e l s t ruc ture is essential for successfully ident i f y ing a s y s t e m . L j u n g ^ i n his book describes techniques a n d procedures about m o d e l s t ruc ture select ion a n d m o d e l v a l i d a t i o n . T h e choice of a m o d e l s t ruc ture should be based on good u n d e r s t a n d i n g o f t h e ident i f i ca t i on m e t h o d and a p r i o r i knowledge of the s y s t e m . T h e r e are three bas ic steps i n choosing a m o d e l s t ructure i.e. 1. C h o i c e of type of the m o d e l set: hnear or nonhnear 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 mode l s , etc. . 2. C h o i c e of the m o d e l size: m o d e l order for hnear systems. 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 : ident i f i ca t i on of m o d e l parameters . T h e q u a h t y of the m o d e l is usua l ly measured by m i n i m i z i n g a c r i t e r i on . T h e r e is a t r a d e - off between flexibihty and pars imony . F l e x i b i h t y w i l l give, for a larger n u m b e r of p a r a m e t e r s , a bet ter fit for the m i n i m i z a t i o n of the c r i t e r i on since the d a t a set is larger . O n the o t h e r h a n d it is i m p o r t a n t , i n pract i ce , to employ the smallest n u m b e r of parameters for adequate representat i on of phys i ca l systems. T h e t y p e of the chosen m o d e l is usua l ly based on a p r i o r i knowledge about the s y s t e m a n d i n t u i t i o n . GeneraUy it is adv isable to start w i t h the s implest possible m o d e l . O r d e r e s t i m a t i o n of a hnear sys tem is usua l ly based o n p r e U m i n a r y d a t a ana lys i s . T h e relevant m e t h o d s fa l l into the next categories: • S p e c t r a l analysis of the transfer func t i on . • T e s t i n g ranks of sampled covariance 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 methods can be as follows: • C o m p a r i n g hnear models w i t h a p r i o r i knowledge o f the sys tem. • C o m p a r i n g measured and s imula ted o u t p u t s . • T e s t i n g residuals for independence of past i n p u t s . • C o m p a r i n g c r i t e r ia fit ob ta ined i n different m o d e l s t ruc tures . Of f - l ine methods are capable of choosing 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 ts p a r a m e t e r s . T h e r e are m a n y cases where on-hne ident i f i ca t ion of the m o d e l is needed. T h e r e are t w o disadvantages to recursive (on-Hne) ident i f i ca t i on , i n contrast to off-hne i d e n t i f i c a t i o n . T h e first is that the decis ion on w h i c h m o d e l s t ruc ture to use has to be m a d e a n d f i xed a p r i o r i , before s t a r t i n g the recurs ive ident i f i ca t ion procedure . It is we l l k n o w n that p a r a m e t r i c m o d e l s can give large errors when the order of the m o d e l does not agree w i t h the order of the process (see Astrom ^^). I n the off-hne s i t u a t i o n , different types of models can be e x a m i n e d . T h e second disadvantage is that recurs ive methods general ly are not as accurate as off-hne ones ( L j u n g 3 ) . F o r off-hne ident i f i ca t i on , the basic three steps are: • data recording - good choice of the d a t a record makes i t m a x i m a l l y i n f o r m a t i v e . • a set of candidate models - the most i m p o r t a n t a n d dif f icult choice of the 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 the a c t u a l d e t e r m i n a t i o n of the m o d e l s t r u c t u r e . • determining the best model in the set - guided by the d a t a ( this is the i d e n t i f i c a t i o n m e t h o d ) . A f t e r h a v i n g sett led on the preceding three choices, one must val idate the m o d e l , i . e . es tabhsh a c r i t e r i on to examine i f the m o d e l accepted is good enough. If i t is not , t h e e n t i r e procedure is repeated again as shown 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 the sys tem model ' s s t ruc ture , the user m a y choose f r o m the i d e n t i f i c a - t i o n a n d adapt ive contro l techniques that are avai lable . Cons iderab le work i n these areas has been done a n d w i U not be discussed at this po int , but some i m p o r t a n t references are L j u n g a n d Sôderstôm ^, L j u n g Astrom a n d E y k h o f f Âstrôm Âstrôm a n d W i t t e n m a r k ^, Stre j c G o o d w i n a n d P a y n e . 2.5 R e v i e w of P r e v i o u s W o r k in O r d e r D e t e r m i n a t i o n T o emphas ize the effect of incorrect m o d e h n g , we refer to R o h r s et a l w h i c h a n a l y z e d t h e effect of u n m o d e l e d d y n a m i c s o n the robustness and s t a b i h t y of cont inuous t i m e a d a p t i v e c o n t r o l a lgor i thms . T h e conc lus ion i n this paper is that adapt ive a lgor i thms p u b h s h e d i n t h e h t e r a t u r e are hke ly to produce unstab le contro l systems i f they are i m p l e m e n t e d o n p h y s i c a l systems d i rec t ly as they appear i n the h te ra ture . U n f o r t u n a t e l y , s t a b i h t y proofs of a l l those a lgor i thms have i n c o m m o n a very res tr i c t ive a s s u m p t i o n that the order of the s y s t e m is k n o w n . So i f there are errors i n the s t r u c t u r e assumpt ions , i n s t a b i l i t i e s can occur . It is also n o t e d that this p r o b l e m can be p a r t i a l l y a l l ev ia ted by sufficient e x c i t a t i o n , b u t the a m o u n t of m o d e h n g error or the a m o u n t of d i s turbance for w h i c h the a d a p t i v e sys tem can m a i n t a i n s t a b i h t y m a y be ex t reme ly s m a l l . I n the 1970's, several works were p u b h s h e d , such as A k a i k e a n d A k a i k e , I s e r m a n n 2^, Schwarz , p r i m a r i l y es tabhshing est imates of a measure of fit o f the m o d e l . T h o s e procedures are off-hne ones a n d are par t of the m o d e l v a h d a t i o n techniques . r TASK ECONOMY FINAL C O A L DESIGN OF exPERIMENTS SIGNAL Oi M E A S U R E * O A T * STC •NERATION. ENT AND «ACE paoccss PHYSICAL LAWS P«EHEASLIAEHENTS O n U A T I N C CONDITIONS APPLICATION OF IDENTIFICATION , ."£THOp ASSUMPTION Cf MODEL STRUCTURE H__rwOC£SS»<OOEL N O N . P A R A M E T B C - f -•PASAMETRIC MODEL S T R u a U S E DETERMINATION MODEL VERIFICATION MO F i g u r e 2.2: G e n e r a l procedure of process ident i f i ca t i on , ( I s e rmann *̂̂ ) A s m e n t i o n e d , several c r i t e r i a for order e s t i m a t i o n were pub l i shed . R i ssanen ^9 proposes the foUowing cost func t i on : ' + ( n + m + l)\n{N + 2) + 21n(n + l ) ( m + 1) (2.14) U{x,n,m,() = Nlnr + ^ W h e r e ' j ô^ ln f da J X is the m o d e l : x{t) = / , [ x ( f ^l),...,x{t- n), e{t),e(i - m)] iV - n u m b e r of sampled observations, e - d i s turbance . ^ - consist of real valued parameters . AT log f - the m i n i m i z e d log h k e h h o o d func t i on . T h e first and t h i r d terms together v i r t u a l l y co inc ide w i t h a c r i t e r i on der ived b y S c h w a r t z . T h e most c o m m o n l y used c r i t e r i o n , der ived by A k a i k e ( " ) , is the A I C ( A k a i k e ' s I n f o r m a t i o n T h e o r y C r i t e r i o n ) w h i c h is : A I C = ATlogr^ + 2Jb (2.15) W h e r e Nlogr - the m i n i m i z e d l og -hkehhood f u n c t i o n . k - the n u m b e r of parameters i n the m o d e l . M o r e c r i t e r i a were i n t r o d u c e d by A k a i k e i n 1977 a n d by 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 new cr i t e r i on for order e s t i m a t i n g of a C A R A M A m o d e l . T h e f ina l conc lus ion is that est imates presented are non-recurs ive a n d require a v a i l a b i h t y of upper bounds for u n k n o w n orders; therefore fur ther research is r equ i red . A p p r o a c h e s for different s tructures have been exp lored such as canon i ca l s t ruc tures ( G u i d o v z i a n d ^^). These t r y to close the gap by e x p l o r i n g a class of state space c a n o n i c a l mode ls w i t h p a r t i c u l a r l y s imple relations to i n p u t / o u t p u t difference descr ipt ions t h a t can be d i - r e c t l y ident i f i ed f r o m i n p u t / o u t p u t sequences. However , i n order to avo id e r ror , the prev ious es t imate of the order a n d s t ructure of the process, out of a class of models , is r e q u i r e d . F u r t h e r off-hne methods , i n c l u d i n g M I M O ( M u l t i Input M u l t i O u t p u t ) s y s t e m s , h a v e been p u b h s h e d a n d wiU not be discussed here. M o s t of the recursive 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 ruc ture a p r i o r i . S u c h a m e t h o d by G a u t h i e r e x t e n d s t h e use of i n p u t / o u t p u t descr ipt ion i n t e rms of p o l y n o m i a l matr ices for recurs ive i d e n t i f i c a t i o n i n canon i ca l state space f o r m . However , before ident i f y ing the p o l y n o m i a l s ' coef f ic ients , one must define the s t ructure and the p o l y n o m i a l s ' degree or at least, the i r u p p e r b o u n d s . L j u n g a n d Sodes t rom discuss i n the i r book the concept of i dent i f y ing 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 and the choice of a m o d e l order . T h e choice of a m o d e l order is a dehca te t rade off between good descr ipt ion of the d a t a a n d the m o d e l complex i ty . M o s t m e t h o d s for m o d e l order selection are developed for off-hne techniques . T h e basic approach is t o c o m p a r e per f o rmance of models w i t h different orders a n d test whether a higher order m o d e l is w o r t h whi le . Recurs ive a lgor i thms i n on-hne apphcat ions require ident i f i ca t ion of several m o d e l s s imul taneous ly . A m o d e l set is said to be ident i f iable i f i ts parameters can be i d e n t i f i e d i .e . parameters can be un ique ly d e t e r m i n e d f r o m the d a t a . L a c k of i dent i f i ab i l i t y can be caused by n o n - e x c i t i n g i n p u t s and o v e r p a r a m e t r i z a t i o n . M I M O systems, can be p a r a m e t r i z e d depend ing on the choice of s t r u c t u r e of the s y s t e m . T h e p r o b l e m has been avoided by some researchers, assuming that the designer has e n o u g h a p r i o r i knowledge of the s t r u c t u r e of the sys tem to select stable p a r a m e t r i z a t i o n . O v e r b e e k a n d L j u n g suggested a procedure t h a t provides a means of o b t a i n i n g the best m o d e l s t r u c t u r e . T h e m o d e l S t r u c t u r e Select ion ( M S S ) a l g o r i t h m , the s t r u c t u r e dealt w i t h i n t h a t w o r k , is the p a r a m e t r i z a t i o n of systems and can be per formed i n a n u m b e r of ways . T h e technique does not deal w i t h how to select an appropr ia te order . T h e a l g o r i t h m receives as i n p u t a g iven sys tem w i t h a g iven p a r a m e t r i z a t i o n . It tests whether this p a r a m e t r i z a t i o n is we l l c o n d i t i o n e d for ident i f i ca t i on purposes . If i t is not wel l c o n d i t i o n e d , another s t r u c t u r e is cons idered . T h e best s t r u c t u r e of a possible set is dec ided u p o n a p r i o r i . N a g y a n d L j u n g describe i n the ir paper the subject of c o m p u t e r - a ided m o d e l s t r u c t u r e se l e c t i on . In order to 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 , an appropr ia te m o d e l s t r u c t u r e should be chosen. A c o m m o n feature for these m e t h o d s is that they m i x extens ive n u m e r i c a l c o m p u t a t i o n s , code generat ion a n d s y m b o h c a lgebra . D a v i s o n i n his paper describes a m e t h o d of m o d e l size r educ t i on . M a n y p h y s i c a l p l a n t s can be represented by s imultaneous hnear d i f ferent ia l equations w i t h constant coeff ic ients, o f the f o r m : X = A x -|- B u W h e r e the order of m a t r i x A can be large , for e x a m p l e c h e m i c a l p lants or nuc lear reac tors , w h i c h can pose n u m e r i c a l prob lems , the m e t h o d suggests the r e d u c t i o n of the r a n k of s u c h matr i ces by c o n s t r u c t i n g a m a t r i x of lower order w i t h the same d o m i n a n t eigenvalues a n d eigenvectors as the o r i g i n a l sys tem. T h e paper by N i u , X i a o and F i s h e r presents a s imul taneous recursive e s t i m a t i o n of the m o d e l parameters a n d loss funct ions for aU possible m o d e l orders f r o m zero t h r o u g h n is done by us ing augmented 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 za t i on a l g o r i t h m . T h e A I M m a t r i x is : k 3 = 1 (2.16) W h e r e is the regression vector a n d C „ is the 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 the o u t p u t , v is the noise a n d u is the i n p u t . T h e a l g o r i t h m is reported to be c o m p u t a t i o n a l l y efficient a n d have good n u m e r i c a l p r o p - ert ies due to the use of the UDU^ a l g o r i t h m . In a second paper N i u a n d F i s h e r '̂ ^ repor t o n a M I M O sys tem ident i f i cat ion technique us ing augmented U D fac tor i za t i on . T h i s w o r k e x t e n d s the A U D a l g o r i t h m from S I S O systems to M I M O systems. It is based on the c a n o n i c a l s ta te space representat ion be G u i d o r z i the B i e r m a n ' s U D fac tor i za t i on . T h i s a l g o r i t h m too , is r epor ted to poses exceUent n u m e r i c a l propert ies . T h e work reported next , ( 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 error m o d e l o rder e s t i m a t i o n procedure . T h e e s t i m a t i o n is based on s a m p h n g a s ignal a n d c a l c u l a t i n g i t s a u t o c o r r e l a t i o n func t i on : Rin) = j^'e'<JMJ + ( 2 1 7 ) k = l,2,---,Lk < L W h e r e x{t — j) are sampled values of the s ignal x(t) a n d L is the n u m b e r of samples . A s y s t e m of n hnear equations is generated: W h e r e A is a nonzero vector of coefficients a n d R „ is a m a t r i x . T h e order is e s t i m a t e d by e x a m i n i n g the de terminant of the m a t r i x R„ : If det R „ = 0 then n> N a n d If det R „ 7̂  0 t h e n n< N W h e r e n = 1 ,2 ,3 , a n d N is the correct order . A m e t h o d for s imul taneous ly se lect ing the order a n d i d e n t i f y i n g the p a r a m e t e r of a n A u t o r e g r e s s i o n ( A R ) m o d e l , has been developed, ( K a t s i k a s et a l '^^). T h e A R m o d e l is def ined as: y{k) = J2'^iik)yik-i)+v{k) «=1 W h e r e a ; are the coefficients, y{k — i)) are previous outputs and v{k) is zero m e a n w h i t e noise. T h e order of the system N i n u n k n o w n but i t is i n the range of 1 < AT < M , w h e r e M is k n o w n . T h e true m o d e l wiU be one of a f a m i l y of models w i t h the above range of o rder . A paper by ( B i r c h , Lawrence 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 to E E G (E lec t roencepha lography ) signals. T h e E E G s ignal is mode led w i t h an A R m o d e l t y p e a n d a spec t ra l e s t i m a t i o n procedure is per formed . T h e selection of the proper m o d e l order is done by some a p r i o r i knowledge of expected results . V e r y h t t l e work has been pubhshed on s t ruc ture d e t e r m i n a t i o n for nonHnear s y s t e m s . O n e w h i c h treats m a t h e m a t i c a l models for representat ion of the d y n a m i c of sh ip r u d d e r - yaw a n d r o l l mot ions is i n Z h o u et a l T h e work checks suggested mode l s g i v e n i n t h e Hterature for the p r o b l e m w i t h the Recurs ive 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 an off-Hne m e t h o d that finds the best nonHnear terms for the m o d e l . A n o n Hnear on-Hne m e t h o d has been described by Zervos a n d D u m o n t T h e p lant is m o d e l e d by a n o r t h o g o n a l L a g u e r r e network put into state space f o r m . T h e n u m b e r of the L a g u e r r e f i l ters used depends of the presence of t i m e delay a n d u n d a m p e d modes . T h e a c t u a l p lant o r d e r does not inf luence the n u m b e r of Laguerre f i lters ( N ) used. U s u a l l y the choice is 5 < AT < 10, a n d N can be changed on Hne. H e m e r l y '̂ ^ presents a m e t h o d for order a n d parameter ident i f i ca t i on of i n d u s t r i a l p r o - cesses. T h e processes to be ident i f ied are descr ibed by A R X mode : y(t) = -aiy(t - 1) a „ y ( i - n) + biu{t - 1) -f • • • + fc„w(i ~ n) + w(t) W h e r e y(t) is the systems o u t p u t , u{t) is the i n p u t , a n d bi are the coefficients a n d w{t) is w h i t e noise. T h e parameters are ident i f ied by Recurs ive Least Squares ( R L S ) a l g o r i t h m a n d for the order e s t i m a t i o n the P r e d i c t i v e Least Squares ( P L S ) c r i t e r i o n is used , ( R i s s a n e n PLS{n.t)=y'£e'{n,i + l) i=0 T h e best est imate of the order should be w i t h where n(t) = argmin PLS(n,t) e{n, t + l) = y{t + l ) - 0 ^ ( n , t)^n, t) = [yU),---,y{J - n + l),u{j),--- ,u{j - n + 1) T h e P L S cr i t e r i on is h igh ly i n t u i t i v e a n d at t i m e t the order es t imate n(t) is the order of the m o d e l w h i c h has given the least m e a n square pred i c t i on error u p to t h a t t i m e . T h e process can be ident i f ied for different opera t i ng po ints by v a r y i n g the e x c i t a t i o n a m p h t u d e a n d therefore ge t t ing several hnear mode ls . A control ler can be designed for each m o d e l a n d changed i n real t i m e i f necessary. Mede i ros a n d H e m e r l y in tegra ted l a t t i c e f o r m for c o n s t r u c t i n g a m i n i m u m variance a d a p t i v e contro l ler w i t h parameter a n d order e s t i m a t i o n . A s descr ibed i n the former paper the order is e s t imated w i t h the 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 ter is a way to parametr i ze as foUowing: ê n + l ( 0 = ê „ ( f ) + ^^n(<- l ) Vn+lit) = r-nit - 1) + pèn{t) where: ê4t) = y{t)~Ut/®n Kit - 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 parameters for order n , y(t) is the m e a s u r e d o u t p u t , y „ is the e s t imated o u t p u t for m o d e l order n , a n d ê„ is the error . A set of p r e d i c t i o n errors is ca l cu la ted and the e s t i m a t e d order of the m o d e l is the one that has a m i n i m a l least squares error . A s can be in ferred f r o m the prev ious discussions, m u c h w o r k has been p u b h s h e d o n ident i f i ca t i on a n d adapt ive contro l a lgor i thms , mos t ly for f ixed order 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 research work has been done for flexible m o d e l s t ructures p a r t i c u l a r l y for on -hne methods a n d there is def ini te ly a need for research i n the area. T h i s work cons t i tu tes a c o n t r i b u t i o n to the research for on-hne m o d e l s t ructure d e t e r m i n a t i o n . C h a p t e r 3 S I N G L E F L E X I B L E L I N K M A N I P U L A T O R 3.1 O u t l i n e : I n this C h a p t e r , we present the work done w i t h the 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 equations of m o t i o n have been deve loped . N u m e r i c a l results for the c o n t r o l of the m a n i p u l a t o r are presented, as wel l as the effects of m i s - m o d e h n g on the m a g n i t u d e o f a cost f u n c t i o n a n d an outhne for an i t e rat ive m e t h o d for the order es t imator . Sec t i on 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 invo lved i n the 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 the hnear equat ions of m o t i o n a n d contro l s t ra tegy . Sec t i on 3.3 presents the n u m e r i c a l results a n d analysis for the flexible h n k a p p h c a t i o n . 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 3.2.1 I n t r o d u c t i o n A robot is a complex sys tem to contro l not on ly because i t is a nonhnear s y s t e m a n d has var ia t ions i n the m o m e n t of i n e r t i a , but also because 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 , such as hysteresis or back lash . T h e r e are two approaches for the design of control lers for s u c h systems: i .e. to design one that w iU not exc i te the p o o r l y d a m p e d modes , or one t h a t a c t i v e l y d a m p s osc i l l a tory modes . T h e second op t i on is not used i n i n d u s t r i a l robots . S u c h c o n t r o l systems are c o m p h c a t e d , since the frequencies of the osc i l la tory modes vary w i t h o r i e n t a t i o n a n d l oad . T h e var iat ions i n the osc i l la tory modes are the reason for choos ing the flexible h n k as an a p p h c a t i o n . N e w v i b r a t i o n modes that arise m e a n that the s t r u c t u r e of the m o d e l has changed on hne, so the m o d e l a n d the contro l sys tem should be u p d a t e d . D a t a for t h e chosen f lexible hnk can be found m 50 a n d 19. T h e a r m is a 1 meter long , f lexible m e c h a n i c a l s t r u c t u r e w h i c h can b e n d freely i n the h o r i z o n t a l p lane but is stiff i n ve r t i ca l b e n d i n g a n d i n tors ion . Its m o t i o n is on ly i n the hor i zonta l plane i .e. g rav i ty effects are not 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 the Single F l e x i b l e L i n k T h e f lexible a r m is comparab le to a cant i lever b e a m . F i g u r e 3.1 describes the flexible h n k conf igurat ion . W h e r e : XQ - is the reference axis . X - is the pos i t i on of a r i g id a r m at 0 [rad.] f r o m XQ. w{x,t) - is the deflection f r o m the r i g id body. 1B - is the moment of i n e r t i a about the hub [kg • m^' IH - is the motor ' s m o m e n t of i n e r t i a [kg • . TH - is the torque apphed by the m o t o r [N • m]. E - is Y o u n g ' s module [N/m^]. I - is the b e a m cross sect ional m o m e n t of i n e r t i a [m"* T h e d isp lacement of any point P a long the b e a m at a d is tance x f r om the h u b is g i v e n b y 0{t) a n d the deflection w(x,t), measured f r o m the hne Ox w h i c h w o u l d be the a r m , h a d i t been r i g i d . T h e assumpt ions m a d e are: • the def lect ion is smaU - w{L,t) <^ L • shear de f o rmat i on a n d r o t a r y i n e r t i a effects are neglected. • g r a v i t a t i o n effects for def lect ion a n d movement i n a h o r i z o n t a l p lane are neglected . T h e d i sp lacement y(x,t) of a po int p along the a r m is defined 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 a r m y{x,t) = w{x,t) + xe{t) (3.1) L e t : wix,t) = J2M^)<liit) (3.2) i=l W h e r e ^i{x) is the i " * mode shape and qi{t) is the i * ' ' m o d e generahzed c o o r d i n a t e , n = 1,2, 3 , 4 , . . . is the n u m b e r of v i b r a t i o n modes . ( N o t e that w h e n i = 0, (l>o{x) = x, "^"^"^ = 1, qo{t) = 0 a n d qa{i) = 9 are the parameters for a r i g id body. ) A U der ivat ives as 6 are w i t h respect to t i m e , a l l der ivat ive as Q' are w i t h respect to x. T h e k i n e t i c energy of the sys tem is : 2T, = I,0^ + m f \ ' ^ + xèYdx Jo at U s i n g E q u a t i o n 3.2 the k i n e t i c energy is: 2n = hé' + è hi qUt) + 2èJ2lu Qiit) (3.3) «=1 t=l T h e i n e r t i a integrals In, hi, hi are descr ibed i n A p p e n d i x B a n d the i r values are presented i n Tab le B . l . ( T h e n u m b e r indicates 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 the i shows for w h i c h mode the in tegra l is ca l cu la ted ). T h e p o t e n t i a l energy is s t r a i n energy due to b e n d i n g de format ion is : 2 ^ = E E r EIcl/:cl>';qiqjdx i=i j=i U s i n g o r thogona l i ty re lat ions : EI 4>'l cl>'! dx = the p o t e n t i a l energy becomes: f (j)] pdx = hi u;^ i = j 0 i ^ j (3.4) 2 P = E ^3 . ql i = l I n t r o d u c i n g a d i ss ipat ion funct i on D that m a y be defined as: the d a m p i n g force wiU be: 2 n n ^ = 9 E E ^ i j Mi i = i C o m b i n i n g a l l together and a p p l y i n g Lagrange ' s e q u a t i o n , the equat ions of m o t i o n are: for 1 = 0 - r i g i d b o d y : (3.5) (3.6) lT'é^-Y.hiqi{t)^rH~cJ »=i for i = 1 , 2 , 3 , . . . : (3.7) lu 0 + h^ qi{t) = - / a . g i ( i ) - Co qi{t) (3.8) W h e r e : IT = IB + IH is the t o t a l moment of i n e r t i a , and Cj where i = 1 , 2 , 3 , . . . , n is t h e d a m p i n g coefficient. 3.2.3 T h e State Space M o d e l E q u a t i o n s 3.7 a n d 3.8 can be put in to a m a t r i x f o r m , such as: M x = K X + hTH W h e r e x is the state space vector defined as: x = [0, qi, 92, ••• gn, 0, qi, qz, ••• Çn M , K a n d b are matr i ces def ined as: (3.9) 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 / i 2 ha • • hn 0 0 0 ••• . . . 0 hi hi 0 . . . • • 0 0 Û 0 ••• . . . 0 1X2 0 h2 • • • • . 0 u 0 0 ••• . . . 0 / i n 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 - / 3 1 0 •• . . . . 0 0 -- C l 0 ••. . . 0 0 0 0 •• • . . . 0 0 0 - C 2 • • • • • 0 0 0 0 •• 0 0 0 ••• . • - c „ b = - 0 0 0 ••• 1 0 0 0 > . . T h e state space m o d e l i n its final f o r m is: X = A • X + B • r/i y tip = C • X W h e r e : A = . K (3.11) (3.12) (3.13) B = b C = f l 1 1 ••• 0 0 0 (3.14) T h e order of the sys tem depends on the n u m b e r of modes 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 iU be an 8"* order m o d e l . Tab le 3.1 shows the order of the sys tem versus the n u m b e r of modes considered. 3.2.4 T h e D i s c r e t e T i m e M o d e l T h e system was converted w i t h a zero order h o l d s a m p h n g f r o m a cont inuous s t a t e space f o r m into a discrete t ime f o r m . For s a m p h n g w i t h per i od h, the t i m e is: tk — k • h T h e state space discrete t i m e m o d e l has the fo l lowing s t r u c t u r e : x{{k + l)h) = ^x{kh) + ru{kh) y{kh) = Cx{kh) (3.15) W h e r e : * = e x p ^ ' ' (3.16) J\xp^^ dSB I n order to s imphfy ca lcu lat ions , we expanded i t in to a series, i .e . : * . ^ e x p - . 5 = M + ^ + ^ + . . . + ^ + . . . (3.17) N o w the matr i ces are g iven by: * = 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 Different N u m b e r of M o d e s T h e discrete t i m e models for ytip ver. Th were developed for cases w i t h a different n u m b e r o f modes . T h e s t ruc ture of the discrete m o d e l is i n the f o r m of E q u a t i o n 2.6, where t h e no ise sequence is equa l to zero. T h e s a m p h n g per i od is = 0.01 [sec.]. T h e d a m p i n g fac tors are , Ci = 0.05 . R e m a r k : A U the foUowing discrete t i m e models for the different orders were checked w i t h cont inuous t i m e s i m u l a t i o n for the flexible h n k w i t h the same n u m b e r of v i b r a t i o n m o d e s ( i .e . the same m o d e l order) . T h e results show the same behavior for the cont inuous t i m e a n d t h e discrete t i m e models . For r i g i d b o d y - i = 0, m o d e l order is 2 : ytip(tk)/o = +I.m2ytip{tk - 1) - 0.99822/tip(<fe - 2) (3.19) +0.17774 • 1 0 - ' TH{tk - 1) + 0.1773 • rnitk - 2) For one v i b r a t i o n mode z = 1, m o d e l order is 4 : ytip(tk) / i = +3.14233/eip(ffc - 1) - WUyupitk - 2) (3.20) +2.S781 ytipitk - 3) - 0M73yupitk - 4) +0.1458 • 10-Vj,(ifc - 1) + 0.0173 • lO'^THitk - 2) - 0 . 0 4 0 5 • 10-^TH{tk - 3) + 0.1344 • lO'^rnitk - 4) For two v i b r a t i o n modes i = 2, m o d e l order is 6 : y t i p ( t k ) / 2 = + 1 . 5 5 0 6 2 / a p ( i f e - l ) - 0 .48743/t ip( f fc -2) (3.21) +0.5763yup{tk - 3) - 1.2837y,ip(«fc - 4) + 1.03112/t.p(ife - 5) - 0.3870ytip{tk ~ 6) +0.0534 IQ-^THitk - 1) + 0.4899 IQ-^Tsitk - 2) +0.1043 lO-^mitk - 3) - 0.1471 IQ-^sitk - 4) +0.1764 lO-^Hitk - 5) + 0.0187 1 0 - ^ ^ ( 4 * - 6) For three v i b r a t i o n modes i = 3, m o d e l order is 8 : ytip(tk) /3 = + 0 . 0 6 6 2 - 1) + 2.0013 ytip(<fc - 2) (3 .22) +0.6734y,ip(ffe - 3) - 1.70432/tip(<fc - 4) ~1.15Uytij,{tk - 5) + 0.9074ytip(<ife - 6) +0.6223 ytipitk - 7) - 0.4120 yn^it^ - 8) +0.0576 • lO'^THitk - 1) + 0.5583 • 10-\Hitk - 2) +0.8196 • lO'^Tnitk - 3) - 0.1301 • lO-^Tsitk - 4) - 0 . 6 1 1 3 • 10'\H{tk - 5) + 0.0127 • lO-^rnitk - 6) +0.1971 • lO'^rnitk - 7) + 0.0217 • lO'^ffitk - 8) F o r four v i b r a t i o n modes i — 4, m o d e l order is 10 : ytip(tk ) /4 = +l.mOytip{tk - 1) + 1.24332/iip(<fc - 2) (3.23) -l.m7yup{tk - 3) - 1.10162/tip(<fe - 4) +0.95442/tip(^fc - 5) + 0.84Myup{tk ~ 6) - 1 . 0 6 4 5 3/t.p(ifc - 7) - 0.3661 ytip(<fc - 8) +0.8878 y,ip(i;fe - 9) - 0.3062 2/,ip(<fc - 10) +0.0583 • \0~^rii{ik - 1) + 0.4965 • lO'^Tuitk - 2) +0.2602 • \0~'^Tii{tk - 3) - 0.6346 • \0~^Tu{tk - 4) +0.0219 • \0^^rii{tk - 5) + 0.4743 • \0~^rii{tk - 6) - 0 . 2 8 4 6 • \0-^rH[tk - 7) - 0.1749 • lO-^rnitk - 8) +0.1224 • 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 models is such that yup at the present value i n t i m e depends o n a s equence of prev ious measured outputs ytip{tk — i) and previous 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 the F lex ib le L i n k M a n i p u l a t o r Self t u n i n g adapt ive contro l a lgor i thms are the contro l s trategy used i n this w o r k . T h e s e a lgor i thms can be direct S T R ' s or ind irec t ones. I n the direct a lgor i thms , the c o n t r o l l e r parameters are e s t i m a t e d d i re c t l y whereas the e s t i m a t i o n for the ind irec t ones is d o n e o n the p l a n t ' s m o d e l parameters ra ther t h a n the regulator ' s parameters , 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 already been i m p l e m e n t e d , is the 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 its results are presented i n Sec t ion 2.3. A s was presented i n the prev ious sect ion, the basic s t r u c t u r e of the hnear m o d e l for the f lexible h n k is ( w i t h o u t noise) : A{q-')yit) = biq-')u{t) (3.24) F i g u r e 2.1 presents a b lock d i a g r a m of the contro l w h i c h was designed to deal w i t h one s t r u c t u r e of the p lant ' s m o d e l at a t i m e . W e w i l l e x a m i n e the effects of changes i n t h a t s t r u c t u r e . A c t u a l changes should be done i n the parameter e s t i m a t i o n b lock once t h e i r n u m b e r changes and i n the contro l ler ca l cu lat ions , since the d imens ions of a l l p o l y n o m i a l s a n d matr i ces w i l l change. 3.3 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 a n d C o n t r o l W o r k P e r f o r m e d 3.3.1 I n t r o d u c t i o n T h e results presented i n this sect ion are for n u m e r i c a l s imula t i ons of the single f lexible h n k cont ro l l ed w i t h the 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 so lv ing the d y n a m i c equations for the flexible h n k as presented i n Sect ion 3.2 . T h o s e e q u a - tions are referred to as the " r e a l p l a n t " whose m o d e l is to be e s t imated . T h e u n k n o w n m o d e l is chosen i n the f o rm of E q u a t i o n 2.6 (w i thout the noise t e r m ) . Dif ferent mode ls for t h e different n u m b e r of modes are presented i n Subsect ion 3.2.5. I n a d d i t i o n , the p a r a m e t e r s of those models are presented, even though i n a c t u a l s i tuat ions they are u n k n o w n s . T h e y are found by an e s t i m a t i o n technique fitting for the p lant to be e s t imated . T h e m e t h o d used i n this work so far has been Recurs ive Least Squares ( R L S ) . 3.3.2 EflFects of 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 on the C o n t r o l l e d F l e x i b l e L i n k T h i s sect ion w i l l present the s i m u l a t i o n results of inves t iga t ing under a n d over m o d e h n g o f the a c tua l p l a n t , w i t h regard to the s t ruc ture . It means that for each figure, the a c t u a l n u m b e r of modes are shown, i .e. modes t h a t are used i n the d y n a m i c equat ions ( S e c t i o n 3.2 ), as we l l as the n u m b e r of modes t a k e n in to account i n the e s t imator m o d e l . W i t h e v e r y choice of e s t i m a t e d s t ruc ture for the p lant ' s m o d e l , the n u m b e r of parameters to be e s t i m a t e d changes. I n F i g u r e 3.2, the flexible h n k was m o d e l e d w i t h two v i b r a t i o n modes , w h i c h means t h a t the " r e a l " m o d e l of the fink is of 6" ' order (see Tab le 3.1). T h e es t imated s t r u c t u r e for t h e d iscrete t i m e hnear m o d e l was also of order 6. A s ment i oned before, the t i p p o s i t i o n of t h e flexible h n k is contro l led w i t h the G P C a l g o r i t h m . T h e set po int is a square wave b e t w e e n the values of ± 1 . T h e o u t p u t , the t i p p o s i t i o n , follows i t very wel l . F i g u r e 3.3 shows t h e b e h a v i o r of the angle 9 and its derivatives for the same condi t ions as i n F i g u r e 3.2. S ince i t is a hnear m o d e l a n d the l ength of the a r m is 1 m . , and the effect of the v i b r a t i o n modes is s m a l l , ytip a n d 9 appear to be the same. F igures 3.4 and 3.5 present the v i b r a t i o n m o d e s generahzed coordinates qi a n d q2 a n d the ir der ivat ives , a n d F i g u r e 3.6 shows the t o rque 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 mode flexible l ink w i t h two mode e s t i m a t o r m o d e l 2 mode link - 2 mode estimotor - GPC control h 0.4 H 1 1 L flex 2 mode link - 2 mode estimotor - GPC control 1 ^ L 0.3- 15 10- o 5- 'ë- 0- - 5 - -10 flex 2 mode link — 2 mode estimator — GPC control - J 1- 0.04 -I 1 ^ 4 6 TIME [sec] 0.02- 0.00- -0.02 - - -0.04 - -0.06 10 —I 1— 4 6 TIME [sec] 10 CM -10.- TIME [sec] flex 2 mode link - 2 mode estimator - GPC control F i g u r e 3.6: T h e torque i n p u t T h e effect of under-modeling c an be seen i n F i g u r e 3.7, where a " r e a l " f lexible t w o m o d e h n k (ô"* order p lant ) is e s t i m a t e d for a m o d e l w i t h a s t ruc ture of one v i b r a t i o n m o d e (4"* order e s t i m a t e d mode l ) . T h e result is unstable . T h i s unstab le result was e x p e c t e d s ince s tab ih ty analyses usuaUy assume that the es t imator m o d e l shou ld be at least as c o m p l e x as the p lant itself . F i g u r e 3.8 presents the oppos i te effect of over-modeling where a f lexible one m o d e l i n k ( 4*'' order) is mode led w i t h a four mode es t imator ( 10*'' order ) . 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 diverge a n d ends w i t h ins t ab i h t y . T h e frequency of osc i l lat ions is about 14 [ r a d . / s e c ] w h i c h is the frequency of the first modes . W h e n the over -modehng is closer, as i n F i g u r e 3.9, where a 2 m o d e f lexible h n k (ô"* order ) is mode led w i t h an es t imator m o d e l of 3 modes (8" ' o rder ) , the response is not u n s t a b l e . H e r e the o u t p u t t racks the square wave of the set po int as shown i n F i g u r e 3.2, w h e r e the s i m u l a t e d m o d e l and the es t imator ' s m o d e l m a t c h i n s t r u c t u r e . These results are also checked i n C h a p t e r 5 for the two h n k r i g i d m a n i p u l a t o r w i t h the h y d r a u h c ac tuators . T h e conclusions so far are that u n d e r - m o d e h n g of a p lant w i t h an adapt ive c o n t r o l s y s t e m w i l l most p r o b a b l y result i n an unstab le sys tem. T h i s is expec ted , i f one observes a s s u m p t i o n s m a d e i n s tab ih ty proofs i n the h t e r a t u r e (Edgare a n d A s t r o m w h i c h say t h a t the e s t i m a t e d m o d e l shou ld be at least as c omplex as the rea l p lant m o d e l . T h e r e is m o r e f reedom i n the choice of s t ruc ture for over -modehng a p lant . If the e s t i m a t o r is close to the 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 the sys tem behaves very we l l , b u t w h e n the difference grows, as i n F i g u r e 3.8, i n s t a b i l i t y can occur . It seems t h a t the ins tab i l i t i es i n the over -modehng case are due to d y n a m i c s i n t r o d u c e d i n the contro l a l g o r i t h m through the 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 not a c t u a l l y exist i n t h e sys tem b u t w h i c h are present for the contro l a l g o r i t h m , since the e s t i m a t o r w i l l g ive non-zero values to over mode led m o d e l parameters . T h e r e is an effort m a d e t o contro l a n ent i re ly different p lant t h a n the a c t u a l one, w h i c h is p ro j e c ted t h r o u g h the " m e a s u r e d " values flex 2 mode link - 1 mode estimator - GPC control F i g u r e 3.7: T h e effect of under -modeUing flex 1 mode link - 4 mode estimotor - GPC control F i g u r e 3.8: T h e effect of over -mode lhng F i g u r e 3.9: O v e r - m o d e l l e d 2 mode h n k w i t h 8" ' order e s t i m a t o r of ytip. T h e next subsect ion describes the b e h a v i o r of a cost func t i on and a poss ib le m e t h o d of de tec t ing the need to change the s t r u c t u r e a n d integrate the new one, on hne , i n t o the contro l sys tem. 3.3.3 U s e of an 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 - t u r e of the P l a n t ' s M o d e l T h e cost f u n c t i o n is a t oo l for a t t a i n i n g a n o p t i m a l behavior of a p h y s i c a l p r o p e r t y of the sys tem. O n e m a y want to op t imize a t r a j e c t o r y for a robot a r m or to o p t i m i z e t i m e or o u t p u t error , as m a y be log ical for this 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 the o u t p u t error between the t i p pos i t i on ytip, the " 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 equat ions of m o t i o n , and the one f r o m the e s t i m a t o r m o d e l ye*t- t J{ytip,ye.t) = ^[ytip - yest? (3.25) k=0 F i g u r e 3.10 presents the behavior of the cost f u n c t i o n i n E q u a t i o n 3.25. Its va lue rises i n i t i a l l y w h e n there is a difference between the m o d e l a n d p lant d y n a m i c s , a n d t h e n w h e n the error goes to zero , i t settles to a constant va lue . T h e r e are , of course, a d d i t i o n a l possibiht ies for the choice of a cost f u n c t i o n w h i c h are not h m i t e d t o the one ment ioned . A n o t h e r in teres t ing cost f u n c t i o n , descr ibed i n Sect ion 2.3, is the one used for the G P C a l g o r i t h m ( E q u a t i o n 2.7). T h e r e an error is also m i n i m i z e d ; however, the o u t p u t s are the ones p r e d i c t e d . N o t only should the present o u t p u t t rack a set p o i n t , b u t the values to be m i n i m i z e d are pred i c ted ones, so the cost f u n c t i o n ensures t h a t the f u t u r e error w i U be m i n i m a l . In order to have reasonable c o n t r o l i n p u t s a n d not to d e m a n d , for e x a m p l e , e x t r e m e l y h i g h i n p u t signals that m a y dr ive the sys tem to s a t u r a t i o n , the 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 result is m i n i m a l o u t p u t error , 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 behav ior of the two cost funct ions for the 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 Effects of O n L i n e C h a n g e s in M o d e l order C h a p t e r 5 w i l l present a fuU discussion on the effects of under a n d o v e r - m o d e h n g o n the cost f u n c t i o n for b o t h apphcat ions of this work , the flexible h n k m a n i p u l a t o r a n d the h y d r a u h c a U y a c t u a t e d two hnk m a n i p u l a t o r . It w i U also present a m e t h o d to detect s t r u c t u r e m o d e h n g errors a n d correct t h e m . In this sect ion , p r e h m i n a r y discussion on the effects of the change o f the e s t i m a t o r m o d e l s t r u c t u r e on the controUed t i p pos i t i on of the flexible h n k is p resented . T h e h n k itself , the " r e a l " m o d e l , was chosen to have 2 v i b r a t i o n modes ( i .e . 6*^ o rder ). T w o cases are presented; i n the first, the es t imator 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*'' order ) a n d is then changed in to a 1 mode m o d e l ( i .e. 4*'' o rder ) . I n the second case, the 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 mode m o d e l ( 4*'' order) a n d is t h e n changed i n t o a 2 m o d e m o d e l ( 6"* order) to m a t c h the " r e a l " m o d e l m e n t i o n e d above. T h e next 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 on hne m o d e l changes. For each case, the t i p p o s i t i o n , the e s t i m a t o r cost f u n c t i o n , a n d the es t imator o u t p u t error behav ior w i U be presented . It s h o u l d be n o t e d that the c r i ter ion used to change the s t ruc ture of the e s t i m a t o r i n the cases presented was t i m e , w h i c h is not the final one ( C h a p t e r 6 presents the fuU c r i t e r ia ) . T h e a c t u a l c r i t e r i a are the changes i n the values of the chosen cost f u n c t i o n a n d i ts der ivat ives . In F i g u r e s 3.11 a n d 3.12 the 6" ' order m o d e l converges to the set p o i n t , a n d after 5 s e c , the e s t i m a t o r m o d e l has been changed to an under -mode led s i t u a t i o n (order = 4) w h e n the who le s y s t e m goes unstab le . In F i g u r e 3.11 yup reaches i n s t a b i h t y after the change of the 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 the changes i n the 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 the process s tarts w i t h the w r o n g e s t i m a t o r m o d e l a n d is changed on hne to the correct one.   flex 2 mode link - ^ & 2 mode estimator - GPC control 0 2 3 TIME [sec.; In the examples presented, the o u t p u t converges and t racks the set po in t . I n F i g u r e s 3.13 the change i n models is done at an ear ly stage (0.1 sec w h i c h are 100 s a m p h n g s teps ) , 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, but the results are st iU sa t i s fa c to ry . T h e es t imator cost f u n c t i o n ( F i g u r e 3.14) converges to a h igher value ( order of m a g n i t u d e of 10""*) t h a n the one ( order 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 models m a t c h . I n F igures 3.15 and 3.16 the correc t i on of the m o d e l is done la ter , so the sys tem ga ins m o r e error f r o m the w r o n g e s t i m a t o r m o d e l . T h e convergence takes longer t h a n i n F i g u r e 3.13, a n d the es t imator cost f u n c t i o n has m u c h larger values. 3.3.5 C o m p a r i s o n B e t w e e n the 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 It is of interest to compare the e s t i m a t o r cost f u n c t i o n behavior as presented i n E q u a t i o n 3.25 a n d the G P C contro l a l g o r i t h m cost f u n c t i o n as presented i n E q u a t i o n 2.7. F i g u r e s 3.17 3.18 3.19 present such comparisons for different m i s - m o d e h n g cases. W h e n w r i t i n g G P C error or e s t imator o u t p u t error , the ca lcu lat ions are of the t e r m s w i t h i n the s u m symbols i n b o t h E q u a t i o n 2.7 a n d 3.25, respect ively . T h o s e terms are c a l c u l a t e d at each t i m e step. W h e n w r i t i n g the G P C cost func t i on or the e s t i m a t o r cost f u n c t i o n , the ca l cu lat ions 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 the cost f u n c t i o n for the c h a n g i n g e s t i m a t o r m o d e l . It starts w i t h the correct 6"" order m o d e l that changes to a 4" ' o rder one after 5 seconds, as i n F i g u r e 3.11. F igures 3.18 and 3.19 present the oppos i te case, where the difference between the two is the t i m e of swi t ch ing models (as i n F igures 3.13 a n d 3.15 ). T h e two cost funct ions are funct ions of different variables . 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 the s u m of the square es t imator error , w h i c h is the difference be tween ytip, the measured value , a n d y^.t , the es t imator o u t p u t . T h e G P C cost f u n c t i o n is the s u m of the square error between the p r e d i c t e d output and future set po ints a n d the s u m of the we ighted square future contro l effort. Y e t even t h o u g h the behav ior of b o t h cost funct ions is very s i m i l a r , the ir values are different. T h i s m a y promote the use of different possible cost .0007 c .0006 c .0005 •a^ -0004 o o ^ .0003 . J .0002 (0 flex 2 mode link - 1 à 2 mode estimator - GPC control . 0 0 0 1 - .0000 0 4 6 TIME [sec. 8  flex 2 mode link - ^ ic 2 mode estimator - GPC control flex 2 mode link - 2 & 1 mode estimator - GPC control 10 '=1 1 1 \ 1 1 F 0 1 2 3 4 5 6 TIME [sec] flex 2 mode link - 1 2 mode estimator - GPC control 4 6 TIME [sec] 4 6 TIME [sec] flex 2 mode link — 1 & 2 mode estimator — GPC control 10000 * 10 Z 1500 d " 1000 o " 500 -1 r 4 6 TIME [sec] funct ions for the m o d e l s t ruc ture changes a l g o r i t h m . 3.4 C o n c l u s i o n s A n interest ing po int of this research is the response of a d a p t i v e l y contro l led s y s t e m s t o on-hne changes i n the m o d e l s t ruc ture due to var iat ions i n operat ing condi t ions . A d a p t i v e a lgor i thms usua l ly use an es t imat i on procedure for the p lant or controUer parameters i n w h i c h the s t r u c t u r e of the p lant ' s m o d e l is assumed to be fixed. E s t i m a t e d values are c ons idered t o be correct , a n d uncerta int ies i n those values are ignored ( the cer ta inty equivalence p r i n c i p l e ) . Re l iance on that p r inc ip l e can lead to i n s t a b i h t y i n the s y s t e m . A good e x a m p l e is a flexible h n k m a n i p u l a t o r , where changes i n l oad d u r i n g a w o r k i n g cycle can result i n the rise of v i b r a t i o n modes w h i c h were not there before. T h i s c h a p t e r presents the equat ions of m o t i o n for a single flexible 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 w i t h a 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 the parameters e s t i m a t e d w i t h the Recurs ive Least Squares ( R L S ) a l g o r i t h m . S i m u l a t i o n results of the controUed s y s t e m are presented. U n d e r - m o d e h n g of the p lant ' s d y n a m i c s (i.e. the order of the e s t imator is s m a l l e r t h a n the " r e a l " order ) leads to i ns t ab i h t y . O v e r - m o d e h n g c o u l d also l ead to i n s t a b i l i t y w h e n the gap between the es t imated m o d e l order a n d the a c t u a l system's m o d e l is too large . However , there are condi t ions under w h i c h the sys tem behaves weU w i t h over -modehng . It is also shown t h a t a sys tem w h i c h begins w i t h an u n d e r - m o d e l e d es t imator p l a n t , a n d is t h e n changed to the correct one, w iU not become unstable under the right cond i t i ons , as i t w o u l d have i f the change i n the es t imator h a d not been done. It is suggested t h a t the cost f u n c t i o n presented i n E q u a t i o n 3.25 m a y be a c r i t e r i o n to detect the need for the es t imator ' s order change. A change of the es t imator ' s m o d e l s t r u c t u r e o n Une requires a change i n the controUer '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 and execute i t w h e n the sys tem is contro l led 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 chapter shows the results of c o r rec t ing 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 that cou ld be unstable , but w i t h the correct ion has an acceptab le p e r f o r m a n c e . No. of Modes 0 1 2 3 4 5 Order of the system 2 4 6 8 10 12 Tab le 3.1: N o . of flexible modes vs . order of sys tem C h a p t e r 4 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 A T O R S O u t l i n e R o b o t i c m a n i p u l a t o r s consist of Unks ( r ig id or flexible), connected by jo ints t h a t c o n t r o l t h e re lat ive m o t i o n of ne ighbor ing h n k s . T h e j o ints usua l ly have pos i t i on sensors w h i c h m e a s u r e the re lat ive m o t i o n a n d are ac tuated by e lectr ic , p n e u m a t i c or h y d r a u h c dr ives T h e s e systems are subject to nonhnear i t ies such as c ouphng , coulomb f r i c t i o n and b a c k l a s h . T h e i r i n e r t i a l character ist ics and loads vary d u r i n g opera t i on and are not 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 ide ly used i n i n d u s t r y today. H y d r a u h c s y s t e m s have re la t ive ly large torque to weight rat ios , higher loop gains a n d wider b a n d w i d t h s c o m - p a r e d w i t h e lec t r i ca l motors H y d r a u h c robots are used for heavy d u t y tasks r e q u i r i n g p o s i t i o n accuracy , r a p i d d y n a m i c s a n d r a p i d start a n d stop. However , h y d r a u h c sys tems are c o m p l e x , nonhnear a n d dif f icult to analyze for c ontro l purposes , I n the design of a m a n i p u l a t o r contro l s trategy two 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 cons idered , those that can be d e t e r m i n e d accurate ly w i t h the values r e m a i n i n g r e l a t i v e l y constant , a n d those t h a t vary w i t h i n a range of values d u r i n g a w o r k i n g cyc le . T h e second t y p e of quant i t ies cannot always be avo ided i n a contro l sys tem a n d m a y require an on -hne change of the controUer parameters . E x a m p l e s i n h y d r a u h c systems w o u l d be 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 iscos i ty changes of the h y d r a u h c fluid, d a m p i n g coefficient etc. I n the m o d e h n g of the h n k s , one can find changes i n the m o m e n t s of i n e r t i a d u r i n g a w o r k i n g cyc le w h e n an e x t e r n a l l o a d ( in some cases a n u n k n o w n e x t e r n a l load) is b e i n g p i cked u p and put d o w n . C o m p l i a n c e i n the h n k s m a y give osc i l latory d y n a m i c s w i t h low d a m p i n g , i.e. exc i te v i b r a t i o n modes t h a t change the m o d e l of the contro l led sys tem. M o s t of the techniques for contro l sys tem design assume the p lant 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 however, this is not so, since the p lant m i g h t be too c o m p l e x , the m o d e l not fu l l y unders tood , or the process a n d the d is turbances c h a n g i n g w i t h o p e r a t i n g cond i t i ons . W h e n a system'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 the p o s s i b i h t y of c h a n g i n g its parameters on-hne , adapt ive contro l m a y be considered. C o n t r o l of r obo t i c systems has been w i d e l y discussed i n the h t e r a t u r e before, ( F u et a l A n et a l A s a d a , C r a i g ^1 a n d others. ) T h e d y n a m i c s of the a c t u a t o r s are usua l ly i gnored a n d the h n k m o t i o n provides second order equations w i t h c o u p h n g effects 1^. Sepehr i et a l show a contro l s trategy i n w h i c h the 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 variance contro l ) , and the hydrauhcs is c ont ro l l ed b y a c lass i ca l c o n t r o l a l g o r i t h m . 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 contro l led by an adapt ive a l g o r i t h m was presented by V a h a i n 1988 'j '^g contro l a l g o r i t h m was based on a one-step-ahead s e l f - t u n i n g c o n - troUer proposed by C l a r k e et a l i n 1975 A n in tegra l t e r m was i n t r o d u c e d to a q u a d r a t i c per f o rmance c r i t e r i o n w h i c h was m i n i m i z e d to find a contro l law that was a p p h e d to 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 per formed i n order to eva luate the a p p h c a b i l i t y of the adapt ive a l g o r i t h m to contro l the movement of the m a n i p u l a t o r ' s l i n k s . T h e autoregressive m o d e l chosen for the adapt ive contro l process exper i enced d i f f i cu l - ties caused by m e c h a n i c a l a n d p h y s i c a l character ist ics a n d measurement noise. A s weU, t h e s t u d y cons idered s i m u l a t i o n eva luat ion of the p r o b l e m . T h e m o d e l chosen to s i m u l a t e the a c t u a l m a n i p u l a t o r was a hnear ized second order m o d e l . A s prev i ous ly m e n t i o n e d , i n 1987 C l a r k e et a l ^ developed the 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 t e chn ique w h i c h m a y have advantages for the contro l of c omplex systems such as heavy d u t y m a n i p u l a t o r s . T h e present work apphes the G P C a l g o r i t h m to an extensive m e c h a n i c a l a n d h y d r a u h c sys tem m o d e l of an 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 , to assess the contro l c a p a b i h t y of t h i s more recent a l g o r i t h m on such a nonhnear s y s t e m , a n d to s tudy the effect of t h e des ign parameters , K o t z e v et a l '̂̂  . F i r s t the d y n a m i c m o d e l of the m a n i p u l a t o r is presented as wel l as the e q u a t i o n s o f m o t i o n of the h y d r a u h c ac tuator i n c l u d i n g comphance , dead t i m e a n d f u l l d y n a m i c s of t h e servovalve, resu l t ing i n a ra ther c omplex nonhnear sys tem i n w h i c h the order of the e s t i m a t e d hnear m o d e l for the G P C m a y vary f r o m 6 to 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 de ta i l . It uses for contro l purposes a hnear i zed m o d e l of the sys tem. T h e c o n t r o l law d e r i v e d depends on values of the measured o u t p u t f r o m the nonhnear sys tem, a n d uses a s s u m e d a n d e s t i m a t e d parameters for the hnear m o d e l . T h e c o n t r o l s trategy i n th is work , consists of two adapt ive loops, i n w h i c h the process m o d e l contains the m a n i p u l a t o r h n k w i t h the h y d r a u h c ac tuator . T h e r e is a n advantage i n c o m b i n i n g aU the sys tem states in to one contro l l oop , where the sys tem is represented by a n i n p u t / o u t p u t m o d e l i n the G P C , since the e s t i m a t e d parameters can reflect a l l changes i n the s y s t e m as wel l as the uncer ta int ies , d i s turbances , nonhnear i t ies a n d c o u p h n g , p r o v i d e d t h a t safety l i m i t s on the required sys tem variables exist . T h i s approach can be i m p l e m e n t e d on any 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 actuators as r e q u i r e d . It can also be i m p l e m e n t e d on m a n i p u l a t o r s w i t h other ac tuators such as e lectr ic motors . T h e results show the effects of the different contro l t u n i n g parameters o n the contro l led s y s t e m p e r f o r m a n c e . T h e G P C has an inherent in tegrator w h i c h helps overcome offsets b u t results i n undes i rab l e overshoot w h e n opera t ing robot m a n i p u l a t o r s . I n a d v a n c i n g the state of the art of pred i c t ive c o n t r o l , i n th is w o r k spec ia l a t t e n t i o n is g iven to the 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 larger values (i.e. larger 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 and damps the o u t p u t behav ior but slows the t rans ient response. T h e work also introduces an on- l ine a u t o m a t i c change of the 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 the transient response can be suff ic iently fast a n d undes irable overshoots avo ided . F u r t h e r advances are also made i n the selection of other G P C design parameters . T h e d y n a m i c equations of the m a n i p u l a t o r a n d i ts ac tuators have been s i m u l a t e d i n a F O R T R A N p r o g r a m along w i t h the contro l a l g o r i t h m a n d the n u m e r i c a l s i m u l a t i o n s were per f o rmed on a V A X 3200 computer . 4.1 R i g i d 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 a t o r s 4.1.1 I n t r o d u c t i o n A two h n k r i g i d m a n i p u l a t o r is a c omplex and nonhnear sys tem. T h e c o u p h n g be tween t h e m o t i o n of the a rm ' s components introduces nonhnear i t ies . T h e h y d r a u h c a c t u a t o r s consist o f servovalves a n d cyhnders a n d m a y be descr ibed as a t h i r d or a fifth order s y s t e m . T h e n e x t two sections wiU present the equations of m o t i o n of the d y n a m i c s of the m a n i p u l a t o r h n k s , a n d the equat ions of m o t i o n of the h y d r a u h c ac tuators , w h i c h w i l l be expressed i n equat i ons for so lv ing Pii,,,Piaut T h e state space vector , of order 8, for th is speci f ic s y s t e m is : X = Ql 01 P2.„ P2. (4.1) 4.1.2 E q u a t i o n s of M o t i o n for the 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 the conf igurat ion of the two h n k m a n i p u l a t o r . U s i n g a genera l f o r m u l a t i o n , , the d y n a m i c equations of m o t i o n der ived v i a L a g r a n g e ' i s a p p r o a c h are : 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 the two h n k m a n i p u l a t o r W h e r e n is the n u m b e r of degrees of f reedom, 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 int i due to h n k j m o t i o n , Dijk t erms for the Cor iohs a n d c e n t r i p e t a l forces at j o i n t i as a result of m o t i o n i n hnks j a n d k, a n d Di are terms for g rav i ty l oad ing at j o i n t i. T h e k i n e t i c energy of the sys tem is : (4.3) +7712/1/2 cos ^ 2 ( ^ 1 + ^ 1 ^ 2 ) T h e p o t e n t i a l energy of the system is : 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 a p p l y i n g Lagrange ' s equat i on , the nonhnear equations of m o t i o n are : T i = (77ii + 7712) ll + 7712/2 + 27712/1/2 cos 62 di (4.5) + 77X2/0 + "^2/1/2 cos 02 9, -277x2/1/2 s in 02^1^2 - «12/1/2 sin ^2^2 + ("'•1 + "^2) gh s in 9i + 77125/2^171(^1 + Ô2) T2 = 7712/2 + «12/1/2 C0SÔ2 9i (4.6) + m 2 / 2 ^ 2 + 7712/1/2 s in ^2^1 1̂̂  + 77125/2 s i n ( e i + ^2) W h e r e r i a n d r2 are i n p u t torques to the j o ints . 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 of the m a n i p u l a t o r are a c t u a t e d by h y d r a u h c ac tuators . E a c h h n k is a c t i v a t e d by a h y d r a u h c m o t o r w h i c h is connected to a servovalve t h r o u g h expandab le hoses. T h e servovalve m o n i t o r s the flow of the h y d r a u h c h q u i d . F i g u r e 4.2 describes a c r i t i c a l center s y m m e t r i c valve. T h e supp ly pressure is kept constant w h i c h allows each servovalve to 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 the a tmospher i c pressure, since i t is connected to a storage t a n k . C o m p o n e n t s such as check valves a n d rehef valves are for mach ine safety. T h e servo- valves c o n t r o l the f lu id power. T h e most w i d e l y used valve has a spoo l valve t y p e c o n s t r u c t i o n , a n d is classified by the way flow goes t h r o u g h the valve . T h e valve variables are t h e s p o o l d isp lacement ( A V J , the flows i n a n d out of the valve qi-^ a n d g^^, , the supp ly pressure (P,up)) the r e t u r n pressure (Près), a n d the hne pressures (P,-.^ a n d T h e equat ions d e s c r i b i n g the equat ions of m o t i o n for the valves are nonhnear . T h e flow equat ions are > 0, (positive direction) g..„ = K.ai.e Xv, v^P,up - Pi,„ (4.7) qia^t = K^cilve Xv, \JPi^t ~ Pre. (4.8) < 0, (negative direction) = K,al.e Xv, ^JPi,^ - Pre. (4.9) g.„„. = K,al.e Xv, ^P.up - Pi^, (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 tuator L i n e a r i z a t i o n of tliese equations w i t h a T a y l o r series expans ion about zero spoo l d i sp lace - m e n t , for i n i t i a l design purposes only, gives: = if . . Xv, - i^p. P,.„ (4.11) = ^ F . + ^ P . (4.12) W h e r e K^i a n d Kp- are the flow ga in and the flow pressure coefficients, respect ive ly . A first order m o d e l , descr ib ing the equat ions of the pipes m o d e l are: K.-^{<li..-D^éi) (4.13) ^ w = ^ ( ^ m ^ i - 9 w ) (4.14) W h e r e Dm is the v o l u m e t r i c d isplacement of h y d r a u h c m o t o r , a n d ^ is the h y d r a u h c c o m p h a n c e . 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 the first t e r m expresses the movement of the h y d r a u h c m o t o r , the second is a d a m p i n g t e r m , Ti expresses e x t e r n a l l oad by the finks movement a n d Ti is the a p p h e d t o r q u e t o the h n k i. 4.2 C o n t r o l S trategy 4.2.1 I n t r o d u c t i o n A s m e n t i o n e d before, the equat ions of m o t i o n of a robot 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 , i n e r t i a l character is t i cs a n d d is turbances t h a t vary d u r i n g a w o r k i n g cycle a n d m a y n o t a l w a y s be pred i c tab le . L a t e l y self t u n i n g pred i c t ive a lgor i thms have been used since the resu l t s have better robustness c ompared w i t h other self t u n i n g contro l a lgor i thms such as P o l e P l a c e m e n t a n d G e n e r a h z e d M i n i m u m V a r i a n c e ( A s t r o m T h e robustness of pred i c t ive a l g o r i t h m s is due to the 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 pred ic t ive m e t h o d has t h e fo l lowing steps: 1. P r e d i c t i o n of the output i n the future . 2. C h o i c e of the future set po ints , 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 future errors , between the future o u t p u t s a n d future set po ints , y ie lds a set o f f u t u r e contro l signals. 3. T h e first element of the contro l signals is a c t u a l l y used a n d the whole p r o c e d u r e is repeated . T h i s is a receding-hor izon controUer. T h e t y p e of controUers m e n t i o n e d above consider the o u t p u t at one po in t of t i m e i n t h e fu ture . 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 considers the future pred i c ted o u t p u t s j steps ahead , the fu ture set po ints a n d f u t u r e c o n t r o l s ignals . 4.2.2 C o n t r o l S t r a t e g y for the T w o l ink 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 block d i a g r a m for controUing the t i p p o s i t i o n of the two h n k m a n i p u l a t o r w i t h the h y d r a u h c actuators . Process parameters Design Regulator parameters Regulator Estimation Process F i g u r e 4.3: C o n t r o l s trategy for the two h n k m a n i p u l a t o r T h e t i p l o cat i on error is t rans la ted to angle changes i n the jo ints . T h e c o n t r o l consists of two adapt ive loops, i n w h i c h the process m o d e l conta ins the 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 ac tuator . E a c h of the j o int h n k s is controUed separate ly w i t h the genera l p r e d i c t i v e contro l a l g o r i t h m . T h e m o d e l for each l oop wiU be an I n p u t / O u t p u t t y p e of m o d e l i n the f o r m of ; Aiq-')y{t)=^b{q-')u{t) + c{q-')e{t) (4.16) W h e r e : y(t) is the o u t p u t - jo int angle. u{t) is the i n p u t to the process - spool d isp lacement e{t) is the noise sequence. 4.3 A n a l y s i s a n d Resul ts of S i m u l a t i o n 4.3.1 S y s t e m P a r a m e t e r s T h e l i n k parameters are: h = 50 c m . I2 = 50 cm. m i = 1 kg. m 2 = 1 kg. T h e h y d r a u h c ac tuator parameters are: K.al.e - 243. kg Dm = 8.2 cm^ ^ ^ o t i l V ' cm^ kgf Psup = 65. - 250. cm' P = 0 M ^ res — o K^, = 1387. cm Kp. - 4.65 ^ cm? T h e range of the spoo l d isp lacement is : - 0 . 5 cm. < Xvi > 0.5 cm. W h e r e i = 1,2 for the n u m b e r of h n k s . 4.3.2 O p e n L o o p A n a l y s i s T h e h n e a r i z e d equations 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 hnear i zed E q u a t i o n 4.5, E q u a t i o n 4.6 produces the f o l l owing s tate space f o r m , for a single hnk a n d are used on ly for p r e h m i n a r y s t u d y a n d design : X = A X + BXv, where the state space vector for one loop is : A U o w P L , = Pi„^ - P.„.. , a n d : A = 0 0 0 \ Jii - 2 f 1 0 W h e r e the eigenvalues are: (4.17) (4.18) 0.9997413 1.072178 0.9996837 B = (2^K ) 0 0 / T h e transfer f u n c t i o n between the angle a n d spoo l d i sp lacement is : (4.19) Oi{s) 2 f Dm K,, (4.20) Xv, s {Jli . 2 + JU l,Kp, 3 + 2 f D ^ ) T h e s y s t e m was converted w i t h a zero order h o l d s a m p h n g in to a discrete t i m e f o r m , t h e i n p u t / o u t p u t m o d e l i s : Oiitk)^ -ai Oiitk - 1) - a2 Oi{tk - 2) - Biitk - 3) (4.21) +h Xv,{tk - l ) + b2 Xv,{tk - 2) + fci Xv,{tk - 1) + ei{tk) For s a m p l i n g per i od oî h = 0.005 [sec] the parameters of the A a n d B p o l y n o m i a l s are: a l = - 2 . 9 3 2 6 , a2 = 2.8652, a3 = - 0 . 9 3 2 6 , 61 = 0.000000554, 62 = 0.00O002177, 63 = 0.0000005348. W h e r e e{t) is the u n m e a s u r e d d i s turbance t e r m w h i c h i n c l u d e s t w o components , the first, d y n a m i c couphng between the 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 second an uncorre la ted r a n d o m noise sequence i f exists . 4.3.3 S i m u l a t i o n S t u d y a n d R e s u l t s Subsec t i on 4.3.2 developed the open loop analysis w h i c h was used to de termine the o r d e r o f the i n p u t / o u t p u t m o d e l for the G P C a l g o r i t h m , a n d the i n i t i a l values for i d e n t i f i c a t i o n of i t s parameters . T h e a c t u a l sys tem is a nonhnear sys tem resu l t ing f r o m couphng t e r m s d u e t o re lat ive m o t i o n of the h n k s , the 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 on var iables , the h y d r a u h c s y s t e m etc. T h e nonhnear i t ies were i n c o r p o r a t e d in to the s i m u l a t i o n m o d e l wh i l e t h e m o d e l e s t i m a t e d a n d used by G P C is hnear by i ts na ture . 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 the contro l l ed nonhnear s y s t e m , (the two 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 ac tuators ) w h i c h e x h i b i t good per formance i n o u t p u t t r a c k i n g of the g i v e n set po in ts . T h e nonhnear i t ies are t rea ted as u n k n o w n de te rmin i s t i c d i s turbances i n t h a t the G P C assumes a hnear m o d e l for the a c t u a l system. F i g u r e 4.4 a n d F i g u r e 4.5 show the behavior of the o u t p u t s 6i, a n d 02 a n d the i r der iva t ives to square wave setpoints . F i g u r e 4.6 presents the spoo l valves d isp lacements a n d the c o n t r o l a c t i o n Au(t) for b o t h hnks a n d actuators . F i g u r e 4.7 presents the hose pressures for b o t h ac tuators . T h e design parameters used for t u n i n g the G P C are noted at this po i n t . N2, the m a x i m u m o u t p u t h o r i z o n , is changed on hne. Ni, the m i n i m u m o u t p u t h o r i z o n , N^, the c o n t r o l h o r i z o n a n d A , the we ight ing factor are a d d i t i o n a l design parameters . A t first a lower value of N2 {N2,. )was used , to achieve a faster t rans ient response a n d later i t was increased to iV2,_ to a v o i d overshoots . In the case presented 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 square wave i n p u t TIME [sec] F i g u r e 4.5: ^2,^2,^2 for square 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 on and spool d isp lacement for 9i a n d 82 GPC control — 2 link monipulotor with hydraulic octuotors 25- 10 15 TIME [sec] 1 r 10 15 TIME [sec] 20 10 15 TIME [sec] 10 15 TIME [sec] F i g u r e 4.7: Pressures for Oi a n d 62 the sys tem ' s per formance was achieved w i t h the fo l lowing design parameters : 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 steady state there is a chatter i n some of the parameters , such as t h e s p o o l valve d i sp lacement , the contro l increment s ignal , the hose pressures a n d the acce lerat ions . In s teady s tate the spool valve chatters a r o u n d the zero value w h i c h it cannot m a i n t a i n due to nonhnear i t i es i n the sys tem 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 imulates the a c t u a l sys tem a n d the represent ing hnear m o d e l used by the 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 the results for the same design parameters 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 this case the m o d e l of the h y d r a u h c ac tuator sys tem was h n e a r i z e d . T h e nonhnear i t i e s due to coup l ing between the movement of the hnks or due to s a t u r a t i o n i n the d i sp lacement of the spoo l valve r e m a i n i n the s i m u l a t e d m o d e l of the sys tem. T h e c h a t t e r i n g has been r e d u c e d s ignif icantly . Changes i n the values of the design parameters above w i l l change the b e h a v i o r of the s y s t e m . T h e effects of the o u t p u t hor i zon N2 were checked i n b o t h loops. T h e larger the value of N2, the slower the response. L a r g e r values have the t endency to s tab ihze the s y s t e m since i t uses pred i c ted errors over a larger p e r i o d of t i m e . O n the other h a n d fewer p r e d i c t i o n steps w i U result i n a more r a p i d contro l a c t i o n , a n d the inherent i n t e g r a t i o n t e r m of the m o d e l used for G P C causes the response to have more overshoot a n d more osc i l la t i ons . 0i i n the u p p e r part of F i g u r e 4.10 is the same as the one i n F i g u r e 4.4 b u t the setpo int is constant i n s t e a d of a square wave. T h e lower part shows 61 i n response to the same setpo int , b u t the o u t p u t h o r i z o n is constant a n d its values are the h igher values for iVz,^ = 100 a n d N2,^ = 70 used i n F i g u r e 4.4. T h e response for the larger values is slower. GPC control - 2 link naonipulotor with hydroulic octuotofs 1.0 10 TIME [sec] 3* 0.5- 10 TIME [sec] 15 20 10 15 TIME [sec] E o . CM 1.0 T a u u _ç o c o u 0.5- - i2 0.0 - T - 5 10 TIME [sec] 15 20 F i g u r e 4.8: C o n t r o l a c t i on a n d spoo l d isplacement for 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 15 TIME [sec] F i g u r e 4.9; Pressures for di a n d 62 for h y d r a u K c l inear i zed m o d e l . F i g u r e 4.10: T h e effect of h igher values of ATj ( lower case) P A G I N A T I O N E R R O R . E R R E U R D E P A G I N A T I O N . T E X T C O M P L E T E . L E T E X T E E S T C O M P L E T . N A T I O N A L L I B R A R Y O F C A N A D A . C A N A D I A N T H E S E S S E R V I C E . B I B L I O T H E Q U E N A T I O N A L E D U C A N A D A . S E R V I C E D E S T H E S E S C A N A D I E N N E S .  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: and 62 for N^^^ ^ 3 a n d N^^^ = 3 GPC control — 2 link monipulotor with hydraulic actuators 0.6-1 ' 1 ' h 0 5 10 15 20 TIME [sec] F i g u r e 4.14: 9^ a n d ^2 for N^^^ = 10 a n d = 10 4.3.4 Effects of O n - l i n e C h a n g e s in M o d e l O r d e r A n analysis of the effect of under 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 iven i n C h a p t e r 5. T h e analysis is based on the 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 correct , under a n d over -modehng . 4.4 C o n c l u s i o n s I n this chapter a r i g i d 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 actuators contro l l ed b y a G P C a l g o r i t h m is presented. T h e sys tem is h i g h l y nonhnear , c ontroUed w i t h a G P C a l g o r i t h m t h a t assumes a hnear 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 , whose p a r a m e t e r s are e s t i m a t e d onhne . T h e c o n t r o l s trategy treats the contro l of each h n k a n d i ts a c t u a t o r as one m o d e l . T h e changes i n the s y s t e m parameters are h a n d l e d w i t h o u t the need to i d e n t i f y the exact cause of the change . S u c h a n approach m a y need safety measures a n d b o u n d e d values on some of the s y s t e m var iables . T h e sys tem is weU behaved w h e n controUed w i t h the set of des ign p a r a m e t e r s shown 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 overcomes the effects of the nonhnear i t i e s i n the a c t u a l m o d e l of the s y s t e m , ( couphng between h n k s , sa turat i ons etc . ) I n t e r m s of advanc ing the state of the art of p red i c t ive c o n t r o l , the inf luence of the m a i n G P C design parameters , N2 and N^, was s tud ied . It was r e c o m m e n d e d , by C l a r k e et a l , t h a t the c o n t r o l hor i zon N^, shou ld be chosen as h igh as the n u m b e r of p o o r l y d a m p e d poles of the s y s t e m . I n this s t u d y there was no signif icant effect on the o u t p u t by choos ing 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 the o u t p u t s have a more osc iUatory n a t u r e . A s weU, the m a x i m u m o u t p u t h o r i z o n N2 is shown to have a s t a b i h z i n g effect i f the p r e d i c t i o n m a r g i n is w i d e enough. T h e larger the value N2, the slower and more d a m p e d the response. T h e o u t p u t h o r i z o n was found to p l a y a role i n r educ ing the inherent overshoot of the G P C a l g o r i t h m ( F i g u r e 4.10 c ompared to F i g u r e 4.11 ). I n th is work , i ts value was changed on hne a n d resul ted i n a re la t ive ly q u i c k e r response at first a n d s igni f i cant ly reduced overshoot (if at a l l ) as shown 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 to F i g u r e 4.10. T h e sluggish response to h igher values of N2 a n d the o s c i l l a t o r y n a t u r e of i t for lower values was shown i n F i g u r e 4.12 c o m p a r e d to 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 represented by 2 h n k h y d r a u h c a U y a c t u a t e d mechanisms of the t y p e i n this work . T h i s work studies the behavior of th is c a t e g o r y of m a n i p u l a t o r w h e n contro l led by a n adapt ive contro l a l g o r i t h m . T h e s t u d y can be e x p a n d e d to 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 hnks . H y d r a u h c a U y a c t u a t e d machines are h i g h l y n o n h n e a r systems a n d the ir parameters m a y vary onhne d u r i n g a w o r k i n g cycle . W h e n c o n t r o U e d b y G P C a good per formance of the o u t p u t i n t r a c k i n g a sequence of set po ints was a c h i e v e d . T h u s i n a d d i t i o n to advanc ing the state of the art i n ce r ta in areas of p r e d i c t i v e c o n t r o l r e la ted to design parameters , the work descr ibed i n this chapter has also e x a m i n e d some c o m p l e x considerat ions such as the effect of h a r d nonhnear i t i es i n the a p p h c a t i o n of G P C t o a b r o a d category of h y d r a u h c a U y ac tuated m a n i p u l a t o r s . C h a p t e r 5 M O D E L O R D E R D E T E R M I N A T I O N 5.1 I n t r o d u c t i o n A d a p t i v e contro l a lgor i thms are designed assuming that the p lant m o d e l is def ined b y a fixed s t r u c t u r e . A quest ion asked i n this work is how w i l l an a d a p t i v e c o n t r o l a l g o r i t h m ( the G P C i n th is case) behave w h e n the t rue p lant is not per fect ly descr ibed by any m o d e l o f a g i v e n class. T h e behav ior of a specific a l g o r i t h m is unders tood 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 usua l ly require restr i c t ive assumptions , for e x a m p l e , a s s u m p t i o n o n the s t r u c t u r e of the m o d e l (number of poles, n u m b e r of zeros, t i m e delay, etc . ) . I n m a n y research studies , s tab ih ty proofs for adapt ive contro l led systems are done for examples w h i c h dea l p r i m a r i l y w i t h hnear systems, a n d the 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 modehng errors are suff ic iently s m a l l , robust s t a b i h t y of 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 ). M o d e h n g errors ( s u c h as u n m o d e l e d or over -modeled d y n a m i c s , nonhnear i t ies , etc . ) appear as d i s turbances i n t h e a d a p t i v e process. T h e present research deals w i t h changes i n the s t ruc ture of the p l a n t , a n d therefore changes i n the m o d e l order . W h e n the m o d e l order is not accurate , the m o d e h n g error can be large a n d ins tab iht i es can appear . B o t h apphcat ions used i n th is work 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 hnear ized m o d e l of the sys tem for contro l purposes . A n y d ivers i on f r o m the hnear f o rm is as a d i s turbance to the 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 igh ly nonhnear s y s t e m w h i c h is represented by a hnear m o d e l for the G P C . T h e m o d e h n g errors act as d is turbances caused p a r t l y by the nonhnear i t ies . C h a p t e r 4 a n a l y z e d th is sys tem a n d good per formance is achieved. W h e n the order of the sys tem is c h a n g e d , for e x a m p l e w h e n t i m e delay or valve d y n a m i c s are i n t r o d u c e d , a n d the contro l a l g o r i t h m has not been u p d a t e d w i t h the changes, the m o d e h n g error is large a n d ins tab i l i t i e s o c c u r ( the nonhnear terms r e m a i n unchanged) . T h e single flexible 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 hnear m o d e l that matches the order of the one for the adapt ive a l g o r i t h m . G o o d p e r f o r m a n c e is shown i n C h a p t e r 3. W h e n the order of the sys tem does not m a t c h the one r e p r e s e n t i n g it for the adapt ive a l g o r i t h m due to v i b r a t i o n modes , the m o d e h n g errors are large a n d ins tab iht i es occur . T h i s chapter presents a m e t h o d for de tec t ing and correc t ing the m o d e l order a n d hence m i n i m i z i n g the modehng error . It evaluates a cost f u n c t i o n a n d its der ivat ive . I f necessary, the represented m o d e l order for the adapt ive a l g o r i t h m is changed on-hne to reduce m o d e h n g errors a n d the uncer ta in parameters i n the m o d e l e s t i m a t e d as is n o r m a l l y done i n a n a d a p t i v e a l g o r i t h m . 5.2 C o s t F u n c t i o n - F o r D e t e c t i o n O f T h e M o d e l S t r u c t u r e T h e goal , i n order to achieve good per formance , is to reduce m o d e h n g errors . A s a measure of the m o d e h n g error , we choose a cost f u n c t i o n w h i c h is the square of the difference be tween the o u t p u t measured f r o m the a c t u a l sys tem a n d the one of the hnear m o d e l as used 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]^ (5-1) A cost f u n c t i o n is usua l ly chosen as a c r i t e r i on to be m i n i m i z e d a c co rd ing t o the final target . For the e s t imat i on p r o b l e m , for e x a m p l e , the process output a n d the e s t i m a t e d m o d e l o u t p u t are compared a n d some o p t i m a l ad justment between the two s h o u l d b e f o u n d . T h e o p t i m u m is defined us ing a c r i t e r i on w i t h respect to o u t p u t signals or t o t h e e x p e c t e d error of the e s t imated parameters values. F o r contro l purposes the c r i t e r i on is o p t i m i z e d i n order to achieve a desired c o n t r o l l a w . T h e c r i t e r i on can be a quadrat i c f o rm w h i c h , for example , cou ld be a f u n c t i o n o f t h e s tate vector and the i n p u t s ignal (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 ^^). I n C h a p t e r 3 the behav ior of the cost func t i on of the 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 the one presented i n E q u a t i o n 5.1 a n d was f ound to be s imi lar . T h e behav ior of the cost f u n c t i o n i n E q u a t i o n 5.1 was s tud ied w i t h o p e n loop a n d c losed loop contro l i n order to de termine a m e t h o d of detec t ing a n on-hne change i n t h e a c t u a l system's m o d e l a n d of chang ing the m o d e l for the contro l a l g o r i t h m accord ing ly . T h e cost f u n c t i o n behav ior was s tud ied i n two ways , first as a func t i on of order changes ( F i g u r e s 5.1 a n d 5.2 ) i .e . the value of the cost f u n c t i o n was recorded at a ce r ta in 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 orders (the p lant ' s m o d e l r e m a i n e d unchanged) a n d second , as a f u n c t i o n of t i m e . I n each r u n the p lant m o d e l a n d order of the 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 the desired p e r i o d of t i m e . T o e x a m i n e the open loop behav ior of the cost f u n c t i o n as a f u n c t i o n of the o rder , the e s t i m a t e d m o d e l order was changed whi le the a c t u a l sys tem s t ruc ture 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 the f lexible h n k (as descr ibed i n C h a p t e r 3 ), the a c t u a l s y s t e m h a d two v i b r a t i o n modes (p lant m o d e l order of 6) , but the e s t i m a t e d m o d e l order was changed ( f r o m 2 w h i c h is a r i g i d body , to 10 w h i c h is four v i b r a t i o n modes) . F i g u r e 5.1 shows the 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 the m o d e l order . T h e values at each order are re la t ive ly s m a l l , a n d i t is h a r d to dif ferentiate between t h e m . O n the other h a n d . F i g u r e 5.2 shows the b e h a v i o r of the same cost f u n c t i o n w i t h the same m o d e l order changes to a closed loop 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 , the results for each chosen order of the e s t i m a t e d m o d e l differ F i g u r e 5.1: Cos t f u n c t i o n behavior for open loop flexible h n k F i g u r e 5.2: C o s t f u n c t i o n behavior for closed loop flexible h n k extens ive ly , a n d the cost funct i on can i n d i c a t e the size of the m o d e h n g error . 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 the i d e n t i f i c a t i o n of the 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 215B excavator , w h i c h is a two h n k m a n i p u l a t o r a c t u a t e d by h y d r a u h c actuators . T h e ident i f i ca t ion done was a n open loop one i n w h i c h no a d a p t i v e contro l a l g o r i t h m was i n t r o d u c e d . T h e results i n A p p e n d i x A show that after the cost f u n c t i o n becomes flat at h igh orders i t is easy to mis-choose the order for the sys tem 's m o d e l , (see F igures A . 8 a n d A . 9 ) O b s e r v i n g the t i m e behavior of the cost f u n c t i o n reveals three parameters t h a t can be u s e d for de te c t ing mis -modehng . T h e parameters are: the R i s e Slope J r , the Zero Slope Jz, a n d C h a n g e T i m e T^. F i g u r e 5.3 also shows a behav ior f ormat of the cost f u n c t i o n ( C . F . ) ( w h i c h are backed u p , la ter i n the work , w i t h figures showing the a c t u a l behav ior of the C . F . ) for several con f igurat ions , for the two apphcat ions , where the three parameters are s h o w n . I n a l l of the conf igurat ions shown i n F i g u r e 5.3 the m o d e l of the a c t u a l sys tem matches the o rder of the e s t i m a t e d m o d e l by the G P C . O n e a n d two mode flexible h n k s , a n d a hnear a n d n o n h n e a r m o d e l of the h y d r a u h c a U y ac tuated m a n i p u l a t o r a l l have the same n a t u r e of b e h a v i o r , i .e . a l l three parameters , for hnear ized models of the systems, have a s i m i l a r behav io r . T h e R i s e S lope JR has the order of m a g n i t u d e of 10"^ [deg/sec or c m / s e c d e p e n d i n g o n the s y s t e m ) , the C h a n g e T i m e Tc, f rom 1.5 [sec] to 5.4 [sec] depend ing on the system's n a t u r e a n d the c o n t r o l parameters . T h e Zero Slope stabihzes on different values w i t h an order of m a g n i t u d e of 10"^ [deg. or c m . . T h e values of the parameters do not necessari ly have the same order of m a g n i t u d e , as i t aU depends o n the nature of the sys tem be ing controUed. However the na ture of the b e h a v i o r is the same. T h i s is i m p o r t a n t , since regions can be defined for each of the parameters for a speci f ic s y s t e m so that detect ion of m i s - m o d e h n g can be achieved . It w iU be shown below t h a t the b e h a v i o r of the C . F . is different w h e n a m i s - m o d e h n g occurs . F i g u r e 5.3 also shows t h a t the behav io r of the C . F . for a nonhnear i zed m o d e l of the 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: Schemat ic desc r ip t i on of the C F . behavior for the different apphcat i ons W h e n the system is under or over -modeled , the behavior of the cost f u n c t i o n changes extens ive ly i n some cases, a n d m o d e r a t e l y i n others. Since those changes can be d e t e c t e d , the assumpt ions about the s t ruc ture of the es t imated m o d e l w i l l be u p d a t e d a n d the resu l ts i m p r o v e d . 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 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 L i n k W h e n a f lexible two mode h n k ( order 6 ) is e s t imated by a two m o d e h n k , the m o d e h n g error is s m a l l a n d good c o n t r o l is achieved , as m e n t i o n e d i n C h a p t e r 3. F i g u r e 5.4 shows the behav ior of the cost f u n c t i o n , w h i c h has low values i n the order of m a g n i t u d e of 10~^°, the C h a n g e T i m e is T^ = l.5[sec.] to a zero slope, w h i c h means that for t > Tc, the error has very s m a l l values ( F i g u r e 5.5). M i s - m o d e h n g can be classif ied i n t o two categories: 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 . W h e n this sys tem is u n d e r - m o d e l e d w i t h a one mode e s t i m a t e d m o d e l (order 4) , i n s t a b i h t i e s o c c u r , since the contro l a l g o r i t h m does not account for the u n m o d e l e d d y n a m i c s a n d c a n not overcome i t as a d i s turbance . F i g u r e 5.6 presents the behav ior of the cost f u n c t i o n i n t h i s case, showing t h a t i ts values rise very h i g h even before the C h a n g e T i m e (1.5 s e c ) ; thus t h e m i s - m o d e h n g c a n be detected a n d changed on-hne. F i g u r e 5.7 produces v e r y s i m i l a r resu l ts for u n d e r - m o d e h n g of the e s t i m a t e d m o d e l w i t h the order of 2. W h e n the sys tem is over -mode led , the react ion to the m i s - m o d e h n g is m o r e m o d e r a t e . W h e n the e s t i m a t e d m o d e l is a 3 mode one (order 8), the cost f u n c t i o n a n d the o ther p a - rameters behave as i f there is no mis -modeHng ( 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 , the m o d e h n g error is larger , a n d the cost f u n c t i o n value rises b e y o n d the des i red value ( order of m a g n i t u d e goes to 10"* ins tead of 10 '^° ); after Tc i t keeps o n r i s i n g a n d does not achieve the zero slope. ( F i g u r e 5.9). * 10^ flex 2 mode link - 2 mode estimator - GPC control 4 6 TIME [ s e c ] 8 10 * "10*10 flex 2 mode link - 2 mode estimator - GPC control 2 . 0 o 1.5 H Q. 1.0 H - 4 - ' o o D 0.5 H E '•4-' CO 0.0 0 4 6 TIME fsec. 8 10 F i g u r e 5.6: Cos t func t i on behav ior for 2 mode h n k a n d 1 mode e s t i m a t e d m o d e l w i t h l oga - r i t h m i c ax is F i g u r e 5.7: Cos t func t i on behavior for 2 mode 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 conc lus ion 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 the per formance a n d s t a b i h t y o f the sys tem faster a n d i n a more intensive m a n n e r t h a n the over- m o d e h n g does. T h i s fac t w i U help the on-Hne m o d e l s t r u c t u r e detec t ion dif ferentiate between under a n d o v e r - m o d e H n g . These conclusions wiU be backed up by fur ther results . F l e x i b l e O n e M o d e L i n k 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 mode e s t i m a t e d m o d e l , p roduces a controUed sys tem w i t h good per formance . F i g u r e 5.10 shows the behavior of the cost f u n c t i o n for these condi t ions . T h e C . F . ( Hke the one i n F i g u r e 5.4) has a zero slope a n d s tab ihzes at the order of m a g n i t u d e of 10~® a n d = 2sec. . T h e behavior of the two cost f u n c t i o n s is the same but the values are different. Here too , the mis -modeHng is addressed b y the t w o categories: u n d e r - m o d e l i n g a n d over -modeHng. 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 uns tab le response i f an e s t i m a t e d m o d e l of a r i g id b o d y is chosen. It produces a n o s c i U a t i o n frequency of about li.bHz., the f requency of the first mode not accounted for b y the c o n t r o l a l g o r i t h m (see Tab le B . l ) . F i g u r e 5.12 shows that the cost f u n c t i o n grows, a n d at Tc = 2, i ts value is a p p r o x i m a t e l y 50, whereas i n F i g u r e 5.10 i t was at the order of m a g n i t u d e of 10"^. T h i s is a c lear i n d i c a t i o n t h a t the m o d e l chosen was not the r ight one. These results m a t c h the behav ior of the 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 presented 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 more modera te response. A two m o d e es t imated m o d e l ( F i g u r e 5.13) behaves l ike the 1 mode m o d e l , but a 3 mode es t imated m o d e l ( F i g u r e 5.14) at Tc has a value o f the order of m a g n i t u d e of 10"^ where the 1 mode e s t i m a t e d m o d e l h a d the value of 10"^. T h i s case shows that the C . F . cont inues increas ing , and the whole process becomes u n s t a b l e . T h e rate of approach ing b a d per f o rmance or even ins tab iHty is m u c h slower t h a n the one for u n d e r - m o d e h n g . A 4 mode 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 behav io r to the prev ious case, b u t the cost func t i on value rises quicker . A t Tc the C . F . has the order of m a g n i t u d e  flex 1 mode link - 0 mode estimator - GPC control 0.0 0.5 1.0 1.5 TIME fsec. 2.0 F i g u r e 5.12: C o s t f u n c t i o n behav ior for 1 mode 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 l o g a r i t h m i c axis of 10 ^ a n d at < = 5sec. the order of m a g n i t u d e of 10 where the order of m a g n i t u d e for t h e prev ious case is 1 0 ' ^ . T h e conc lus ion that can be d r a w n so far f r o m a n a l y z i n g the flexible h n k , is t h a t for u n d e r , over or correct m o d e h n g , the cost f u n c t i o n behav ior is s igni f i cant ly di f ferent . So a m o d e l p lant m i s - m a t c h cou ld be detected by e x a m i n i n g the behavior of the cost f u n c t i o n a n d i ts der ivat ive . 5.2.2 T h e C o s t F u n c t i o n for the 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 two h n k m a n i p u l a t o r is h i g h l y nonHnear , w ide ly used i n the i n d u s t r y , a n d is there fore of interest i n this inves t igat i on . C h a p t e r 4 presents a t h o r o u g h discussion of such a s y s t e m controUed w i t h the G P C a l g o r i t h m w h i c h achieves good per formance . I n th i s c h a p t e r , t h e behav ior of a cost f u n c t i o n for such a sys tem w i U be 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 robot i c Hnk is basicaUy a t h i r d order s y s t e m , order of 2 for the d y n a m i c s of the H n k s , a n d order of 1 for the h y d r a u h c systems. T h e order of such a sys tem can change i f the H n k is not r i g i d , b u t flexible w i t h an u n k n o w n n u m b e r of v i b r a t i o n modes , or i f the h y d r a u H c s y s t e m contains a t i m e delay ( w h i c h wiU a d d an order of one to the basic sys tem) , or i f the s p o o l va lve d y n a m i c s inf luence the process ( then an order of 2 is added to the basic process b r i n g i n g i t t o order 5). I n this discussion the order change wiU be i n the h y d r a u h c p a r t , where the resul ts w i U be d i v i d e d in to two categories. F i r s t a hnear i zed m o d e l of the m a c h i n e is i n t r o d u c e d a n d the behav io r of the cost f u n c t i o n s t u d i e d , a n d then the fuU nonHnear m o d e l for the s y s t e m is used i n the s i m u l a t i o n a n d the behavior of i ts C . F . s tud ied as weU. F i g u r e 5.16 shows the behav ior of 6i and 82 to a step f u n c t i o n w h e n the h n e a r i z e d m o d e l of the s y s t e m a n d the m o d e l assumed for G P C m a t c h and are b o t h of order 3. F i g u r e 5.17 shows the behav ior of the cost f u n c t i o n and its der ivat ive for b o t h Hnks T h e same p a t t e r n o f behav ior can be observed (as i n the flexible h n k ) . T h e R i s e Slope, the T i m e C h a n g e Tc,    a n d the Zero Slope are f o u n d i n this case too (schematic descr ip t i on of these p a r a m e t e r s a n d the i r values are descr ibed i n F i g u r e 5.3). W h e n the a c t u a l nonhnear sys tem is i n t r o d u c e d ( the order remains 3) , the b e h a v i o r of the cost f u n c t i o n changes and so does the behav ior of the system. In order to m a i n t a i n s teady state values for some of the h y d r a u h c parameters , the spoo l valve chatters a r o u n d i t s zero value. A s a resu l t , the o u t p u t error is constant (not zero) , a n d the cost f u n c t i o n rises constant ly . F i g u r e 5.18 shows that the C . F . behaves s i m i l a r l y to the behav ior i n F i g u r e 5.17 u p to the t i m e w h e n the chat ter ing begins. It can be detec ted c lear ly o n the C . F . d e r i v a t i v e p lo t . S ince good per formance is achieved i n controUing the nonhnear sys tem w i t h G P C (see C h a p t e r 4 ), the cost f u n c t i o n indicates that the order of the e s t imated (hnear ized) m o d e l matches the order of the a c t u a l nonhnear mach ine m o d e l a n d can be used to detect m o d e h n g errors of the sys tem. W h e n the order of the 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 evident i n the cost f u n c t i o n behav ior . 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 ts e s t imated m o d e l was of order 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 order 3. T h e cost funct i on of L i n k 2 has a very s i m i l a r behav io r to the one seen i n F i g u r e 5.18 a n d its o u t p u t ( F i g u r e 5.20 ) stabihzes o n i ts set po in t . H o w e v e r , i t has a larger overshoot due to the c ouphng w i t h h n k 1, w h i c h is under -mode led b y i t s e s t i m a t e d m o d e l . O v e r - m o d e h n g , as i n the f lexible h n k case, reacts i n a more moderate way t h a n the u n d e r - m o d e h n g . I n F i g u r e 5.21 b o t h hnks were over mode led w i t h an e s t i m a t e d m o d e l of o rder 4. T h e cost f u n c t i o n for b o t h hnks grows, i n d i c a t i n g the m i s - m a t c h between the mode ls . T h e der iva t i ve however , decreases eventual ly . T h e cost f u n c t i o n a n d i ts der ivat ive change 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 the u n d e r - m o d e h n g case. I n order to get a reahstic f i f th order m o d e l for the 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 to the servovalve of the sys tem. U s u a l l y , for most p r a c t i c a l purposes , the servovalve d y n a m i c s GPC control - 2 link manipulator with hydraulic octuotors .6 H L_ 1 1 1 . TIME [sec] . T O " 35 2 link monipulotor - lineor plont - linki - order=3 Iink2- order«3 •10« 4 6 TIME [sec] 4 6 TIME [sec] • XT' 40- 1 1 1 1 1 0 0 - 30- 80- > / <s 60- J 2 0 - > 1 c 3 j :5 40- | 1 0 - u i 0- 0 - 4 6 TIME [sec] 10 F i g u r e 5.17: Cos t func t i on behavior for 3 mode h y d r a u h c h n k s (hnear ized p lant m o d e l ) a n d 3 m o d e e s t i m a t e d models 2 link monipulotor — nonlineor plant — linki — order—3 Iink2— order=3 F i g u r e 5.18: Cos t func t i on behavior for 3 mode h y d r a u h c hnks (nonhnear p lant mode l ) a n d 3 m o d e e s t i m a t e d models ' E T 2 link manipulator —linki— order—3 , Rnk2— order»3 —I 1— 4 6 TIME [sec] 10=1^ 10" 10" 10 —I r— 4 6 TIME [sec] 10 « : 10^: 10-° Ê r " 4 6 TIME [sec] 10 b I 10' _, 10"°: 1 0 - i 10-»: 10" : ^ 1 0 - ' \ ^ 1 0 - ^ 10-̂ 10-̂  —I 1— 4 6 TIME [sec] i r r 10 GPC control — 2 link manipulator with hydroulic octuotors —8 H 1 1 1 1 0 2 4 6 8 10 TIME [sec] .8 H ' ' ' <• I I I I I I 0 2 4 6 8 10 TIME [sec] 2 link monipulotor — linki 200-1 1 1 • 1 1 1. > 150- c o 1 100-8 •X " ^ - i 1 \ ^ f- 0 2 4 6 8 10 TIME [sec] 1 1 140- - 120 - ^ 100- 1 80-§ 6 0 - / 1 / o / 20- y 0-J 1 0 2 4 6 B 10 TIME [sec] order=4 Iink2- order=4 25-1 1 ^ 1 L are fast enough to be ignored . I n this case, a n equivalent second order sys tem was a d d e d t o the s t r u c t u r e of the 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 the 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 the d a t a for the servovalve d y n a m i c s w i t h n a t u r a l f requency of 20Hz a n d a d a m p i n g r a t i o of 0.6 (see C a t a l o g , M o o g Inc . ^^). I n F i g u r e 5.22 h n k 1 has an e s t i m a t e d m o d e l of o r d e r 5, w h i c h grows constant ly due to the error between the models . F o r h n k 2, the o rder is 3 a n d the cost f u n c t i o n behaves hke the one i n F i g u r e 5.18. F i g u r e 5.23 shows the b e h a v i o r of t h e o u t p u t s . 9i c an not achieve the goal of i ts set point due to the m i s - m a t c h of the m o d e l s , a n d $2 behaves we l l since the models m a t c h each other. T h e conclusions d r a w n f r o m this sect ion are s imi lar to the ones f r o m the flexible h n k . It is possible to ident i fy , t h r o u g h the cost func t i on a n d i ts der ivat ive , the case i n w h i c h the e s t i m a t e d m o d e l matches the a c t u a l system's m o d e l . T h e cost f u n c t i o n also i n d i c a t e s u n d e r - m o d e h n g a n d over -modehng . T h i s i n f o r m a t i o n forms the basic d a t a for the m e t h o d for de te c t ing on- l ine the order of a sys tem 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 B e h a v i o r M o s t adapt ive contro l a lgor i thms assume that p l a n t d y n a m i c s can be mode led b y one m e m b e r of a speci f ied class of models . U s u a l l y , there are uncerta int ies i n the e s t i m a t e d m o d e l d u e to u n k n o w n b u t e s t imated parameters or d i s turbances . These can be f r o m e x t e r n a l sources or i n t e r n a l ones such as nonhnear i t ies i n the 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 e s t i m a t e d m o d e l . I f the d is turbances are b o u n d e d a n d there is sufficient e x c i t a t i o n b y t h e i n p u t s ignal to es t imate the m o d e l parameters , t h e n the sys tem can be contro l led a n d s t a b i h t y re ta ined . T h i s was demonst ra ted i n C h a p t e r 3 by controUing 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 controUing a h y d r a u h c a U y ac tuated two h n k m a n i p u l a t o r w h i c h is a h igh ly nonhnear sys tem. B o t h systems were m o d e l e d by a hnear m o d e l for c o n t r o l purposes , w i t h parameters e s t imated on-hne, a n d the nonhnear i t ies are considered to 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 a n d good per formance can be ach ieved ( see F i g u r e s 4.4, 4.5, 4.6, 4.7). T h i s work 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 the e s t i m a t e d m o d e l for the 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. hnear a n d different order) f r o m t h a t of the rea l p l a n t . S u c h m i s - m a t c h is another f o r m of u n c e r t a i n t y i n the adapt ive controUer. 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 ts t i m e var iat ions were chosen as a measure of t h a t p h e n o m e n o n . W h e n the p lant a n d m o d e l m a t c h , the cost func t i on rises i n i t i a l l y for the t i m e p e r i o d t h a t i t takes for the e s t i m a t e d m o d e l to ad just , a n d t h e n stabihzes on a close to constant va lue , since the error between the models becomes very smaU. T h e behav ior of the cost f u n c t i o n J as a f u n c t i o n of the e s t i m a t e d m o d e l order ( F i g u r e 5.1, for open loop invest igat ion) shows that u n d e r - m o d e h n g , and over -modehng are h a r d to detect by c o m p a r i s o n w i t h the correct s t ruc ture . O n the o ther h a n d , F i g u r e 5.2 (closed loop calculat ions) shows t h a t m o d e l m i s - m a t c h i n most forms is signif icant and can be detected . T h e real processes have c o m p l e x n o n h n e a r d y n a m i c s ( as i n the 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 the a d a p t i v e c o n t r o U e r a t t e m p t s to contro l the d y n a m i c s by a s imple hnear m o d e l . T h e parameters of the hnear e s t imated m o d e l depend s trongly o n the proper t i e s o f t h e i n p u t s igna l and its frequency content. P r o p e r e x c i t a t i o n is needed for good e s t i m a t i o n r e s u l t s . T h e r e is self ex c i ta t i on w h e n the e s t imat i on is done i n closed loop (as w h e n the a d a p t i v e controUer is used) , since the e s t imat i on process is exc i t ed by the s ignal f r o m the feedback . T h e feedback c o u l d cause dependencies between the e lements of the regression vec tor ( E q u a t i o n 5.4 w h i c h means that the parameters cannot be d e t e r m i n e d un ique ly ( A s t r o m ^). E r r o r s d u e t o m o d e h n g errors arise w h e n the chosen m o d e l does not describe the sys tem comple te ly , i t c a n cause poor per formance depend ing on the value of the m o d e h n g error a n d i ts n a t u r e . I n the foUowing m a t e r i a l , a discussion o n the 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 Squares ) , used 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 the effect of u n d e r , a n d over -mode l ing is g iven . W h e n the plant is hnear a n d its order is k n o w n , i t can be descr ibed by the m a t h e m a t i c a l m o d e l : y{t) = ~aiy{t - 1) - a2y{t - 2) + bou{t - 1) + biu{t - 2) + • • • (5.2) or: y{t) = * ^ ( t ) 0 (5.3) where $ is the regression vector : $ ^ = [ - 3 / ( f - l ) , -y{t-2), u{t-l), uit-2)- a n d 0 is a vector of u n k n o w n parameters : (5.4) 0 = [ a i , 02, • • •, bo, bi. (5.5) T h e e s t i m a t e d m o d e l is: m = $ ^ ( 0 © (5.6) where 0 is a vector of the e s t i m a t e d parameters , a n d err(t) is the error . err{t) = yit)-y{t) (5.7) T h u s the R L S a l g o r i t h m is : 0m è(t u a P j t - m t M t ) - m ) . . . . Pit) Pit ^) - a p j t - m m ^ i m t - ^ ) . . . . Pit) - Pit - 1 ) - ^ + AT^^t)Pit - i ) m (^-^^ where: 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 the s a m p h n g p e r i o d , yit) is the measured o u t p u t , a n d y ( i ) is the es t imated o u t p u t . U n d e r - M o d e l i n g W h e n u n d e r - m o d e h n g is considered the measured o u t p u t can be expressed as: yit) = ^''{t)e + r,{t) (5.10) where T]{t) contains the u n m o d e l e d te rms . r earrang ing 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 ie lds : W h e r e : 0 = 0 - 0 the regression vector , contains i n f o r m a t i o n on previous outputs a n d i n p u t s to the p l a n t a n d therefore i n f o r m a t i o n about the feedback to the control ler . T h e u n m o d e l e d d y n a m i c s , a l t h o u g h u n k n o w n , are part of the p l a n t ' s o u t p u t a n d of the feedback s ignal . It can t h u s be c onc luded t h a t $ a n d rj are dependent . E q u a t i o n 5.11 shows that even w h e n the e s t i m a t e d parameters m a t c h the ones i n 0 t h e t e r m w i t h ^i} i n i t can cause 0 to dr i f t . T h i s effect can also be seen i n the equat ions d e s c r i b i n g the cost f u n c t i o n . B a s i c a l l y w h e n the error grows the cost f u n c t i o n grows i n value i .e . : B a s e d o n equat ions 5.10 5.5 5.6 a n d 5.7, the error i n terms of the regression vector the difference between es t imated a n d t rue parameters and the u n m o d e l e d d y n a m i c s is : W h e r e : err{t) = $^( f ) 0 + 7 / (0 0 = [ôi, • • •, â„^, 6 i , • • •, (5.12) a n d the regression vector for this case is : = [~y{t~l), - y ( t - n „ ) , u ( t - l ) , u ( t - n „ ) ] (5.13) i ts d imens i on is : dim{i^) = 2nu, where is the order of the mode led d y n a m i c s for t h e u n d e r - m o d e h n g case, and the terms y{t — i) are: y{t-i)=.^'^(t~i)e + ri(t-i) (5.14) i = 1,2, •••,7i„ T h e t e r m ^^(t — i)Q is der ived us ing equat ion 5.5: #^(f - z ) 0 = -a^y[t - j) + bAt - j) = ny,n) (5.15) T h e cost func t i on is defined as: t J^Y^err' (5.16) 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 second t e r m on the right h a n d side of E q u a t i o n 5.19 contains the regression vec tor $ , a n d the u n m o d e l e d d y n a m i c s 77, w h i c h as can be conc luded f r o m E q u a t i o n s 5.14 a n d 5.15, are dependent . A s can be seen f r o m E q u a t i o n 5.17 there are two cont r ibut i ons to the error , the m o d e h n g reflected i n the u n m o d e l e d d y n a m i c s t e r m , a n d the e s t i m a t i o n w h i c h is re f lected i n the parameters . B y express ing the regression vector $ t e rms , w i t h E q u a t i o n 5.15 the t e r m $ ^ ( i ) 0 is ca l cu la ted 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 and 5.18 into E q u a t i o n 5.16 the cost f u n c t i o n is d e s c r i b e d by : J = è ( * ' ' ( f c ) 0 f (5.19) k=i +2 Ehàj X: vikHk - j) - àj è r,{k)ny, ^) + h É n{k)u{k - j)] j=i fc=i k=i k=i T h e u n m o d e l e d d y n a m i c s is a p h y s i c a l s ignal 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 p lant ' s o u t p u t . T h e second t e r m i n the right h a n d side of E q u a t i o n 5.17 ^^{k)Q, w i l l be s t rong for u n d e r - m o d e h n g because of 0 a n d because of the corre la t i on between the regress ion vector $ a n d the u n m o d e l e d d y n a m i c s rj. W h e n e v a l u a t i n g the corre la t i on between m e a - surements of pairs of var iables , the corre lat ion is d e t e r m i n i n g whether there exists a p h y s i c a l r e la t i onsh ip between the two , or whether the var iat ions i n the observed values of one q u a n t i t y are corre lated w i t h the var iat ions i n the measured values of the other . I n Press ^2 the d iscre te co r re la t i on of two sampled fanct ions is defined by: Corrig,h)^ = J29U + mk) (5.20) W h e n g a n d h are the same f u n c t i o n the above is the a u t o c o r r e l a t i o n of the s ignal . T h e corre la t i on w i l l be large at some value of k i f the first f u n c t i o n g is a close copy of the second h b u t lags i t by k. In E q u a t i o n 5.19 several terms are s u m m e d w i t h respect to t i m e . S ince r), y a n d u are rea l phys i ca l signals there are terms of a u t o c o r r e l a t i o n a n d cor re la t i on . T h e s e t e r m s do not exist i n the over m o d e h n g case as w iU be shown i n the d iscuss ion on over - m o d e h n g . T h e t h i r d t e r m i n the r ight h a n d side is an autocor re la t i on of two rj signals sh i f t ed i n t i m e . T h e f o u r t h a n d the f i f th terms i n E q u a t i o n 5.19, are corre la t i on t e rms be tween rj a n d the o u t p u t y or 7/ a n d the i n p u t u . In the u n d e r - m o d e h n g case 7/ is a par t of the o u t p u t y a n d since there is feedback of y it is correlated w i t h the i n p u t u a n d therefor the u n m o d e l e d d y n a m i c s 7/ as we l l . I n the u n d e r - m o d e h n g case, the corre lat ion b e t w e e n t h e different variables i n E q u a t i o n 5.19, is the reason for the r a p i d rise i n the cost f u n c t i o n ' s values as was shown i n Sec t ion 5.2 for b o t h apphcat ions . O v e r - M o d e l i n g W h e n over -modehng is concerned y a n d y are: y{t) = $^(00 (5.21) a n d : m = ^''m+7,(1) (5.22) T h e regression vector for the over m o d e h n g case is : = [ - 2 / ( ^ - 1 ) , -yit-2), -yit-n), uit-1), uit-2), uit-n)] (5.23) its d i m e n s i o n is : ( / im($-^(i ) ) = 2 n , where n is the correct order of the s y s t e m , a n d dimirjit)) = Uo ^ n, where Uo is the over -modeled m o d e l order . T h e error is t h e n : errit) = $^(^0 - vit) (5-24) a n d : err\t) = ( ^ ' ^ ( i ) © ) ' " 2$^(O0t,(O + v'it) (5.25) *^(O0 = E -àjyit - 0 + bAt - j) (5-26) S u b s t i t u t i n g E q u a t i o n 5.26 into E q u a t i o n 5.25 a n d in to E q u a t i o n 5.16 results i n the cost f u n c t i o n for over -modehng : J = E ( * ' ^ ( ^ ) ê r (5.27) - 2 E [ - â , y: v{k)yik - j)+bj y: vikHk - j)] j = l fc=l *! = 1 + i:vik)r,ik) k=l I n this case, r) contains a l l the e x t r a terms of the e s t imated m o d e l . These d y n a m i c s are j u s t i n the es t imated m o d e l a n d not i n the rea l sys tem. T h i s means that there is no c o r r e l a t i o n between the regression vector $ and the e x t r a terms T/, i .e. b o t h variables are i n d e p e n d e n t . T h e corre la t ion a n d autocorre la t i on terms i n E q u a t i o n 5.19 do not exist i n e q u a t i o n 5.27, a n d r]{k) i n the t h i r d a n d f o u r t h t e r m of the r ight h a n d side acts as a t i m e v a r y i n g coeff ic ient. 7) therefore influences the contro l parameters w h i c h inf luence the i n p u t to the process ( u ) , b u t not the feedback of the contro l led sys tem. C l o s e d L o o p Poles for U n d e r , O v e r a n d C o r r e c t - M o d e l i n g T h e change i n the contro l ler parameters is a change i n the contro l ler d y n a m i c s w h i c h d e t e r m i n e the l o c a t i o n of the closed loop poles. T h e m i n i m i z a t i o n of the cost f u n c t i o n for the e s t i m a t i o n w i l l de termine the parameters unique ly , on ly w h e n the m o d e l order is correc t . W h e n the m o d e l is over p a r a m e t r i z e d it can result i n any one of several so lut ions , a n d t h e correct parameters cannot be de te rmined . T h e closed loop poles show i n some cases u n s t a b l e modes i n d i c a t i n g that the 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 to the d o m i n a n t poles of the sys tem d r i v i n g the error in to the higher values. T h i s can result e v e n t u a l l y i n poor per formance or i n i n s t a b i h t y (especial ly i f there is not enough e x c i t a t i o n i n the process ) . T h e reac t i on for the over -modeled d y n a m i c s is not as extreme as to the u n d e r - m o d e l e d ones. N e x t we discuss some examples , f rom the flexible h n k a p p h c a t i o n , for the behav ior of the closed loop poles w h i c h show the unstable modes of the contro l led sys tem w h e n m i s - m o d e h n g poses a p r o b l e m . T h e ca l cu lat ions of the closed loop po led are based on C l a r k e ^ a n d o n L a t o r n e U . F i r s t the closed loop poles for correct modehng : c o r r e c t - m o d e l i n g - o r d e r 6 for plant a n d m o d e l p l = +0.9871 + 0.0140J p2 = +0.3955 + 0.4982J p3 = - 0 . 6 1 9 2 + 0.7582J p4 = +0.0360 + O.OOOOj p5 — - 0 . 6 1 9 2 - 0.7582J p6 = +0.3955 - 0.4982J p7 = +0.9871 - 0 .0140i A U poles for the correct m o d e h n g of a two m o d e flexible h n k ( order 6 ) are w i t h i n the u n i t c irc le i n d i c a t i n g a stable sys tem. T h e sys tem behavior is presented i n F i g u r e 3.2 a n d the 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 plant a n d m o d e l pi = +0.9869 + 0.0132J p2 = +0.5720 + 0.7359J p3 = +0.0337 + O.OOOOj p4 - +0.5720 - 0.7359J p5 = +0.9869 - 0.0132J W h e n one mode occurs for the flexible h n k a n d the system is correct ly m o d e l e d t h e G P C c o n r t o l achieves good results a n d the cost func t i on behaves as presented i n F i g u r e 5.10. T h e closed loop poles as shown above are a l l i n the stable region w i t h i n the 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 plant a n d o r d e r 8 for m o d e l p l = +0.9987 + 0.0299J p2 = +0.6295 + O.OOOOj p3 = +0.5045 + 0.6561J pi = - 0 . 1 5 6 7 + 0 .8209i p5 — +0.3912 + 0.5150J p6 = - 0 . 6 1 9 2 + 0 .7582i p7 = - 0 . 6 9 1 0 + 0 .2549i p8 = - 0 . 6 9 1 0 - 0.2549J p9 = - 0 . 6 1 9 2 - 0 .7582i p l O = +0.3912 - 0 .5150i p l l = - 0 . 1 5 6 7 - 0 .8209i p l 2 = +0.5045 - 0.6561J p l 3 = +0.9987 + 0.0299J A s prev ious ly ment i oned w h e n a sys tem is over p a r a m e t r i z e d there is no u n i q u e s o l u t i o n to the ident i f i ca t i on process. If the excess d y n a m i c s a d d closed loop poles t h a t are close to the d o m i n a n t ones it c ou ld dr ive the contro l led sys tem in to ins tab i l i t i e s . In this case a l l c losed loop poles are stable a n d i t was shown that good per formance was achieved. H o w e v e r p6 a n d p9 are close to the c ircle at a radius of 0.97889 a n d pi a n d p l 3 are at the rad ius o f 0 .9915. A n y s m a l l change i n the systems parameters or even i n the contro l parameters ( w h i c h are chosen for correct m o d e h n g of the best sys tem k n o w n to the designer) cou ld dr ive the s y s t e m to ins tab ih ty . o v e r - m o d e l i n g - o r d e r 4 for plant a n d o r d e r 10 for m o d e l pl = +1.2989 + 0.3954; p2 = +0.8569 + 0 .7815; p3 = +0.2784 + 0 .9573; p4 = - 1 . 0 2 0 4 + 0.7592; p5 = - 0 . 0 3 4 8 + 0 .0000; 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 over -modehg case is one w i t h s ix unstab le modes : pi a n d p9 at a rad ius of 1.3577, p2 a n d p8 at 1.1597 a n d p6 a n d p4 at 1.2718, the sys tem is unstables as the cost f u n c t i o n ind i cates 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 plant a n d order 4 for m o d e l p i p2 = +1.2795 + 0.0000; = +0.7764 + 0 .4191; 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 = +1.2795 - O.OOOOj W h e n u n d e r - m o d e h m g occurs the react ion of the cost f u n c t i o n was more r a p i d a n d the sys tem became unstable faster t h a n the under -modehng case. T h e unstab le modes are : p i at radius of 1.2795, a n d p4,p8 at 1.3002. T h e over -modehng of the two m o d e m a n i p u l a t o r presented above was stable , i n th is case the unstable poles are qui te far i n the uns tab le reg ion . u n d e r - m o d e l i n g - o r d e r 4 for plant 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 p3 = +0.7904 + 0.8734J p4 = - 1 . 1 6 4 0 + 0.0ÛÛÛJ p5 = - 0.2321 + O.OOOOj p6 = +0.6089 - 1.3839J p7 = +0.7904 - Û.8734J I n th is under -mode led case the unstab le modes are: pi at radius of 1.8581, p2 a n d p6 at 1.5119, p4 at 1.164, a n d p3 a n d p7 at radius of 1.1779. i n th i s case too , the u n s t a b l e poles are fur ther i n the unstab le zone that the ones for the over -modehng case (for p lant order 4 a n d m o d e l order 10). 5.4 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 Sec t i on 5.2 presents the behavior of the cost f u n c t i o n J a n d its t i m e var ia t i ons for b o t h apphcat i ons , the flexible h n k m a n i p u l a t o r a n d the 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 measure of the a c c u m u l a t e d e r ror between the p lant and the m o d e l d y n a m i c s . T h e difference between under a n d o v e r - m o d e h n g is c lear i n the behavior of J i ts t i m e derivatives as discussed i n Sect ion 5.2. T h i s sec t ion presents a n a l g o r i t h m to detect m i s - m a t c h between plant a n d m o d e l , based o n the resul ts above , a n d to correct the order . It should be noted that correc t ing m i s - m o d e h n g is not a target i n itself , but ra ther , is to detect a possible route to i n s t a b i h t y a n d poor per f o rmance . T h u s , i f an over -modeled sys tem is weU controUed, there is no reason to inter fere . T h e goa l of the m e t h o d presented is to detect p r o b l e m a t i c m i s - m a t c h cases, to ident i f y the i r n a t u r e , a n d to correct t h e m regardless of the ir 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), the adapt ive sys tem contains two loops; one is a n o r d i n a r y feedback l oop , a n d the second loop identif ies the e s t i m a t e d m o d e l parameters a n d updates the parameters of the controUer. F i g u r e 5.24 shows a n a d a p t i v e sys tem b lo ck 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 lock w h i c h is an a d d i t i o n to the two loops m e n t i o n e d above . I n the procedure a feedback loop is added to the ident i f i ca t ion loop. T h i s l oop ca lculates the error between the measured and es t imated outputs and m i n i m i z e s i t w i t h the a l g o r i t h m g i v e n below. F i g u r e 5.25 shows the block d i a g r a m of the order d e t e r m i n a t i o n m e t h o d . 1 1 y RLS A 1 N , (.order) MOD Calculations 1 1 Algorithm y Design Contoller parameters Adaptive controller u F i g u r e 5.24: F l o w chart of an adapt ive contro l loop w i t h m o d e l order d e t e r m i n a t i o n I n i t i a l i z a t i o n : 1. G u e s s in i t ia l s t r u c t u r e ( o rder ) . 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 c o r r e c t , u n d e r - a n d o v e r - m o d e l i n g . C h e c k v a l u e s of C . F . t t c J A d a p t i v e L o o p correct structure mis-modeling Parameter Estimation C h e c k U n d e r - or O v e r - m o d e l i n g I C h a n g e s t r u c t u r e i o f m o d e l i F i g u r e 5.25: F l o w chart of the order d e t e r m i n a t i o n procedure T h e order change is c a l c u l a t e d w i t h the M O D a l g o r i t h m w h i c h is presented n e x t : = iV ( f - 1) + A i V ( J , j , t„a.r. Tu, Tc, NOT) (5.28) W h e r e J is the cost f u n c t i o n w h i c h is descr ibed i n E q u a t i o n 5.1. Its der ivat ive J is the fo l l owing : W h e r e : • torder is the M O D ' s t i m e scale. In the event of several order changes d u r i n g a w o r k i n g cyc le , i n every change tarder is set to zero. T h i s moves the or ig in of the t i m e scale re lat ive to the absolute t i m e t, and enables the t i m e parameters ( that w i l l be s t a t e d nex t ) for each order change to be considered. • Tu is the t i m e for under m o d e h n g detect ion . • Tc is the t i m e w h e n the cost func t i on changes to Zero Slope for correct m o d e h n g . Its value w i l l be w i t h i n the region 7 c , „ i „ ^ Tc > Tc^^^ . • NDT is the n u m b e r of t i m e steps to wait for convergence after an order change. • No is the i n i t i a l guess for the order . • K^ait is the n u m b e r of t i m e steps to wait between i n d i c a t i o n of possible m i s - m o d e h n g and its acceptance. T h e order change f u n c t i o n , A.N based on the behavior 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 over -modehng and for correct m o d e h n g , is as fol lows: AN = < 0 tarder <AT•NDT <Tc J< JR >Tc DU M tarder [ < Tv J > JR^ J > JR^ (5.30) DOM tarder <Tc >Tc where : • jRy a n d JR^ are m i n i m u m values for the cost func t i on a n d i ts d e r i v a t i v e , for i d e n t i f y i n g u n d e r - m o d e h n g . • JR^ a n d JR^ are m i n i m u m values for the cost func t i on a n d i ts d e r i v a t i v e , to i d e n t i f y over -modehng , for t < Tc- • Jz^ a n d Jz„ are m i n i m u m values for the cost f u n c t i o n a n d i ts d e r i v a t i v e , to i d e n t i f y over -modehng , for t > Tc- • JR^^^ a n d are m a x i m u m values for the cost f u n c t i o n a n d i ts d e r i v a t i v e , to i d e n t i f y co r rec t -modehng . • Jz,„ a n d Jz^^ are m i n i m u m values for the cost f u n c t i o n a n d i ts der iva t ive , to i dent i f y c o r re c t -modehng . • D U M is the u n d e r - m o d e h n g a d d i t i o n to the order at each order change step. • D O M is the over -modehng s u b t r a c t i o n f r o m the order at each order change step. T h i s is a gradient a l g o r i t h m designed to m i n i m i z e the n u m b e r of steps to achieve t h e correct order . B a s e d o n the results f r o m the inves t igat i on of the robot i c apphcat i ons p r e s e n t e d i n this work parameters 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 on the f lexible a n d the h y d r a u h c m a n i p u l a t o r a n d was f ound to be stable i n b e h a v i o r due t o several factors: • K^ait is the n u m b e r of t i m e steps to wait a n d ver i fy the need for order change. T h i s prevents a r a n d o m increase i n the values of the parameters and on unnecessary o r d e r change. • tarder a re lat ive t i m e or ig in is used a n d reset after an order change to w h a t is b e h e v e d is the correct value. C h a p t e r 6 I M P L E M E N T A T I O N O F T H E M O D A L G O R I T H M 6.1 I m p l e m e n t a t i o n 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 descr ibed as follows: 1. I n i t i a h z a t i o n : def init ions by the user (a) No : i n i t i a l value for the e s t i m a t e d m o d e l order . (b) ? c , „ i „ <Tc> Tc,^^^ : t i m e region for the t i m e change. {c) Tu : Tu < Tc^i^ : t i m e for u n d e r - m o d e h n g detec t ion (d) Kuiait' n u m b e r of t i m e steps to wait between i n d i c a t i o n of possible m i s - m o d e h n g a n d i ts acceptance. (e) NUDT: the n u m b e r of t i m e steps to wait for convergence, w h e n an order change was done due to u n d e r - m o d e h n g . (f) NODT: the n u m b e r of t i m e steps to wait for convergence, w h e n a n order change was done due to over -modehng . (g) d a t a for correc t -modehng : i . at - i < T c • J H „ : m a x i m u m value for r i s ing cost f u n c t i o n : J < JR,^ • JR„^ '• m a x i m u m value for r i s ing 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 slope J < Jz^, • Jz„i '• inf luence of nonl inear i t ies J ^ ^ , > Jz^ • Jzr,l '• inf luence of nonl inear i t ies 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 func t i on (to ident 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 value for cost f u n c t i o n slope (to ident i fy u n d e r - m o d e l i n g ) : (i) d a t a for over -model ing : i . at -t<Tc • JR^: m i n i m u m value for cost f u n c t i o n (to ident i f y over -modeHng) : J > • JR^/. m i n i m u m value for cost func t i on , slope (to ident 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 value for cost func t i on (to ident i f y over -mode l ing ) : J > • Jzfj'- m i n i m u m value for cost func t i on , slope (to ident i f y u n d e r - m o d e l i n g ) : j > JZO 2. T h e M O D A l g o r i t h m (a) C h e c k t i m e re lat ive to Tc a n d values of the cost f u n c t i o n J , a n d i ts slope J . (b) D e t e r m i n e i f mis -modeHng is i n d i c a t e d , t h e n : • D e t e r m i n e under or over-modeHng • 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 handle m o d e l order changes accord ing to t h e t y p e of m i s - m o d e h n g • Convergence to acceptable s t ruc ture w h e n at i > T c : J < Jz„^ a n d J < Jz^ (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 : • convergence for ver i f i cat ion of the m o d e l m o d e l occurs : w h e n at t > Tc: J < Jz,^ a n d j < Jz^ (d) A t each t i m e step, the contro l a l g o r i t h m is ac t iva ted w i t h the present m o d e l o rder . 6.1.1 T h e M e t h o d F o r 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 D a t a for the F l e x i b l e L i n k T h e values for the parameters presented i n this sect ion are based o n the inves t iga t i on done i n Sec t ion 5.2 for the behav ior of the cost func t i on of the flexible Hnk. I n Sec t ion 5.2 the M O D parameters 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 results . I n o ther apphcat ions of the a l g o r i t h m , such s imulat ions w o u l d first be r u n on-hne w i t h M O D t u r n e d off, the order of the m o d e l w i U be changed a n d based o n the cost func t i on behavior parameters w i l l be d e t e r m i n e d . T h e values for TCmin, Tc„,^,, JR^, JR^, JZ,^, J Z ^ , ^ZnU Jznl were d e t e r m i n e d f r o m correct modeHng results shown i n F i g u r e 5.4 a n d F i g u r e 5.10. Va lues for JRy, JR^J a n d Tu are f r o m d a t a based on u n d e r - m o d e h n g 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 the over- m o d e h n g case, JR^, JR^, JZ^, JZOI was ob ta ined f rom 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 the regions i n w h i c h J a n d J i n d i c a t e d the m i s - m o d e h n g . T h e s h a d e d areas show for the flexible h n k at what values u n d e r , over or correct m o d e h n g o c c u r . 1. T i m e d a t a (a) rc ,„ . „ = lA[sec.] (b) T - c ^ = 2.5[.ec.] (c) Tu = 0.5[sec. 2. D a t a for correct 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^ = 1 0 - " (b) JR, = 1 0 - ^ 4. D a t a for over -modehng (a) J H „ = 10^« (b) JR, - 1 0 - ^ 10" K)"' 10- K'' 10^ 10^ 10-" 10-" H 10-"=- 10-» - 10" / 1 / ! / ' / / ^ / / / / TIME [sec] 10 KT 7 10^ H 10" / - - - / oerrMt fne« 4 6 TIME [see] 10 J io^'^o"^o'=io'"io'"id-id- id-' id- lo"' lo" J 10^'10' '*io'-"ioSo'-"io'-id-io- J '10-'10^10 '10 F i g u r e 6.1: Regions for under , over a n d correct m o d e l i n g (c) Jzo - 10-« (d) Jzo = 10-« 5. D a t a for other parameters (a) DUM = 2 , (4), (6) (b) DOM = 1 , (2) R e s u l t s for the F l e x i b l e L i n k T h e d a t a for b o t h the two m o d e and the one mode examples w h e n mode led correct ly , show t h a t w h e n t < Tc, the 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 the o r d e r of m a g n i t u d e of 10"®. W h e n under -mode led at s m a l l t, the values of J r ise to the order o f m a g n i t u d e of 10"^ a n d higher , a n d so do the values of J. F o r over -modehng the changes are more moderate . In the case of smaU over -modehng (by order of 2 ), the m i s - m o d e h n g c a n not be detected since the contro l a l g o r i t h m works wel l a n d the C . F . der ivat ive ' s values are s m a l l . W h e n over -modeHng is larger , the values of J change i n a more m o d e r a t e s lope . T h e observat ion of the values of J a n d J were m a d e on the basis of results i n Sec t i on 5.2. J accumulates i ts values w i t h t i m e , a n d order changes on-Hne. Its values m a y be h igher , b u t i t s shape remains the same 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 results to be shown describe the behavior of the systems dealt w i t h i n this w o r k i n u n d e r a n d over -modehng for different values for the d a t a needed by the order d e t e r m i n a t i o n a l g o r i t h m . ( N o t e : each t i m e an order change is done, the a l g o r i t h m sets i ts i n t e r n a l t i m e to zero , so a l l parameters for the value regions can be t reated i n the proper t i m e f rame a n d not i n t h e absolute one). F igures 6.2, 6.3, show the behavior of the 2 mode [order = 6) f lexible h n k w h e n u n d e r - m o d e l e d w i t h a second order e s t imated m o d e l . Case A is the behav ior of the sys tem w h e n no order correct ion is done. Y t i p goes unstab le . I n case B , the sys tem converges t o t h e desired set po in t , but has a very h igh overshoot . T h e a l g o r i t h m detects the u n d e r - m o d e h n g soon enough , but does not have the needed t i m e to settle on order = 6. It u n d e r s t a n d s i t as over -modehng again a n d goes to order = 4, w h i c h drives the sys tem in to 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 the order to the value of order = 10, w h i c h stops the r a p i d change i n the values of J. A f t e r Tc,„ax the over -modehng has been detected a n d the order is r e d u c e d to 8, J is reduced to the region accepted as the correct m o d e h n g a n d the sys tem goes t o t h e value of the set po in t . In case C some of the parameters have been ad justed . F i r s t , m o r e t i m e has been g iven for the sys tem to settle after the u n d e r - m o d e h n g was detected a n d c h a n g e d . I n a d d i t i o n , the values for a n d Jza have b o t h been increased to 10~^. A U the changes m a d e i n the parameters of this case were m a d e done order to increase the t i m e of convergence between order changes. A s a resul t , Y T I P has almost no overshoot . It takes a h t t l e m o r e t i m e to converge to the set po int (about 4 seconds ins tead of about 1.5 seconds); yet , i t is far be t ter t h a n the under -mode led response presented i n case B a n d ofcourse case A . T h e next set of results combines under a n d over -modehng i n the process of c o r rec t i on a n u n d e r - m o d e h n g case of a two m o d e {order = 6) flexible h n k i n i t i a U y u n d e r - m o d e l e d w i t h e s t i m a t e d m o d e l of order = 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 show three s u c h cases. Case A shows the unstable behavior of a two mode flexible h n k , controUed w i t h a n e s t i m a t e d m o d e l of order 4. In case B , the o u t p u t converges to the set po int slower (about 5 seconds i n contrast to 1.5 seconds) and has an overshoot of 27 percent , b u t i t does not go uns tab le hke case A . T h e parameters for the order d e t e r m i n a t i o n a l g o r i t h m are DOM = 2, Jza — a n d Jz^ — 10~^. T h e a l g o r i t h m detects the u n d e r - m o d e h n g soon enough so as not to have a very large overshoot , and the order is set to the correct value of 6. There fo re , the cost f u n c t i o n der ivat ive rises at first, but after the order change, i t drops a n d settles o n an order of m a g n i t u d e of 10"^^ w h i c h is an i n d i c a t i o n for convergence. I n case C , DOM = 2, DUM — 6, JR^^ — 1 0 ® a n d JR^^ = 10~®. T h e o u t p u t has a larger overshoot ( F i g u r e 6.4) s ince  flex 2 mode link — under—modelled estimator — GPC control 10-1 1 1 ^ 1.0 1.5 TIME [sec] 8- V Y 6 o 4 - 4 6 TIME [sec] 10 3- 1— 6 [sec] —r 8 4 TIME 10 the order of the e s t i m a t e d m o d e l changes f r o m under -mode led (order = 4) to o v e r - m o d e l e d [order — 10) to (order = 8) a n d is inaccurate for a longer t ime . B u t the o u t p u t converges at the same t i m e as case B . F i g u r e 6.8, F i g u r e 6.9, aga in show the behavior of the 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 — 4) a n d depending on the order change of a l g o r i t h m p a r a m e t e r s , la ter over- mode led . Case A has a larger NUDTu, DUM = 6, DOM ^ 2. NUDTu is t h e p a r a m e t e r def ining the 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 detected . W h e n i t has a larger value, the sys tem is i n the unstable mode longer; thus , there is a large overshoot ( F i g u r e 6.8 case A ) . DUM = 6 changes the order to the value of 10, over -modehng t h a t y i e lds h igher values for the cost f u n c t i o n and its der ivat ive ; w i t h t i m e , the order is r educed to 6. T h e whole procedure resu l ted , as m e n t i o n e d , i n a h igh overshoot a n d i n a slower convergence . In case B the t i m e to change the order after u n d e r - m o d e h n g is detected was r educed , w h i l e aU other parameters r e m a i n e d unchanged . T h e result is a m u c h smal ler overshoot a n d a faster convergence. Case C presents the best result of the three , w i t h an overshoot of a b o u t 15 percent a n d convergence to the setpoint w i t h i n less that 6 seconds. T h e one mode flexible h n k (order = 4) w h e n under -mode led , reacts hke the two m o d e h n k to u n d e r - m o d e h n g . F igures 6.10, 6.11, present 3 cases. Case A is a one m o d e (order — 4) flexible h n k u n d e r - m o d e l e d w i t h a n es t imated m o d e l of order 2. T h e o u t p u t is uns tab le Uke the correspond ing 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 its der ivat ive rise to h i g h values w h i c h i n d i c a t e the responding i n s t a b i h t y . I n case B , the order is changed o n hne to s tab ihze t h e response. W h e n u n d e r - m o d e h n g is detected , the order is changed , by the parameters g iven a p r i o r i , to an over- m o d e h n g value of 8, a n d t h e n goes down g r a d u a l l y to the correct va lue of 4. T h e response has a n overshoot of 83 percent w h i c h is m a i n l y due t o the i n i t i a l i n s t a b i h t y a n d the later over -modehng . B u t the overa l l response, ins tead of go ing uns tab le , converges to the set po int after an acceptable t i m e (about 4 seconds). Case C presents a be t t e r behav io r of the s y s t e m , i n w h i c h the overshoot is smal ler . T h e order change of a l g o r i t h m parameters  0.0 0.2 flex 2 mode link — under—modelled estimotor — GPC control 6.0-4, ^ 0.4 (X6 TIME [sec] 4 6 TIME [sec] 9- 8- o 6- 5-1 4- 4 6 TIME [sec] C: 10  flex 2 mode link — under-modelled estimator — GPC control TIME [sec]  flex 2 mode link - under-modelled estimator - GPC control 4 6 TIME [sec] 0) o 6- 5- 4 4 6 TIME [sec] B: 10 4 6 TIME [sec] was changed, w i t h D U M t h a t has a smal ler value of 4 ins tead of 6. T h u s after d e t e c t i n g the u n d e r - m o d e h n g , the order rises to the value of 6 ( the correct one is 4); yet the overshoot is s t i l l qu i te h igh due to the i n i t i a l ins tab i l i t y . F igures 6.12, 6.13, show 3 cases for i n i t i a l over -modehng (for a one m o d e h n k ) w h e r e the different cases present different parameters of the order 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 the 3 cases, the e s t i m a t e d m o d e l order for the contro 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 vary shghtly i n the overshoot and t i m e of convergence. In case A , DOM = 4, a n d the a l g o r i t h m , after de tec t ing over -modehng , sets the order on 6. I n case B , the p a r a m e t e r s changed a r e . D O M = 2, Tc„,,^ = 2.5 s e c , JR^ = 10"^, JR^ = 10"®. T h e order d e t e c t i o n a l g o r i t h m detects the over -modehng and changes i t to the value of 8, t h e n after 4.3 sec. t o the order of 6. Case C has îbma» = 1-5 s e c , JR^ = 10"^, JR^ = 10~^, w h i c h br ings a q u i c k e r change of the order f r o m 8 to 6. T h e differences between the three results is s m a l l . Cases A a n d C have almost no difference because of the quick change of order = 8 or 10 t o order = 6; Case B takes more t i m e a n d the cost func t i on a n d i ts der ivat ive h m i t s are h igher , so t h e result has a shght ly higher overshoot and takes a h t t l e longer to converge. T h e conclusions d rawn so far f r o m the results of the f lexible h n k (one or two modes ) 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 can be contro l led by detec t ing the u n d e r - m o d e h n g a n d changing it to the correct one or one close to i t . T h e results m a y take longer to converge a n d have an undes ired larger overshoot ( t h a n the correct modehng ) , but the o u t p u t is not unstab le . T h e over -modehng has a m u c h more moderate response, w h i c h is easier to c o n t r o l after de te c t ing i t a n d changing the order of the e s t i m a t e d m o d e l . flex 1 mode link - mis-modelled estimotor - GPC control 0.5 1.0 1^ TIME [sec] 2.0 4 6 TIME [sec] F i g u r e 6.10: T h e o u t p u t behavior of a one mode 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 an order 2 m o d e l 0.0 flex 1 mode link — mis—modelled estimator — GPC control 6 -| r 0.5 1.0 1.5 TIME [sec] 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 beh av ior of a one mode flexible h n k i n i t i a l l y over-modeled. flex 1 mode link - mis-modelled estimator - GPC control -J 1- 10 T r 4 6 TIME [sec] - r ' r 4 6 TIME [sec] 6.1.2 T h e M e t h o d F o r 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 a t o r s D a t a for the T w o L i n k M a n i p u l a t o r T h e values for the parameters presented i n th is sect ion are based o n the inves t iga t i on done i n Sec t ion 5.2 for the behav ior of the cost f u n c t i o n of the T w o L i n k M a n i p u l a t o r . T h e values for ^ c , „ i „ , 2 c , „ „ , , JR„„ JR„,, JZ„,, Jz^, Jz.,1, Jz„t were d e t e r m i n e d f rom correct m o d e h n g resul ts shown i n F i g u r e 5 . 1 7 a n d F i g u r e 5 .18 . Values for JR^^, JR^ a n d Tu are f r o m d a t a based on u n d e r - m o d e h n g , F i g u r e 5 .19 . D a t a for the over -modehng case, J r „ , J r „ , JZ„, JZO, was o b t a i n e d f r o m F i g u r e 5 .21 a n d F i g u r e 5 .22 . 1. T i m e d a t a (a) = 2.0[ .ec . (b) Tc,„„ . = 3.5 (c) Tu = 1.0 sec. sec. 2. D a t a for correct m o d e h n g (a) J« , „ = 5 . 10 -® (b) ^ „ = 5 • 10-® (c) Jz,,„ = 5 . 10-® (d) J^,„ = 10-^^ (e) Jz.i = 1 0 - « (f) Jz,j - 10-« 3. D a t a for u n d e r - m o d e h n g (a) JR, = 1 0 ^ (b) Jn, = 10-== 4. D a t a for over -mode l ing (a) = 10-« (b) Jna = 1 0 - " (c) Jzo = 10° (d) J z „ = 10° 5. D a t a for other parameters (a) DUM = 1 , (2) , (3) (b) DOM = 1 , (2) R e s u l t s for the 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 the h y d r a u h c m a n i p u l a t o r w h e n mode led correct ly , show, as i n the 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 behavior of the cost f u n c t i o n a n d its slope where the values are r e l a t i v e l y s m a l l . Here too , the changes are more r a p i d for the u n d e r - m o d e h n g case t h a n for the over- 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. C a s e A is the one where the order d e t e r m i n a t i o n a l g o r i t h m is not ac t ivated . 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 order 3 m o d e l a n d is weU controUed to foUow a set po 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 order m o d e l , so that the o u t p u t does not detect the set p o i n t . In cases B and C , the order d e t e r m i n a t i o n a l g o r i t h m is ac t iva ted to detect the correct m o d e l a n d contro l the sys tem. T h e difference between the two cases is i n the p a r a m e t e r s used for the order d e t e r m i n a t i o n a l g o r i t h m . I n Case B , DUM = 1 a n d DOM = 1, so the u n d e r - m o d e h n g w h e n detected , is corrected to the correct order i n the first t ry . T h e resul t is that Oi converges to the set po int i n , about 7 seconds i n c ompar i son w i t h 5.5 seconds for the case where b o t h hnks are s t r u c t u r a l l y c o r rec t l y mode led , as descr ibed 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 changed to the value 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 the a l g o r i t h m changes to the order 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 resu l t is even slower t h a n i n case B (about 9 seconds to converge) , but the set po int is t r a c k e d , i n contrast to case A . T h e next three cases present the behav ior o f the h y d r a u h c sys tem w h e n o v e r - m o d e l e d . F i g u r e 6.18, F i g u r e 6.19, present the results . Case A shows the behavior of the s y s t e m w h e r e h n k 1 is correc t ly modeled w i t h a t h i r d order s y s t e m a n d h n k 2 is over mode led (order = 5 ) . 6i does not t rack the set po in t . I n case B , h n k 2 is i n i t i a l l y under -mode led (order = 2) . W h e n the order detect ion a l g o r i t h m is a c t i va ted , DOM = 3 brings the system to o v e r - m o d e h n g (order = 5) , w h i c h gradua l ly is brought d o w n to the correct value (order = 3). A s a r e s u l t , the o u t p u t i n the first 3 seconds goes i n the unstab le d i rec t i on a n d t h e n stabihzes on the set po in t . I n case C , l i n k 2 i n i n i t i a l l y over -mode led (order — 5) , a n d g r a d u a l l y the o rder is changed to the correct one. T h e o u t p u t stabihzes faster (6.5 seconds, as c o m p a r e d to 9.5 seconds) . T h e h y d r a u h c ac tuated m a n i p u l a t o r , hke the flexible h n k , w h e n m i s - m o d e l e d c a n 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 the response of the o u t p u t more t h a n over -modehng a n d is more di f f icult to c o n t r o l , b u t b o t h are solved 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 . F i g u r e 6.20 presents resu l ts s i m i l a r to those presented i n case B of F i g u r e 6.18, but on a larger t i m e scale. O n c e t h e m i s - m o d e h n g is detected a n d corrected , the sys tem w i l l persist w i t h the su i tab le e s t i m a t e d m o d e l s t r u c t u r e and wiU y i e l d the desired response. 0.6 0.3 o ^ 0.0 *''-0.3 - * ' -0 .6 - -0.9 -1.2 2 link monipulotor - linki - order=2 Iink2- order= . . . - Votve 1 Volv» 2 4 6 TIME [sec] 0.6 0.5 P 0.4 H m 0.2 H O .H 10 0.0 . -0.1- - Valve 1 - Volve 2 —I r— 4 6 TIME [sec] 8 10 06- „ 0.4- O " 0.2- -. 0 .0 - - ; 6 -0 .2 - -0.4- . . - Volve 1 Volve 2 - T 1— 4 6 TIME [sec] 8 10 F i g u r e 6.14: T h e o u t p u t behav ior of a h y d r a u h c ac tuated two h n k m a n i p u l a t o r 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=2 Ilnk2- order=3 —1 r - 4 6 TIME [sec] 4 6 TIME [sec] 5.- J •S4.- o « - Vot™ 1 - Vi*«« 2 —1 r— 4 6 TIME [sec] 10 F i g u r e 6.15: O r d e r changes of the es t imated m o d e l for h y d r a u h c a c t u a t e d two h n k m a n i p u - l a t o r i n i t i a l l y under -mode led F i g u r e 6.16: T h e cost func t i on der ivat ive behavior 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 - linki -I L 4 6 TIME [sec] 4 5 TIME [sec] —1 1— 4 6 TIME [sec ] 10 cost function - valve 1 F i g u r e 6.17: T h e cost f u n c t i o n behavior 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 manipulator — linki — order = 5 Iink2— order=3 0.6 0.5 r> 0.4 o •' 0.3 ^ ft ' 0.2 r 0.1 : 0.0 -0.1 -0.2 / / / / / / / . - • • . . . - Votve 1 / / . . -• VoPve 2 4 6 TIME [sec] 0.6 0.5 - T) 0 .4- 0.3- - 0.2- 0.1- - m 0.0 -0.1- -0.2 10 _ voiv« 2 4 6 TIME [sec] 10 0.6 0.5 V 0.4-o t. " 0.3- 0.2- 0.1- <-> 0.0- - ; -0.1- -0.2- 0 . - VoPve 1 volv« 2 4 6 TIME [sec] 10 F i g u r e 6.18: T h e o u t p u t behavior of a h y d r a u h c ac tuated two 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 - Volv« 1 4 6 TIME [sec] 10 6. 5. H ^ 4 . H œ 1. - 0. ^3.4- • 1- •D O 2 . - I - . . . - V a l v t 1 Volve 2 - T 1— 4 6 TIME [sec] 8 10 - Votv« 1 - Votvt 2 —r 8 4 6 TIME [sec] 10 F i g u r e 6.19: O r d e r changes of the 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 - l a t o r over -mode led 1 ' -4 2 link manipulator - linki - order-2 Iink2- order=3 1 1 \ •• / ,-' / \ •. / .•' / \ '•. / .•' / \ •. / / / \ '•, / , • ' / \ • / . • • /.'' . . . - Votn 1 1 1 Vdv* 2 10 15 TIME [sec] 20 25 30 5H fe4H T3 i o 1- 0 . . . - Valvt 1 V o l » 2 10 15 TIME [sec] —I— 20 25 30 F i g u r e 6.20: Per f o rmance of a system i n i t i a l l y m i s - m o d e l e d on a larger 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 Resul ts w i t h O t h e r W o r k T h e m e t h o d developed i n th is thesis , detects, determines a n d executes on-hne changes i n m o d e l order . T h e M O D a l g o r i t h m is a gradient a l g o r i t h m based o n the behav ior of a chosen cost f u n c t i o n a n d its t i m e der ivat ive . T h e cost f u n c t i o n behavior enables the a l g o r i t h m to d i s t i n g u i s h between types of m i s - m o d e h n g of the sys tem (under a n d over -modehng) . W h e n m i s - m o d e h n g creates prob lems i n controUing the sys tem it changes the mode l ' s order . I n i - t i a h z a t i o n of the M O D a l g o r i t h m is based e i ther on a p r i o r i knowledge of the s y s t e m a n d s i m u l a t i o n results or o n p r e h m i n a r y tests of the sys tem. T h e contro l s trategy for a sys tem is designed based on the best knowledge of the s y s t e m avai lable . A m o d e l order change w i U occur i f a n oper a t i ng po int change on-hne a n d there fore the condi t ions change or i f the i n i t i a l ident i f i ca t ion of the sys tem was not accurate . T h e M O D a l g o r i t h m act ivates a m o d e l s t ruc ture change on ly w h e n the 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 of this 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 by us ing i t the desired behavior of the sys tem is achieved . A discussion of the advantages and disadvantages of ex i s t ing techniques w i U n o w be p r e - sented. A more deta i led discussion of the methods , has been rev iewed i n Sect ions 2.4 a n d 2.5. Ident i fy ing a sys tem depends s trongly on the choice of the m o d e l s t r u c t u r e . Of f -hne methods have the advantage of choosing 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 its parameters a n d t h e n v a h d a t i n g the m o d e l . If the results are not satisfactory, different types of mode ls c a n be e x a m i n e d to f ind the best mode l for the sys tem a n d the o p e r a t i n g cond i t i ons . Of f -hne 97 90 9ft m o d e l v a l i d a t i o n techniques were pubhshed by A k a i k e , I s e r m a n n Schwarz a n d others . These works propose different c r i t e r i a as a measure of the fit of the m o d e l . A s weU, different methods for state space representat ion have been deve loped . C a n o n i c a l s t ruc tures were proposed by G u i d o v z i . Dav i son presented a m e t h o d for m o d e l order r e d u c t i o n by select ing the d o m i n a n t eigenvalues and eigenvectors of the sys tem. 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 ignif icant d y n a m i c s is a basis for designing a contro l ler , b u t the m e t h o d does not address the quest ion of changing condit ions a n d therefore changes i n the n u m b e r of d o m i n a n t eigenvalues. T h e r e are also several recurs ive methods . Overbeek a n d L j u n g suggested the m o d e l s t r u c t u r e select ion ( M S S ) a l g o r i t h m i n w h i c h the s tructures differ i n the p a r a m e t r i z a t i o n of the m o d e l , b u t there is no recurs ive order selection t h a t is , the order is chosen a p r i o r i . T h e a l g o r i t h m calculates at each t i m e the ent ire set of s t ructures a n d compares t h e m on-Une resu l t ing i n a poss ib i l i ty of a h igh 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 imul taneous recurs ive e s t i m a t i o n of parameter a n d order . T h e order is f ound 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 up to a k n o w n upper b o u n d . T h e order w h i c h corresponds to the m i n i m a l value of the cost func t i on is the one t h a t is used. F u r t h e r work B y N i u a n d F i s h e r i m p l e m e n t e d the above a l g o r i t h m for M I M O sestems. H e m e r l y presented a m e t h o d for on hne order a n d parameter ident i f i ca t i on us ing the R L S a l g o r i t h m a n d the P L S c r i t e r i o n . M e r e i r o s a n d H e m e r l y integrated the above m e t h o d w i t h la t t i ce f o r m f i l ters for a m i n i m u m var iance contro l ler . T h i s work is the closest i n n a t u r e to the w o r k presented i n th is thesis . However , i t requires c o m p u t a t i o n of a cost f u n c t i o n for aU possible orders ( u p to a n u p p e r b o u n d ) , at each t i m e step. T h e r e is no reference or d iscuss ion of the t y p e o f m i s - m o d e h n g . O t h e r recurs ive and off-hne order a n d parameters ident i f i ca t i on m e t h o d s were r e p o r t e d (open loop m e t h o d s ) , such as the one by W u h c h a n d K a u f m a n a n d K a t s i k a s These methods are based o n a p r i o r i defined c r i t e r i o n , a n d ca lcu lat ions of a l l possible orders a n d the choice of the one w h i c h gives the best per formance of the c r i t e r i o n . T h e r e are several advantages to the M O D m e t h o d presented i n this thesis : 1. T h e m e t h o d is an on-hne m e t h o d that detects the need to change the mode l ' s o rder a n d i m p l e m e n t s i t wh i l e us ing wel l k n o w n methods for the 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 contro l processes. M o s t of the p u b h s h e d works that were presented i n Sec t i on 2.5 discuss m o d e l order d e t e r m i n a t i o n , e i ther off-hne or on-hne but open l oop , w i t h n o contro l a l g o r i t h m i m p l e m e n t e d . • Off -hne methods can est imate a m o d e l t h e n check a n d val idate i t a n d i f the resu l ts do not satisfy, another m o d e l s t r u c t u r e is chosen, u n t i l a good represent ing m o d e l is achieved. If a m o d e l is chosen w i t h one of the off-hne methods i ts s t r u c t u r e is f ixed w h e n used on-hne , for contro l purposes for example . Represent ing w o r k s c a n be found i n A k a i k e I se rmann , Schwartz R i s sanen G u i d o v z i a n d D a v i s o n • Recurs ive ident i f i ca t i on methods a n d s t r u c t u r e selection are m a i n l y p a r a m e t e r selection methods where the order is f ixed . T h e possible s t ructures are s c a n n e d a n d the best chosen, Overbeek a n d L j u n g or s imultaneous order 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 the same pr inc ip l e , a set of possible orders are chosen a n d a cost f u n c t i o n is ca l cu la ted for a l l the set each t i m e step. T h e chosen order is the one that correspond to the m i n i m a l value of the cost func t i on . See N i u , X i a o a n d F i s h e r , N i u a n d F i s h e r , W u h c h a n d K a u f m a n ^ K a t s i k a s a n d H e m e r l y 47 2. Cos t f u n c t i o n behavior indicates the best e s t i m a t e d m o d e l order to the M O D a l g o r i t h m . • It was found that the cost funct i on has a different behav ior for under a n d for over -modehng , but s i m i l a r behavior for the two apphcat ions . • T h e cost f u n c t i o n indicates the best possible order for the present o p e r a t i n g p o i n t a n d does not search for the exact m o d e l . • A n i n i t i a l order is prov ided and there is no need to assume o n u p p e r b o u n d to the order . • T h e cost funct i on behavior indicates indicates the influence of the c l osed l o o p , especial ly i n the mis -modehng cases. • T h e r e is no need to examine and scan a set of models for each t i m e s tep as t h e cost funct i on is ca lcu lated for the present m o d e l order. T h e work done by H e m e r l y , combines a recurs ive ident i f i cat ion process a n d r e c u r s i v e order e s t imat i on for a mode l represented by la t t i ce 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 contro l a l g o r i t h m . T h e upper b o u n d of the order is assumed to be k n o w n a n d the order is e s t imated by scanning a l l possible P L S funct ions a n d choosing the o r d e r t h a t corresponds to the m i n i m a l value. T o the best of our knowledge , there is no other m e t h o d hke the one presented i n t h i s thesis . B a s e d on the t i m e behavior of a cost f u n c t i o n the a l g o r i t h m detects a n d executes on-hne changes of m o d e l order . It was estabUshed t h a t b o t h , under a n d over -modeHng c a n cause poor per formance and i n s t a b i h t y a n d for b o t h , order correct ion is done 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 the cost f u n c t i o n behavior 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 fa i r ly s m a l l a n d the a l g o r i t h m has stable character is t i cs a n d is able based o n some a p r i o r i knowledge w i t h i n a few i terat ions to m a i n t a i n des ired p e r f o r m a n c e of the s y s t e m . 6.3 C o n c l u s i o n s T h i s chapter presented the behavior of two k i n d s of robot 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 the 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 the e s t i m a t e d m o d e l for the c o n t r o l a l g o r i t h m . T h e behav io r of the cost f u n c t i o n a n d its der ivat ive for the one a n d two m o d e flexible h n k a n d for the two 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 showed a p a t t e r n of behav ior for u n d e r , over a n d correct s t r u c t u r e for the e s t i m a t e d mode l . B a s e d on these results , a n order d e t e r m i n a t i o n a l g o r i t h m was presented. D e p e n d i n g on i ts parameters , m i s - m o d e h n g can be cor rec ted t o give an acceptable response f r o m these systems, even w h e n they are i n i t i a l l y uns tab le or have b a d per formance . C h a p t e r 7 C O N C L U S I O N S A N D S U M M A R Y 7.1 M a i n R e s u l t s of the Thes is T h i s work has chal lenged the concept of us ing a fixed s t r u c t u r e m o d e l for a p lant c ont ro l l ed b y an adapt ive contro l a l g o r i t h m ( G P C ) . Genera l ly , i n order to i m p l e m e n t an adapt ive a l g o r i t h m for a s y s t e m , the p lant is m o d e l e d by a hnear m o d e l i n w h i c h parameters are e s t i m a t e d o n - hne . T h i s can result i n uncerta int ies i n parameter values, espec ia l ly w h e n the m o d e l order is i n c o r r e c t l y chosen. T w o robo t i c apphcat i ons chosen for the s tudy were m o d e l e d , s i m u l a t e d a n d controUed . T h e single flexible h n k can give rise to modes of osciUations on-hne d u r i n g a w o r k i n g cyc le , a n d therefore have on-hne changes of the p lant 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 actuators can be m i s - m o d e l e d , but i t was also chosen because of i ts h i g h l y n o n h n e a r na ture a n d i ts extensive use i n indus t ry . C h a p t e r s 3 and 4 present the d y n a m i c m o d e h n g of the systems , i m p l e m e n t a t i o n of the G P C a l g o r i t h m a n d the t u n i n g of the contro l parameters to achieve good per f o rmance contro l . C h a p t e r 5 presents a m o d e l order 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 de tec t ing the need to change the m o d e l s t r u c t u r e , correc t ing the order a n d execut ing i t on -hne . T h e s t u d y t h a t l ed to the above m e t h o d began w i t h the estabhshment of a cost f u n c t i o n as a measure of the error between the p lant and m o d e l d y n a m i c s . T h e m a i n results of this s t u d y are: m o d e l i n g , the cost f u n c t i o n rises i n i t i a l l y ( w h e n the error goes to zero) , a n d t h e n sett les o n a constant value . W h e n under - mode l ed , the cost func t i on ' s i n i t i a l rise is s teeper , go ing to m u c h higher values and leading to i n s t a b i h t i e s . W h e n over -mode led , t h e behavior is m u c h more moderate , but the per formance deteriorates a n d the s y s t e m c a n go unstable . W h e n u n d e r - m o d e h n g is invo lved , the regression vector is cor re la ted t o t h e u n m o d e l e d d y n a m i c s . However , the over -modehng does not inc lude that c o r r e l a t i o n a n d therefore the response is more moderate . T h e excess of d y n a m i c s i n the m o d e l (over- modehng) causes the c o n t r o l a l g o r i t h m to t r y a n d c o n t r o l d y n a m i c s that are not there ; thus , the contro l parameters are no longer wel l t u n e d . B o t h apphcat ions show s i m i l a r cost func t i on behavior . B a s e d on the above resul ts , an on-hne m o d e l order 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 iven to detect the need to change the m o d e l order a n d to correct i t on on-hne . Resu l t s f rom i m p l e m e n t i n g the m e t h o d on b o t h apphcat ions show that for u n d e r a n d over -modehng , i n s t a b i l i t i e s are avoided a n d desired per formance is restored . 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 apphed to heavy 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 nonhnear systems. T h e h y d r a u h c a U y a c t u a t e d heavy d u t y m a n i p u l a t o r s are used extensively i n large resource based industr ies , a n d any i m p r o v e m e n t i n eff ic iency m a y result i n m a j o r financial benefits. Therefore , the results i n C h a p t e r 4 that advance the state of the art w iU be s ta ted next ( K o t z e v et at ^^): — T h i s work e x a m i n e d the effect of nonhnear i t ies i n the a p p h c a t i o n of G P C to a wide range of h y d r a u h c a U y ac tuated m a n i p u l a t o r s . - Spec ia l a t t e n t i o n is g iven to the m a x i m u m o u t p u t hor i zon . T h e w o r k i n t r o d u c e s an on-hne a u t o m a t i c change of the m a x i m u m o u t p u t hor i zon so t h a t the t r a n s i e n t response can be suff ic iently fast a n d undes irable overshoots avoided. T h e se l e c t i on of other G P C design parameters is also addressed. • E x p e r i m e n t a l results f r o m a n open loop exper iment on a heavy d u t y m a n i p u l a t o r , a 215B C a t e r p i l l a r , ind i ca te that the cost f u n c t i o n behavior i n open loop does not v a r y s trongly enough for m i s - m o d e h n g to be rehably de te rmined . T h i s has also been ver i f i ed by n u m e r i c a l s imula t i ons w i t h other app l i ca t i ons . A closed loop approach is n e e d e d . 7.2 Suggestions for F u t u r e W o r k T h e goal of this thesis was to s t u d y the behavior of a sys tem controUed w i t h a n a d a p t i v e contro l a l g o r i t h m w h e n p lant m i s - m o d e h n g occurs , u n d e r s t a n d the behavior , a n d suggest a m e t h o d to overcome prob lems that arise, such as poor per formance a n d i n s t a b i h t y . S u c h a s t u d y was conduc ted a n d a m e t h o d that provides good results is presented. T h e scope of t h e inves t igat i on can be m a d e broader for fur ther genera l i zat ion of the results : • M o r e adapt ive a l g o r i t h m s , pred i c t ive a n d non -pred i c t ive , shou ld be t r i e d . • M o r e systems should be checked, not only f r o m the robo t i c fami ly . T h i s c o u l d general ize the conclusions on the behav ior of those systems i n m i s - m o d e h n g a n d m a y come u p w i t h parameters to character ize i t . For example the t i m e constant of the s y s t e m c o u l d have a n inf luence on the results . • A closed loop exper iment w i t h the order 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 be done w i t h a 2 1 5 5 C a t e r p i U a r or a s imi lar sys tem to f ind advantages a n d d isadvantages , since the 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 ent i on should be g iven to other possible order 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 per formance should be c ompared w i t h the present one. • T h e sens i t iv i ty of the ident i f i cat ion a l g o r i t h m a n d its inf luence on the cost f u n c t i o n shou ld be checked. 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I n Proceedings of the 1991 IFAC International Symposium on Intelligent Tuning and Adaptive Control, pages 293-297. I F A C , 1991. 48] J . R i s sanen . A pred i c t ive least squares pr inc ip le . IMA J. of Math. Control and Infor- mation, 3 :211-222, 1986. [49] A . A . D Mere i ros a n d E . M H e m e r l y . L a t i c e form i n adapt ive c o n t r o l w i t h recurs ive o rder e s t i m a t i o n . In Proceedings of the 1991 IFAC International Symposium on Intelligent Tuning and Adaptive Control, pages 1083-1087. I F A C , 1991. 50] R . H . C a n n o n a n d E . S c h m i t z . Prec ise contro l of flexible m a n i p u l a t o r s . Robotics Re- search, pages 841 -861 , 1983. [51] J . J . C r a i g . Introduction to Robotics Mechanics and Control. A d d i s o n - W e s l e y P u b l i s h i n g C o m p a n y Inc . , R e a d i n g Massachuset t s , 1986. 52] H . E . M e r r i t t . Hydraulic Control Systems. J o h n W i l e y a n d Sons, Inc . , N e w Y o r k , 1966. 53] P . V a h a . A p p h c a t i o n of parameter adapt ive approach to servo contro l of a h y d r a u H c m a n i p u l a t o r . Acta Polytechnica Scandinavia, Mathematics and Computer Science, 5 1 : 3 - 86, 1988. [54] N . Sepehr i , G . A . M D u m o n t , P . D . Lawrence , a n d F . Sassani . Cascade c o n t r o l of 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 . Advanced Systems Institute, 1990. [55] C . H . A n , C . G . A t k e n s o n , , a n d J . M . HoUerbach . Model-Based Control of a Robot Ma- nipulator. T h e M I T Press series i n ar t i f i c ia l inteUigence. T h e M I T Press , C a m b r i d g e , Massachuset t s , 1988. [56] H . A s a d a a n d j . j . e . S lo t ine . Robot Analysis and Control. J o h n W i l e y & Sons , I n c . , N e w - Y o r k , 1986. 57] A . K o t z e v , D . B . C h e r c h a s , P . D . Larence , and N . Sepehr i . G e n e r a h z e d pred i c t ive c o n t r o l of a robot i c m a n i p u l a t o r w i t h h y d r a u h c actuators . Robotica - in press., 1992. 58] J . W a t t o n . T h e d y n a m i c per formance of an e lec tro -hydrauHc servo v a l v e / m o t o r s y s t e m w i t h t ransmiss i on l ine effects. ASME Journal of Basic engineering, 190:14-18, M a r c h 1987. 59] A . A . B a h n a s a w i and M . S . M a h m o u d . Control of Partially Known Dynamical Systems. S p r i n g e r - V e r l a g , N e w - Y o r k , N . Y . , 1989. [60] P . Eykof f . System Identification: Parameter and State Estimation. J o h n W i l e y &: Sons , Inc . , N e w - Y o r k , 1974. 61] M o o g Inc. Moog Type SO Flow Control Servo Valves, v o l u m e C a t a l o g 301385. M o o g Inc . , Aerospace G r o o p , N e w Y o r k , 1988. [62] W . H . Press , B . P . F l a n n e r y , S . A . Teukolsky , a n d W . T . V e t t e r l i n g . Numerical Recipies, The Art of Scientific Computing, cambrige U n i v e r s i t y Press , C a m b r i d g e , 1990. 63] D . J . L a t o r n e l l . Force Control for Robotic Manipulators with Structurally Flexible Links. P h D thesis , D e p t . o f M e c h a n i c a l E n g r g . , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1992. 64] A . O . S t e i n h a r d t . Househo lder transfroms i n s ignal processing. IEEE ASSP magazine, 5(3) :4 -12 , J u l y 1988. 65] N . Sepehr i . Control of a Heavy Duty Hydraulic Machine. P h D thesis , U B C - T h e U n i v e r s i t y of B r i t i s h C o l u m b i a , T h e F a c u l t y O f M e c h a n i c a l E n g i n e e r i n g , 1991. 66] R . L . Bisphnghof f , H . A s h l e y , a n d R . L . H a l f m a n . Aeroelasticity. A d d i s o n - W e s l e y P u b - h s h i n g C o m p a n y , R e a d i n g , Massachuset ts , 1955. 67] F . S . T s e , I . E . M o r s e , a n d R . T . H i n k l e . Mechanical Vibrations Theory and Applications, v o l u m e Second d i t i o n . A i i y n a n d B a c o n Inc . , 1980. A p p e n d i x A E x p e r i m e n t a l R e s u l t s for the 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 I n t r o d u c t i o n A t U B C (the 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 ) an excavator , 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 te leoperat ion pro j e c t , Sepehri ^as avai lable to us for some e x p e r i m e n t s . T h e goal i n the exper iment was to ident i fy the d y n a m i c m o d e l of the m a n i p u l a t o r w i t h o p e n l o o p , check the behav ior of the cost funct i on ( E q u a t i o n 5.1), a n d compare i t w i t h the results 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 contro l led w i t h G P C , as descr ibed i n C h a p t e r 4 a n d C h a p t e r 5. T h e results show that after the cost func t i on becomes flat at h i g h orders , i t is easy to choose a m i s - m a t c h e d order for the sys tem. B u t results i n C h a p t e r 5 for the t w o h n k m a n i p u l a t o r w i t h the h y d r a u h c ac tuator show that i f the order is wrong ly chosen, t h e n t h e ins tab iht i es can occur . A . 2 D e s c r i p t i o n of the S y s t e m T h e C a t e r p i l l a r 215B excavator is a mobi le three degree of f reedom m a n i p u l a t o r . T h e h n k s are the " S w i n g " , w h i c h is the base that rotates . T h e " B o o m " a n d the " S t i c k " are two h n k s operated t h r o u g h h y d r a u h c cyhnders . T h e e n d effector, the " b u c k e t " , w h i c h is used t o d i g a n d car ry heavy loads is also operated w i t h a hydrauhc ac tuator . F i g u r e A . l describes t h e excavator ' s s t ruc ture . T h e mot ions of the " B o o m " and " S t i c k " are coup led by cross-over valves , w h i c h aUow for a faster movement of one hnk when the other is slower. T h e e x p e r i m e n t for th is work is the ident i f i ca t i on of the d y n a m i c s of the " B o o m " a n d i t s a c t u a t o r t h a t were operated . T h e couphng between the valves was e h m i n a t e d . T h e e s t i m a t i o n a l g o r i t h m used was a Recurs ive Least Squares . T h e i n p u t is the s igna l f r o m t h e spoo l va lve , a n d the o u t p u t , f r o m the angular pos i t i on of the h n k . A . 3 R e s u l t s S i x runs were m a d e where in each, the next i n p u t character is t i cs were g iven : • S l f - a m p h t u d e of 1.0 volt and frequency of 4 seconds. • S2f - a m p h t u d e of 1.5 volt a n d frequency of 3 seconds. • S3f - a m p h t u d e of 1.5 volt a n d frequency of 2 seconds. • S4f - a m p h t u d e of 1.3 volt and frequency 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 vo 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 vo l t . F igures A . 2 A . 3 A . 4 A . 5 A . 6 A . 7 present the behav ior of the measured 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 the cost f u n c t i o n w h i c h is an i n d i c a t i o n of the error between the two o u t p u t s . I n aU six cases, the d r a w i n g of the measured o u t p u t vs. the m o d e l o u t p u t show very h t t l e difference between the two, as does the cost f u n c t i o n w h i c h grows fast to a value a n d dri f ts s lowly f r o m it due to the nonhnear i t ies w h i c h are not m o d e l e d i n the h n e a r m o d e l for the 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 the behavior of the cost f u n c t i o n for the four S i f runs . T h e values of the cost f u n c t i o n i n aU runs was t a k e n after 2000 s a m p h n g steps. T h e results con f i rm the discussion i n C h a p t e r 2 that the cost f u n c t i o n for open loop ident i f i ca t ion w i l l have h igh values for under -modehng a n d w i l l reach a p l a t e a u for h igher  orders of the e s t imated m o d e l . T h e same goes for the R i f exper iments , as shown i n F i g u r e A . 9 , the measured a n d the m o d e l o u t p u t agree w i t h each other qui te wel l , and the cost f u n c t i o n , after 2000 s a m p h n g steps, again has h igh values for u n d e r - m o d e h n g and reaches a p l a t e a u for higher order values. In C h a p t e r 5 the cost func t i on for the closed loop a l g o r i t h m behaves differently. T h e r e is a clear difference between the values for u n d e r - m o d e h n g a n d correct modehng . For over -modehng , there cou ld also be a c lear difference f r o m correct m o d e h n g , resu l t ing i n i n s t a b i h t y of the sys tem i f left u n a t t e n d e d . A . 4 C o n c l u s i o n s T h i s exper iment was done w i t h an excavator w h i c h is used extens ive ly i n the 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 sys tem is a h igh ly nonhnear one, a n d it is therefore of interest t o check the behav ior of the cost funct i on w h e n open loop ident i f i ca t ion is done. T h i s c on f i rms the fact that even w h e n a n ident i f i cat ion is done open loop a n d an over mode led m o d e l is chosen the closed loop contro l led sys tem m a y r u n i n t o per formance a n d s t a b i h t y p r o b l e m s . T h u s , there is a need to detect on-hne a signif icant 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 - ter 5. T h e results show (see Figures A . 8 a n d A . 9 ) , t h a t w h e n ident i f i ca t i on is done i n open loop the behav ior of the cost func t i on is not such t h a t an error i n m o d e l s t r u c t u r e c a n be easi ly d e t e r m i n e d . T h u s a closed loop approach is needed. T h i s has also been s h o w n i n s imulat ions i n chapter 5. Caterpillar 215B oxcavotor — one link identification — s1f_3 I " 0.3 u ^ 0.2 H I 0.0-1 I- 1-0.1 - I 3 o. c -0 .2- 500 1000 1500 2000 2500 no. of time steps -0.3 500 1000 1500 2000 2500 no. of time steps 500 1000 1500 2000 2500 no. of time steps .50 .45- ^ . 4 0 - J 3 0 - .25 .20 Coterpillor 215B axcovotor — one link identification — s2f_3 ' J . l ' l l ' . ri 1 Vn o 0.2-f c « E O.H 0 u a 0.0-1 m Ô - 0 - H •» -0.2 H 1 ^ -0.3 H 1000 2000 3000 no. of tinne steps 4000 -0.4 1000 2000 3000 no. of time steps 4000 .200 .195- .190- 8 -185 H o .180- .175 1000 2000 3000 no. of time steps 4000 Caterpillar 215B axcovotor — one link identification — s3f_3 u J I , , - 500 1'' I I L U . 1000 1500 no. of time steps 2000 2500 0.3' o ^ 0.2- "c E 0.1-u u "5. 0.0 tr, '•B 0 « - 0 . 2 H 1 ^ - 0 . 3 H a. c " -0.4 500 1000 1500 no. of time steps 2000 2500 500 1000 1500 2000 2500 no. of time steps Coterpillor 215B axcovotor — one link identification — s4f_3 r-' 0.3- —I 500 1000 1500 2000 2500 3000 no. of time steps ^ 0.2 H g 0.1 3 a. f- -I 0.0-1 § . -0 .1 CI) I ^ -0.2 3 a. c "" -0.3 500 1000 1500 2000 2500 3000 no. of time steps .222- 1 1 1 1 .220- - .218- - J -216- - 1 .214- 8 .210- - .208- - .206- .204- 1 1 1 1 1 0 500 1000 1500 2000 2500 3000 no. of time steps Caterpillar 215B oxcovator - one link identification — r1f_3 1000 2000 3000 4000 no. of time steps 5000 1000 2000 3000 4000 no. of time steps 5000 ^.070 H o .065 - .• • 1000 2000 3000 4000 no. of time steps 5000 Caterpillar 215B oxcavotor - one link identificotion - r2f_3 I I E' u rrr 0.4 1000 2000 3000 no. of time steps 0.2 H 0.0 4 -0.2 H 4000 -0.4 1000 2000 3000 no. of time steps 4000 1000 2000 3000 no. of time steps 4000   A p p e n d i x 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 analysis is based o n Bisphngho f f et. a l a n d Tse et. a l T h e P a r t i a l D i f f e rent ia l E q u a t i o n ( P D E ) for the deflection w(x,t) i n a cant i lever b e a m , cons ider ing b e n d i n g a n d shearing s tra ins a n d neglect ing shear de format ion 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 the cross sect ional m o m e n t of i n e r t i a of the a r m , a n d m is the mass per un i t l e ng th . T h e above P D E is separable , so let : w{x,t) <l>{x)q{t) ( B . 2 ) dp + u;\ = 0 ( B . 3 ) W h e r e 6" = G e n e r a l solut ions for the above equations are: q{t) = Asin{ujt) + Bcos{ut) ( B . 5 ) (f>{x) = Csinh{bx) + Dcosh{bx) + Esin{bx) + Fcos{bx) ( B . 6 ) T h e b o u n d a r y condit ions for cant i lever b e a m are: at a; = 0 : < (̂0) = 0, ^ = 0 dx^ ~~ ^' dx^ " ^ at X = L : S o l v i n g the above equat ions w i t h the b o u n d a r y condi t ions result w i t h : T h e n a t u r a l frequencies by: cos{bL)cosh(bL) = 1 ( B . 7 ) W h e r e bL = 0.59 697r, 1.49427r, |7r, T h e mode shapes by : ^{x) = D[A{sinh{bx) — sin[bx)) + cosh[bx) — cos{bx) W h e r e : A = sin(bL) — sinh(bL) cos(bL) + cosh{bL) D is a n o r m a h z e d coefficient where (j>{L) = 1 so: ^ _ cosh{bL) + cos{bL) ( B . 8 ) ( B . 9 ) ( B . I O ) 2sinh{bL)cosh{bL) A d y n a m i c m o d e l has three integrals as a f u n c t i o n of ^(as), w h i c h are (assuming u n i f o r m mass d i s t r i b u t i o n m ) : 2mD 11 — m • I x^(x)dx = ——- Jo P 12 = m- I (t>'{x)dx = mD'L Jo 73 = EI I Jo ( B . l l ) dx' F o r a r m w i t h the next d a t a : L = 1 [meter E I = 574.024 [N -m'] It = 0.2817 [kg-m' m = 1 m = 0.8451 [kg/m T a b l e B . l presents the d a t a for the first five modes . F i g u r e B . l shows the m o d a l shapes for the cant i lever b e a m . Mode 6/ (rad) w (rad/dec) Delta D h h h 1 1.8752 91.644 -0.7266 0.5 0.2404 0.2113 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 Tab le B . l : D a t a for the first five modes of a cant i lever b e a m m o d e » h o p « n o 1 m o d e s h o p e n o 2 « [ m e t e r s ] » [ m e t e r s ] m o d e s h a p e n o 3 m o d e s h o p e n o 4 > [ m e t e r s ] > [ m e t e r s ]

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