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A study of various computational methods for determining time optimal control of time delay system Morse, James Gregory 1970

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A STUDY OF VARIOUS COMPUTATIONAL METHODS FOR DETERMINING TIME OPTIMAL CONTROL OF TIME DELAY SYSTEMS  JAMES GREGORY MORSE B.Sc,  University  of Ottawa, 1968  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the Department o f Electrical  Engineering  We a c c e p t t h i s t h e s i s as conforming t o the required  standard  Research S u p e r v i s o r Members of Committee  Head of Department  Members of the Department o f Electrical  THE  Engineering  UNIVERSITY OF BRITISH COLUMBIA November,  1970  In  presenting  this  an a d v a n c e d  degree  the L i b r a r y  shall  I f u r t h e r agree for  scholarly  by h i s of  this  written  thesis at  the U n i v e r s i t y  make  that permission  It  by  financial gain shall  of  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Columbia  the  requirements  B r i t i s h Columbia, for  I agree  r e f e r e n c e and copying of  this  that copying or  not  for  that  study. thesis  t h e Head o f my D e p a r t m e n t  is understood  permission.  Department  of  for extensive  p u r p o s e s may be g r a n t e d  for  fulfilment of  it freely available  representatives. thesis  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t my  ABSTRACT In  this  t h e s i s some n u m e r i c a l  t e c h n i q u e s f o r o b t a i n i n g the time  o p t i m a l c o n t r o l o f a c l a s s o f time delay systems The d e l a y s may  be f i x e d o r time v a r y i n g .  are s t u d i e d and  compared.  The d e l a y systems c o n s i d e r e d , w h i c h  need not be l i n e a r o r time i n v a r i a n t , a r e those f o r which  the time o p t i m a l  c o n t r o l i s bang-bang. The o p t i m a l c o n t r o l i s found by c a r r y i n g out a s e a r c h i n s w i t c h i n g i n t e r v a l space.  The method of Rosenbrock  i n t e r v a l s which maximize  (2 3) ' i s used to f i n d  the s w i t c h i n g  a performance i n d e x o f the f i n a l s t a t e s and  terminal  (21) time.  Kelly's  w i t h time v a r y i n g  method of g r a d i e n t s i s shown to be a p p l i c a b l e  time d e l a y s by u s i n g the c o s t a t e e q u a t i o n s o f r e f . [10].  p e r t u r b a t i o n s i n the c o n t r o l a r e chosen i n such a way f u n c t i o n space i s changed  In  to a s t e e p e s t descent i n s w i t c h i n g i n t e r v a l space. (19)  a t h i r d approach, a t e c h n i q u e s i m i l a r  i l l u s t r a t e d by  to t h a t o f Bryson and Denham A l l the methods  examples.  The advantages of the d i r e c t s e a r c h based on Rosenbrock's are  method  a) ease o f programming and b) r a p i d convergence c l o s e to the optimum.  However, i n i t i a l method. was  The  t h a t the descent i n  i s used to account f o r the t e r m i n a l c o n d i t i o n s d i r e c t l y . are  to systems  convergence i s slow when compared to t h a t of e i t h e r  Of the two  gradient  g r a d i e n t methods, that based on a p e n a l t y f u n c t i o n approach  s u p e r i o r i n ease o f programming and convergence c l o s e to the optimum to  that based on a descent to the f i n a l  target set.  N e i t h e r g r a d i e n t scheme  c o u l d match -the r a p i d convergence o f the Rosenbrock method c l o s e to the optimum.  (i)  TABLE OF CONTENTS Page ABSTRACT  .  1  TABLE OF CONTENTS. LIST OF ILLUSTRATIONS  1  .  i i i  ACKNOWLEDGEMENT I.  .  iv  INTRODUCTION  I I . GENERAL BACKGROUND 2.1 2.2  1 .  3  Maximum P r i n c i p l e Proposed Techniques  3 6  I I I . THE EXTENSION OF ROSENBROCK'S METHOD TO FINDING TIME OPTIMAL CONTROLS OF TIME DELAY SYSTEMS 3.1 3.2 3.3 3.4 3.5  D e s c r i p t i o n o f Rosenbrock's method Time Optimal C o n t r o l of Time I n t e r v a l Adjustment Computational A s p e c t s Examples and R e s u l t s Conclusions  IV. A GRADIENT METHOD 4.1 4.2 4.3 4.4 4.5 4.6 V.  8 8 9 11 12 17 18  Development of the P e r t u r b a t i o n E q u a t i o n s P e n a l t y F u n c t i o n Approach Algorithm D e s c r i p t i o n Comments on Computational Examples and R e s u l t s Conclusions  DESCENT TO FINAL TARGET SET 5.1 5.2 5.3 5.4  1  19 24 29 ... 30 30 35 36  Perturbation Analysis Algorithm D e s c r i p t i o n . . . . A D e t a i l e d Example Conclusions  "  .  36 40 41 45  V I . CONCLUSIONS  47  APPENDIX I  48  BIBLIOGRAPHY  51  (ii)  LIST OF ILLUSTRATIONS Figure  .  Page  3.1  Time Delays  15  3.2  Convergence of Rosenbrock's Method,  4.1  Nominal C o n t r o l  4.2  New  4.3  Change i n C o n t r o l  23  4.4  New  23  4.5  Change i n C o n t r o l  4.6  Convergence of G r a d i e n t Method,  2nd Order System  32  4.7  Convergence of G r a d i e n t  7th Order System  34  5.1  % | Error |  7th Order System...  16 23  Control  :  Control  23  23  Method,  vs S t a t e X ( t ) , 2nd Order System  (iii)  46  ACKNOWLEDGEMENT I should  like  to express my g r a t i t u d e t o a l l those w i t h o u t  h e l p t h i s work would not have been p o s s i b l e .  whose  S p e c i a l mention i s deserved  by Dr. S. G. Rao whose a d v i c e was c r u c i a l i n s e l e c t i n g  the problem and by  Dr. A. C. Soudack f o r h i s h e l p i n w r i t i n g t h e t h e s i s . I would a l s o l i k e t o thank Dr. E. V. Bohn f o r r e a d i n g t h e m a n u s c r i p t w i t h such a l a c r i t y . with  Thanks a r e a l s o due to Mr. A. MacKenzie f o r h i s h e l p  the i l l u s t r a t i o n s ,  help with proofreading While doing f o r which he i s d u l y  to Mr. J . J o r d a n  and M i s s T. P. S i l v e s t e r  f o r their  and to M i s s V e r o n i c a Komczynski f o r t y p i n g t h e m a n u s c r i p t . t h i s work t h e author  grateful.  (iv)  was supported  by NRC s c h o l a r s h i p s  1  I.  INTRODUCTION  A time d e l a y system i s one i n V 7 h i c h t h e s t a t e e q u a t i o n s i n v o l v e terms, e i t h e r i n the s t a t e s o r t h e c o n t r o l , w h i c h a r e d e l a y e d by some p o s i t i v e q u a n t i t y , ©(t), c a l l e d t h e time d e l a y .  I f more than one d e l a y e x i s t s i n t h e  system they a r e denoted by G . ( t ) , 0 ( t ) , . . .0, ( t ) . O p t i m a l c o n t r o l o f time d e l a y systems p r e s e n t s d i f f i c u l t i e s n o t found i n problems w h i c h have no d e l a y s i c a l l y of i n f i n i t e o r d e r . A l s o  t  s i n c e such systems a r e t h e o r e t -  t h e s t a t e and c o s t a t e e q u a t i o n s a r e  n e a r l y i m p o s s i b l e t o s o l v e a n a l y t i c a l l y even f o r t h e s i m p l e s t  s y s t e m s .  Therefore numerical o r i t e r a t i v e techniques o f determining the optimal c o n t r o l o f f e r , i n most c a s e s , t h e o n l y p r a c t i c a l methods o f s o l u t i o n .  Unfortunately  some n u m e r i c a l methods which work w e l l f o r systems w i t h no d e l a y s , such as descent i n c o s t a t e space, cannot be a p p l i e d f o r time d e l a y systems;  the c o s t a t e  e q u a t i o n s by t h e i r n a t u r e b e i n g i n s o l u b l e o t h e r than backwards i n t i m e . ^ ^ I f t h e c o n t r o l i s r e s t r i c t e d by magnitude c o n s t r a i n t s and, as i s o f t e n the case f o r time o p t i m a l problems,  l i e s on t h e boundary o f t h e con-  s t r a i n t s e t , then the d e r i v a t i v e o f t h e H a m i l t o n i a n w i t h r e s p e c t t o t h e (22) o p t i m a l c o n t r o l i s n o t n e c e s s a r i l y zero (13) Raphson a l g o r i t h m , and the min H cannot be used.  .  Thus methods such as t h e Newton-  s t r a t e g y which make use o f t h i s  fact  A s e a r c h i n f u n c t i o n space i s a l s o r e s t r i c t e d s i n c e t h e  allowable p e r t u r b a t i o n s i n the c o n t r o l are not f r e e . In  t h i s t h e s i s t h r e e n u m e r i c a l methods d e s i g n e d t o overcome these  problems a r e p r e s e n t e d : determine  (2) a) Rosenbrock's method o f m a x i m i z i n g a f u n c t i o n i s used t o the time o p t i m a l c o n t r o l o f time d e l a y systems. (21) b) K e l l y ' s method o f g r a d i e n t s  systems w i t h time v a r y i n g time d e l a y s .  i s shown t o be a p p l i c a b l e t o The r e s u l t i n g descent i n f u n c t i o n  space i s m o d i f i e d by an u n u s u a l c h o i c e o f p e r t u r b a t i o n i n the c o n t r o l t o  2  ensure t h a t t h e new c o n t r o l i s a d m i s s i b l e .  F o r such a c h o i c e o f p e r t u r b a t i o n s ,  the descent  i n f u n c t i o n space reduces to s t e e p e s t  space, w i t h  attendant  reduction of storage  T h i s r e s u l t i s important  and computation  i n switching  o f Banks,  the f i n a l  interval  requirements.  because the g r a d i e n t o f any f u n c t i o n w i t h  to the s w i t c h i n g i n t e r v a l s i s n o t , i n g e n e r a l , c) The t h e o r y  descent  respect  available. t a r g e t s e t approach o f Bryson and  (19) Denham, bined  and the aforementioned p e r t u r b a t i o n i n the c o n t r o l a r e a l l com-  to y i e l d  a steepest  descent  f o r the t e r m i n a l c o n d i t i o n s  i n switching  i n t e r v a l space which accounts  directly.  Comparisons o f these methods d i s c u s s e d i n the c o n c l u s i o n s .  v i a examples are made, and' the r e s u l t s  3  II.  GENERAL BACKGROUND  A maximum p r i n c i p l e , which was  derived by Banks  for systems  with time varying time delays, may be used to show that the time optimal control i s bang-bang.  In such cases the control can be represented by a  f i n i t e number of parameters,  the switching instants.  Ordinary descent i n  function space has d i f f i c u l t y i n dealing with such controls; however, the descent i n function space can be changed to a descent i n parameter space.  2.1  The Maximum P r i n c i p l e Consider a system defined by x(t) = f ( x ( t ) , x(w(t)), u ( t ) , t)  where the dot denotes the derivative  t  Q  * t <  t f  with respect to time, t, the  (2.1) independent  v a r i a b l e and x(t) i s an n dimensional column vector, x(t) = [x ( t ) , x ( t ) , . . . x ( t ) ] , i i. n T  f i s an n dimensional vector function, u i s an m dimensional control vector, and w(t) i s the delayed time defined by 0(t) i s the time varying time delay.  w(t) 4 t-O(t) where  The time delay i s usually r e s t r i c t e d by physical considerations so that 0(t) > 0  -/ t  0(t) < 1  V-t  (2.2a) .  (2.2b)  The reason for the f i r s t r e s t r i c t i o n i s clear enough, for i f the delay becomes negative then w(t) = t-O(t) i s no longer a delayed time but an advanced time. The reasoning behind the second r e s t r i c t i o n i s s l i g h t l y more obscure.. time delay may  The  decrease very rapidly, provided that i t i s always continuous  and everywhere greater than zero.  However, i f the value of the time delay  4  i s i n c r e a s i n g f a s t e r than time i t s e l f , not be m o n o t o n i c a l l y i n c r e a s i n g , may not e x i s t .  and r ( t ) , the i n v e r s e  S i n c e the e x i s t e n c e equations^ ^,  o f the c o s t a t e  then the d e l a y e d time, w ( t ) ,  function of w(t),  o f r ( t ) i s a p r e r e q u i s i t e to the f o r m a t i o n  Q(t) must always be s t r i c t l y  1 0  will  A system o f the form 2.1 r e q u i r e s x(t) = g(t)  initial  l e s s than one.  conditions  w(t ) < t < t Q  (2.3)  D  where g ( t ) i s an n - v e c t o r f u n c t i o n o f time. It i s desired  to maximize some performance i n d e x  <Kx  f  ,t ) + f  t  subject  to c o n s t r a i n t s  2.1 and t e r m i n a l  f (x,u,t)dt  (2.4)  n  conditions  *(x ,  t ) = 0 .  f  f  Here ,<f> i s a s c a l a r f u n c t i o n , I i s a v e c t o r means t h a t  tf  J o  the c o r r e s p o n d i n g q u a n t i t i e s  f u n c t i o n , and the s u b s c r i p t " f "  are evaluated  a t the f i n a l  time.  When the problem c o n s i s t s o f c o n t r o l l i n g the s t a t e x ( t ) = x Q  the o r i g i n i n minimum time the t e r m i n a l  f  time i s f r e e , the f i n a l p o i n t  t ) = f  X  f  i s the o r i g i n ) a n d  Banks i n [10] has shown t h a t the c o s t a t e T  = 0 the performance  ^ 1 dt.  function i s J = [  are p ( t )  to  conditions are  *(x , (the f i n a l  Q  e q u a t i o n s o f system 2.1  = - p ( t ) | | ( x ( t ) , x(w(t)> , u ( t ) , t ) T  •  3f  T  - P (r(t))  3  ^-  t  )  )  (x(r(t)),  x(t), u(r(t)), t  • T P  m  S  T  =  c;  V  A  (x(t), x(w(t)) t  }  5  D  r(t))r(t)  f t < w(t )  u ( t ) , t) .  8x(t) w(t ) f  £ t < t  ( - ) 2  f  5  5  r ( t ) i s a s c a l a r function defined  as the i n v e r s e o f w(t)  t = w ( r ( t ) ) , o r i n s h o r t n o t a t i o n r ( t ) = w "*"(t) . not  such t h a t r ( w ( t ) ) =  The n o t a t i o n w  to the r e c i p r o c a l , b u t to the f u n c t i o n a l i n v e r s e , o f w ( t ) .  n-dimensional costate  1  (t) refers,  p ( t ) i s the  vector p(t) = [ p - ^ t ) , p ( t ) , . . . p ( t ) ] .. T  2  The u(t),  t) with  n  3f — — 7 . means the f u n c t i o n a l d e r i v a t i v e o f f ( x ( t ) , 9x(,w(.t;;  term  x(w(t)),  N  respect  to i t s second argument.  The H a m i l t o n i a n  i s then  defined  as H(x,p,u,t) = p ( t ) T  f ( x ( t ) , x(w(t)), u ( t ) , t) - f ( x ( t ) , x ( w ( t ) , u ( t ) , t ) . o  In a l l subsequent work i t i s assumed t h a t : 1)  the o p t i m a l  control exists  2)  the o p t i m a l  c o n t r o l i s unique  3)  the maximum p r i n c i p l e  as d e r i v e d by B a n k s h o l d s ,  and hence  t h a t a l l assumptions made by him about c o n t i n u i t y , d i f f e r e n t i a b i l i t y , e t c . o f f ( x ( t ) , x ( w ( t ) ) , u ( t ) , t ) , x ( t ) , and u ( t ) (See  appendix I f o r more d e t a i l s , . )  T h i s l a s t i s n e c e s s a r y f o r i f Bank's maximum p r i n c i p l e does n o t h o l d there trols.  hold.  then  i s no t h e o r e t i c a l j u s t i f i c a t i o n f o r c o n s i d e r i n g bang-bang o p t i m a l A l l t h e examples c o n s i d e r e d  con-  in- the t h e s i s were chosen to s a t i s f y  these  assumptions. A)  The v e c t o r  ~T T f u n c t i o n f = ( f j f ) has the form f ( x ( t ) ,  x(w(t)),  1  u(t),  t) = A ( x ( t ) , x(w(t)), t) + B ( x ( t ) , x(w(t)),  A(x(t), x(w(t)), is 5)  t ) u ( t ) , where  t) i s an (n + l ) x l v e c t o r , B ( x ( t ) , x ( w ( t ) ) , t )  an (n + l ) x m m a t r i x  and u ( t ) i s an m x 1 v e c t o r .  There i s no f i n i t e s u b i n t e r v a l ( t ^ j t ^ ) o f the i n t e r v a l T  [ t , t^] Q  o v e r which any element o f the v e c t o r p , ( t ) B ( x ( t ) , x ( w ( t ) ) , t ) i s zero.  6  6)  The  c o n t r o l i s r e s t r i c t e d by  form  |u_^| < $^  |u| < 8.  S i n c e the n u m e r i c a l methods c o n s i d e r e d l a t e r o p t i m i z e by  i + l,...m o r i n v e c t o r  the performance  a c o r r e c t choice of switching i n t e r v a l s i t i s necessary  index  to show t h a t  the  o p t i m a l c o n t r o l i s bang-bang. Lemma:  Under the above assumptions the o p t i m a l c o n t r o l i s bang-bang.  Proof:  By  the maximum p r i n c i p l e o f Banks  maximize the  Hamiltonian  H(x,p,u,t) = p ( t ) A ( x ( t ) ,x(w(t)) ,t) + T  By  the Scharwz i n e q u a l i t y u(t)  for  = -SGN  then  the o p t i m a l c o n t r o l , must  p (t)B(x(t),x(w(t),t)u(t). T  the above i s a maximum i f .  [B (x(t), x(w(t)) T  s  t) p ( t ) ]  (2.6)  the second term o f H i s • p o s i t i v e w i t h  the  largest  p o s s i b l e v a l u e o f u. The  above lemma, which i s a s p e c i a l case o f Bank's maximum p r i n c i p l e , d e t e r -  mines the form o f the time o p t i m a l c o n t r o l .  The next  section details  a com-  p u t i n g scheme which makes use of t h i s knowledge.  2.2  Proposed Techniques In o r d e r to s y n t h e s i z e the o p t i m a l c o n t r o l t h e r e f o r e t h r e e t h i n g s  must be known: a)  the number o f switching, i n t e r v a l s  b)  their  c)  the s i g n of the c o n t r o l i n the f i r s t  durations interval.  A common method o f d e t e r m i n i n g  the o p t i m a l c o n t r o l i s to assume some  c o n d i t i o n s on  to s o l v e equations  the c o s t a t e s and  2.1  w h i l e s y n t h e s i z i n g the c o n t r o l at each s t e p from 2.6.  and  2.5  forward  initial i n time  Such a method w i l l  not work f o r the time delay problem because of the form o f e q u a t i o n s  2.5.  7  The  assumptions made about the d e l a y  r(.t) = w "*"(t) i s g r e a t e r 2.5  ensures t h a t w(t)  than or equal to t [ r e f . 1 0 ] .  forward i n time r e q u i r e s knowledge of  i s bang-bang one drive  cannot use  the g r a d i e n t  the f u t u r e .  schemes such as  If  the p e r t u r b a t i o n s  in  f u n c t i o n space the magnitude c o n s t r a i n t s on  I t i s shown i n Chapter IV i n such a way and  I t i s proposed t h a t one performance i n d e x as mize i t by  correct choice  to the c o n t r o l to  i n the the  zero.  f o r descent be v i o l a t e d . chosen  c o n t r o l are not v i o l a t e d  i n t e r v a l space r e s u l t s .  the s w i t c h i n g  of same.  control  c o n t r o l can be  method of s o l u t i o n would be  a f u n c t i o n of  equations  I f the o p t i m a l  the c o n t r o l may  that the p e r t u r b a t i o n s  descent i n s w i t c h i n g  Thus to s o l v e  chosen i n the u s u a l way  t h a t the magnitude c o n s t r a i n t s on  that steepest  that  the min-H s t r a t e g y which t r y to .  o f the H a m i l t o n i a n w i t h r e s p e c t i n the c o n t r o l are  < t and hence  Any  i n t e r v a l s and  to t r e a t  the  to t r y to  attempt to do so however must  recognize  t h a t i n p r a c t i c e the d e r i v a t i v e s of the performance index w i t h r e s p e c t switching  i n t e r v a l s are  to  the  unavailable.  F l e t c h e r h a s  surveyed a number of methods of maximizing a f u n c t i o n  without evaluating  derivatives.  scheme as  one  the b e s t  mini-  to use  His  results tentatively identify  for functions  of a l a r g e  '(>  Rosenbrock's  4) number of  var-  (3) iables.  Indeed, D a v i s o n and  to n o n l i n e a r  time v a r y i n g  of some l a r g e o r d e r  Monroe  systems and  have a p p l i e d Rosenbrock's p r o c e d u r e identified  the  time o p t i m a l  systems w i t h a good degree of s u c c e s s .  I t has  control not  shown however that such a method w i l l work f o r systems which c o n t a i n or time v a r y i n g  time d e l a y s .  The  good around the optimum.  t h a t convergence i s  However, the i n i t i a l  method i s slow when compared to those methods which use In t h i s t h e s i s t h i s has  constant  r e s u l t s p r e s e n t e d i n the next c h a p t e r show  t h a t t h e i r a l g o r i t h m w i l l work f o r such systems and especially  been  a l s o been shown to be  convergence of  gradient  the  information.^  the case f o r time d e l a y  systems.  1  8  III.  THE EXTENSION OF ROSENBROCK S METHOD TO FINDING 1  TIME OPTIMAL CONTROLS OF TIME DELAY SYSTEMS (2) I n t h i s c h a p t e r Rosenbrock's outlined. described 3.1  method o f maximizing a f u n c t i o n i s  The a p p l i c a t i o n o f t h i s method t o the time o p t i m a l and i s i l l u s t r a t e d by s e v e r a l  problem i s  examples.  D e s c r i p t i o n o f Rosenbrock's method The  Rosenbrock method i s a s e a r c h i n g  a l g o r i t h m which maximizes a  f u n c t i o n J o f s e v e r a l v a r i a b l e s I , , i = 1 t o n when the T.'s • l i  are s u b j e c t t o  constraints  s^VV'-V <W 1' 2'--- n> T  The  search  T  T  i- i ^ l ' V - V  T  h  " V j  ± n j  ' »---  < 1» 2""V  h  T  +  i s c a r r i e d out by t a k i n g s t e p s  that J i s extremized,all  i = 1  T  i n the T.'s  the w h i l e e n s u r i n g  2  c - ) 3  n  lrl,2,...  1  (3.2)  V  along v e c t o r s V_^, so  t h a t the c o n s t r a i n t e q u a t i o n s  are not v i o l a t e d . The  s i z e o f the s t e p s  m a t i c a l l y by the a l g o r i t h m  and  taken i n each d i r e c t i o n i s determined autothe s t e p d i r e c t i o n s are a d j u s t e d p e r i o d i c a l l y .  Whenever s u f f i c i e n t p r o g r e s s has  been made, the d i r e c t i o n v e c t o r s  r o t a t e d by the Schmidt p r o c e d u r e .  Rosenbrock and F l e t c h e r a r g u e  a f t e r s e v e r a l r o t a t i o n s the d i r e c t i o n o f search Because b o t h Rosenbrock's contain misprints here.  (2)  i s that o f steepest  and Davison and  k  i n the d e s c r i p t i o n o f the Schmidt procedure i t i s  k  e t c . , where  A  th i s the k c h o i c e of'V]_, then l e t  =  +  d  2 2 V  k +  • • ••  + d  that descent.  (3) Monroe's papers  Suppose t h a t d^ i s the a l g e b r a i c sum o f a l l s u c c e s s f u l s t e p s  direction  are  n „ V  k  rederived e^ i n  9  A. =  d V. 9  2  A  n  2  + . . . + d V n n  k  k  2  =  d V n n  K  Thus A^ i s the v e c t o r j o i n i n g  the i n i t i a l  and f i n a l p o i n t s o b t a i n e d by use  k k k o f v e c t o r s V.. , V„ ...V , A„ i s the v e c t o r sum o f a l l advances made i n i z n 2 d i r e c t i o n other ^  than f i r s t e t c .  ,  -,  .  x,  Orthogonal u n i t v e c t o r s ,  k+1  k+1  „  ,  „ ...V  ,  k+1  ,  . .  a r e now o b t a i n e d by  forming V  k  +  1  %  1  1^1  B  =  2  v  k  +  v  -  2  "  1  (A 2  =  n  k  +  1  V  ^  1  ^  1  | |  2  A - "ll] (A • V . n j=l n j  B  )  2  l | B  2  B  A  k + 1  )V. j  k + 1  n  Il nlI B  The  m a x i m i z a t i o n o f the f u n c t i o n , J , i s c o n s i d e r e d  when the change i n t h e v a r i a b l e s T^ i s l e s s  3.2  Time o p t i m a l It  to be complete  than some s p e c i f i e d  c o n t r o l by s w i t c h i n g i n t e r v a l  value.  adjustment  i s d e s i r e d to b r i n g the s t a t e v e c t o r whose dynamics a r e c o n s t r a i n e d  by x(t) to  = A ( x ( t ) , x ( w ( t ) ) , t ) + B ( x ( t ) , x ( w ( t ) ) , t) u ( t )  the o r i g i n , x ^ f ) = 0, r  from the i n i t i a l  under the assumptions made i n c h a p t e r appendix I t h e c o n t r o l which w i l l  2.  s t a t e , x ( t ) = x , i n minimum time o o As was shown i n chapter  do t h i s i s bang-bang.  2 and  Therefore,  the number  and  l e n g t h s of the s w i t c h i n g i n t e r v a l s are r e q u i r e d a l o n g w i t h  the c o n t r o l i n the f i r s t Since  the s i g n of  interval.  the s w i t c h i n g i n t e r v a l s are to be adjusted,and  since  they  are c o n s t r a i n e d by 0 < T.,and s i n c e the d e r i v a t i v e s of the performance -  function with  1'  r e s p e c t to the s w i t c h i n g i n t e r v a l s are u n o b t a i n a b l e ,  c l i m b i n g method of Rosenbrock seems i d e a l l y  s u i t e d to the  the  hill  task.  (3) Davison brock's  and Monroe  method which w i l l  . n o n - l i n e a r systems.  have r e p o r t e d an a l g o r i t h m based on Rosen-  find  the time o p t i m a l  Because i t would be u s e f u l to know i f the time  c o n t r o l s o f time delay systems can be results obtained algorithms  c o n t r o l of time v a r y i n g o r  i n this  found  i n t h i s way,  and because  any  f a s h i o n would form a b a s i s f o r comparison w i t h  d e r i v e d l a t e r i n chapters  IV and  V,  i s o n and Monroe a l g o r i t h m c o n s i s t s of two the c o n t r o l i n the f i r s t  the  t h e i r a l g o r i t h m , and some  examples o f i t s a p p l i c a t i o n to time delay systems are p r e s e n t e d  of  optimal  here.  main p a r t s : a) d e t e r m i n i n g  s w i t c h i n g i n t e r v a l and  i n t e r v a l s by maximizing a composite performance  The  Dav-  the s i g n  the c o r r e c t number of s w i t c h i n g  index  T -'-C b)  - x  determining  ( t ^ ) x ( t ^ ) where C i s a w e i g h t i n g  the s i z e s of the s w i t c h i n g i n t e r v a l s ,  number of i n t e r v a l s has been found squared  intervals equivalent forward  =  x  and  a f t e r the c o r r e c t  i n p a r t a, by m i n i m i z i n g  the  distance  assuming a number (NSW)  of  switching  T (t^) x ( t ^ ) .  The  a l g o r i t h m i s s t a r t e d by  and  the c o n t r o l (+6)  to c h o o s i n g  i n the f i r s t  a nominal c o n t r o l .  interval. The  evaluated  at the assumed f i n a l  by  the Rosenbrock h i l l  time and  climbing procedure.  These two  s t a t e equations  i n time u s i n g the assumed nominal c o n t r o l .  is  and  f a c t o r , o f t e n =1  The  choices  are  performance  the s w i t c h i n g i n t e r v a l s The  the s w i t c h i n g i n t e r v a l s are a d j u s t e d u n t i l  s t a t e equations  then  are solved  index are-adjusted  are s o l v e d  the performance index J  has  11  been maximized. Once = x  has been maximized the square o f the d i s t a n c e to the o r i g i n ,  ( t ^ ) x ( t ^ ) may  zero t h i s i m p l i e s  o r may  not have approached  i n the f i r s t  t h i s i s the assumed uniqueness  approached  too s m a l l o r  o f the o p t i m a l c o n t r o l ) .  In e i t h e r case the  and the m a x i m i z a t i o n o f  climbing procedure i s repeated.  If T_^ may  has not  i n t e r v a l was wrong. (The reason f o r  number o f s w i t c h i n g i n t e r v a l s i s i n c r e a s e d by one the h i l l  If  t h a t the number o f s w i t c h i n g i n s t a n t s was  t h a t the c h o i c e o f u = +3  by  zero.  the f u n c t i o n  o r may  has  approached  zero some s w i t c h i n g i n t e r v a l ( s )  not have tended to d i s a p p e a r (by t e n d i n g to zero) .  I f some  s w i t c h i n g i n t e r v a l s have tended to d i s a p p e a r , the uniqueness o f the o p t i m a l control implies  that t h e r e are too many s w i t c h i n g i n t e r v a l s  and the  correct  number o f i n t e r v a l s i s then o b t a i n e d by e l i m i n a t i n g the v a n i s h i n g ones, c o u n t i n g the remainder. then the f i r s t When t h i s  Note t h a t i f the i n i t i a l  time i n t e r v a l w i l l  3.3  was  climbing  wrong  proceeds.  v a l u e o f u ( t ) w i l l be  correct.  t h i s s t a g e the c o r r e c t number o f s w i t c h i n g i n t e r v a l s i s known,  as w e l l as t h e i r approximate intervals  tend to d i s a p p e a r as h i l l  time i n t e r v a l i s e l i m i n a t e d the • i n i t i a l At  c h o i c e of u = +6  and  values.  More a c c u r a t e v a l u e s f o r the s w i t c h i n g  are then found by maximizing  T the f u n c t i o n J„(x..) = - x . x^. 2 f f f  Computational a s p e c t s Because o f the u n a v o i d a b l e e r r o r s i n computation  that J  can ever be made e x a c t l y  zero.  above treatment i s t h a t a c o n t r o l which p r a c t i c a l purposes,  i t i s unlikely  Hence an i m p l i c i t  assumption  reduces J  t o " zero i s , f o r  "close  the o p t i m a l c o n t r o l and i s not too d i f f e r e n t  i n the  from the  t h e o r e t i c a l optimum. One  shortcoming o f t h i s method i s t h a t t h e r e i s no r u l e to h e l p  i n d e c i d i n g , a t the end o f the m a x i m i z a t i o n o f J , whether J  i s unacceptably  12  large or not.  One way o f s o l v i n g  t h i s problem i s to i n c r e a s e the assumed  number o f s w i t c h i n g i n t e r v a l s by one, and r e p e a t t h e m a x i m i z a t i o n o f one o r more o f the s w i t c h i n g i n t e r v a l s In  the maximizing o f  tends t o zero a t the maximum.  a c o r r e c t c h o i c e o f the w e i g h t i n g f a c t o r C,  can do much to speed up convergence. are  3.4  the o n l y g u i d e s to a c o r r e c t  However, e x p e r i e n c e o r t r i a l  and e r r o r  c h o i c e o f C.  •  Examples and r e s u l t s 1..  the  until  The f i r s t  system f o r which  the above a l g o r i t h m was implemented  was  one s t u d i e d t h e o r e t i c a l l y by W e l l s and Kashewagi i n [ 5 ] .  k (t)  = x (t)  X (t)  = - X ^ t - 0(t)) + U ( t )  ±  2  with i n i t i a l  2  data x (s) = 2.0 + 2s s < 0 x (s) 2  In  = 2.0  the W e l l s and Kashewagi system 0 ( t ) was a c o n s t a n t e q u a l to —6  0.3.  The time taken to d r i v e the s t a t e s to w i t h i n 10  seconds.  From the s t a t e t r a j e c t o r y s u p p l i e d by W e l l s and Kashewagi the ap-  p r o x i m a t e t h e o r e t i c a l minimum time i s 7.7 seconds. i n t e r v a l s was c o r r e c t l y i d e n t i f i e d as f o u r . find The  o f the o r i g i n i s 7.9  The number o f s w i t c h i n g  The a l g o r i t h m was a l s o used to  t h e time o p t i m a l c o n t r o l o f the above system when,0(t) was time v a r y i n g . results  f o r d i f f e r e n t d e l a y f u n c t i o n s are  0(t)  = 0.3 - t/100  t  f  = 6.636  d i s t = 2.5 x 1 0 ~  0(t)  = 0.3 e  t  f  = 5.577  d i s t = 1 x 10~  0(t)  =  t  f  = 6.2115  d i s t = 1.5 x 1 0 ~  _ t  0.2 + 0.1 s i n t  5  4  2  13  2.  A l a r g e o r d e r system The next problem c o n s i d e r e d was a m o d i f i e d form o f the seventh  o r d e r system d e s c r i b e d i n [3]  k (t) x  =x (t) 2  i ( t ) = x (t-eCt)) 2  3  x (t) = x (t-e(t)) 3  4  i (t)  =x (t)  x (t)  = x (t)  4  5  '  5  6  i . ( t ) =-x (t) . 6  x  ?  ?  ( t ) = -720 x ( t ) - 1 7 6 4 2  x (t)-1624 3  x ( t ) - 7 3 5 x (t)-17.5 x ( t ) 4  5  -21 with i n i t i a l  f u n c t i o n o f zero and i n i t i a l  T •x (0) = (0,-10,0,0,0,0,0).  g  x ( t ) + u(t) ?  c o n d i t i o n s at t = 0 o f o  The c o n t r o l u i s r e s t r i c t e d by |u|  < 1000.  Again i t i s d e s i r e d to d r i v e the s t a t e v e c t o r to the o r i g i n i n minimum time.  The r e s u l t s  i n the case o f two d i f f e r e n t  time d e l a y s a r e  g i v e n below. Delay  0(t)  number o f intervals  = 0.2  d i s t a n c e from o r i g i n a t f i n a l time  trials  11  1.7  . 79  11  1.4  92  minimum time 36.86  0.5 0(t) =  t+1  36.1  These r e s u l t s  a r e o f the same o r d e r o f magnitude as those r e p o r t e d i n [3] f o r  the non d e l a y  case.  F i n a l l y i t was d e c i d e d to t r y to determine i n m o d e l l i n g the delay would have on system response. c o n t r o l which drove  what e f f e c t  approximations  A c c o r d i n g l y the o p t i m a l  the s t a t e s o f the f o l l o w i n g system to w i t h i n 1.2 x 10  -4  14  of the o r i g i n i n 8 - 2 seconds was found. i  (t) = x ( t ) 2  = -x ( t - e ( t ) ) + u ( t )  x„(t) L The exact  time delay  1  •  0 ( t ) was chosen as 3ir  0 ( t ) = 0 . 2 + 0 . 1 sinC-^-t)  0 < t < 4 sec  8  0 ( t ) = 0 . 2 .+ 0 . 3 5 ( l - e ~  ( t _ 4 )  )  t > 4 sec  (see f i g . 3 . 1 ) The  same c o n t r o l was then a p p l i e d to the above system w i t h  the exact  delay  approximated by 0(t) = 0.36  0  0(t) - 0.25  2.6  < t < 4.6  0(t) = 0.4  4.6  < t <; 5 . 4  5.4  < t  0(t)  = 0.525  The  r e s u l t s were n o t encouraging  one  to s u s p e c t  response.  < t < 2.6  (see f i g .  3.1)  as the miss d i s t a n c e was 0 . 5 5 .  This  t h a t s m a l l changes i n time d e l a y w i l l g r e a t l y a f f e c t  T h e r e f o r e when m o d e l l i n g  leads  the system  time d e l a y systems i t i s important to  a) determine the s e n s i t i v i t y o f the system to v a r i a t i o n i n d e l a y and b) i f t h i s s e n s i t i v i t y i s h i g h , t h a t i s a s m a l l change i n d e l a y r e s u l t s i n a l a r g e change i n system response,  ensure t h a t the d e l a y i s modelled as e x a c t l y as p o s s i b l  As a n o n - l i n e a r example, D u f f i n g ' s e q u a t i o n w i t h d e l a y i n the cont r o l a c t i o n was chosen.  The s t a t e equations  f o r t h i s system a r e  x (t) = x (t) x  2  x (t) = -x^t) 2  with and  initial  -x  3 1  ( t ) -x (t) + u(t - 0.3) 2  |u| < 1  conditions  x^(0) = 1 x (0) 2  =  1  17  The  hill  climbing  the system to the  t e c h n i q u e i n 270 final  t r i a l s produced a c o n t r o l which drove  state  X ; L  ( t ) = 0.006  x ( t ) = -0.8  x  2  i n 4.75  3.5  Conclusions  switching  examples p r e s e n t e d here show t h a t the i d e a o f a s e a r c h  i n t e r v a l space f o r an o p t i m a l  time v a r y i n g  time d e l a y s  l o c a l minima and  has  and Kashewagi i n [5]  that increases  started. trial  time d e l a y s . The and  convergence r a t e . i n the  also v e r i f i e d  The  s p e c u l a t i o n by  i n d i r e c t l y , even i n the  Therefore,  even s m a l l e r r o r s i n m o d e l l i n g  must be  system o u t p u t s .  time  i n getting  taken  a l s o shown t h a t f o r at l e a s t one  the d e l a y  Wells  d e s c e n t d i r e c t i o n i s found  many i n e f f i c i e n t steps  I t was  any  increases  case o f  Rosenbrock scheme showed some d i f f i c u l t y  r a p i d convergence r e s u l t s .  between the model and  The  intercept  time delay would l e a d to  r e a s o n f o r t h i s i s t h a t the s t e e p e s t error.  in  c o n t r o l o f a c l a s s o f systems w i t h  r e s u l t s i n an a l g o r i t h m which does not  an a c c e p t a b l e  the minimum time was  varying  by  6  seconds.  The  in  10~  l e a d to q u i t e l a r g e  before system,  differences  18  IV. The scheme can be  previous  A GRADIENT METHOD  chapter  showed how  the Rosenbrock h i l l  adapted to f i n d a time o p t i m a l  f o r which t h i s o p t i m a l  climbing  c o n t r o l of a c l a s s of systems  c o n t r o l i s bang-bang.-  While the computing time  increases  (3) only l i n e a r l y with  system o r d e r ,  Schmidt p r o c e d u r e and of s w i t c h i n g  storage  S i n c e an i n c r e a s e i n system o r d e r n e a r l y always  b r i n g s about an i n c r e a s e i n s w i t c h i n g  as  systems the s t o r a g e  the system o r d e r  instants especially for non-linear^ "^ 2<  r e q u i r e m e n t s i n c r e a s e at l e a s t as  i s known from an a n a l y t i c a l v i e w p o i n t  about  vergence of the Rosenbrock procedure or about the e f f i c i e n c y of the method.  F l e t c h e r b e l i e v e s that with  gradient  i n f o r m a t i o n i s always f a s t e r than non  approach developed i n t h i s chapter stages.  However, i t was  In o r d e r urbations  sufficient  gradient procedures.  utilizing The  slower near the optimum.  to be  i n f u n c t i o n space the p e r t -  chosen to ensure t h a t the performance  minimized at each step and  admissible.  T h i s l a s t r e s t r i c t i o n not u s u a l l y encountered i n s t a n d a r d  i n f u n c t i o n space was  circumvented by  t h a t the r e s u l t i n g c o n t r o l s were  choosing  kept the c o n t r o l i n s i d e the c o n s t r a i n t s e t .  o n l y those  The  space.  descent  p e r t u r b a t i o n s which  d e r i v e d a l g o r i t h m makes  i n f u n c t i o n space p o s s i b l e f o r bang-bang c o n t r o l s and  c o n t r o l p e r t u r b a t i o n s a r e chosen c o r r e c t l y , interval  gradient  d i d i n f a c t converge more r a p i d l y i n the  to implement a g r a d i e n t s e a r c h  i n the c o n t r o l had  c a r e an approach  con-  searching  f u n c t i o n was  descent  fast  squared.  Further, nothing  initial  the  d i r e c t i o n v e c t o r s i n c r e a s e as the square of the number  instants.  or time d e l a y ^ " ^  r e q u i r e m e n t s because of  l e a d s , i f the  to a s t e e p e s t descent  i n switching  19  4.1  Development o f the p e r t u r b a t i o n As b e f o r e ,  i t i s desired  equation  to minimize a performance index J ( x , t ) f f  subject  to the c o n s t r a i n t s i(t)  with i n i t i a l  = f ( x ( t ) , x(w(t)),  data x(s) = g(s)  u ( t ) , t ) -/ t t <  t < t  Q  f  (4.1)  w(t ) < s < t o  o  ijj(x^, t^) = 0.  and  terminal  constraints  All  D e f i nused e yabove = x (have w ( t ) )been d e f i n e d the terms  i n c h a p t e r 2.  <5y = <$x(w(t)) = " 6x ( s ) s=t- (t) 0  if 8x If  2£ 9x  ( x ( - ) , t)  (x(t), x(w(t)),  u ( t ) , t)  the c o n t r o l u ( t ) i s changed by an amount 6 u ( t ) then the performance i n d e x  J will  change by an amount  8J  6J =  SxT  6  f  x  '  To r e l a t e 6x^ and 6 u ( t ) c o n s i d e r 8f 6x = — 9x  af  3f 6x + —  6y + — . Su  dy  (4.1)  6u  J  and  I n t e g r a t i n g both s i d e s from t  D  to t  "f P (t) 6x(t)  [(p (t) + T  T P  gives 8f ^)6x]dt  +  1  t  P  T  < * >  o  p  T  ( t )  ^  8  7  D  T  +  ^y  _ 3  5u  dt  (4.2)  u  C o n s i d e r now the second i n t e g r a l on the r i g h t hand s i d e o f the above e q u a t i o n . A ' I f w = t - O ( t ) then l e t t = r ( w ) . I f the i n t e g r a t i o n w i t h r e s p e c t to t i s changed to i n t e g r a t i o n w i t h r e s p e c t  to w the second term becomes  20  p (r(w)) —  (r(w)) 6x(w) —  (w) dw  Jt-e(t)' o  o  Because the i n i t i a l data i s fixed and therefore 6x(w) = 0 on the i n t e r v a l w(t ) < s < t , the above i n t e g r a l can be rewritten as o - o r t -0(t ) f  f  p (r(w)) | ^ (r(w)),sx(w) ^ T  dw  The dummy v a r i a b l e of i n t e g r a t i o n , w, can be replaced by another dummy variable t.  Equation 4.2 then becomes  tf-0(t ) _  „  f  P (t)6x(t)  [p (t) +  t  +  rp  f  t  9  f  [p T (t) + P T ( t ) — (t)] 6x (t) d x tf-e(t )  r  t  af  (t) + p V C t ) ) —  P  f  r  „  .  (r(t))]6x(t)  dt  dt  3f P^t) —  8u  (t) 6u(t) dt  In order to relate the change i n performance  (4.3)  function to the change i n control  d i r e c t l y , p(t) may be chosen a r b i t r a r i l y as 9J  p(tJ  "f  T = -p (t) — 3  p(t)  p(t)  f  =  9x,  T (t) - p ( r ( t ) ) — 8  T = -p (t) A  f  (r(t)) t  Q  £ t < w(t ) f  U  (t)  t -0(t ) < t < t f  f  f  (4.4) On closer examination i t can be seen that the equations 4.4 equations derived by Banks i n [10].  are the costate  21  If  the H a m i l t o n i a n  i s defined  as T  . and  equations  H(x,y,p,u,t) = p  4.4used to e l i m i n a t e the  rt  terms i n <5x(t),  equation  f  t  MacAuley  h  (4.5)  o  a s  given a s i m i l a r proof  f o r systems w i t h more than one  change i n performance i n d e x  6u i n the c o n t r o l i s g i v e n  by  AJ that r e s u l t s  where  6J = — — 3x  f  and J  i s the t o t a l d e r i v a t i v e of J w i t h  r M  +  3 t  I t has  from a p e r t u r b a t i o n  (23)  AJ = 6J _ + J St t  x  respect  f  f  to time  t, 'f  3t  been shown e a r l i e r t h a t the o p t i m a l For  6x f  9x  9 J  8  under d i s c u s s i o n i s bang-bang.  c o n t r o l f o r the c l a s s of systems  t h i s reason  the u s u s a l c h o i c e o f  cannot be used because the p e r t u r b a t i o n 6u must a l l o w f o r the t h a t the c o n t r o l i s bounded.  However, use may  about the form of the o p t i m a l  d i s c u s s i o n r e f e r to diagrams 4.1 been assumed bang-bang. the new  Since  through 4.5.  the o p t i m a l  the e q u a t i o n  the p e r t u r b a t i o n s  u = u + <5u.  The  The  and  -kHu  fact  In the f o l l o w i n g  nominal c o n t r o l u  c o n t r o l i s known to be The  p e r t u r b a t i o n 6u  diagrams 4.1  i n the c o n t r o l are e s s e n t i a l l y  the s w i t c h i n g i n t e r v a l l e n g t h s  6u =  be made o f the a. p r i o r i know-  c o n t r o l to choose 6u.  c o n t r o l u i s a l s o assumed bang-bang.  d e f i n e d by  times.  constant  delay. The  in  becomes  (6)  ,  ledge  4.2  H u ( x , y , p , u , t ) S u ( t ) dt .  p(t )6x(t ) =  time  (t) f(x,y,u,t)  through 4.5  has  bang-bang i s then show t h a t  changes, p o s i t i v e or  negative  t h a t <5u(t) e x i s t s o n l y around the -  switching  22  Lemma:  The e q u a t i o n  d e s c r i b i n g 6u when u  6u  (u T. l  , where:  is  +  e  max  °  mm  h  the T  < u < u - max  1 -u . ) • sgn [u(T. ) - u ( T . )] sgn [.2L .  .th „ y T, xs the l a) T. = , ., k l k=l N  . mxn  u i s r e s t r i c t e d by  being  l  -l  switching 6  the s w i t c h i n g  6  J=l J  ]  time, '  intervals  b) £j i s t h e amount o f change i n the s w i t c h i n g i n t e r v a l T c) u ( T ^ ) , u ( T ^ ) are the c o n t r o l u e v a l u a t e d +  to the l e f t  and r i g h t o f  the s w i t c h i n g i n s t a n t T_^ r e s p e c t i v e l y . Proof: 4.2.  Consider  first  the case i l l u s t r a t e d by diagrams 4.1 and  Here the s w i t c h i n g i n t e r v a l T^ i s to be lengthened  by the amount e^.  T h i s means t h a t s w i t c h i n g time T^ w i l l be changed by an amount e^.  The c o n t r o l  in  s w i t c h i n g i n t e r v a l T., i s u and i n s w i t c h i n g i n t e r v a l T„, u . . I n the 1 max ° 2 mm s u b i n t e r v a l [T.. , T.. +e. ] the d i f f e r e n c e between the new c o n t r o l , u , and 1 1 1 max the o l d one u . mm  i s (u -u . ) . max mm  Had the i n t e r v a l T^ been s h o r t e n e d subinterval new  max  the new  c o n t r o l i n the  [T,-e_ , T, 1 would have been u . and the d i f f e r e n c e between the 1 1 1 mm  and o l d c o n t r o l s (u . -u ) or -(u -u . ) . mm max max mm  considered, (u  by  t h a t i s s w i t c h i n g time T  -u . ) sgn e . mm 1  Thus f o r the case  changed by an amount  A s i m i l a r argument h o l d s  just  6u(T^) =  i n the case where the c o n t r o l i n  the f i r s t  i n t e r v a l i s u . . The change 6u(T..) i s then g i v e n by (u . -u ) mm r mm max ( e ^ ) . These two r e s u l t s may be combined i n t o one e q u a t i o n thus: b  sgn  6 u (  V  The diagrams 4.1, interval  will  =  <W min> U  J  « 8 n [ u ( - ) - u d ^ ) ] sgn  .  T l  4.2 and 4.3 a l s o show t h a t a change i n the l e n g t h of s w i t c h i n g affect  a l l subsequent s w i t c h i n g times T^, T  3  etc.  The  j u s t g i v e n f o r e v a l u a t i n g 6u(T^) when T^ has been s h i f t e d by an amount  argument can  23  I1  —  i  F i g . 4.1  Nominal C o n t r o l  T 1 —  i  - i Fig.  4.2  New  Control with  Switching  Interval  Perturbed  1  '—I  -l  -2 — ' Fig.  4.3  ''j-'The Change i n the C o n t r o l  23a  F i g . 4.4  New  C o n t r o l with a l l Switching I n t e r v a l s  Perturbed  e, —  '  F i g . 4.5  The  "  u  •  Change i n the C o n t r o l of F i g . 4.1  C o n t r o l of F i g .  4.4  to O b t a i n  the  24  be a p p l i e d i n these  A change o f  cases.  Therefore  i n t h e l e n g t h o f s w i t c h i n g i n t e r v a l T^ w i l l n o t a f f e c t  time.., T^, b u t i t w i l l  a f f e c t s w i t c h i n g times  T^,Ty  e t c . ( s e e f i g . 4.4 and 4.5). I f  th ese s w i t c h i n g . times have a l r e a d y been s h i f t e d by Since  they a r e now s h i f t e d bye^+e^.  need not be o f the same s i g n o r magnitude as  the t o t a l change,  + e^, must be c o n s i d e r e d .  switching  the s i g n o f  Otherwise t h i s c a s e , and the  cases when the o t h e r s w i t c h i n g i n t e r v a l s  are a d j u s t e d , a r e the same as when  o n l y one s w i t c h i n g i n t e r v a l i s changed.  Thus the change i n t h e c o n t r o l 6u,  at s w i t c h i n g time T_^ i s g i v e n by  <V  6u  In most cases '.  =  max- min  ( u  U  ) s  ^  [ u ( T  i  "  _ )  ^\ ^ +  [  jh  £  u > 0 and u . < 0 and the above e q u a t i o n max mm  j  <&>  ]  s i m p l i f i e s to  « .. i 6u(T.) = (u -u . ) s g n [u(T. )'] sgn [.Z •£.] . x max mxn I ° •j=l j  I f u . = -1, and u = mxn max  +1  6u(T.) = (2) sgn[u(T " ) ] s g n [ . 1| • e,] . i i 3 3 1  4.2  The p e n a l t y  f u n c t i o n approach  When t r y i n g to  to d r i v e the system 4.1 from the i n i t i a l  the f i n a l s t a t e x ( t ^ ) = x^ = 0, i t i s o f t e n convenient  state x ( t ) Q  n  J = -K.E x. i=l x For  this  choice of  2  ( t j where t = f f  performance index  T  f  ±  f  c  f  AJ becomes, u s i n g eq. 4.6,  AJ = - 2 ^ ^ x ( t ) 6 x ( t ) + [-2K x ( t ) i  t + f  f  f(x(-),  t ) +. 0] 6t f  x  Q  to t r y to maximize  (21) a penalty function  =  (4.7)  25  In order to make use of equation 4.5 to relate Su to 6x(t^) i t i s necessary to evaluate the i n t e g r a l Hu(x,y,u,t)6u(t) dt  Because of the s p e c i a l form of Su, a) t ^ i s not i n general the same as t ^ and b) the i n t e g r a l from t integrals  to t ^ can be broken up into the sum of several  around the switching instants. i T. + <=• j 1 1=1 Adt T. x  Let . *  Adt denote  (  e  J  and  .  i f .E, e. i s p o s i t i v e or zero . J=l 3  Adt  if  VI  e. i s negative . 1=1 . J  j=l Then-  C  +  r t  „  r  Hu(x,y,p,u,t)  Hu(x,y,p,u,t)[u^  iucix  rp  Hu(x,y,p,u,t) [ " m a x ^ m i j ^ s g n t " ^ ) ] s g n [ e ] ,  1  - u ] sgn[u(T nix IT  Z,  )] sgn[e  J.  + e ] dt + z.  - 2  (  •+  c: insw_l  Hu(x,y ,u t)[u j P  5  m a x  -u  m i n  ]  nsw-1 sgn [.^ - ] dt  sgn [ u C T ^ )  e  where nsw i s the number of switching i n t e r v a l s . Because t  f  i t  f  Sx(t ) = Sx(t ) + f  .  x (t ) 5 t + 0(6t ) 2  f  f  « Sx(t ) + x ( t ) 5 t f  f  To s a t i s f y the l i n e a r i z a t i o n requirements nsw  2 E  3  2 - A  f  f  f  l e t the e.'s be r e s t r i c t e d by  2 where A i s chosen small enough so that terms of order (e. ) are 1  n e g l i g i b l e with respect to terms of order (e.). To evaluate the integrals i y consider f i r s t the case when . • / > 0. Then by 'definition E  dt  r  becomes  f.+. , ' j E  . n I f A i s s m a l l and . £. 3=1 e  ^ can exceed  e  Hu(x,y,p,u,t) 2 sgn [u(TT)] (+1) d t  x E 3=1  2  j <  e  then  A  J  e. i s 3  probably  them's need  ^ i n magnitude and a l l  small,  f o r no  n o t be o f the same s i g n ,  t h e n t h e above i n t e g r a l can be a p p r o x i m a t e d , u s i n g t h e mean v a l u e theorem, by  , .+ ^ x T. x T  j 3= 1 E  e  n  Hu(x,y,p,u,t) 2 sgn [ u ( T  = Hu(x y  u,T )2  s g n [u(T. ) ] - [ T . +  .•* Hu(x,y,p,u,T.)2  1 sgn [u(T. ~) ] • [..^  )  i ^  I f on t h e o t h e r hand  C  becomes  T.  ) ] dt  i  E  ( P l  i  T.]  e  j ]  j i s negative the i n t e g r a l  - _ i , Hu(x,y,p,u,t) 2 sgn [ u ( T i ^, j J=l e  ±  ) ] [-1] dt  J  A g a i n i f lj-£-j_ J I I e  s m a l l enough the i n t e g r a l can be a p p r o x i m a t e d by  s  L^jl]  Hu(x,y,p,u,T.) 2 sgn [ u ( T . ~ ) ] [ T . - T . +  ~ Hu(x,y,p,u,T.) 2 sgn [u(T. ) ] [- | ^ I  1  since  I e. < 0 J=l J  -  •  1  e' | ]  1  j=l J  =  i.  e  3=1  J  Therefore, i n e i t h e r case  I  ^ (. T.  Hu(x,y,p,u,t) 2sgn[u(T "] sgn[ J -* = HuCx.y.p.u.T^.) 2 s g n [ u ( T . ) ]  D e f i n e now  e  j ] dt  (4.8)  [ j ^ ^ j ]  f o r convenience W(T ) ±  = Hu(x,y,p,u,T.) 2 s g n [ u ( T  i  )]  (4.9)  Then u s i n g  e q u a t i o n 4.6, 4.8 and 4.9  p (t )6x(t ) = H T T  f  and  f  1  )  h  + W(T )[ 2  + e)-  £ l  2  +  _ _ i f p ( t ^ ) i s chosen as -2K x ( t ^ ) = -r--— x  AJ = W(T ) e i x n  n s w  _  l ) [ e i +  . .  then  + W(T.)[e +e ]+. ..+W(T .) [e. + . . .-+e 1 /. x z nsw—x x nsw-1 t t  • + [^ ~ 3x, r Remark:  . . +W(T  Note t h a t  e  = ST^,  x ( t ) + | f - ] [e. + c . +. . .+£ ] f 9t,. 1 2 nsw f +  v  (4.10) '  - <$T e t c . 2  E q u a t i o n 4,10 can then be r e w r i t t e n as  AJ = W(T )6T. +.W(T_)6T_ + ... +W(T  J6T  n  1  S e t t i n g 6 t = -6T f  1  . = 0  2  2  nsv7-l  nsw-1  ^ i yields  AJ = W(T.) ST; . i l  D i v i d i n g both s i d e s by ST_^ and t a k i n g the l i m i t  W(T.)  3T: x and  = Hu(x,y,p,u,T.)-[u  •  one  - u . ] sgn [u(T.~) - u ( T . ) ] +  m a x  m  n  t h e term on the r i g h t hand s i d e i s the g r a d i e n t  index with respect - -  as 6T^ tends to zero y i e l d s  By  obtains  AJ  A  =  to s w i t c h i n g  o f the performance  i n s t a n t T^.  2 2 2 2 a d j o i n i n g the c o n s t r a i n t E , + e„ +...+ e = A to eq, ° 1 2nsw n  an augmented  +  function  W(T>. . +W(i :  n s w  _ ) g - i (i ) 1  +  f  i  +  f - } •+ f  ^  4.10, '  28  + 2 £  { W ( T  2  )+  - ' •- < nsw-1 W  lf  )  +  1x7 *  (  t  f  -3T:  )+  ^ 2  } +  £  +  '' '  2 £  n s w { - — x ( t j + T r — > + n nsw dX^ f 9t^' E  -n A  (4.11)  In the above n. i s a s c a l a r Lagrange m u l t i p l i e r .  I n o r d e r to choose the  which maximize the i n c r e a s e AJ i n the performance index s u b j e c t nsw 2 2 straint ^ A I wade o f the f o l l o w i n g e q u a t i o n s e  =  u  s  e  s  SAJ  „ [-57- x ( t , ) + ~ - ] + 2 n e = 0 9 x ^ f 9t.. nsw f f 3  A  9E  = 0 implies  that  nsw  9AJ 9e  to the con-  „  T  J. ,  _  (4.12)  ST  -= 0 implies that W(Tnsw-1) '+ [—3x^ i C O f nsw-1 f A  + -£7, = 0 S.t,.]'+ 2n e nsw-1 f  o r making use o f 4 . 1 2 W(T  In g e n e r a l ,  .) - 2q E + 2n £ = 0 nsw-1 nsw nsw-1 n  3AJ s e t t i n g t h e p a r t i a l —5 1  t-r^ ox^  (4.13)  A  = 0 yields  £ ( t ) + i i - ] + 2n f ot^  E  nsw  W(T.) - 2n E . _ + 2r) e = 0 . 1 i+± 1 The r e q u i r e d  ,  =0  i = nsw  i - 1 , 2 , to nsw-1  (4.14)  e's can be found from t h e f o l l o w i n g r e c u r s i v e r e l a t i o n s  E  _ 1 9J • /7 \ 1 9J •) = TT- [ X (t,) + ~T~ J 2n 9x.f 9E,. t f r  nsw  e. = 1  L  1 ~ - - r - W(T.) l + l 2r\ 1  1=1,2, to nsw-1  (4.15)  I n o r d e r to s o l v e f o r the Lagrange m u l t i p l i e r , n , observe t h a t 4.15' gives  the E ' S i n the form •  E . _ _ 1_ g 2n i  29  Thus  , „ 1 „ „ - = ~ T [Bi +...+B ] = A^ j-1 j A ^ 1 nsw nsw •-i  e  Then  / 1  ^  _  rt  =  0  "  2  • / _ _ _ _ _  *V  2 B l  +  ( -16) 4  ... B  2  +  1  nsw  S i n c e J i s always n e g a t i v e and i s t o be maximized the s i g n o f 0 must be chosen to make AJ  positive.  2 2 + B„e_ + 2n e. +...+B e + 2n e 1 2 2 2 nsw nsw nsw 2  A J . = B ^ , + 2r\ A 11  then by the Schwarz i n e q u a l i t y -sgn[B^].  4.3  t o maximize A J ^ the s i g n o f the  s h o u l d be  Thus a s h o u l d be chosen w i t h a p l u s s i g n i n 4.16.  Algorithm  description  A s h o r t d e s c r i p t i o n o f how the above a n a l y s i s can be a p p l i e d as a numerical algorithm f o l l o w s . its  use.  S e v e r a l examples are then g i v e n t o • i l l u s t r a t e  The a l g o r i t h m c o n s i s t s o f 9 steps'1.  Assume an i n i t i a l p o i n t i n the s w i t c h i n g i n t e r v a l  2.  S o l v e the s t a t e e q u a t i o n s  3.  A t the f i n a l  4.  space.  forward i n time.  time compute J ,  9J 3x  £  S o l v e the c o s t a t e e q u a t i o n s backwards i n t i m e w i t h  the  correct  terminal conditions. 5.  W h i l e doing s t e p 4 c a l c u l a t e and s t o r e W(T_^) a t v a r i o u s p o i n t s a l o n g the t r a j e c t o r y .  6.  S o l v e the r e c u r s i v e r e l a t i o n s 4.15 f o r the e's and e q u a t i o n 4.16 f o r the Lagrange m u l t i p l i e r .  7.  Update the s w i t c h i n g i n t e r v a l s .  8.  Check to s e e i f the s t o p p i n g c o n d i t i o n i s s a t i s f i e d .  30  9.  I f i t i s not s a t i s f i e d  r e p e a t s t e p s 2 to 8.  p r i n t o u t the r e s u l t s and s t o p . '  4.4  Comments on Computational  If i ti s satisfied  .  aspects  Two s u i t a b l e s t o p p i n g c o n d i t i o n s i n s t e p 8 a r e , e i t h e r t h e p e n a l t y f u n c t i o n i s c l o s e to zero o r the s t e p s i z e A  2  has become s m a l l .  While  doing  s t e p 3 t h e newest v a l u e o f the performance index i s compared t o the b e s t one 2 obtained  to d a t e .  I f i t i s g r e a t e r than  i s m u l t i p l i e d by two; i f i t i s l e s s termed a f a i l u r e , date and A  2  the p r e v i o u s b e s t , the s t e p s i z e A  than the o l d e r maximum, the t r i a l i s  the newest T.'s d i s c a r d e d i n f a v o u r o f the b e s t ones t o l  i s d i v i d e d by 10.  The adjustments o f A  2  ensure t h a t the s t e p  s i z e does n o t s t a y s m a l l when l a r g e s t e p s c o u l d be taken and thus speeds up convergence. I n c o n t r a s t , to the u s u a l descent  i n f u n c t i o n space,  for linear  systems i n f o r m a t i o n need o n l y be s t o r e d a t the s w i t c h i n g times, n o t a l l a l o n g the t r a j e c t o r y . costate equations  F o r n o n - l i n e a r systems o n l y the s t a t e s r e q u i r e d t o s o l v e the need be s t o r e d .  o n l y l i n e a r l y w i t h system o r d e r . determined  by the time i t takes An  area o f d i f f i c u l t y  A g e n e r a l i z e d method o f f i n d i n g extreme cases fitting 4 .5  Otherwise s t o r a g e space requirements  increase  The computing time p e r i t e r a t i o n i s l a r g e l y to i n t e g r a t e the s t a t e and c o s t a t e  equations.  i s the f u n c t i o n r ( t ) i n the c o s t a t e e q u a t i o n s . r ( t ) i n c l o s e d form does n o t e x i s t .  i t may be n e c e s s a r y  to f i n d  In  an e x p r e s s i o n f o r r ( t ) by curve  o r to s t o r e i t as a s e r i e s o f p o i n t s .  Examples and R e s u l t s  Example I The  n u m e r i c a l procedure  system d e s c r i b e d i n c h a p t e r I I I .  o u t l i n e d above was f i r s t That i s  a p p l i e d to the  31  x (t)  = x (t)  x (t)  = -x (t-O(t)). + u(t)  x  2  2  X ] L  ( s ) = 2.0 + 2.0si  x (s)  with i n i t i a l data  s < 0  =2.0  2  The results f o r several d i f f e r e n t time delays are summarized i n the table below and i n f i g .  4.7.  Delay 0(t)  distance to o r i g i n at f i n a l time  =0.3  1.7 x l O " 2.9 x 1 0  5  - 5  0(t)  = 0.3- /100 ' .  2.1 x 10~ .1.7 x IO" '  t  3  3  i r i  _  a l s  3 0 0  minimum time 7  '  8 2  275 425 400  6.684  These figures are i n close agreement with the results of Rosenbrock's method i n chapter III. They also i l l u s t r a t e the drawback of any  technique  based on small scale l i n e a r i z a t i o n namely that convergence may be l i m i t e d by l i n e a r i t y requirements.  From the very f i r s t , however, the switching i n t e r v a l s  are adjusted i n an optimum manner and i n i t i a l convergence i s better than that of the method of chapter I I I .  For example the gradient method reduces  the square of the miss distance to 0.05  from 1.7  i n only 10 t r i a l s while the  Rosenbrock scheme required 18.  A large order system Next the same 7th order system considered i n chapter III was In  this case the state equations are given by i (t)  = x (t)  ±  2  • x ( t ) = x ( t -0(t)) 2  .  3  x (t) = x (t-0(t)) 3  4  x (t)  = x (t)  4  x  5  5  (t) = x ( t ) 6  .  used.  TRIALS  F i g . 4.6  Convergence of Gradient Method for a Second Order System with a delay of 0.3  33  x ( t ) = x (t) 6  x ( t ) = -720  x ( t ) -1764  ?  The  initial  2 different  The  time d e l a y s  the i n i t i a l  condition vector x |u| < 1000,  The  o  was  results  are g i v e n below. D i s t a n c e to o r i g i n at f i n a l time  Trials  Minimum time  0(t) =  0.2  1.4  28  36.1  6(t) =  "j^f • '  1.76.  35  35.4  an example o f the e f f o r t  consider  (t)+ u(t)  6  c o n t r o l u i s r e s t r i c t e d by  Delay  As  x (t)-175 x ( t ) - 2 1 x  4  f u n c t i o n s were a l l zero and  T (0,-10,0,0,0,0,0) . for  x (t)-1624 x ( t ) - 7 3 5  2  r e q u i r e d to o b t a i n the i n v e r s e f u n c t i o n r ( t )  the case when ©(t)  =  a "TTT  a>0  , \ . ' w(t) i s t h e r e f o r e t -  a  r  b > 0  t + b  t + b  S i n c e w ( r ( t ) ) = t then a r(t)  = t  or  r(t)+b [r(t)]  and  Since  r(t)  r ( t ) > t, t  o  =  -  2  + r ( t ) [b-t] -  [ ^  < t < t . f  ]  [a+bt] = 0  ±.\J  +  Then i f t  o  [a+bt]  (4.19)  = 0  1 r  S i n c e b o t h a,b If e q u a t i o n has  (0)  =  ~^ 2  + "  ./ V  > 0 the p l u s s i g n s h o u l d be  2 k_  +  a  must be  >  0.  4  chosen i n  4.19.  0 ( t ) = ae . t h e n r ( t ) i s a s o l u t i o n to r ( t ) - a e ~ ^ ^ t  no  c l o s e d form s o l u t i o n f o r r ( t ) .  r  = t.  This  2000  •| 0  I -5  I 10  1 "15  1 20'  I 25  .  I 30  TRIALS  . F i g . 4.7 Convergence of Gradient Method f o r a Seventh Order System with . a Delay .of 0.2 "' ' •  35  4.6  Conclusions A comparison between the r e s u l t s o f t h i s c h a p t e r and those of  c h a p t e r I I I f p r the l a r g e o r d e r system shows the marked s u p e r i o r i t y o f the g r a d i e n t method i n the i n i t i a l constant  stages.  s t a r t e d from the same p o i n t .  ly  the h i l l  F o r the  to reduce the  c l i m b i n g scheme took  70 when  C l o s e to the optimum the Rosenbrock scheme was  s u p e r i o r to the g r a d i e n t method f o r l a r g e o r d e r systems and marked-  s u p e r i o r f o r the low o r d e r c a s e s .  F o r the example I w i t h c o n s t a n t  —5 to  3.2 and 4.7).  d e l a y system the g r a d i e n t method took o n l y 29 t r i a l s  miss d i s t a n c e from 1760 to 1.4 w h i l e  slightly  (Compare f i g s .  reduce the miss d i s t a n c e from 10  only 9 t r i a l s while from 3 x 1 0 ~  5  delay  -8 to 10  the Rosenbrock method r e q u i r e d  the g r a d i e n t approach took 25 to reduce the miss d i s t a n c e  to 1.7 x 1 0 ~ . 5  36  V.  DESCENT TO FINAL TARGET SET (19)  B r y s o n and Denham  have d e r i v e d a g r a d i e n t method o f descent  i n f u n c t i o n space i n which the steps instead of minimizing  a penalty  the descent i n s w i t c h i n g  a r e taken t o reach  function.  interval  In t h i s  the f i n a l  chapter  target set  i t i s shown t h a t  space can a l s o be made so t h a t  terminal  c o n d i t i o n s a r e accounted f o r d i r e c t l y . Once a g a i n  the s t a t e v e c t o r x ( t ) must be brought to the o r i g i n ,  x ( t ^ ) = 0 , from the i n i t i a l p o i n t x ( t ) index J ( t ^ ) t f dt = t ^ - t . S i n c e  =  Q  to s t i l l h o l d the o p t i m a l  control w i l l  must be d r i v e n t o the o r i g i n  q  a g a i n be bang-bang.  = 0  f  '  ^  W  5.1  Perturbation If  When the s t a t e v e c t o r  a s u i t a b l e choice of terminal conditions i s iKx )  where  X w h i l e m i n i m i z i n g the performance the assumptions o f chapter I I , s e c . 2.1  =  x  i  ( t f  ) =  0  analysis  the nominal c o n t r o l u i s changed by &U so t h a t u = u + 6u  h^f  +  6  t  then f  x(t ) = x(t ) + f  f  6x(t ) f  x ( t ) = x(t ) + 6x(t ) + i ( t ) 6 t f  f  J(x,t  f  f  ) = J(t ) +  6x  + j 6t  8x i j i ( t ) = 6<Kt ) + i ( t ) 6 t f  f  f  f  f  37  In general 6u(t) must be chosen so that a) J i s decreased and b) the terminal functions sec.  ^(x^,  t^) change by some s p e c i f i e d amount 3^.. As before (chapter IV,  4.1) the change 6x(t^) = x(t^) - x(t^)  i s related to the change i n con-  t r o l by the e q u a t i o n 4.5  " p(t )  4 Hu(xjy,p,u,t)  6x(t ) =  f  f  6u(t)  dt  (4.5)  t Adopting  the a d m i s s i b l e <$u o f c h a p t e r IV, s e c . 4.1, e q u a t i o n 4.5 becomes  p(t )6x(t ) = Hu(x,y,p,u,T )-(u f  +  f  1  M A X  -U  )-sgn[u(T ~)-u(T )] +  m i n  1  1  ...+Hu(x,y,p,u,T _ ) ( u -u . )-sgn[u(T " . ) - u ( T ,)]•[£,+£,+..•+£ J nsw-1 max mxn nsw-1 nsw-1 1 2 nsw-1 +  (5.1) where a l l t h e terms used above have been d e f i n e d i n c h a p t e r IV. determine  <$x.. ( t ) p ( t ) i s chosen f  }  f  I n order to  so t h a t p (t ) = 0 ±  f  V i 4 j  P.(t ) = l f  L e t ^ p ( t ) denote the s o l u t i o n t o 4.4 w i t h the above i n i t i a l  6x.(tJ l f  nsw-1 = . Hu(x,y, p k=l E  i  1  r  conditions,  „ „ _ „ , k ,u,T ) (u -u . ) sgn [u(T ) - u(T, ) ] E e. ' max mxn k., k j-1 j 6  k  + x . ( t j [e +...+e ] x f 1 nsw Define  then  . Hu(x,y, ^ u . T ^ ^ - u ^ )  - _ sgn [ u ^ " )  - (T  +  U  K  )]  A =  C.  K  Then 6x, = C.,e. + C . ( e + e ):+...+C. ( e . +...+e ) i x l1 x2 1 2 xnsw-1 1 nsw-1 0  1  0  n  + x,(t.)[£.+...+£ ] x f 1 nsw  Rearranging  terms 'fix.(t-) = e,[C.. + C.„ +...+C. . + i.(t,)] i f 1 i l i2 msw-1 1 f 2  i2  i3  insw-1  i  f  + e [i.(t.)] nsw l f  Let  G. msw  and  G..  = =  x.(t ) i f  i = 1 to n  £  G....+C..  i = 1 to n j = 1, nsw-1  Then 6x. ( O i f  = G.. L + G . „ e + . . . + G. e. i l1 i22 msw insw  or i n matrix  notation 6  In o r d e r  (5.2)  0  to s a t i s f y  X  =  G-e  (5.3)  t h e t e r m i n a l c o n s t r a i n t s a c e r t a i n change  f o r each fix.. E q u a t i o n i  i s specified  5.3 can then be w r i t t e n G-e = B  The change i n performance index  " (5.4)  6 J i s g i v e n by  nsw 6J =fit.= .E. e. f 1=1 l where I i s the u n i t column v e c t o r c  = e l c  (5.5)  T [ 1 1 1 1 ...1] .  The e's a r e r e s t r i c t e d by T e i n order  f o r equation  4.1 to h o l d .  to 5.4 and 5.6 c o n s i d e r  2  We =  To f i n d  t h e augmented  A  (5.6)  the e's which maximize fiJ s u b j e c t  change  i n performance i n d e x  fiJ = e I + v [ G e -3] + n(e We - A ) A c T  T = e I  T  c  T T T + e G v - v g + n  where n i s a s c a l a r and v i s an n x 1 v e c t o r .  T  2  T 2 (e We -A )  39  •36J  S e t t i n g the d i f f e r e n t i a l  equal to zero y i e l d s  I  -1 Solving  f o r e gives  e. = —  By s u b s t i t u t i n g 5.7  + G v + 2nWe = 0 T  c  T [ I + G v]  i n t o 5.4  one  obtains  -  3 = Isolating  (5.7)  c  -1  1  — GW 2n •  T + G v]. „  [I C  terms i n v on the l e f t hand s i d e o f the e q u a t i o n  ^(GW-V)^-  [ 3 + ^  gives  GW" ^] 1  -1 T Define  Z = GW  G , the Z has dimension n x n  and  therefore  v = - Z [ 2 n 3 + GW I - 1  S o l v i n g e q u a t i o n 5.7  f o r e using w !! - 1  e  =  2n —1  e = ~  2n  —1  + G [ - Z ( 2 r | 3 + GW I _ 1  R_ = {I-W G  Then  Rg = W  e  -1 2n  = —  Next eq. 5.9  (5.8)  T  -1  + W  c  T -1 G Z G}W  —1  -1  )] } c  T —1 G Z 3 •  -  I• c  G Z  Note t h a t i f G i s square then W G Z hence  .]  -1 T -1  +  "G 1  _1  T —1 ) —1 G Z G W I  For b r e v i t y l e t  and  c  5.8 y i e l d s T  c {I-W  - 1  Rg3  "G = W G [ ( G )  3  1  T  T  V i s s u b s t i t u t e d back i n t o 5.6  1  WG  1  ]G  =  I  and  -  (5  to s o l v e  9)  T 2 f o r n.. e WE = A becomes  40  A  ' = 4^  Let  W R  G " ^  * G  V  - r ^ — = a and observe 2n .  V  3  V  W  B  +  V  W  V  ( 5  '  1 0 )  that  R  WRgg =  T G  T  R  = b  WR  Q  WR  .G  = a  a scalar  a scalar  G  RgB E q u a t i o n 5.10  G  R  -A  = .  2  c  a scalar  then becomes aa  Therefore  =  a  2  D  -2ba + c = 0  b ± V—b" I  V  A  D  -  (5.11)  a c  a In o r d e r to minimize <5J. w i t h r e s p e c t to e the second A  derivative  T  ^ " A  ,—rv  =  2n W  s h o u l d be >  0  (3e> T h e r e f o r e i n e q u a t i o n 5.11  Note a l s o t h a t i n o r d e r f o r the step s i z e c o n s t r a i n t a to be  f o r maxima. r e a l , ac < b  the p l u s s i g n i s chosen f o r minima, the minus s i g n  2  .  I f the 3's  a r e chosen s m a l l enough i n r e l a t i o n to the A  C  =  P  V B R  i s l e s s than zero and a i s always r e a l . m a t r i x which may  be  adjusted,.by  trial  2  then  S  The W i s a p o s i t i v e d e f i n i t e and e r r o r i f n e c e s s a r y ,  weighting  to improve  the convergence r a t e of the a l g o r i t h m . 5.2  Algorithm  .  description  The  computing a l g o r i t h m may  be d e s c r i b e d by  the 9 f o l l o w i n g s t e p s :  1.  Assume a c o n t r o l and s o l v e the s t a t e e q u a t i o n s  2.  Solve  the c o s t a t e equations backwards n times  necessary  forward. storing  i n f o r m a t i o n at the s w i t c h i n g p o i n t s .  the  41  3.  2 Choose values for g. and A .  4.  Form the G matrix.  l  -1 T 5.  Find the inverse of Z = GW  6.  Form the matrices R^,  7.  Calculate the step s i z e constraint o>  8.  Calculate the changes i n the switching  9.  Check to see i f the stopping condition i s s a t i s f i e d .  R^.  not, repeat steps one  Comments  G .  intervals. If i t i s  to eight.  The 3's are usually chosen to be some percentage of the f i n a l states. 2 • The step s i z e constraint (A ) i s i n i t i a l l y chosen a r b i t r a r i l y and then adjusted during computation i n a manner s i m i l a r to that outlined i n chapter IV.  The  2 procedure may  be stopped therefore when A  becomes so small that errors i n  the simulation become the l i m i t i n g factors. 5.3  A detailed example Consider  the system (t) = x (t)  X ; L  2  x ( ) = -x (t) 2  t  x (0)  = -0  x (0)  = -2  x  2  with u(t)  =  -1  0 < t < T  ' +1  T< 1  -1 The s o l u t i o n to 5.12  T  f  f  x ( t ) = -2 cos t 2  for  t  > T  2  (5.12)  1  t < T < t < T  2  3  i s given by  x ( t ) = -2sin t -2 x  2  + u(t)  ±  f  f  cos(t -T )+ f  2 cos(t -T )-cos(0)+  1  f  2  + 2 sin(t -T )-2 sin(t -T )-sin t f  1  f  2  cos  t  f  ( - ) 5  13  1^ become T  Let the times 1^,  + ST^  ±  T  2  + 6T , T 2  changes i n the switching i n t e r v a l s become e^= 6T^, 2 G  =  + ST.^. Then the  3  <  ^^2 ^'''l' 3 _  e  ^2~^2  =  Then expanding x ^ ( t ^ ) , x ^ t ^ ) by Taylor's series one obtains 6  X]L  ( T ) = -2 c o s ( T ) 6 T 3  3  -2 s i n [ T  6x (T ) = 2 sin(T )6T 2  3  3  3  + 2 sin^-T^  [ 6 ^ - 6 ^ ] - s i n ( T ) 6T 3  3  - T ] [6T - 6T ] + 0(6 )  3  2  3  2  + 2 cos(Tg-T^ [ 6 ^ - 6 ^ ] - cos T  3  6T  3  3  -2costT -T ][6T -6T ] + 0(6 ) 2  3  2 where 0(6 ) represents  2  3  2  2 a l l the terms of order 6 or higher. T  For convenience l e t  =  3  3TT/2  TT/2  x ^ ) x  l (  x ( T ) = -1  - 3  2  +  T ) = -1  3  x ( T ) = -4  3  2  3  The exact changes i n the states at the f i n a l time are, therefore, 6  X;L  ( T ) = -6T + 26T + 0(6 ) 3  3  2  6 x ( T ) = -A6T + 25T 2  The costate equations  3  3  + 0(6 )  (5.14)  are given by pV^t) = p ( t ) 2  p (t) = - p ^ t ) 2  so that  P (t) x  =  2  fix (T ) 1  3  put  sih(t-T )  -sin(t-T )  cos(t-T )  3  P (t)  to f i n d  cos(t-T )  _J  P (T ) = 1 X  3  P (T ) - 0 2  3  (T )  P l  3  _P (T )_ 2  3  43  HJx/p.u,^) = P (T ) 1  2  Vv  = - H r -'^h. - ° sin  V (T: )  = -sin(Tr- ^ r ) = +1  X  2  C  C  ll  HuCx.Vu,^) ( u  -  1 2  =  2  m a x  K  - u . ) sgn [ u O f ) - u d / ) ] m  =  n  +2  0  G  13 = -  G  1  - +1  1 2  '  G -+1 u  One can f i n d s i m i l a r i l y  that  G^ G  22 = "  G Evaluating fix^ and 6x  = -4  2 1  4  =-2  by means of equation  2  5.3  6x. = G.-.e. + G.„e + G..e x i i1 i2 2 i3 3 0  (5.3)  0  Yields 6  l  x  6x  =  l 2  e  = -2  2  ~ 3  + £  e;L  e  -4e  - 4e  2  (5.15)  3  The change i n switching i n t e r v a l s i n terms of the switching times are £  1  =  E  2  =  £  3  6 T  6 T  6  1  2  T  "  l"  e  6 T  2  "  3  " 1 "2 =  1  "  £  £  6 T  6 T  1  3  "  6 T  2  Therefore equations 5.15 become  6  x  l  =  &  \  +  5 T  2  "  6 T  6 T  3  +  6 T  2  =  2 6 T  2 '  6 T  3  „  1 A  , (5.16)  6x  2  = 26  - 46T  44  If terms of order (6T)  may be neglected (this i s true i f 6T's are s u f f i c i e t l y  small) equations 5.16 are the same as equations For s i m p l i c i t y  Then Z = GG  -1 and Z  let W = I 1  1  -1  -2  -4  -4  =  36 104  104  -2 .104  3 104  becomes  JL 6  26  1  26  1  T  1  1  I ] = 1 104  -2  3  1  -4  -2  -1  -4  52 52  -48  ~4  c  I  4 I F ^2  -11 26 [I-G Z G][W  5.14.  16 ~  ~1  -48  36  -12  1  16  -12  4  1  4/13 -3/13 1/13  Then  6 T  1  =  6 T  2  =  6T  3  i e  i  IT °  =  +  e  2=  = _ , £]  +  2  +  +  i l e  3  =  8 / 2 6 B  0  +  l"  i  1 / 2 6  3  2  ei-'ir |^  3  X  6 2  Z|  B  2  -2 36  45  Since 6x  = -6T  +  3  26T  2  2 Then men  6x cx  = — 1  ±  and  6x  2  3  a — 2  = -46T  8_ 13  The a l g o r i t h m w i l l ,  &  3-, - + I- —  ^  +  3  a  2  P32 Z  6  - - —a 6 + — + 3i 26 — ^ $2 ~ $1 2  1  +  3  2  26^  -16 g 26  1  +28 26  g  +  13  2  16 3 26  -± 3 = 3 26  z  i n t h e o r y , g i v e the r e q u i r e d change i n t e r m i n a l  states  2 p r o v i d e d the terms  0(6 ) a r e s m a l l .  The a l g o r i t h m when t r i e d w i t h the  above system reduced the magnitudes o f the s t a t e s a t the f i n a l iteration until  the miss d i s t a n c e a t the f i n a l  time was reduced below 0.2. 2  F u r t h e r improvement was n o t p o s s i b l e f o r around so s m a l l f o r e q u a t i o n 4.1 to h o l d l e s s importance  time on each  the optimum A  must be made  t h a t the changes i n t h e i n t e r v a l s were o f  than the e r r o r s i n the n u m e r i c a l i n t e g r a t i o n scheme. (See f i g .  5.1 f o r b e h a v i o u r of % | r e l a t i v e e r r o r | vs f i n a l  state).  The same s o r t o f b e h a v i o u r was observed when c o n t r o l o f the system d e s c r i b e d i n r e f . 5 and o f a Vander P o l e q u a t i o n were t r i e d . drawback the a l g o r i t h m i s n o t s u i t a b l e f o r s e a r c h i n g around 5.4  Because o f t h i s the optimum.  Conclusions A new n u m e r i c a l method f o r f i n d i n g  time o p t i m a l c o n t r o l o f time  d e l a y systems, based on Bryson and Denham's approach was developed. showed poor f i n a l , b u t good i n i t i a l  convergence.  The method  T h e r e f o r e , i f i t c o u l d be  i n c o r p o r a t e d i n a two s t a g e a l g o r i t h m and used to get c l o s e to the optimum , the method o f c h a p t e r I I I c o u l d  then be used to o b t a i n  the f i n a l  results.  47  VI.  CONCLUSIONS  Three d i f f e r e n t methods o f d e t e r m i n i n g bang-bang o p t i m a l c o n t r o l s have been s t u d i e d .  A l l the methods are based on the assumption t h a t a bang-  bang o p t i m a l c o n t r o l can be d e s c r i b e d by intervals. meter  the number and s i z e s o f s w i t c h i n g  V a r i o u s types o f s e a r c h e s can then be c a r r i e d out i n t h i s p a r a -  space. The Rosenbrock method o f maximizing a f u n c t i o n can be used to s e a r c h  the  parameter space w i t h o u t e v a l u a t i n g g r a d i e n t s .  T h i s p r o c e d u r e shows good  convergence around the optimum, and i s e a s i l y used w i t h a wide v a r i e t y o f time v a r y i n g time d e l a y s .  I t s main drawback i s slow i n i t i a l  convergence.  A g r a d i e n t method which minimizes a p e n a l t y f u n c t i o n can a l s o be used.  T h i s method cannot be e a s i l y programmed w i t h some time v a r y i n g  d e l a y s because o f the need f o r the i n v e r s e r(t which may  Convergence scheme.  i s v e r y good and the s a v i n g s i n computer order  function  -Q(t)) = t  have no c l o s e d form s o l u t i o n .  poor by comparison w i t h the Rosenbrock  around the optimum i s  However, i n i t i a l  convergence  time a r e s u b s t a n t i a l f o r l a r g e  systems. A g r a d i e n t approach which s a t i s f i e s  showed good i n i t i a l the  time  convergence.  the t a r g e t s e t d i r e c t l y  However, f i n a l  programming so much more complex  convergence was  also  so poor and  t h a t there would be no advantage i n  using i t . In  a l l cases t e s t e d no problems w i t h m u l t i p l e minima were encountered.  S i n c e the performance i n d e x may  thus be assumed unimodal i n the. parameter  (25) space ' o t h e r . s e a r c h i n g a l g o r i t h m s such as a F i b b o n a c c i mini-max s e a r c h c o u l d a l s o be  tried.  48  BIBLIOGRAPHY  1.  M. 0. O g u z t o r e l i ,  "Time l a g c o n t r o l systems," Academic P r e s s , New  York  1966. 2.  H. H. Rosenbrock,  "An automatic method f o r f i n d i n g  the g r e a t e s t o r l e a s t .  v a l u e of a f u n c t i o n " , Computer J o u r n a l , V o l . 5, No. 3, 1963 » pp. 175-184. 3.  E. J . D a v i s o n and D. M. Monroe, "Computational  technique f o r f i n d i n g  time  o p t i m a l c o n t r o l s of n o n - l i n e a r time v a r y i n g systems"', JACC, 1969, pp. 270280. 4.  S. H i r a t s u k a and A. Ichikawa,  "Optimal c o n t r o l of systems w i t h  transport-  a t i o n l a g " , IEEE T r a n s a c t i o n s on Automatic C o n t r o l , V o l . AC-14, No. 3, June 1969. PP5.  237-247.  W. R. W e l l s and Y. K a s h i w a g i , " S y n t h e s i s of a time o p t i m a l c o n t r o l w i t h d e i a y " , IEEE T r a n s a c t i o n s on Automatic  problem  C o n t r o l , V o l . AC-14, No. 1,-  F e b r u a r y 1969., PP- 99-100. 6.  E. B. L e e , "Optimal c o n t r o l of systems w i t h time d e l a y s " , JACC, 1969, pp.  7.  451-452.  B. C. Ragg, "Necessary c o n d i t i o n s f o r the o p t i m a l c o n t r o l of a system w i t h time v a r y i n g  t r a n s p o r t l a g s " , IEEE T r a n s a c t i o n s on Automatic  Control,  V o l . AC-11, October 1966 , pp. 738-740. 8.  R. J . McAulay, "A g r a d i e n t method f o r systems w i t h time d e l a y s and i t s a p p l i c a t i o n to waveform d e s i g n " , IEEE T r a n s a c t i o n s on Automatic V o l . AC-14, No. 3, June 1969, PP- 231-237.  9.  .  Andrew B u c k a l o , " S t a t e e q u a t i o n s f o r d e l a y systems", A u t o m a t i c C o n t r o l ' , V o l . AC-14, No. 4, August  10.  Control,  IEEE T r a n s a c t i o n s on  1969, PP-  H. T. Banks, "Necessary c o n d i t i o n s f o r c o n t r o l problems  419-420. with v a r i a b l e  time l a g s " , SIAM J o u r n a l of C o n t r o l , V o l . 6, No. 1, 1968, pp. 9-47.  49  11.  A. T. F u l l e r ,  "Optimal  n o n - l i n e a r c o n t r o l o f systems w i t h pure d e l a y " .  I n t e r n a t i o n a l J o u r n a l of C o n t r o l , V o l . 8, No. 2, 1968, pp. 145-168. 12.  A. Halanay, "Optimal  n o n - l i n e a r c o n t r o l o f systems w i t h pure d e l a y " ,  SLAM J o u r n a l o f C o n t r o l , V o l . 5, No. 2, 1968, pp. 215, 234. 13.  J . W. S u t h e r l a n d , " S y n t h e s i s o f o p t i m a l c o n t r o l l e r s f o r a c l a s s of aerodynamical  systems, and the n u m e r i c a l  s o l u t i o n of n o n - l i n e a r  optimal  c o n t r o l problems", Ph.D. t h e s i s , UBC, May 1967. 14.  R. B e l l m a n and K. L. Cooke, "Delay  d i f f e r e n t i a l equations",  Academic  P r e s s , New York, 1963. 15.  G. L. H a r a t i s v i l i ,  "The maximum p r i n c i p l e i n the t h e o r y of o p t i m a l  p r o c e s s e s w i t h d e l a y " , S o v i e t Mathematics, Dokady, V o l . 2, 1961—1962, pp. 39-42. 16.  D. MacKinnon, "Optimal  c o n t r o l o f systems w i t h pure time d e l a y s u s i n g  v a r i a t i o n a l programming approach", IEEE T r a n s a c t i o n s on A u t o m a t i c C o n t r o l , V o l . AC-12, No. 3, 1967, pp. 255-262. 17.  R. F l e t c h e r , " F u n c t i o n a l m i n i m i z a t i o n w i t h o u t . a review,"  18.  Leonard  Computer J o u r n a l , V o l . 8, A p r i l  evaluating derivatives -  1965, pp. 33-41.  Weiss, "An a l g e b r a i c c r i t e r i o n f o r c o n t r o l a b i l i t y o f l i n e a r  systems w i t h d e l a y , " IEEE T r a n s a c t i o n s on Automic C o n t r o l , August 1970, pp. 19.  443-444.  A. E. Bryson  and W. F. Denham, " S t e e p e s t  a s c e n t method f o r s o l v i n g  optimum programming problems", J o u r n a l of A p p l i e d Mechanics, V o l . 29, 1962, 20.  pp. 247-257.  E. B. Lee and L. Marcus, "Foundations Chapter  21.  7, W i l e y  of o p t i m a l c o n t r o l  theory",  and Sons, New York, 1967.  J . K e l l y , "Method of- g r a d i e n t s - t o p i c s i n o p t i m i z a t i o n " , Ed. G. Lutman, Academic P r e s s , New York, 1962.  22.  M. Athans and P. F a l b , "Optimal  c o n t r o l " , McGraw-Hill,  New York, 1966.  23.  V. Eveleigh, "Adaptive c o n t r o l and optimization techniques", McGraw-Hill, New York, 1967.  24.  A. P. Sage, "Optimum systems c o n t r o l " , Prentice H a l l , Englewood C l i f f s , New Jersey, 1968.  25.  Douglas Wilde, "Optimum searching methods",Prentice H a l l , .Englewood C l i f f s , New Jersey, 1964.  51  APPENDIX I .  THE MAXIMUM PRINCIPLE WHEN THE TIME DELAYS ARE  The  f o l l o w i n g appendix i s not i n t e n d e d  p r o o f , a t a s k admirably intended without  TIME VARYING  to be a r i g o r o u s mathematical  done by H. Banks i n r e f e r e n c e  [10].  Rather i t I s  to o u t l i n e t h e assumptions he has made and the r e s u l t s the complex i n t e r m e d i a t e The  obtained  steps.  r e s u l t s a r e then a p p l i e d to q u i c k l y show that under the f u r t h e r  assumptions made i n chapter  2, the o p t i m a l  c o n t r o l f o r the g i v e n problem i s  bang-bang. The problem c o n s i d e r e d by Banks. Given: . • I. The Th state equation  x(t) = f ( x ( t ) ,  T  x(t)  x(w ( t ) ) , . . . x ( w ( t ) ) , u , t ) 1 v  = g(t)  t > t - o  w (t\ < t < t v o o t  where  0 (t) = 0 o ©^(t)  < 0 (t) ±  < ©  i + 1  V - i  (t)  i=l,2,...v  0 (t)  < 1 A w ^ t ) = t-© (t) ±  and  x(t)  i  T [ x _ ( t ) , x ( t ) , . . . x ( t ) ] and f a v e c t o r I l n T t ) , . . . f (x(')> t ) ] where x ( ' ) i s s h o r t n o t a t i o n f o r n  an n - d i m e n s i o n a l  state vector  f u n c t i o n f = [f.(x(-)» i  0  the s t a t e v e c t o r i n a l l i t s d e l a y e d  and undelayed  forms.  And i f 1. . 2.  f(x(t),  3m(t) . (  3.  where f . xCw^t)) argument.  x(w ( t ) ) , i - l , v , u , t ) i s continuous e L  1  || f  x  (  w(  t  )  )  t 0  »  T f  )  s-t  (x(t) , x (  W ; L  f  ( (')> x  i n a l l i t s arguments,  u,t) < m(t).  ( t ) ) , . . , x ( w ( t ) ) , u,t)( | < m(t) v  denotes the f u n c t i o n a l d e r i v a t i v e w i t h . .  i-0...v  r e s p e c t t o the i + l v  s t  II.  The performance  index  r j  where f a l s o s a t i s f i e s o  III.  That  the i n i t i a l  ^ f  =  Q  ,(x(-)» u , t ) d f  the above assumption 1 to 3.  and f i n a l  c o n s t r a i n t s e t s form a m a n i f o l d T o f dimension  < 2 n - l , and T i s a t t a i n a b l e .  IV. The c o n t r o l u where u i s c o n s t r a i n e d by u e .Q where Q s a t i s f i e s  certain  common assumptions about compactness and c o n v e x i t y .  Define  t h e f u n c t i o n s r_^(t) such  that t  *.(w.(t)) = t i 1 are always a b s o l u t e l y c o n t i n u o u s  o  _  < t <• T . r •  These f u n c t i o n s  _  and one t o one because o f t h e n a t u r e of w ^ ( t ) .  Then t h e f o l l o w i n g maximum p r i n c i p l e  holds.  I f u* i s the o p t i m a l c o n t r o l and x* and f * a r e the c o r r e s p o n d i n g optimal 1)  trajectories  then  there e x i s t vector functions X ( t ) = [X Ct) , X ( t ) ] d e f i n e d on [t , o o  2)  t j f  satisfying 1) X ( t ) = c o n s t a n t 2)  X (t) = -X(t) f "Jl  here  f i s the augmented v e c t o r  and  x^(t) i s the c h a r a c t e r i s t i c  < 0  x  ( x ( - ) , t)  X^OMr.Ct))  f  x  (  W  i  (  t  )  f u n c t i o n of the i n t e r v a l  )  ( x ( . ) , r(t))  [ t , w ( t * )] Q  53  that i s  =  0  w  .(t*) < t < t*  x  x  .X (t) =1  t  ±  3)  r  r  < t < w.(t*)  Q  The 2n-l dimensional vector (-X(t ), X(t|), -X(t|) f ( t * ) ) i s orthogonal to the tangent Q  plane to T at  4 )  1=0  {  V i  ( x * ( t ) , x ( t * ) , t*) . o  (  t  )  )  *x(v  . ( t ) (  (  V i  (  t  )  )  V i  (  t  )  a  0  a  ' '  °  e  n  f V l - j ^ c ^  v-i . 5)  .  .  .  . . .  w ' .(t ) ] v-j o  X ( t ) f * ( x * ( * ) , u*, t) > . X(t) f * ( x * ( t ) ,..x*(w (t)) ,. . .x*(w ( t ) ) , u, t) ... V^u e « . 1  This l a s t condition i s the maximum p r i n c i p l e .  In chapter II i t was assumed  that f(x(.«), t) i s of the form A(x(-), t) + B(x(-), t) u  and  |u| < 3  Then i n order to maximize X(t)A(x('), t) + X( )B (x(-), t) u t  by Schwartz inequality u must be u = - | B | sgn [X(t) B (x(-), t ) ] and hence  u i s bang-bang.  

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