Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The derivation of optimal control laws and the synthesis of real-time optimal controllers for a class.. Chan, Wah Chun 1965-12-31

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1965_A1 C5.pdf [ 4.92MB ]
Metadata
JSON: 1.0302284.json
JSON-LD: 1.0302284+ld.json
RDF/XML (Pretty): 1.0302284.xml
RDF/JSON: 1.0302284+rdf.json
Turtle: 1.0302284+rdf-turtle.txt
N-Triples: 1.0302284+rdf-ntriples.txt
Original Record: 1.0302284 +original-record.json
Full Text
1.0302284.txt
Citation
1.0302284.ris

Full Text

The  U n i v e r s i t y of B r i t i s h  Columbia  FACULTY OF GRADUATE STUDIES  PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  of  WAH-CHUN  B. S c , M.Sc,  CHAN  N a t i o n a l Taiwan U n i v e r s i t y ,  1958  The U n i v e r s i t y o f New Brunswick, 1961  TUESDAY, FEBRUARY 23, 1965, AT 3:30 P.M. IN ROOM 208,  MacLEOD BUILDING  COMMITTEE IN CHARGE Chairman: I . McT. Cowan E. V. Bohn C. B r o c k l e y E. Leimanis-  F. Noakes A. C, Soudack Y. N. Yu  E x t e r n a l Examiner: National Division  I . H. M u f t i  Research C o u n c i l  of M e c h a n i c a l Ottawa,  Engineering  Ontario  THE DERIVATION OF OPTIMAL CONTROL LAWS AND THE SYNTHESIS OF REAL-TIME OPTIMAL CONTROLLERS FOR A CLASS OF DYNAMIC SYSTEMS ABSTRACT A method f o r t h e s o l u t i o n o f a c l a s s of optimal c o n t r o l problems based on a m o d i f i e d method i s d i s c u s s e d . solution  descent  T h i s method i s s u i t a b l e f o r the  of problems i n v a r i a t i o n a l  Mayer type^  steepest  c a l c u l u s of t h e  and can be used t o r e a l i z e  comparatively  simple o n - l i n e optimal c o n t r o l l e r s by means of analogue computer The  essence  techniques.  of t h e m o d i f i e d s t e e p e s t descent  method  i s t o s e a r c h f o r the optimum v a l u e of a performance f u n c t i o n by r e p l a c i n g a s e a r c h i n f u n c t i o n space by a s e a r c h i n parameter space.  I n g e n e r a l , an i t e r a t i v e  type of s e a r c h f o r t h e optimum v a l u e of t h e p e r f o r mance f u n c t i o n i s r e q u i r e d .  However, i n c e r t a i n  c l a s s e s of problems the optimal c o n t r o l v a r i a b l e can be  expressed  as a f u n c t i o n of t h e system s t a t e  v a r i a b l e s and no i t e r a t i o n i s n e c e s s a r y . S e v e r a l optimal c o n t r o l problems f o r t h e r o c k e t f l i g h t problem a r e s t u d i e d and optimal c o n t r o l  laws  are d e r i v e d as f u n c t i o n s of t h e system s t a t e v a r i a b l e s . Experimental satisfactory. to  r e s u l t s show t h a t t h e method i s v e r y A PACE 231-R analogue computer i s used  s o l v e t h e sounding  r o c k e t problem.  problem, the two-dimensional  zero-lift  A more complex rocket  flight  problem, i s s o l v e d u s i n g t h e m o d i f i e d method of s t e e p e s t descent lator. flight  and an e l e c t r o m e c h a n i c a l f l i g h t  The e x p e r i m e n t a l  simu-  r e s u l t s obtained with the  s i m u l a t o r show t h a t the m o d i f i e d  steepest  descent method i s p r a c t i c a l  and show promise of  b e i n g u s e f u l i n the d e s i g n of r e a l - t i m e optimal controllers.  GRADUATE STUDIES  Field  of Study:  E l e c t r i c a l Engineering  . Analogue Computers Electronic  '  E. V. Bohn  Instrumentation  N o n l i n e a r Systems  F. K. Bowers A. C.  Soudack  A p p l i e d E l e c t r o m a g n e t i c Theory G. B. Walker  Related  Studies:  Theory and A p p l i c a t i o n s of D i f f e r e n t i a l Equations  J . F. Scott-Thomas  F u n c t i o n of a Complex V a r i a b l e Dynamical Systsms I  Hsin  Chu  E. L e i m a n i s  THE DERIVATION OF OPTIMAL CONTROL LAVS AND THE SYNTHESIS OP REAL-TIME OPTIMAL CONTROLLERS FOR A CLASS OF DYNAMIC SYSTEMS  by VAH-CHUN CHAN B.Sc,  National  Taiwan U n i v e r s i t y , 1958  M . S c , The U n i v e r s i t y of New Brunswick, 1961  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Electrical We accept t h i s required  Engineering  t h e s i s as conforming to the  standard  Members of the Department of E l e c t r i c a l Engineering The U n i v e r s i t y of B r i t i s h JANUARY, 196 5  Columbia  In the  r e q u i r e m e n t s f o r an  British  Columbia, I  available mission  for extensive be  of  written  Department  of  and  by  for  Library I  Head  o f my  llsi*Columbia,  fulfilment  University  shall  further  agree for  that  of  not  per-  scholarly  Department  shall  of  make i t f r e e l y  or  t h a t , c o p y i n g or  f i n a n c i a l gain  E(z,Cstsi*-*i~*-^  the  this thesis  permission*  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, Canada Date  the  study.  the  in partial  degree at  I t i s understood  this thesis  w i t h o u t my  advanced  copying of  granted  representatives.  cation  this thesis  agree that  for reference  p u r p o s e s may his  presenting  be  by publi-  allowed  ABSTRACT  A method f o r the s o l u t i o n of a c l a s s of optimal problems based on ^ m o d i f i e d  control  steepest descent method i s d i s c u s s e d .  This method i s s u i t a b l e f o r the s o l u t i o n of problems i n v a r i a t i o n a l c a l c u l u s of the Mayer type, and can be used to r e a l i z e comparatively  simple o n - l i n e optimal c o n t r o l l e r s by means of  analogue computer techniques* The essence to  of the m o d i f i e d steepest descent method i s  search f o r the optimum v a l u e of a performance f u n c t i o n by  r e p l a c i n g a search i n f u n c t i o n space by a search i n parameter space.  I n g e n e r a l , an i t e r a t i v e type of search f o r the optimum  value of the performance f u n c t i o n i s r e q u i r e d *  However, i n  c e r t a i n c l a s s e s of problems the optimal c o n t r o l v a r i a b l e can be expressed  as a f u n c t i o n of the system s t a t e v a r i a b l e s and no  i t e r a t i o n i s necessary© S e v e r a l optimal c o n t r o l problems f o r the r o c k e t f l i g h t problem are s t u d i e d and optimal c o n t r o l laws are d e r i v e d as f u n c t i o n s of the system s t a t e v a r i a b l e s *  Experimental  show t h a t the method i s v e r y s a t i s f a c t o r y .  A FACE 231-R  analogue computer i s used t o solve the sounding A more complex problem^ the two—dimensional  results  rocket problem*  zero-lift  rocket  f l i g h t problem, i s s o l v e d u s i n g the m o d i f i e d method of steepest descent and an e l e c t r o m e c h a n i c a l f l i g h t s i m u l a t o r . mental r e s u l t s obtained w i t h the f l i g h t  The e x p e r i -  s i m u l a t o r show t h a t the  m o d i f i e d steepest descent method i s p r a c t i c a l and show promise of being u s e f u l i n the d e s i g n of r e a l - t i m e optimal  controllers*  TABLE OF  CONTENTS Page  LIST  OF  ILLUSTRATIONS  vi  ACKNOWLEDGEMENT 1.  v i i  INTRODUCTION 1.1  ..  Historical Processes  Note on the T h e o r y  of  The  Principle  1.3  The  Method o f Dynamic Programming  The  Problem  1.3.3  The  Euler-Lagrange Equations  1.3.4  The  Legendre-Clebsch  1.3.5  1.7  2 as a 4  o f D i m e n s i o n a l i t y .......  5  ........  6  C o n d i t i o n ......  7  The  W e i e r s t r a s s C o n d i t i o n ...........  8  1.3.6  The  Transversality  8  1.3.7  The V e i e r s t r a s s - E r d m a n n C o r n e r Conditions  9  1.3.8  The  Inequality  10  1.3.9  The  Lagrange  The  Constraint  Multipliers  10  The Dynamic Programming A p p r o a c h f o r the Case o f Two F i x e d End P o i n t s  Numerical Computation e s t D e s c e n t Method  by the  Steep20  The S t e e p e s t D e s c e n t Method f o r F i n d i n g the Minimum o f a F u n c t i o n a l  The A d j o i n t Equation  System and  15 16  The C a l c u l u s o f V a r i a t i o n s of Optimal C o n t r o l  The  C o n d i t i o n ........  G r a d i e n t Method  1.4.2  1.6  2  1.3.2  1.4.1  1.5  of O p t i m a l i t y  The P r i n c i p l e o f O p t i m a l i t y Numerical Technique  1.3.10 1.4  Optimal 1  1.2  1.3.1  1  and  the  .  20  Theory 21  the E u l e r - L a g r a n g e 24  Maximum P r i n c i p l e  iii  25  Page 1.8  The F i r s t I n t e g r a l  28  1.9  The M o d i f i e d Steepest Descent Method  29  1.10 2.  36  2.1  Introduction  36  2.2  F o r m u l a t i o n of Rocket F l i g h t Problems by Means of the C a l c u l u s of V a r i a t i o n s  36  2.2.1  Basic Assumptions and Equations of Motion  37  2.2.2  Formulation of the Rocket F l i g h t Problem  38  A n a l y t i c a l Study of Optimal C o n t r o l f o r the Sounding Rocket Problem  40  OPTIMAL FEEDBACK CONTROL SYSTEMS  61  3.1  Introduction *  61  3.2  The Concept of Optimal Feedback C o n t r o l and the Synthesis of Optimal C o n t r o l l e r s .......  61  3.3 3.4  3.5 4.  34  OPTIMAL CONTROL PROCESSES FOR ROCKET FLIGHT PROBLEMS  2.3 3.  Remarks  3.2.1  A M u l t i v a r i a b l e Optimal C o n t r o l System  Feedback  3.2.2  Synthesis of Optimal C o n t r o l Laws f o r Rocket F l i g h t  64  Analogue Computer Technique f o r the Synthesis of Optimal C o n t r o l l e r s  64 73  Analogue Computer Study of the Sounding Rocket Problem  75  Some Other P o s s i b l e Optimal C o n t r o l l e r s ....  81  THE MODIFIED STEEPEST DESCENT METHOD 4.1  Introduction  4.2  B a s i c Concept of the M o d i f i e d Steepest Descent Method  83 .  83  ....  83  4.3  P o s s i b i l i t y of P r a c t i c a l A p p l i c a t i o n s ......  88  4.4  F u r t h e r I n v e s t i g a t i o ni vs  88  .«  Page 5.  6.  FLIGHT SIMULATOR AND ANALOGUE SIMULATION 5.1  Introduction  5.2  B a s i c Components of a F l i g h t Simulator  91 ..  91  ......  91  5.3  Simulation of the Optimal C o n t r o l Law .......  92  5.4  A n a l y s i s of a Test Problem ..................  93  5.5  Experimental Test of the M o d i f i e d Descent Method  96  Steepest  CONCLUSION  104  REFERENCES APPENDIX .  106 The Euler-Lagrange Equations F l i g h t Problems  v  f o r Rocket 107  LIST OF ILLUSTRATIONS Figure  Page  1.1  The F i n a l Stage and the Terminal C o n d i t i o n ..  16  1.2  A General Optimal Process  31  1.3  True Minimum and L o c a l Minimum  33  2.1  The Forces A c t i n g on a Rocket  37  2.2  The State V a r i a b l e s  60  2.3  The Lagrange M u l t i p l i e r s  60  3.1  A General M u l t i v a r i a b l e Optimal Feedback Cont r o l System  63  3.2  A M u l t i v a r i a b l e Optimal Feedback System  65  3.3  Control  The Modes of C o n t r o l f o r Optimal Rocket Flight  74  3.4  Synthesis of Optimal C o n t r o l l e r s by Means of Analogue Computers  76  3.5  Analogue Computer Program f o r the Sounding Rocket Problem  77  3.6  Experimental R e s u l t s f o r the Sounding Rocket Problem  79  4.1  An Optimal C o n t r o l l e r f o r a General Process .  89  5.1  Three Modes of Thrust C o n t r o l  94  5.2  A Subclass of Admissible T r a j e c t o r i e s  5.3  Determination of Approximate I n i t i a l Values of the State V a r i a b l e s  101  A P a r t i c u l a r Set of Approximate I n i t i a l Values of the State V a r i a b l e s  102  Optimum Performance F u n c t i o n  103  5.4 5.5  vi  .......  101  ACKNOWLEDGEMENT  The his  author wishes to express h i s profound g r a t i t u d e to  s u p e r v i s i n g p r o f e s s o r , Dr. E.V. Bohn, f o r continuous  guidance and a s s i s t a n c e throughout the r e s e a r c h p r o j e c t and also f o r great patience  i n reading the manuscript of t h i s  thesis. In the progress indebted  of the p r o j e c t , the author i s deeply  to Dr. Y.N. Yu who has always given him encouragement.  The  author would l i k e to thank Dr. E. Leimanis of the  Mathematics Department f o r many h e l p f u l  suggestions,  Mr. F.G.  Berry f o r a s s i s t a n c e w i t h the m o d i f i c a t i o n of the CF-100 f l i g h t simulator, and Mr. J.W* Sutherland  f o r a s s i s t a n c e i n the  s o l u t i o n of the sounding rocket problem on the PACE 231-R analogue computer. Sincere studentship  a p p r e c i a t i o n s are expressed f o r the award of a  f o r 1961-1964 by the N a t i o n a l Research C o u n c i l and  Dr. F. Noakes, Head of the E l e c t r i c a l Engineering  Department.  Sincere  a p p r e c i a t i o n i s a l s o expressed f o r the j o i n t  support  of t h i s p r o j e c t by the N a t i o n a l Research C o u n c i l term  grant A.68 and the Defe nee  financial  Research Board grant No. 4003—01.  1.  1.1  INTRODUCTION  H i s t o r i c a l Note on the Theory of Optimal The  Processes  c l a s s i c a l theory of the c a l c u l u s of v a r i a t i o n s was  developed by E u l e r and Lagrange at the end of the eighteenth century.  E u l e r obtained the necessary c o n d i t i o n f o r a r e l a t i v e  weak minimum i n the form of an equation, now known as the E u l e r equation,  Lagrange i n t r o d u c e d the Lagrange m u l t i p l i e r to  f a c i l i t a t e the f o r m u l a t i o n of minimum problems s u b j e c t to constraints.  The Lagrange equation i n mechanics has the same form  as the E u l e r equation.  The E u l e r equation i s , t h e r e f o r e , also  r e f e r r e d to as the Euler-Lagrange the name Euler-Lagrange  equation.  In t h i s  thesis  equation i n s t e a d of E u l e r equation i s  used. The method of dynamic programming was developed by B e l l man i n the l a s t decade and i s e s s e n t i a l l y a numerical suited f o r d i g i t a l  technique  computation.  Recently P o n t r y a g i n developed  a mathematically r i g o r o u s  theory of optimal c o n t r o l which i s c a l l e d the maximum p r i n c i p l e . A f u r t h e r computational  technique a v a i l a b l e to solve  minimum problems i s the g r a d i e n t method or the method of steepest descent.  The g r a d i e n t method has been a p p l i e d by K e l l e y f o r  s o l v i n g optimal f l i g h t path problems  A s i m i l a r scheme has  been developed by Bryson and h i s c o l l e a g u e s ^ ^ • presented a m o d i f i e d approach f o r s o l v i n g optimal problems which appears  B o h n ^ * ^ has control  s u i t a b l e f o r computing the instantaneous  c o n t r o l p o l i c y i n r e a l time  .  This t h e s i s i s concerned with the  development of t h i s method which, f o r reasons t h a t w i l l be given l a t e r i n the t h e s i s , i s c a l l e d the modified steepest descent method.  Chapter  1 g i v e s a b r i e f review of the v a r i o u s techniques  2 mentioned above. 1.2  The P r i n c i p l e of O p t i m a l i t y (5)  The p r i n c i p l e of o p t i m a l i t y policy  ' s t a t e s t h a t "an optimal  has the property that whatever the i n i t i a l  the i n i t i a l  state and  d e c i s i o n a r e , the remaining d e c i s i o n s must con-  s t i t u t e an optimal p o l i c y  with regard to the s t a t e r e s u l t i n g  from the f i r s t  This p r i n c i p l e p l a y s the fundamental  decision".  r o l e i n the theory of dynamic programming.  1.3  The Method of Dynamic Programming  (6) .  The theory of dynamic programming i s based on the p r i n c i p l e of o p t i m a l i t y . determining a numerical  I t gives a systematic approach f o r  s o l u t i o n to minimum problems.  In  theory, dynamic programming i s a very general approach, however, i n p r a c t i c e , i t has r e s t r i c t e d a p p l i c a b i l i t y because of the problem of d i m e n s i o n a l i t y . In  t h i s s e c t i o n the b a s i c technique of dynamic  programming i s d i s c u s s e d . Consider the problem of minimizing the f u n c t i o n a l J  J ( x ) =\  F ( t , x , x ) dt  (1.1)  where the v e c t o r n o t a t i o n / \ x — v ^»»«»» y» x  * _ dx x —  x  n  and x(0) = c = i s used.  (c »..,c ) 1t  n  The dynamic programming approach to minimizing J i s to  consider  the f u n c t i o n F(r,x, *£) dT  f (t,x) = Min \  I t i s evident  (1.2)  that f(T,x(T)) = 0  and that  f ( 0 , c ) = Min J ( x )  The p r i n c i p l e of o p t i m a l i t y a p p l i e d to (1.2) -T  ,-t+At F<T,x,  f ( t , x ) = Min x  F ( , x , | ^ ) dT  )dT +'  t  l J  yields  T  >+At (1.3)  F ( t , X j ) A t + f(t+At,x+xAt) + O(At)  Thus f ( t , x ) = Min  X  (1.4)  f *\ where O(At) i n d i c a t e s terms of the order  of (At) .  Expanding  ( l . 4 ) i n a power s e r i e s about (t,x) and l e t t i n g At —»- 0, y i e l d s  0 = Min  F(t,x,x)  ^  +  +  2^  ^  .1=1  \ •J  X  j  (1.5)  3  The s o l u t i o n of (1.5) must s a t i s f y the f o l l o w i n g two n o n l i n e a r p a r t i a l d i f f e r e n t i a l equations  & F + 6rYt t  n  +  V  f _ O6 f £—> 6: OX. 3=1  ;  =  1  0  (1.6)  and 6F_ ^ 6 f 6x.  6x.  = 0,  j =  1 , 2 , ..  *  (1.7)  4 Thus the o r i g i n a l problem of minimizing the f u n c t i o n a l J of (1*1) i s transformed  i n t o the problem of s o l v i n g the n o n l i n e a r  p a r t i a l d i f f e r e n t i a l equations, (1.6) and  (l*7) for f .  In  general these n o n l i n e a r p a r t i a l d i f f e r e n t i a l equations can not be s o l v e d d i r e c t l y . 1•3•1  The P r i n c i p l e of O p t i m a l i t y as a Numerical  Technique  Most problems i n optimal c o n t r o l are f a r too complex f o r an a n a l y t i c a l s o l u t i o n . the use of d i g i t a l  A numerical  computers.  s o l u t i o n may  be obtained by  In order to employ d i g i t a l com-  puters f o r the numerical s o l u t i o n of ( l . 6 ) and  (1.7), i t i s  necessary to convert the n o n l i n e a r p a r t i a l d i f f e r e n t i a l i n t o a f i n i t e - d i f f e r e n c e equation.  A more convenient method of  s o l u t i o n i s to solve f o r the f u n c t i o n a l a d i s c r e t e approximation  of the  equations  f of (1.2) by minimizing  form  N-1 ) At  (1.8)  i= k where x ( i A t ) and where the d e r i v a t i v e x i s  approximated  by ( x  Let  (i+l) _  u (i) _ ; ( i ) —  X  ,  X  ( i )  )  /  A  t  and i n t r o d u c e the sequence of f u n c t i o n s  ( i ) )At  (1.9)  'for  -  oo<c <oo , k = 0 , 1 ,  jNjyl  f (T,c) =  .Then  0  'N M  and  (1.10)  •  •  N - l  f ( k A t , c ) = Min F ( k A t , c , u  i=k+l  [u)  = Min F(kAt ,c,u {  Equation  F(iAt,x^ jU^^At  )At + ^  k  )At + f  k  +  1  ( (k+l)At,c+uAt)  W  (1.11)  (l.ll)  i s the b a s i s of the dynamic programming (5)  method f o r the 1*3.2  .  The Problem of D i m e n s i o n a l i t y The  and  s o l u t i o n of minimum problems  storage  introduces  numerical  s o l u t i o n of ( l . l l ) r e q u i r e s the t a b u l a t i o n  of sequences of f u n c t i o n s of n v a r i a b l e s . some c o m p l i c a t i o n s .  To i l l u s t r a t e  This  t h i s , consider  the  case of a two-dimensional problem where c  ( l* 2^  =  c  c  u = (u ,u ) 1  Assume t h a t c^ and  a  r  e  2  both allowed  to have one hundred values*  Since the number of d i f f e r e n t values f o r c^ and  i s 1 0 ^ , the  t a b u l a t i o n of the v a l u e s of f ( c ^ , C 2 » T ) f o r a p a r t i c u l a r value of 4 T r e q u i r e s a memory capable since the recurrence while  of s t o r i n g 1 0 numbers.  Moreover,  r e l a t i o n requires that f(c,T) i s stored  the values f o r T+At are c a l c u l a t e d , and since the  of u^ and  values  must a l s o be s t o r e d , the memory must be capable of 4  storing at l e a s t 4 x 1 0  numbers.  G e n e r a l l y speaking, with current d i g i t a l  computers  having memories of 32,000 words, only two-dimensional minimum problems can be handled  unless some method f o r reducing  d i m e n s i o n a l i t y i s found.  The problem becomes d i f f i c u l t to cope  with f o r higher dimensions. dimensional  As p o i n t e d out by Bellman, a t h r e e -  t r a j e c t o r y problem i n v o l v i n g three p o s i t i o n v a r i a b l e s  and three v e l o c i t y v a r i a b l e s , t r e a t e d by the dynamic programming approach r e s u l t s i n f u n c t i o n s of s i x s t a t e v a r i a b l e s . case, even i f each v a r i a b l e i s allowed to take only  In t h i s  10  9  d i f f e r e n t v a l u e s , t h i s leads to 10 extremely 1.3.3  values r e q u i r i n g an  l a r g e computer memory.  The  Euler-Lagrange  All  the necessary  Equations c o n d i t i o n s i n the c l a s s i c a l theory of  c a l c u l u s of v a r i a t i o n s can be d e r i v e d from the p r i n c i p l e optimality.  Consider  S e c t i o n 1.3.  of  the v a r i a t i o n a l problem d i s c u s s e d i n  The p r i n c i p l e of o p t i m a l i t y y i e l d s the n o n l i n e a r  p a r t i a l d i f f e r e n t i a l equations  (1.6) and  (1.7).  Differentiating  (1.7) with r e s p e c t to t , gives n fx ( F . ) + A ^ A T dt  J ±/  +  / ^ i=l  J  Ox-Ot  Z_j  l \ d  i.  = 0  6x,6x  (1.12) i  i  and p a r t i a l d i f f e r e n t i a t i o n of (1.6) w i t h r e s p e c t to x . gives  x.  /  ,  Ox •  O"to .  T  x  Thus  /  n  6 f 2  ^  .ST'tft  V O x . Ax.  x  v  V  Substituting  (1.13) i n (1.12) y i e l d s 6F 5x~  d_ / 5P_  N  j =  °>  =  which are the Euler-Lagrange  equations.  I t i s a l s o p o s s i b l e to derive partial differential  (1.14)  1,2,. .. , n,  equations  (1.14) from the  nonlinear  f o r f using the method of  characteristics. 1.3.4  The Legendre-Clebsch C o n d i t i o n The  necessary  c o n d i t i o n f o r a minimum of (1.5)  i s that  the second d e r i v a t i v e of the square brackets with respect to x^ must be p o s i t i v e .  This leads to the Legendre-Clebsch  condition n  n  .2 I  Ox.Ox. i=l  3=1  j  3  or  6F 2  6%  6F  >  o,  2  6 F  2  2  6^6x2  dx-j&x^  > 6 F  6 F  2  2  6]  dx 0x 1  ^F  6 F 2  2  1  ••••  dx dx 1  6 F_ 2  n  _  o,  (1.15)  1»3.5  The Weierstrass  Condition  The Legendre-Clebsch C o n d i t i o n does not r u l e out the poss i b i l i t y of a r e l a t i v e minimum.  I f F ( t , x , x ) i s an absolute  minimum, i t f o l l o w s from (l»6) that the f o l l o w i n g i n e q u a l i t y must s a t i s f y  j=l  j=l  J  °  or (Xj " x j ) ^ 7  F(t,x,X) - F ( t , x , x ) + 3=1  ~ 0  (1.16)  j  f o r a l l f u n c t i o n s X. From  and  (1.7),  (1.16) y i e l d s the Weierstrass  c o n d i t i o n f o r an absolute  minimum. n F(t,x,X)  1.3.6  ^  - F(t,x,x) - ^  The T r a n s v e r s a l i t y  .  X  " *j> of?  ~  ^•  0  1 7  ^  Condition  So f a r the d i s c u s s i o n of the m i n i m i z a t i o n  of a f u n c t i o n a l  i s r e s t r i c t e d to the case of f i x e d end p o i n t s . Suppose now that the end p o i n t s are v a r i a b l e .  The  necessary c o n d i t i o n f o r a minimum of the f u n c t i o n a l i s that the d i f f e r e n t i a l of the f u n c t i o n f ( t , x ) must v a n i s h .  Therefore  Thus •x-r dt oT  Multiplying  = -  / x dx L-J ox. i  (1.6) by dt g i v e s n  Of Pdt +  Of  Substituting  * x.dt = 0  dt + j=l  3  (1,7) and (l«18) i n the above equation y i e l d s n  (F -  n x  j «  )  F  d  t  F,  +  This holds at both end p o i n t s *  -i T F. )dt x. 3  dx. + (F -  4  3  dx. = 0  Thus  n  n 3=1  (1.18)  3=1  = 0  Equation (1.19) i s c a l l e d the t r a n s v e r s a l i t y 1«3.7  (1.19)  condition.  The Veierstrass—Erdmann Corner Conditions Many v a r i a t i o n a l problems of e n g i n e e r i n g i n t e r e s t have  s o l u t i o n s which may have a f i n i t e number of corner p o i n t s , where one or more of the d e r i v a t i v e s x. have a d i s c o n t i n u i t y .  •  .  Suppose  5f  that x^ i s discontinuous» then^ since ^  i s continuous, i t  x  OF f o l l o w s from (1.7) t h a t — j — must be continuous at a c o r n e r .  6x  k  Of Similarly, ^ i s continuous and s u b s t i t u t i n g yields  P  (1.7) i n (1.6)  - E • =n  F  Of  6T  3=1  J  which i s a l s o continuous a t a corner. =  F  Therefore (1*20)  10  and n  n  (F - V " x F* )_ = (F x F. ) j=l j j=l j  (1.21)  where the negative and p o s i t i v e signs denote t r a j e c t o r y p o s i t i o n s immediately before and a f t e r a corner p o i n t ,  respectively.  Equations ( l . 2 l ) and (l*20) are c a l l e d the Weierstrass-Erdmann corner 1•3•8  conditions. The I n e q u a l i t y  Constraint  In many problems there may be i n e q u a l i t y c o n s t r a i n t s on the  independent v a r i a b l e u of ( l . l l )  variable).  (the s o - c a l l e d  control  I f , f o r example^ lul <  U  where U i s the upper bound f o r the magnitude of u, then the choice of u^ a t each i t e r a t i o n stage i n the dynamic  programming  approach i s r e s t r i c t e d and the computational aspect of the problem i s thereby s i m p l i f i e d *  1,3.9  The Lagrange M u t l i p l i e r s  (6) '  The Lagrange m u l t i p l i e r method i s the most s u i t a b l e means f o r handling a minimum problem subject  to c o n s t r a i n t s .  Two  d i f f e r e n t k i n d s of Lagrange m u l t i p l i e r s which depend on the type of c o n s t r a i n t s  are d i s c u s s e d  i n this section.  Consider the problem of minimizing the f u n c t i o n a l J ( x ) =\  H(t>x»x)dt, x(0) = c  Jo subject  to the c o n s t r a i n t  (1*22)  11  (1*23)  G(t,x,x)dt = y '0 where y i s a g i v e n v a l u e *  To solve the minimum problem the  lower l i m i t i s considered v a r i a b l e so t h a t the minimum f of J ( x ) becomes a f u n c t i o n of three v a r i a b l e s , t , x , and y.  In  other words, y i s considered as an a d d i t i o n a l v a r i a b l e .  The  s o l u t i o n of the minimum problem i s given by  f ( t , x * y ) = Min  where y i s determined  H(T,x, f*: )dT  (1.24)  by the equation of c o n s t r a i n t  G(T,x, f ^ ) d T = y  Equation  (1*25)  (l.24) can be t r e a t e d i n the same manner as was  done  previously f o r (l.2) y i e l d i n g  f ( t , x , y ) = Min  H(t,x>x)At + f ( t + A t , x+xAt, y-G(t,x,x)At)+ 0(At) (1*26)  Proceeding  as b e f o r e , the f o l l o w i n g f u n c t i o n a l equation f o r  f ( t , x , y ) i s obtained  t  0 = Min H(t,x.x) + f *\  Of  n + ^  x  j  6f ^ -  - G(t,x,x)  dt^  L  (1.27) The s o l u t i o n of (1.27) must s a t i s f y the equations  n TJ U = h* x  j  +. Of' A ~ G P  Ox. 3  x  6f v—  • Oy  j  (1.28)  12 and  j=i  J  Differentiation  of (l«28) "with respect to t , and p a r t i a l  differentiation  of (l»29) with respect to x. y i e l d s  Partial differentiation following  of (l»29) with respect to y y i e l d s the  results:  6 f 2  ,V  " *  6 f 2  „ 6 f 2  or 0  = It  d.32)  = constant  (l.33)  Thus  I t can be seen from (l»30) t h a t i f a new v a r i a b l e (1-34)  i s introduced,  (1.30) r e s u l t s  - F  x  =0,  i n the Euler-Lagrange  j = 1, 2,..., n  equations  (1.35)  13 where P = H + KG  (1.36)  Of This shows that — ^ multiplier.  plays the r o l e of the Lagrange  In the case of the c o n s t r a i n t being  form of ( l . 2 3 ) , the Lagrange m u l t i p l i e r i s a  an i n t e g r a l  constant.  In g e n e r a l , the Lagrange m u l t i p l i e r i s not a  constant.  Consider the c o n s t r a i n t to be of the form h(t,x,x*u) = 0  (1.37)  x = g ( t , x , u ) , x(0) = c  (1.38)  or  where the c o n t r o l v a r i a b l e u = u ( ^ , , , , u ) m  i s to be chosen so  as to minimize the f u n c t i o n a l J(x)„  In t h i s case, the Lagrange  m u l t i p l i e r i s no longer  For example,  a constant.  consider the  problem of m i n i m i z i n g the time r e q u i r e d to t r a n s f e r the system d e s c r i b e d by (l.38) from the i n i t i a l final  state  (b^ , • • • ,b ).* n  state  (c^,...,c ) n  to the  The f u n c t i o n a l T = T(u) to be  mini-  mized i s subject to the c o n s t r a i n t s x,(T) =b.,  •J  j = 1,2,...,n.  (1.39)  J  This i s a minimum-time problem.  By i n t r o d u c i n g the f u n c t i o n  f ( t , x ) = time r e q u i r e d to t r a n s f e r the system d e s c r i b e d by (1.38 ) from x to b and applying  the p r i n c i p l e of o p t i m a l i t y the equation f ( t , x ) = Min  i s obtained.  At+f(t+At,x+gAt)+0(At)  (1.40)  Expanding the second term i n a power s e r i e s and  l e t t i n g the l i m i t as At  0 y i e l d s the r e l a t i o n  14 n 0 = Min  1 + f  t  +  (1*41) 3=1  3  The s o l u t i o n of (l.41) must s a t i s f y the equations n 0 =  i f +  t  (1.42)  +  3=1 and n 0  =  f  x. 5 u t '  3=1  2  (1.43)  = 1.2,...*m.  1  1  P a r t i a l d i f f e r e n t i a t i o n of (1.42) with respect to x. y i e l d s 3  o f  6 f  2  V - 6f  2  5t5x7  +  Z_  k=l  1  Sx^oir k g  +  2_,  <L_ dt  f  6  x.  5T  3  E  J  =  5x7k oTT = ^ j  n  x.  6T^t  i t f o l l o w s by s u b s t i t u t i n g  +  k=l  v  +  ;  dt  2 s x 7k x ,j k ( f  ) g  A  k=l  (1.45)  (1.45) t h a t  ^  f  k=l  f  ^ k  f t x. E5T x ^' ' f  (1.44)  n  0  4_ < o) 6x, x.  (l»44) i n t o  n ^  g  v  k=l  Since  ^> k  =0  k  =1  2  '  n  (1.46)  I n t r o d u c i n g the Lagrange m u l t i p l i e r s *3 = f x.  into  (1.46) y i e l d s  (1.47)  k=l  J  The s o l u t i o n of the 2n+m equations (1.38),  (1.43) and  (l.48) gives the 2n+ra unknown f u n c t i o n s which are A., x. and u^.  The t r a j e c t o r y d e f i n e d by these v a r i a b l e s s a t i s f i e s the  necessary c o n d i t i o n s f o r a minimum-time t r a j e c t o r y . 1.3.10  The Dynamic Programming Approach to the Case of Two F i x e d End P o i n t s The numerical  technique d i s c u s s e d i n S e c t i o n 1.2.1  allows  a problem w i t h two f i x e d end p o i n t s to be r e p l a c e d by an i n i t i a l - v a l u e problem* Consider  the problem of minimizing  J(x) = \  the f u n c t i o n a l  F(t,x,x)dt  (1.49)  ^0 subject to the two end c o n d i t i o n s x(0) = a, x(T) = b  (1.50)  Proceeding as i n S e c t i o n 1*3*1 where u = x y i e l d s the r e l a t i o n f ( c + uAt t+At) = Min y  F(c,u)At  +  f(c,t)  (1.51)  M The c o n d i t i o n t h a t the f i n a l values signed values b must be s a t i s f i e d . at the l a s t stage of the p r o c e s s ,  of x ( t ) be the a s -  This means i n e f f e c t t h a t f o r any values  of x., the  choice of the c o n t r o l v a r i a b l e s u. must be such as to r e s u l t i n x.(T) = b j * Consequently, the t e r m i n a l  c o n s t r a i n t s f i x the f u n c t i o n  f ( c , T ) given by the  relation (1.52)  f (c, (N-1) Ait) '= F(c,u) where u = thus  b-c At  f(c,(N-l)Ajfc)  (1.530  Here, b i s taken to be f i x e d and This i s shpwn i n P i g .  c i s considered  (N-1)At  The  to be v a r i a b l e ,  1*1«  0  F i g . 1.1.  (1.5 4)  = * ( c , ^ rAt' r)  final  stage  and  T = NAt  t  the t e r m i n a l  condition  In dynamic programming the t e r m i n a l c o n s t r a i n t s i m p l i f i e s the computation.  Since f(c»T) i s determined by the  terminal  c o n d i t i o n s , the remaining f u n c t i o n s of the sequence f(c+uAt, t+At) are determined by means of the t e r m i n a l 1.4  The  to  conditions.  Gradient  The  ( l . 5 l ) with no f u r t h e r reference  Method (7)  g r a d i e n t method or the method of steepest descent i s  an elementary concept s u i t a b l e f o r the s o l u t i o n of minimum problems*  In recent years the computational  convenience of the  g r a d i e n t method has l e d to a v a r i e t y of a p p l i c a t i o n s . In order to present the b a s i c i d e a of the g r a d i e n t method* consider the problem of minimizing  a continuous  function  f = f ( x ^ * . . ,x ) n  I f an arc l e n g t h i s d e f i n e d by n  - £  ds'  3=1  dx. 3  (1.55)  the d e r i v a t i v e of f along the arc i s  df ds  n dx, V " Of / , Ox. * ds  =  3=1  (1.56)  3  Introducing the c o n s t r a i n t  E  dx. LLS. .  - 0  (1.57)  3=1  by means of a Lagrange m u l t i p l i e r X y i e l d s n df GTS  Z.  6f  iO  3=1 n  x  d  •  i  x  y  i  +  x  1  as  -i  dx.  - > , <a^> 3=1  J  " Of , 5x-^  A +  3=1 where  n  1  r  * L  1  n  - 2_, 3=1  dx. _ l ~ ds  P a r t i a l d i f f e r e n t i a t i o n of  df  with respect to y . y i e l d s  (1.58)  18  6  6f  /dfx  oy" fe) = 53ET "  01  2 X  (1.59)  yj  df  For  -jj-^-  to be a maximum, the above equation must v a n i s h : £*--2Xy.  Hence  y-;= j  Substituting y. into 3  (1.60)  =0  6f  (1.61)  2X * 5 x ~  (1.57) y i e l d s n  Hence  I n  * = ± 2  j  ~ ds  1  ~  (1.62)  (l.6l) yields  dx. y  2  J  3=1  Substituting X into  2  X-p  6f  ±  •1 2  n  , j = l * 2 , . . . , n . (1.63)  5x.  i=l  and the maximum d e r i v a t i v e of f with r e s p e c t to s i s 1 ds  —  (  3=1 For the steepest descent  (1.64)  Z _ oT7> J  d i r e c t i o n , the negative  s i g n i s taken,  while the p o s i t i v e s i g n i s taken f o r the steepest direction.  Now consider x. as components of a v e c t o r x, the d  directions  ascent  x  i  J  as components  of the u n i t v e c t o r  dx  and the  19 Of p a r t i a l d e r i v a t i v e s pr— as components Ox.  of a gradient v e c t o r ,  then ff =  ^  r  a  d  d- 5)  • i f  f  6  Introducing the f u n c t i o n ds  v = dT where T i s a parameter i n t o (l.55) y i e l d s 1 n dx. 2 V = (dT ) 1  (1.66)  Since dx. dT  dx ~ ds  1  ds * dT  i t f o l l o w s from (l.63) and (1.66) t h a t n  dx, dT  ~ i  Ox. i = l  (1.67)  If V = k i=l where k i s a p o s i t i v e constant, i t f o l l o w s t h a t dx.  dT =  ±  For the steepest descent,  Of k  (1.68)  5  the negative  s i g n i s taken.  r e l a t i o n i s the b a s i c c o n d i t i o n of the steepest descent ection f o r f .  This dir-  20 1.4*1  Numerical  Computation by the Steepest Descent Method  The numerical computation  of the minimum of the f u n c t i o n  f(x^,»••yX )  r e q u i r e s t h a t the equation of steepest descent be  approximated  as a f i n t e - d i f f e r e n c e equation, t h a t i s , (1*68)  n  i s w r i t t e n as  6f Ax.. * - k A T j g -  ,  j = l,2,*.*,n.  The p r o p o r t i o n a l i t y constant k can be absorbed  by the step  s i z e AT, hence x. may be w r i t t e n as 3  6f_  x,<  1+1  > S x ^  - U><gL)<i> h  ,  j  =  1  ,  2  , . . . ,  n  .  (1.69)  where h = kAT  and h ^ ^ = k ^ ^ A T .  The process i s repeated  until  (m) a minimum of f ( x ^ , . . . , x ) i s obtained at. ^ ( x n  (1.69) i s a general formula f o r i t e r a t i o n * be a d j u s t e d to reduce 1.4*2  ).  Equation  The step s i z e h may  the number of steps r e q u i r e d .  The Steepest Descent Method f o r F i n d i n g the Minimum of a Functional Consider the problem of minimizing the f u n c t i o n a l  f  T  J(x) = I  F ( t , x , x ) d t , x(0)-= c  (1.70)  Jo where x belongs  to a c l a s s of admissible f u n c t i o n s .  L e t x ( t ) = y ( t ) + h u ( t ) , u(0) = u(T) = 0  (l.7l)  21 where h i s a parameter, y ( t ) i s a f i r s t approximation and where ;, u i s to be found so that J ( x ) <C J ( y ) . Equation (1.7.0) can be w r i t t e n as  J(h) = \  F(t,y+hu, y+hu)dt  (1.72)  JO The d e r i v a t i v e of J ( h ) w i t h respect to h i s  5E = f E V (  J  0  j=l  3+F  J  x^ X  )dt  '  (1  j  73)  I n t e g r a t i n g the second term of (1.73) by p a r t s y i e l d s  I = i Z < v » v* JO  i=l  3  x  (i  i  -  74)  For the path of steepest descent (1.74) must be negative which i s the case i f u. i s chosen so that •0  - V  u (t) x.  (1.75) j  At the minimum of J , u . ( t ) = 0. 1.5  The C a l c u l u s of V a r i a t i o n s and the Theory of Optimal Control The general problem of the c a l c u l u s of v a r i a t i o n s can be  formulated as a problem of B o l z a , Lagrange or Mayer.  These  three  formulations are t h e o r e t i c a l l y e q u i v a l e n t and the problem of Lagrange and Mayer can be considered as p a r t i c u l a r cases of the (8) problem of B o l z a .  22 The problem of B o l z a can be formulated  as f o l l o w s :  Consider the s e t of f u n c t i o n s Xj(t),  satisfying  j  =1,2,...,n.  the s e t of c o n s t r a i n t s ^ ( t j X j x )  which i n v o l v e s  =  0, i = 1,2,...,  m < n  (1.76)  (n-m) degrees of freedom.  Assuming that the f u n c t i o n s x . ( t ) and t are c o n s i s t e n t w i t h the boundary c o n d i t i o n s  at t=0 and a t t=T, that i s ,  0,x(0)  =0, r = l , 2 , . . . , q .  (1.77)  T,x(T)  =0, p = q+1, . . . ,s ^2n+2  (1.78)  then the problem i s to f i n d the s p e c i a l s e t of f u n c t i o n s x . ( t ) which r e s u l t s i n a minimum f o r the f u n c t i o n a l T J =  G(t,x)  +\ - 0  H(t,x,x)dt  (1.79)  JO  I f the f u n c t i o n G of (l.79) i s i d e n t i c a l l y zero, that i s if, G(t,x) = 0 then the f u n c t i o n a l of (1.79) reduces to -T (1.80)  H(t,x,x)dt  This i s the problem of Lagrange. On the other hand, i f the integrand identically  zero, that i s i f , H(t,x,x) = 0  of (1.79) i s  23 then the f u n c t i o n a l of  (1.79) becomes -i T J =  G(t,x) ->0  This i s the problem of Mayer. I t i s of primary interest, to i n t e r p r e t the  general  problem of B o l z a from the p o i n t of view of optimal c o n t r o l . e s s e n t i a l d i f f e r e n c e between the  c a l c u l u s of v a r i a t i o n s and  The the  theory of optimal c o n t r o l i s that the d e r i v a t i v e s i n the integrand replaced  of the f u n c t i o n a l J i n the by the  c a l c u l u s of v a r i a t i o n s  are  c o n t r o l v a r i a b l e s u, ( t ) .  Thus, i n s t e a d  of c o n s i d e r i n g  the m i n i m i z a t i o n of  the  functional  subject  to the  constraints  (1.81) the m i n i m i z a t i o n of the  subject  to the  Functional  constraints  0  0  of the  form  x. - f . ( t , x , u ) , j = l , 2 , . . . j n .  i s considered.  Where u i s the  In general the follows?  set  ( u, . , .,,u 1 m  (1.82)  )«  optimal c o n t r o l problem can be  Given an i n i t i a l  state  (0,x(0)), f i n d the  stated  as  corresponding  admissible  c o n t r o l v a r i a b l e s u^ d e f i n e d  :j.n the i n t e r v a l [|O,TJ  f o r which the f u n c t i o n a l J assumes i t s minimum. I f the s e t of c o n t r o l v a r i a b l e s u^ can be determined as functions  of the state v a r i a b l e s x. so that the f u n c t i o n a l J 3  i s minimum, then the set of c o n t r o l v a r i a b l e s u^ can be obtained by feedback from the state v a r i a b l e s at the output. In t h i s case the c o n t r o l v a r i a b l e s are of the form u  k  = L ( x ) , k=l,2,...,m.  and  the f u n c t i o n s L^(x)  The  problem can therefore  c o n t r o l problem;  (1.83)  k  are r e f e r r e d to as the c o n t r o l laws. be stated as an optimal feedback  F i n d the c o n t r o l laws such that when  (l.83)  i s s u b s t i t u t e d i n (1.82), the f u n c t i o n a l J assumes i t s minimum w i t h regard 1.6  to the s e t of a l l admissible  c o n t r o l laws.  The Ad.joint System and the Euler-Lagrange Equation The  equations of c o n s t r a i n t s  f i r s t order nonlinear  (1.82) a r e , i n g e n e r a l ,  d i f f e r e n t i a l equations.  I f these non-  l i n e a r d i f f e r e n t i a l equations are l i n e a r i z e d , one obtains a system of l i n e a r d i f f e r e n t i a l equations of the form  Sx. = y ^ s r ^ Sx. + y j=l  ^-i-  k=l  J  Su.  (1.84)  K  where the p a r t i a l d e r i v a t i v e s are evaluated on the^ optimal trajectory. The  a d j o i n t system of (1.84) i s d e f i n e d by *i  n ^ = " E 3=1  6f *j 5^S 1  1  = 1.2  n.  (1.85)  Consider now the problem  of Mayer of S e c t i o n 1.5, where the  Euler-Lagrange equations are given by | T (F. ) - F x.  and where  = 0, j = 1,2,...,n.  x  (1.86)  i  n F  =X X [ ii=l X  f  s u b s t i t u t i n g t h i s f u n c t i o n F i n the Euler-Lagrange equations yields X  ±  = - E  5 ^  \j  j=l  f  i=l,2,..*,n.  (1.87)  1  The equations of (1.87) are e x a c t l y the same as equations of (1.85), thus the Euler-Lagrange equations i n the c a l c u l u s of v a r i a t i o n s are the same as the a d j o i n t system f o r the l i n e a r i z e d equations of c o n s t r a i n t s .  I t should also be noted that the  equations of (1.48) are the Euler-Lagrange equations, where the Lagrange m u l t i p l i e r s have the s p e c i a l meaning i n dynamic programming given by (1.47). (9) 1.7  The Maximum P r i n c i p l e Pontryagin and h i s co-authors have s t a t e d i n the book  "The Mathematical  Theory of Optimal Processes" that the method  of dynamic programming l a c k s a r i g o r o u s l o g i c a l b a s i s i n those cases where i t i s s u c c e s s f u l l y made use of as a h e u r i s t i c The maximum p r i n c i p l e gives a r i g o r o u s mathematical optimal p r o c e s s e s .  Therefore, i t i s of t h e o r e t i c a l  to d i s c u s s b r i e f l y the minimum problem the maximum p r i n c i p l e .  tool.  theory f o r interest  as i t i s formulated by  26 Consider the f u n c t i o n a l J = \  F(t,x,x)dt  (1.88)  Jo where  x = (x^,...,x ) n  and the problem i s to f i n d the minimum of J f o r a l l the admissible  c o n t r o l v a r i a b l e s u^ which t r a n s f e r the p o i n t from  x (0) to x ( T ) .  J  J  Let  x  = P(t,x u)  0  (1.89)  f  x- = u.,  and  j = l,2,...,n.  (1.90)  form the H-function n H(p,x,u) = p F + Q  Pj  u..  (1.91)  j=l where the v a r i a b l e s p are d e f i n e d by the r e l a t i o n s d  P i  6  dt  Hence  r  5"x.  ~  ,  s  i = 0,1,...,n,  (1.92)  P , i = 0,1,...,n,  (1.93)  x  dp. x ^ = -  p Q  then the r e l a t i o n of (1.93) gives dp dt  £ = 0  i dt d  p  =  (1.94)  OF ~ 0 5x~" ' p  '  J  =  1»2,..•»!!.  (1.95)  The maximum p r i n c i p l e s t a t e s that i n order f o r u and x to  27 d e f i n e an optimal  t r a j e c t o r y i t i s necessary that there e x i s t s  a continuous v e c t o r f u n c t i o n p = (pQ,...,p ) n  corresponding  to u and x, such that 1.  f o r every t , 0 < t < T , the f u n c t i o n H a t t a i n s i t s maximum at the p o i n t u, M(p,x) = Sup H(p,x,u)  (1.96)  M 2.  at the terminal time T, the r e l a t i o n s p (T)<:0, M [ p ( T ) , x(T) Q  are The equation  = 0  (1.97)  satisfied.  of (1.96) i m p l i e s that  5H = 0, 5u-  j = 1,2,...,n.  (1.98)  P a r t i a l d i f f e r e n t i a t i o n of (1.96) w i t h respect to u. y i e l d s  OTT  =  p  0  By the equation P  p  +  j '' 3  =  of (1.98),  0 5 u ~  +  j  P  =  lt2,...,n.  the above equation ° '  J  =  1f2,.••,n.  (1.99)  becomes (1.100)  J  I t f o l l o w s from (1.100) that P Q ^ 0, otherwise a l l the p^ = 0, i  = Ojl^.B^.n.  negative  I t i s seen from (1.94) and (l.97) that P Q i s a  constant.  I t i s convenient to choose p  = -1  Q  so that (1.100) becomes OF Pj ^ Q ^ J  -  »  3 = 1,2,...,n.  (1,101)  28 On the other hand, i f P Q = -1 i s s u b s t i t u t e d i n (1.95) and then i n t e g r a t e d , i t gives  P ; j  = Pj(O) .+\  g £ - ds, j = l , 2 , . . . n . r  Jo  (1.102)  3  «  r e p l a c i n g u, by x. i n (1*101) and s u b s t i t u t i n g i n t o yields  g- = ^  f i:  (0)+  d s  (1.102),  -  (1  l03)  D i f f e r e n t i a t i n g t h i s equation with r e s p e c t to t y i e l d s the Euler-Lagrange equations fr  (P. ) - P X  1.8  The F i r s t  j  = 0,  x  j = l>2,...,n.  (1.104)  j  Integral  The s o l u t i o n of the Euler-Lagrange equations  satisfies  the r e l a t i o n ,  n"-E i i. -^ ;  f  )  (1  a  0=1  -  105)  I f F does not depend on the independent v a r i a b l e t e x p l i c i t l y , OF  5t  = 0  and the f o l l o w i n g f i r s t i n t e g r a l i s o b t a i n e d . n . , 0=1  x. F. = C x. J  (1.106)  3  where C i s the constant of i n t e g r a t i o n .  This r e l a t i o n i s c a l l e d  29 the f i r s t 1.9  i n t e g r a l of the Euler-Lagrange  The M o d i f i e d Steepest The  equations.  Descent Method  essence of the modified  steepest descent method f o r  s o l v i n g minimum problems i s to consider a general process i s d e s c r i b e d by a system of o r d i n a r y d i f f e r e n t i a l  which  equations of  the form x = f(x,u), x (0)  = c  i  where  and  The  x =  (x^,«»*,x )  u =  (u ,...,u )  f =  (f^,.».,f )  ±f  i = 1,2,...,n.  (1.107)  n  1  m  n  system under c o n s i d e r a t i o n i s assumed to move from a  p o i n t x(0) to another terminal p o i n t x ( T ) .  Some of the  c o n d i t i o n s of x(T) may  problem i s to m i n i -  be u n s p e c i f i e d .  The  mize the performance f u n c t i o n P(T,x(T)) by choosing set  of c o n t r o l v a r i a b l e s u^»  b a s i c i d e a of the modified  terminal  a special  This i s a problem of Mayer.  The  steepest descent method i s to con-  s i d e r the f u n c t i o n P as a f u n c t i o n of a set of unknown parameters which are f u n c t i o n s ofthe unknown i n i t i a l the state v a r i a b l e s and  the Lagrange m u l t i p l i e r s . P = P(a)  where  a = =  c o n d i t i o n s of Thus (1.108)  (a ...,a ) l t  n  X (0),...,X (0) 1  and where X^(0)  r  >  x  r + 1  (0),...,x (0)]  are the unknown i n i t i a l  n  c o n d i t i o n s f o r the  Lagrange m u l t i p l i e r s . The problem under c o n s i d e r a t i o n follows:  The  can be formulated as  function n E  =  Z ^ d  (  X  J -  a  f  )  (  i  a  0  9  )  i s formed where X. are the Lagrange m u l t i p l i e r s . At a minimum, the Euler-Lagrange equations ft'(F.  ) = F  ,  j = l,...,n.  (1.110)  and  0 =  must be  ,  V  k = 1,...,m.  ( l .111)  satisfied. Substituting  the  F  (1.109) i n t o  (1,110) and ( l . l l l ) ,  yields  f o l l o w i n g equations  ax, dt  1 =  ^ Of.  " Z^h i=l  n  0=  E^ i=l  i 0  5^  '  d  =  1  n  '  ( i a i 2 )  J  lO fT »  k = l,...,m.  (1.113)  k  By s o l v i n g the system of (2n+m) d i f f e r e n t i a l equations of Xj,  (1.107), (1.112) and (1.113), the (2n+m) unknown v a r i a b l e s X j , and u^ can be determined.  The general scheme f o r the  s o l u t i o n i s represented i n F i g . 1.2.  The i n i t i a l values are  sampled and introduced i n t o a high speed r e p e t i t i v e t r a j e c t o r y computer.  The performance f u n c t i o n P i s determined and the  u(t)  x(t) Process  f  (o)  Traj e c t o r y  Iteration  M9l  Procedure  Computer (Analogue)  Fig. 1 . 2  A general optimal process  P£a)  unknown i n i t i a l values are adjusted by an i t e r a t i v e to  minimize P.  process.  The  procedure  sampled value of u i s introduced i n t o the  I f there are no d i s t u r b a n c e s the s t a t e x ( t ) of the  process w i l l  correspond  i n r e a l time to the computed t r a j e c t o r y .  In the above system the i n i t i a l values f o r the t r a j e c t o r y are the r e a l - t i m e values of the process  variables.  In most problems not a l l the i n i t i a l and t h e r e f o r e a search procedure P must be employed. est  The  c o n d i t i o n s are given  f o r the minimum of the f u n c t i o n  important  i d e a of the modified  descent method i s to solve the preceeding  steep-  (2n+m) equations  subject to the c o n d i t i o n that the d e r i v a t i v e s of the p e r f o r mance f u n c t i o n P w i t h respect to the parameters a^ are always n e g a t i v e , that i s ,  6P  5T:  < 0,  j = l  f t  ..,n,  (1.114)  The values of a. are unknown and can be determined iteration.  by  For each i t e r a t i o n the c o n d i t i o n of (1.114) must  be s a t i s f i e d .  The modified steepest descent method does not  r u l e out the p o s s i b i l i t y of a l o c a l minimum unless the range of parameter values are used which may (see F i g . 1.3  not be  entire  practical  where a^ r e s u l t s i n a true minimum and a^ r e s u l t s  i n a l o c a l minimum). As f o r the numerical  computation,  i t i s assumed that  the computation s t a r t s from a p o i n t AQ = ( & J Q ) which may arbitrary.  The parameter a-^Q i s adjusted so that P  to  a minimum.  The  remaining  parameters can then be  in  sequence i n the same manner.  Proceeding  be  decreases adjusted  i n t h i s way,  a  new  33  Fig.  1.3  True minimum and l o c a l  p o i n t A^ = ( j i ) i - obtained. a  The general step may be  s  summarized i n the f o l l o w i n g way* to the next p o i n t A step  From a p o i n t A  = ( -j( i)) i a  r + 1  minimum  s  r +  = (a. )  found by a step-by-  procedure. 1.  Adjust a-^ by a small amount to have, a smaller P r  u n t i l P s t a r t s to i n c r e a s e . 2.  Repeat  1  for 2 »•••» a  r  a n r  »  e a  ch  time- a d j u s t i n g  one parameter only. 3.  Now a new p o i n t A -^ = ( - j ( + i ) ) I a  r+  the steps  1  and  r  2  s  obtained and  are repeated u n t i l  a minimum  of P i s obtained. I t i s important L  j(r)  to note t h a t f o r the adjustment of each  34 P(a  P ( a  •  •  P ( a  l ( r + 1 )  ,  a ,...,a )<P(a ,...,a ) 2 r  n r  l ( r + l ) ' 2(r+l)» a  o  •  •  •  o  o  o  l(r+l)  o  »  0  o  •  , , , , , a  •  »  •  •  3r**" nr  a  •  l r  a  • ft »  n(r+l)  #  ) <  •  •  P  »  o  o  •  n r  ) < P ( a  *  A  o  «  ^ l(r+l) a  l(r+l)'  P  e  •  , 0 , a  e  «  e  •  •  s  a  «  2r'**- nr a  o  •  •  •  )  •  ' (n-l)(r+l)' nf a  a  )  apply. 1.10  Remarks I t i s of i n t e r e s t to compare the modified steepest descent  method s t u d i e d i n t h i s t h e s i s w i t h other computational The  techniques.  standard v a r i a t i o n a l technique of the c a l c u l u s of v a r i a t i o n s  transforms the o r i g i n a l v a r i a t i o n a l problem i n t o a problem i n the s o l u t i o n of o r d i n a r y d i f f e r e n t i a l equations i n v o l v i n g twop o i n t boundary c o n d i t i o n s .  To solve a two-point  boundary value  problem i s u s u a l l y d i f f i c u l t from the computational  pdiftt of view*  Dynamic programming, i n theory, e l i m i n a t e s the two-point boundary value problem*  However, i t i n t r o d u c e s a new d i f f i c u l t y ,  the problem of d i m e n s i o n a l i t y , which means that an extremely large d i g i t a l  computer memory i s r e q u i r e d .  The g r a d i e n t method or the steepest descent method was developed by Cauchy and has been independently a p p l i e d to v a r i a t i o n a l problems d e a l i n g w i t h f l i g h t paths by K e l l e y and Bryson.  This technique has been very s u c c e s s f u l .  r e q u i r e s extensive d i g i t a l  However, i t  computing f a c i l i t i e s and does not  appear s u i t a b l e f o r developing comparatively simple r e a l - t i m e  35 optimal  controllers. The m o d i f i e d steepest descent method i s p a r t i c u l a r l y  s u i t a b l e f o r the s o l u t i o n of c e r t a i n c l a s s e s of minimum problems by means of d i g i t a l or analogue computers.  The analogue com-  puter i s very convenient f o r s o l v i n g t r a j e c t o r y problems. Another advantage of employing the analogue computer i s that i t i s then p o s s i b l e to c o n s t r u c t comparatively simple r e a l - t i m e optimal c o n t r o l l e r s .  Since the analogue computer solves  problems  i n a continuous manner, i t i s s u i t a b l e f o r high-speed comp u t a t i o n and feedback methods can be used f o r o b t a i n i n g solutions•  iterative  36 2.  2.1  OPTIMAL CONTROL PROCESSES FOR ROCKET FLIGHT PROBLEMS  Introduction A n a l y t i c a l s t u d i e s may f a c i l i t a t e  s o l u t i o n f o r optimal c o n t r o l problems.  the computation of the The i t e r a t i v e  approach  used i n the modified steepest descent method may a l s o be g r e a t l y s i m p l i f i e d i f an a n a l y t i c a l expression f o r the optimal  control  law i n terms of s t a t e v a r i a b l e s can be found. The  c a l c u l u s of v a r i a t i o n s i s the only s u i t a b l e method f o r  o b t a i n i n g a n a l y t i c i n f o r m a t i o n about the p r o p e r t i e s of the optimal c o n t r o l law and the optimal t r a j e c t o r y and i s t h e r e f o r e , of fundamental importance.  This chapter i s devoted  to the a p p l i c a t i o n  of the c a l c u l u s of v a r i a t i o n s to the problem of rocket f l i g h t and to a n a l y t i c a l s t u d i e s f o r d e r i v i n g optimal c o n t r o l  laws.  I t i s a l s o of t h e o r e t i c a l i n t e r e s t to have a complete analytical  s o l u t i o n of a problem.  This allows a study of the  p r o p e r t i e s of the Lagrange m u l t i p l i e r s which p l a y an important r o l e i n the determination of optimal c o n t r o l laws.  On the other  hand, the a n a l y t i c a l s o l u t i o n can serve as a means f o r checking the accuracy of the analogue computations used i n the modified steepest descent method d i s c u s s e d i n Chapter 2.2  3.  Formulation of Rocket F l i g h t Problems by Means of the C a l culus of V a r i a t i o n s The determination of optimal t r a j e c t o r i e s f o r m i s s i l e s ,  a i r c r a f t s and s a t e l l i t e s i s an important z a t i o n theory. an important  a p p l i c a t i o n of o p t i m i -  Goddard recognized the c a l c u l u s of v a r i a t i o n s as  t o o l i n the a n a l y s i s of rocket performance i n 1919.  A general theory of rocket f l i g h t problems was r e c e n t l y developed  37 by B r e a k w e l l ,  F r i e d , Lawden, M i e l e , L e i t m a n and o t h e r s .  review  of the r o c k e t f l i g h t  2.2.1  B a s i c Assumptions For  the  the g e n e r a l  and E q u a t i o n s  The r o c k e t  now be  a r e made  of Motion  (see F i g .  i s considered  brief  given.  f o r m u l a t i o n of the r o c k e t f l i g h t  f o l l o w i n g assumptions (1)  problem w i l l  A  problem,  2.l):  as a p a r t i c l e  or a p o i n t  mass . (2)  The power p l a n t o f t h e r o c k e t an  ideal  V  f o r the f u e l  e  i s a constant.  velocity  The t h r u s t i s t a k e n  The E a r t h i s assumed t o be f l a t , due  (4)  so t h a t t h e e q u i v a l e n t e x i t  as  where P i s a c o n t r o l p a r a m e t e r .  as (3)  engine,  engine i s c o n s i d e r e d  to g r a v i t y i s taken  The r o c k e t moves  t o be  in a vertical  and t h e a c c e l e r a t i o n  constant. two—dimensional  0  Fig.  2.1  The f o r c e s a c t i n g on a r o c k e t  plane.  38 By these hypotheses the equations of motion f o r a rocket can be w r i t t e n ^ ^  as  x - V cos 0 = 0 h - V sin 0 = 0 . D-V V + g sin 0 + 6 +  f  m +  0  cos Q -  (2.1)  0  (2.2)  cos ft) = 0  (2.3)  L + V P s i n ft) ^ = 0  (2.4)  = 0  (2.5)  where x i s the range, h i s the a l t i t u d e , V i s the v e l o c i t y , g i s the a c c e l e r a t i o n due to g r a v i t y , L i s the l i f t ,  D i s the drag,  m i s the mass, 0 i s the path i n c l i n a t i o n , and (o i s the angle between the t h r u s t and the v e l o c i t y .  The drag i s assumed to have  the general form D = D(h,V,L)  (2,6)  and the engine c h a r a c t e r i s t i c s of the rocket are represented as a f u n c t i o n of a parameter a, that i s , the c o n t r o l parameter i s  0 = 0(a) 2.2.2  Formulation of the Rocket F l i g h t  (2.7)  Problem  The set of f i v e equations of motion, (2.1) to (2,5), i n v o l v e s one independent v a r i a b l e , the time t , and e i g h t v a r i a b l e s , they a r e :  dependent  x, h, V, 0, m, <o, L and 0 . Thus, the problem  under c o n s i d e r a t i o n has three degrees of freedom, and three c o n d i t i o n s f o r optimal performance can be imposed.  In t h i s con-  n e c t i o n , the optimal c o n t r o l problem of Mayer type, can be stated as f o l l o w s : Among a l l sets of f u n c t i o n s x ( t ) , h ( t ) , V ( t ) , 0 ( t ) , m(t),  39  (2.1)  co(t),L("t) and p ( t ) , s a t i s f y i n g the equations of motion, to  and c e r t a i n p r e s c r i b e d end c o n d i t i o n s , to determine the  (2.5),  s p e c i a l set which minimizes the performance f u n c t i o n r i * f where P = P(x,h,V,©,m,t) The end c o n d i t i o n s  are c o n s t r a i n t s imposed on the i n i t i a l and  the f i n a l values of x, h, V, 0, m and t . end c o n d i t i o n s  In g e n e r a l , not a l l the  are known.  In the case that two a d d i t i o n a l c o n s t r a i n i n g equations of the form ^f  6  = §>(x,h,V,0,m,L,p\a,t)= 0  (2.8)  ip  7  = Y(x,h,V,0,m L,P,a>,t) = 0  (2.9)  f  are present, the problem has only one remaining degree of freedom, and one c o n d i t i o n f o r optimal performance can be imposed. By i n t r o d u c i n g a s e t of Lagrange m u l t i p l i e r s X ^ ( t ) , i  =  1 , 2 , . . . , 7 ,  the s o - c a l l e d augmented f u n c t i o n can be formed 7 P  =  ^  (2.10)  X.  i=l and the Euler-Lagrange equations are  where x^= x,  x  As discussed function  F  of  (2.10)  3  =  ^'  x  4~ ®'  x  5  =m  '  x  6  =  ^»  x  7  =a  '  a  n  d  x  8  '  = to  i n the l a s t chapter, i f the augmented does not depend on the time t e x p l i c i t l y ,  the f i r s t i n t e g r a l 7 k  i=l  "  OF 0  1  =  d ix .  C  (2,12)  40 exists. The Euler-Lagrange equations and the f i r s t the rocket f l i g h t problem  integral for  are given i n the Appendix,  Several p o s s i b i l i t i e s e x i s t f o r modifying the t r a j e c t o r y of a r o c k e t .  The e l e v a t o r p o s i t i o n , the t h r u s t magnitude, and  the t h r u s t d i r e c t i o n can be c o n t r o l l e d .  Thus, f o r a given set of  end c o n d i t i o n s , an i n f i n i t e number of t r a j e c t o r i e s e x i s t which are mathematically and p h y s i c a l l y p o s s i b l e .  Among a l l the  p o s s i b l e t r a j e c t o r i e s i t i s of i n t e r e s t to f i n d those which meet a requirement f o r optimal  performance.  P a r t i c u l a r forms of the performance (1)  P =  trajectories  function P are:  , problems of minimizing the f u e l con-  —m  t o  sumption,  (2)  (3)  2.3  , problems of minimizing the f l i g h t  P =  o t,f  P =  -X  L Jt  , problems of maximizing  the range.  o  A n a l y t i c a l Study of Optimal C o n t r o l f o r the Sounding Problem  ( 1 1  '  f  Rocket  1 2 )  The equations of motion f o r the rocket f l i g h t , (2.5)  time  t  ( 2 , l ) to  are n o n l i n e a r d i f f e r e n t i a l equations, and the a s s o c i a t e d  Euler-Lagrange equations, ( A . l ) to (A.8), are l i n e a r  differential  equations whose c o e f f i c i e n t s are f u n c t i o n s of the state v a r i a b l e s . I f the equations of motion can be solved so that the state v a r i a b l e s are f u n c t i o n s of time, the Euler-Lagrange equations may be considered as l i n e a r d i f f e r e n t i a l equations with time v a r y i n g  41 coefficients• Since there i s no systematic a n a l y t i c a l method f o r s o l v i n g n o n l i n e a r d i f f e r e n t i a l equations, the d e t e r m i n a t i o n of an analytical  s o l u t i o n f o r the rocket f l i g h t problem i s extremely  d i f f i c u l t and, i n g e n e r a l , i s not p o s s i b l e . s o l u t i o n s may be obtained i n s p e c i a l A problem of i n t e r e s t r e s i s t i n g medium*  However, a n a l y t i c a l  simple cases.  i s the case of rocket f l i g h t i n a  This problem can be solved a n a l y t i c a l l y i n the  case of v e r t i c a l f l i g h t w i t h a drag f u n c t i o n of the form D = kV  2  exp (-ah)  (2.13)  where k and a are constants. The sounding rocket problem has been s t u d i e d by many s c i e n t i s t s , such as, Hamel (1927), Oberth (1929), M a l i n a and Smith (1938), T s i e n and Evans  (1951), and Leitmann  (1957), e t c .  Much work, both numerical and a n a l y t i c a l , has been done on t h i s problem.  However, w i t h the exception of t r i v i a l  complete a n a l y t i c a l  c a s e s j no  s o l u t i o n has y e t been obtained.  The p a r t i a l  a n a l y t i c a l r e s u l t s p u b l i s h e d i n the l i t e r a t u r e w i l l t h e r e f o r e be extended as f a r as p o s s i b l e i n an attempt to o b t a i n a complete analytical  solution.  I t i s assumed that the f o l l o w i n g end c o n d i t i o n s are specified: h(t )  = h  Q  V ( t  o  }  =  V  o  =  m(t ) = m Q  = 0  Q  Q  0  , '  V  = unknown,  (  h ( t ^ ) = hp = f i n a l a l t i t u d e t  f  = )  V  f =  (given)  0  m(t^) = m^ = payload (given)  where m i s the i n i t i a l mass which i n c l u d e s the mass of the f u e l , ' o The problem i s to minimize the f u e l consumption r e q u i r e d to  reach a s p e c i f i e d a l t i t u d e by c o n t r o l l i n g the t h r u s t . formance f u n c t i o n P i s (m - m^).  Since m^ i s f i x e d , the problem  i s e q u i v a l e n t to minimizing the i n i t i a l mass The Euler-Lagrange  ^  (2)  X - X  = 0, subarcswith constant V —  c  0  5  J  m. Q  equation (A.17) shows that two d i f f e r e n t  c l a s s e s of s u b a r c s e x i s t f o r the optimal (1)  42 The per-  trajectory: thrust.  = 0, subarcswith v a r i a b l e  thrust*  m  For the sounding r o c k e t problem i t can be shown t h a t imp u l s i v e b o o s t i n g i s always r e q u i r e d . of the  motion (A.12) may be approximated  In t h i s case the equation f o r the b o o s t i n g p e r i o d by  equation  where t  o  i s the i n i t i a l  time and t, i s the end of the b o o s t i n g i n 1 b  terval. Solving  (2.14) together with (A.13) y i e l d s m * m  Q  where m  o  boosting  exp (- |-) , t < t < t . e 1  (2.15)  i s the i n i t i a l mass and m, i s the mass at the end of the 1  interval.  The b o o s t i n g i n t e r v a l i s o f t e n very short and the impulsive t h r u s t i s extemely l a r g e . may then be taken as t-^- t  The t o t a l time f o r the b o o s t i n g p e r i o d = At, and the v e l o c i t y V i s suddenly  i n c r e a s e d from zero to V. while the mass decreases from m to m,» 1 o 1 The  e n t i r e optimal t r a j e c t o r y has only three subarcs:  The boosting  subarc, the v a r i a b l e t h r u s t subarc, and the c o a s t i n g subarc thrust). Integrating h  ( A . l l ) from t ft l = V dt = ^© 1  r  to t, y i e l d s t V dt *o  0  (zero  43  since At i s very small and V i s f i n i t e , the above i n t e g r a l i s n e g l i g i b l e and = Ah =• 0 Let  (2.16)  the mass flow of the impulsive boosting be p ^ . I n -  t e g r a t i o n of ( A . 1 3 ) g i v e s m, — m 1  o  =  p dt m r  (2.17)  •m Since P  m  i s extremely l a r g e , the product P A t i s a f i n i t e m  quantity. S o l v i n g the Euler-Lagrange equations ( A . 1 4 )  to ( A . 1 6 )  yields 3 6D  X  A-„, = X ™ + 2 1 ~~ 2 0 \  " X  31  -  *30  2 0 •',  m Oh  ^ " 2 A  +  +  X  3 6D"  ~m" S T . dt  = X  (2.18)  (2.19)  30  o -t, and  X  51 = 5 0 \ X  ~2 <VnT > D  +  dt  m  .ID,  X D 3  x  5 0  +^  - V m  'm  = X_  5 0n  <- > -  + X., V 30  N  e  dt  DM  (^- - — ) m, m 1 o  m  (2.20)  where the second s u b s c r i p t denotes the value of X ^ at the time t = t ^ , that i s , ^("k^)  =  ^ik*  •'•be above approximations are  44  obtained by n e g l e c t i n g time i n t e r v a l t ^ — t its derivatives, This can be  a l l i n t e g r a l s w i t h respect to t since is negligible.  Q  and  are  seen from the  and  the  first The  be  end  where C i s the  and  (2.13).  c o n d i t i o n s of the  obtained from the  Lagrange  transversality  condition  integral.  t r a n s v e r s a l i t y condition dm  drag f u n c t i o n  f i n i t e during t h i s i n t e r v a l .  drag f u n c t i o n  Information about the m u l t i p l i e r s may  The  the  + X d h + X^dV 0  2  t  f  + X dm + C dt  = 0  5  c  3  first  is  (2.21)  t  integral.  Since m , t , and o o  t„ are f  f r e e , the  transversality  con-  J  dition yields X  = -1  5 0  (2.22)  and C = 0  The  (2.23)  transversality condition  does not  give any  information  about the  f i n a l v a l u e s of the Lagrange m u l t i p l i e r s f o r t h i s  problem.  However, the  first X  For  the v a r i a b l e  from (A.17) that the  integral  = 0  3 f  thrust  must be cut  off.  5  (2.24)  subarc, ^  ^ 0, and  it  follows  condition V  X  (A.18) gives  " 3lT A  =  °'  t  i< < 2 '  s a t i s f i e d , where t ^ i s the  t  t  time at which the  ( 2  '  2 5 )  thrust  is  45 The f i r s t i n t e g r a l XY  - \  2  I t i s obvious that where  (A.18) now  reduces to  (g + |) = 0, t < t < t  3  1  2  .  (2.26)  (2.26) also holds f o r the c o a s t i n g subarc  0=0. Differentiating m \  5  Substituting  + Xm  - X  5  V  3  e  = 0  (2.27)  (2.5), (A.15) and (A.16) i n t o  X  Substituting  (2.25) with respect to t y i e l d s  2  " m  1  f ~ e  (  (2.13) i n t o  \r = m l 3 (  Eliminating ^  a n  mg - D ( l  +  ^  +  = °' i < t  t  (2.27) gives  <  •  t 2  < * ) 2  28  (2.28) y i e l d s  H'  r  d ^  e  3  e  t <:t<t x  between  0,  2  .  (2.29)  (2.26) and (2.29) gives  t <t<t 1  2  .  (2.30)  Equation (2.30) shows that the v e l o c i t y V can not be zero during the  variable thrust period.  required. the  Therefore impulsive b o o s t i n g i s  Moreover, equation (2.30) can be used to determine j  switching time t ^ f o r the a c t u a l f l i g h t ,  and i t w i l l be used  as  a c o n t r o l law i n the next chapter f o r the analogue computation  of  the sounding rocket problem. Differentiating  (2.30) with r e s p e c t to t y i e l d s  Substituting  x V  Let v = rp— e  46  (2.30) and (2.3l) i n t o (A.12) gives  +  V  a  e  tl-  }  a V e 2 V ^  +  4  2K. aV  £_) JL 2 V e aY  (2.32)  + 2  T~ e  and b = —^> , then (2.32) can be w r i t t e n as aV ^  • _ j * v _ v ^ + (l-b)v-2b bV 2 v + 4v + 2 D  (2.33)  V  e  or e  b V  dt =  v  2  + 4v + 2  5  v [ v + (l-b)v-2b[  g  , dv  (2.34)  I n t e g r a t i n g t h i s equation from t-^ to t gives  t = t + — 1 g  l  ! i  n  li±bi  +  l  n  v  n  +  (l-b)v^2b v ^+(l-b)v -2b 1  r2v  K In 2  +  2  I  1+ ( i - b ) K 2v + (1-b) + K +  2 V + ( 1  _  2v + 1  1  B )  -K  (1-b) -K (2.35)  where  K = J  (1-b)  Since  + 8b  h = V = vV  i t follows that Substituting  2  dh = V v d t e  (2.34) i n t o t h i s equation d  h  =  1 a  ?  v  v  +  4v  +  2  yields d  y  + (l-b)v-2b  47  I n t e g r a t i n g t h i s equation from h, to h gives  31b  +  v +(l-b)v-2b 2  l  n  v^ + ( l - b ) v - 2 b  1  1  _ ) 1 2v + ( l - b ) - K J  K . (•2v + (l-b)4K 2 ^ 2 v + (l-b)+K 1  +  2  l n  v+ ( l  b  K  1  (2.36) The mass m can be determined as a f u n c t i o n of v and t byrewriting  ( A . 1 2 ) i n the form  a m  =  _ i (v V  +  G  D  +  e  }  nr  e  and then s u b s t i t u t i n g (2.30) f o r ^ i n t o the l a s t equation. Thus m ~ ~  da .  or  =  V U+v) e  V~ e  V  ( d T+ f  _  d t )  Now s u b s t i t u t i n g (2»34) f o r dt i n the above equation gives  dm in  /,  g V  -, . \ 7  e  b v (1 +v) ' v  v + 4v + 2 2,/ ,\ v +(l-b)v-2b n  which can be i n t e g r a t e d to the form m ln m  (v m.  or  + fe  t)  + In  2^ v + v v + z  (l-b)v-2b  48  .2 m=ra  v ^  1  v +  (l-b)v -2b  2  +  y  1  + v  1  (2.37)  exp  v +(l-b)v-2b 2  1  To sblve the Euler-Lagrange equation (A.16), the f o l l o w i n g equations .  Tp-D  = V +  m x  x  3  5  m which are obtained from  (A.12) and  s t i t u t i n g these two equations i n t o X  = X (v  5  5  (2c25) are r e q u i r e d .  Sub-  (A.16) gives  +^-) e  where v = V/V" « e  I n t e g r a t i n g t h i s equation from t ^ to t y i e l d s X  5  = X  exp  (v  t)  (2.38)  e Substituting this into  (2.25) gives mX  x The Lagrange integral  3  multiplier X  =  2  —  51 exp  (v  t)  (2.39)  can be determined by the  first  (2.26): mX c -i X„ = — ^ « v V 2  (g +  -n  exp  (v + ^  e  t)  (2.40)  For the c o a s t i n g subarc, the t h r u s t i s cut o f f , so that (3=0.  Thus m = 0 and the mass m i s constant.  Let m =  at  t = i<2> then  = m^,  and the equations of motion and the E u l e r -  Lagrange equations become h - V = 0  (2.41)  Y + g + ~— = 0 m  (2.42)  m  (2.43)  f  " =J  and *2  5E  < -^) 2  <' >  ^=- 2 m70T X  *5  =  2 45  +  " 3 ~^2 m^ X  (  2  e  4  6  )  2 where Since 5  i  n  c  e  D = k V V  exp(-ah)  v _ dV _ dV dh - dt ~ dh dt  dh  V  =  w  2  e  s u b s t i t u t i n g t h i s equation i n t o  T  °  r  3 E  +  f 2 e  ^(v ) + ^ 2  *Z dh  (2.42) gives  + ^ « P ( - ^ )  v  2  =  °  exp(-ah)+2 ab = 0  (2.47)  where  Equation 2 to v .  (2.47) i s a l i n e a r d i f f e r e n t i a l equation with r e s p e c t / 2k — ah\ I t has an i n t e g r a t i n g f a c t o r of the form exp(- — e ),  and can be w r i t t e n as  50 d_ dh  2 / 2k -ahx _vexp(- — e )  2ab exp (- —  e ^) a  (2.48) In order to i n t e g r a t e  the r i g h t hand side of t h i s equation, l e t y = e~ ,  dy = -aydh  a h  / 2k - a h \ , \ / 2k % dy exp (- — e )dh = - exp (- — y) -± J " ' ay  and  u  The i n t e g r a t i o n can be performed by expanding the exponential f u n c t i o n i n a Taylor  series.  Thus  r J  n  0 0  exp (c y ) ^ = l n (c y) + y  £-i  V  n=l  and  integrating v  2  (2.48) y i e l d s  /2k —ah\ = 2 b exp ( e ) -ah + am ^ / - . I .  n.n 1  /_ 2k \ am^'  «*»  n g  -anh  _  +  r  C  n . n ;  n=l -f(h) where  (2.49)  i s the constant of i n t e g r a t i o n and i s given by ( 2k ^n - a n h f am. = ahj — n . n ! n=l g  Thus  i s a known constant since h^ i s g i v e n . dt =  and  (2.49) gives  thus  dh  (2.41)  dh V v e  v = V f (h) dt = ± e V  Integrating  From  t h i s equation t = t  dh VHhj  gives  2  dh + ±e J h . V f (y)  S u b s t i t u t i n g the f i r s t i n t e g r a l (2.26) i n t o  (2.50)  (2.45) f o r \  0  yields  51  « , 3  Since  A  dX~ dX j , dX _ 3 _ 3 dh _ 3 ~ dt dh ° dt dh 0  ax.  thus  dh  X _  V  ax.  or  0  v  -  _  3  / „ . kV  L_ m  e  "  a h d  h  -ahs  2  _ s _ ah V v e 2  f  But  v  v  (2.49) g i v e s  2  = f(h)  Z  thus ax.  k_ ~ a h m^ e e  d d  h h  _ _g_ dh ~ 2 fThT  I n t e g r a t i n g t h i s equation y i e l d s  r-^  X.j = exp  L_  -ah  ]+  _g_ V e 2  am.  c.  = F(h)  where C  9  (2.51)  i s the constant of i n t e g r a t i o n and i s given by h (2.52) e  The Lagrange m u l t i p l i e r X  2  Substituting  1  #  (2.5l) i n t o  L  2  can be obtained from the f i r s t i n t e g r a l kv^  *o =  h  (g  (2.46) gives  2  , e" ) a h  (2.53)  52 X  5  =  •  5 and  .  P ( h  e  dA_  dX,  dt  dh  -ah dXc  dt  dh  V = V v = V \J f ( h )  thus  dX  -kV  c  ,  = — f v/ilM 2  ^1  dh  m  _ X  r  k V  P(h)  , e"  a h  f  I n t e g r a t i n g t h i s equation  where  S !  )  gives  h  5 = ^ p \ h  P(y) V f ( y ) e " y dy + C  (2.54)  a  2  i s the constant  of i n t e g r a t i o n .  c u s s i o n i t w i l l be convenient  3  For the f u r t h e r d i s -  to give a summary f o r the s o l u t i o n  of the sounding rocket problem. (l)  F o r the boosting  subarc h  ( 0 - ^ t < t ^ ) , where t  Q  = 0*  = 0  1  (2.16)  V suddenly i n c r e a s e s from zero to V  l  m = m exp(- % — ) , where m i s o * V ' o e unknown. t  x  = A t = 0  X  2  = X  ^  3  = A  X  5  S  X  (2.15)  (2.18)  2 Q  50  (2.19)  3 Q  +  X  30 e V  £  ' m">  ( 2  '  2 0  >  o where  X  = -1  5 Q  (2.22)  The f i r s t i n t e g r a l i s A  2  V  - x (g 3  +  2) - U  - *  It) = o  53 (2)  For the v a r i a b l e t h r u s t subarc, y - v  + L±±  v  l  v'+  n  v  v  b  +  x  2  v  m = m 1  l  V  + v  2  +  V ; L  .+ (1 -  (2.36)  2b  b)v  1  exp  l  v  x  2v + ( l - b) - K 11 * 2v + (1 - D) - K J J  K  n  2  (1 - b)v -2 b + (1 - b ) v - 2b  2 '^  1 2  , K . f i + - ) 2 l 2 v + (1 - b) + K 2  +  l  (t^^t-^t2),  v  + ( l - b)v - 2b  2  (2.37)  e t  !e  =  [  L  g  K i n  +  I l ^ _ b i  v  [2v  l  n  v  +  (1 - b) + K  1 +  (1 - b)v  2  v,  2  f2v  2  +  Ii  l n  2 v  + (1 - b) + K  y  -  2b  + (1 - b ) v - 2b  2  x  +  ,  1  +  ( l  _  b ?  _  (l-b) - K (2.35)  where  b = — ^ aV.  t  K =./(l - b )  + 8b  2  and  (2.25)  '3 m m X  51  v V m X  A  3  -  y  5  = A  5 1  (g + £) exp (v + m  ;  51  exp (v  exp (v + ^|- t ) e  The f i r s t i n t e g r a l i s X V 0  - X~(g + -) = 0 m  mg - D ( l + f-) e (3)  t)  For the c o a s t i n g subarc,  = 0  (t ^t^t„), 0  t)  (2.40)  (2.39) (2.38)  (2.26) (2.30)  54  v  2  /2k = 2b exp (  e  /_ 2k \ am „  oo  -ahx  )  -ah + n  n=l  . n J  = f(h)  + C\  (2.49) ~  where  —anh  n  / 2k \n - isr:  (  )  -anh„ e  f  = ah^ n  n=l h  , n  dh  (2.50)  m = m^ = constant  x A  2  -  _ lihl ( v V  V g  5  k  +  +  e  2 e  e  e  i  -ah  -ah I  (2.53)  }  dy  __g_ e  J  fTTT.  h,  + C,  F(h)  (2.51)  -kV \ -kV. — I _ 2  =  ~iin  r— L am„  X- = exp A  kv V 2  s  i  F(y) V f(y)  e~  a y  dy + C-  (2.54)  where C  2  = - exp  _ JL_ - a h am. p  f  dy  _ _£ e  J h  1  2  (2.52) The f i r s t i n t e g r a l i s  X  2  V  I t i s evident that  - *3 <«  +  m~)  =  0  (2,26)  the form of the a n a l y t i c a l s o l u t i o n i s  55 very complicated.  On the c o a s t i n g subarc, the a n a l y t i c a l  s o l u t i o n cannot be expressed use of d i g i t a l  i n a c l o s e d form.  computers an accurate numerical  However, by the s o l u t i o n may be  (12) obtained.  For example, Leitmann  has obtained the optimal  t h r u s t program as a f u n c t i o n of time, u s i n g a d i g i t a l and the a n a l y t i c a l r e s u l t s to o b t a i n the optimal  computer  trajectory.  In Leitmann's method the t r a j e c t o r y was solved i n reverse  time,  s t a r t i n g at the f i n a l p o i n t . Although it  the a n a l y t i c a l  s o l u t i o n has a complicated  form  s t i l l y i e l d s i n t e r e s t i n g i n f o r m a t i o n about the optimal  t r a j e c t o r y of the sounding  rocket problem.  This w i l l be d i s -  cussed i n the f o l l o w i n g s e c t i o n , (l)  The Optimal The  Controller  e n t i r e optimal t r a j e c t o r y has three subarcs (the  impulsive b o o s t i n g subarc, the v a r i a b l e t h r u s t subarc c o a s t i n g subarc) and a s s o c i a t e d with these d i f f e r e n t types of t h r u s t programs. v a r i a b l e t h r u s t and zero t h r u s t .  and the  subarcs are three  These are impulsive t h r u s t ,  This means that the optimal  c o n t r o l l e r has three modes of o p e r a t i o n .  The f i r s t and the l a s t  modes are ones of maximum and zero t h r u s t r e s p e c t i v e l y . v a r i a b l e t h r u s t mode i s c o n t r o l l e d by the optimal  The  controller  which must a l s o determine the i n s t a n t s at which modes are switched. of  (2.30).  A p o s s i b l e optimal c o n t r o l l e r can be obtained by means The method whereby (2.30) i s used to o b t a i n the  optimal c o n t r o l law i s to consider (2.30) e • = mg - D ( l + ^-) e  (2.55)  56 as an e r r o r  signal.  The s i g n a l e i s f e d i n t o a high g a i n ampli-  f i e r and the a m p l i f i e r A detailed  output i s used to c o n t r o l  discussion  and some other p o s s i b l e  the f u e l flow,  optimal  control  laws w i l l be s t u d i e d i n the next chapter, (2)  The I n i t i a l Values of the Lagrange M u l t i p l i e r s The  Lagrange m u l t i p l i e r s p l a y an important r o l e i n the  present study of optimal c o n t r o l l e r s .  In the general case^ the  control  law depends on the Lagrange m u l t i p l i e r s .  initial  c o n d i t i o n s of Lagrange m u l t i p l i e r s  and  U s u a l l y the  are not a l l known  the c o n t r o l l e r must then compute the unknown i n i t i a l  conditions. The  sounding rocket problem has two unknown i n i t i a l  Lagrange m u l t i p l i e r s , X  O  and X 30*  A  zv  I t f o l l o w s from the a n a l y t i c a l study that both X^Q X-JQ are n e g a t i v e .  a n <  i  This statement can be proved by the f o l l o w i n g  argument: At the end of b o o s t i n g , that analytical solution  gives X X X  and  The  i s at the time t-^, the  X  2 1  3 1  5  1  5  1  = X  2  Q  (2.18)  = X  3  Q  (2.19)  - - l - X  +  3 ;  X  3  ^  0  V  e  ( i - - ^ ) 1 o  = 0  l a s t equation can be approximated: V A  51  = 3 0 m^ X  (2.20) (2.25)  57  S u b s t i t u t i n g the above equation i n t o  ( 2 . 2 0 ) and s o l v i n g f o r  A o yields m SO--V  2 ,  ( 2  -  5 6 )  e E q u a t i o n ( 2 . 5 6 ) shows that X-JQ must be n e g a t i v e , since m  and  V  X^  Q  G  are p o s i t i v e q u a n t i t i e s .  must be n e g a t i v e .  I t follows  from ( 2 . 1 9 ) that  Furthermore, the f i r s t  i n t e g r a l ( 2 . 2 6 ) shows  that *21 l-*31<« «f>-° T  where  , g,  and m^ are p o s i t i v e , and X ^ i s n e g a t i v e .  X ^ - m u s t be negative and from conclusion,  +  ( 2 . 1 8 ) X^q must be n e g a t i v e .  a l l the Lagrange m u l t i p l i e r s i n the sounding  problem must have negative i n i t i a l (3)  Thus In  rocket  values.  A Q u a l i t a t i v e Study of the Motion of the Sounding Rocket Problem A q u a l i t a t i v e study o f t e n gives  a problem.  a b e t t e r understanding of  The general behaviour of the state v a r i a b l e s and the  Lagrange m u l t i p l i e r s may be obtained from the a n a l y t i c a l solution.  The a l t i t u d e h i s always i n c r e a s i n g along the e n t i r e  traj ectory. For the b o o s t i n g subarc, the a n a l y t i c a l s o l u t i o n shows that V i s i n c r e a s i n g and that both m and X ^ are d e c r e a s i n g , but X  2  and X ^ are almost constant. For  the v a r i a b l e t h r u s t  an optimum v e l o c i t y .  subarc, the optimum t h r u s t  Equation ( 2 . 3 6 ) shows that V must  since h i s i n c r e a s i n g a l l the time. by the equation (see ( 2 . 3 0 ) ) .  gives increase  The mass m i s determined  58  - = | d + fo e  s  k  =  ff  e exp(ah)  Since m i s d e c r e a s i n g , i t f o l l o w s from the above equation that the denominator, ge *\ i n c r e a s e s f a s t e r than the numerator k V ( l a  +  2  The Lagrange m u l t i p l i e r s X„ and X,, i n c r e a s e because they  e  have p o s i t i v e time d e r i v a t i v e s and X,. decreases a negative d e r i v a t i v e w i t h respect to t (see and  because i t has  (A.14),  (A»15)  (A.16)). For the c o a s t i n g subarc, the drag i s small at high  a l t i t u d e , and the t h r u s t i s zero, thus the v e l o c i t y i s a p p r o x i mately equal to V  ~ g ( " t - t ) (see  2  2  i n c r e a s e s u n t i l V becomes zero.  (A.12)).  The  altitude h  The Lagrange m u l t i p l i e r s  and X^ remain almost constant f o r the c o a s t i n g subarc, t h e i r time d e r i v a t i v e s are n e g l i g i b l e  (see  and X-j i n c r e a s e s to i t s f i n a l value X^^ mately equal to -X c o a s t i n g subarc  (see (A« 15))«  2  The  contains an i n t e g r a l *  with a slope  .h  f  /ty  nature  W'^N  -  The  (A.16)) approxi-  f  integrand i s l / f ( h ) = 0 .  are, however, f i n i t e .  The  integrals  The s i n g u l a r  2  of the i n t e g r a n d makes a d i r e c t d i g i t a l computation u s i n g  the a n a l y t i c a l r e s u l t s d i f f i c u l t . V = V_-  since  f  and \  2  2  a n a l y t i c a l s o l u t i o n f o r the  and i s i n f i n i t e at h = h^ since f ( h ^ ) = v h  (A.14) and  X  g(t - O  f o r the c o a s t i n g subarc  - -~ v (t - t r  f (h) = v 2 = V above two  I f the  •  2r  e integrals,  approximation i s made, the f u n c t i o n  .—12  g  2  2  can be used to compute the  59  i  The f o l l o w i n g curves i n F i g . 2.2 and F i g . 2.3 the general behavipur of the state v a r i a b l e s and multipliers.  illustrate  Lagrange  F i g . 2.3  The  Lagrange  multipliers  61 3.  3.1  OPTIMAL FEEDBACK CONTROL SYSTEMS  Introduction The general problem i n optimal c o n t r o l i s the  determination of the inputs to a system subject to c e r t a i n cons t r a i n t s so that the state of the system f o l l o w s a t r a j e c t o r y r e s u l t i n g i n the o p t i m i z a t i o n of a given performance  criterion.  In other words, the problem i s to determine the c o n t r o l v a r i a b l e as a f u n c t i o n of time so that the system the s p e c i f i e d c r i t e r i o n .  satisfies  This i s e s s e n t i a l l y an open loop  c o n t r o l system and, from the c o n t r o l engineering p o i n t of view, may not be s a t i s f a c t o r y .  The c o n t r o l v a r i a b l e r e s u l t i n g i n optimum  performance can be determined a n a l y t i c a l l y only f o r very simple systems, f o r example, the constant c o e f f i c i e n t l i n e a r  system.  Furthermore, the open loop c o n t r o l has the disadvantage that d i s t u r b a n c e s e x i s t i n g i n a p h y s i c a l system r e s u l t s i n nonoptimum performance.  Therefore, a c l o s e d loop feedback c o n t r o l  system i s d e s i r a b l e . This chapter i s devoted to the study of feedback optimal c o n t r o l systems.  S p e c i f i c problems are s t u d i e d and the optimal  c o n t r o l f o r each case i s d e r i v e d as a f u n c t i o n of the system state v a r i a b l e s . 3.2  The Concept of Optimal Feedback Control and the Synthesis of Optimal C o n t r o l l e r s :  Optimal c o n t r o l l e r s synthesized by use of the c a l c u l u s of v a r i a t i o n s r e s u l t i n a m u l t i v a r i a b l e type of c o n t r o l  systems.  In g e n e r a l , a m u l t i v a r i a b l e optimal c o n t r o l system c o n s i s t s of  62  two subsystems. system.  These are the p l a n t and the s o - c a l l e d  The p l a n t i s u s u a l l y described  adjoint  by a set of d i f f e r e n t i a l  equations and the a d j o i n t system corresponds to the E u l e r Lagrange e q u a t i o n s .  The i n t e r r e l a t i o n s h i p between these two  subsystems i s shown i n F i g . 3.1. The system i l l u s t r a t e d  i n F i g . 3.1 may be considered as  an n by m optimal feedback c o n t r o l system, where n r e f e r s to the number of the state v a r i a b l e s x ( t ) , and m r e f e r s to the number of the c o n t r o l v a r i a b l e s u ( t ) .  The f o l l o w i n g matrix  notations  are used i n F i g . 3.1. x (t) 2  x(t)  *  n x 1 matrix of state  variables.  x (t) n _ x (t) x  X(t)  n x 1 matrix of the Lagrange m u l t i p l i e r s .  u (t) x  u(t)  m x 1 matrix of c o n t r o l v a r i a b l e s .  m x 1 matrix of the terminal v a l u e s  P(a P m  of x ( t ) and t .  The performance f u n c t i o n P i s to be o p t i m i z e d .  The  number of elements of the u ( t ) matrix i s always the same as that  x(t„)  Plant  j d t l  Performance Criterion  )  ,  Adjoint System  F i g . 3.1  Optimal  u(t)  Controller  A general m u l t i v a r i a b l e optimal feedback c o n t r o l system  64 of the P m a t r i x . 3.2.1  A Multivariable  Optimal Feedback Control  System  In some cases the optimal c o n t r o l law may not contain the Lagrange  The c o n t r o l  variable  u ( t ) may then be determined as a f u n c t i o n of the state  variable  x(t).  m u l t i p l i e r X(t) e x p l i c i t l y .  In t h i s case the general m u l t i v a r i a b l e  system d e s c r i b e d 3.2.  i n F i g . 3.1 reduces to the form shown i n F i g .  The f o l l o w i n g  sections d i s c u s s  type f o r a v a r i e t y of f l i g h t 3.2.2  feedback c o n t r o l  Synthesis,of  optimal c o n t r o l l e r s of t h i s  conditions.  Optimal C o n t r o l  Laws f o r Rocket F l i g h t  In the study of optimal c o n t r o l systems the synthesis of the  optimal c o n t r o l l e r i s a major problem.  In the case of  optimal feedback c o n t r o l systems the determination of the optimal c o n t r o l law i s of primary importance. The  s i m p l i f i e d problems of rocket  f l i g h t have been  formulated i n the Appendix, and they w i l l be s t u d i e d section.  i n this  These s i m p l i f i e d prdblems have one degree of freedom.  Thus there e x i s t s only one optimal c o n t r o l v a r i a b l e i n these problems• (l)  The V e r t i c a l F l i g h t (Sounding Rocket) Problem -  It follows  from Chapter 2 that optimal c o n d i t i o n f o r the  v a r i a b l e t h r u s t subarc i s  X  5 " 3 S- = X  2  0  (2  - > 25  A c t u a l l y , t h i s c o n d i t i o n holds true f o r a l l the four problems  Optimal Controller u(t)  (t)  Performance Criterion  3.2  A multivariable  optimal feedback c o n t r o l  system  d i s c u s s e d i n t h i s chapter. to  Differentiating  (2.25) with respect  ti y i e l d s m X  5  + m A  I t f o l l o w s from Chapter  2,  - V  e  X  3  = 0  (3.1)  S e c t i o n 2.3  that (3.1) leads to  equation (2.30)j t h a t i s A  f  = mg - D ( l  V  e  = 0  (3.2)  where f  i s c a l l e d the switching f u n c t i o n . s stage terminates when f goes through zero. s (3.2) w i t h r e s p e c t to t gives  mg - D-|- - (1 e  e  (|2 Y - aDh)  The b o o s t i n g Differentiating  = 0  The equations of motion, ( A . l l ) , (A.12) and to  (3.3)  (A.13) can be  used  e l i m i n a t e m, V and h i n the above equation r e s u l t i n g i n  u = (3 D  (g + |)  (2 +  _  21)  +  2 a  V  (1  +T-;)  e  g V + | (2V  e  +  3V) (3.4)  which gives the optimal c o n t r o l v a r i a b l e as a f u n c t i o n of the state v a r i a b l e s f o r the v a r i a b l e t h r u s t (2) The  subarc.  The H o r i z o n t a l F l i g h t Problem, equations f o r optimal h o r i z o n t a l f l i g h t are d e r i v e d  i n a manner s i m i l a r to the problem of v e r t i c a l f l i g h t .  After  67 substituting  (Ao23),  (A.25) and (A.26) i n t o  (3.1) the f o l l o w i n g  equation r e s u l t s  X„ =W + X, V 4 mV 1 e  The f i r s t  3  m  i n t e g r a l f o r the v a r i a b l e  m  (3.5)  o»  thrust  subarc i s  X,V - X, - = 0 1 3 m Solving  (A.29)  t h i s equation f o r X^, and (A.27) f o r X^ and then sub-  s t i t u t i n g into  (3.5), y i e l d s the c o n d i t i o n  6:  which must be s a t i s f i e d by the optimal v a r i a b l e t h r u s t  subarc  L = mg and D = D ( V , L )  Here Expressing  (3.6) i n the form  D  ( V - V ) +V V e  6D  6B  - mgV§£ = 0  e  and then d i f f e r e n t i a t i n g with r e s p e c t to t y i e l d s V D + (V - V ) ( ^ V  - mgV  Substituting  L  n  ^ , 6D r  5L "  (  5LOV  V  +  ~  6D  + ^ L ) + V V  e  6v  L  L = mg i n t o the above equation gives  M  G  6D  5L  +  „ d D e ^2 "  , d p " 5L5V_ 2  2  v  V  V  r  m  g  v  +  V  V  e 5VOL "  - mg m  g  V  e 6L ^2_j  = 0  68 Let  A(m,V,L) = D + V ^  ,+ „.V ,,V  n( v -ir 0D B(m,V,L) = -V ^ A  e  and  substituting  - .g ^  V V  +  ^6 D  - ,g T  ^  - mg ,V Q D—  2  e  ^  fi  2  R  (A.22) and (A.23) i n t o the previous equation  y i e l d s the optimal c o n t r o l v a r i a b l e u = 0 A  (3.7)  D  AV" - mgB e  (3)  The A r b i t r a r y I n c l i n e d R e c t i l i n e a r F l i g h t Problem,  This i s a more general case and includes h o r i z o n t a l f l i g h t problems. v a r i a b l e i s the same. into  (3.1) and using  The d e r i v a t i o n of the optimal  Substituting  +  Y  e  C  0  the optimal c o n d i t i o n  °  S  +  X  2 e V  s i n  Q  control  (A.34), (A.37) and (A.38)  v a r i a b l e t h r u s t subarc, the f o l l o w i n g  *4-mT h  th€ v e r t i c a l and  equation %s obtained.  " 3 m" X  (2.25) f o r the  (  +  lr  5T> =  0  (3.8) The  f i r s t i n t e g r a l f o r t h i s problem along the v a r i a b l e  thrust  subarc i s g i v e n by (A.4l) X  ±  The  cos 0 + X  A-, JJ  2  sin © - ^  (^ + g s i n Q) = 0  Euler-Lagrange equation (A.39) gives  X  4 = 3 X  V  5T  I t f o l l o w s from the above two equations and (3.8) t h a t the optimal v a r i a b l e t h r u s t subarc must s a t i s f y the c o n d i t i o n  69  f  g  = D(V - V ) + Y V e  e  | - m g  (V  e  s i n 0 + V cos 0 ^ )  = 0 (3.9)  where L = mg cos 0, D = D(h,V,L) and 0 i s a constant.  I t can be  seen t h a t (3.2) and (3.6) are s p e c i a l cases of (3.9). Differentiating  (3.9) w i t h respect to t y i e l d s  + V cos 0 £°j)  mg V cos 0 ^  - mg V cos 0 ( g ^ V  By means of ( A . 3 2 ) , ( A . 3 3 ) ,  h  +  g | £) = 0  +  (A.34) and the equation  L = m g cos 0 The previous e x p r e s s i o n can be solved f o r 0 y i e l d i n g the optimal c o n t r o l v a r i a b l e u  =  P  _ mC - A(mg s i n 0 + D ) mB - V A e ,  where  6 D A = D + VV — » -A «  _,_  2  mg cos 0  E  B = g cos 0  - e V  .  ( Y  t  a  " V n  dn  n  w  9  "  6T V  6T  +  W  e  /_  N  \  U.iO;  „ u p - mg V cos 0 Q ^ J J 2  w  5VOL "  m  g  C  0  S  °^2  70 2  C = (T-  V )V sin ©  +  e  (4)  2  sin 0 ^  - mg V  s i n © cos ©  2  The Z e r o - l i f t F l i g h t Problem.  Substituting  (A.48) and (A.53) i n t o  (3.1) y i e l d s the  equation *3 = " 3 m V X  (  3  a  i  )  e  The optimal c o n d i t i o n f o r the v a r i a b l e t h r u s t  subarc i s  given by.(A.54) V Ac - X- ~ 5 3 m Substituting this into  h  = 0  (A.53) gives  = ^  (  V  ">  ( ' >  D  3  12  e It follows  from (A.5l) that A^ = -Aj  Substituting - A  COS  © - A  sin  6  2  + ~  y " ~ ^4 7  2  C  S  0  (3,11) i n t o the above equation y i e l d s =S-  3 mv  e  + A, V cos © + A V s i n © - A;, — 1 2 3 m  + A. # cos © 4 V  0  = 0 The f i r s t  0  i n t e g r a l f o r the v a r i a b l e t h r u s t  Aj V cos © + A  2  (3.13) subarc i s  V s i n © - A ( ^ + g s i n ©) - A 3  4  ^ cos © = 0 (3.14)  The Euler-Lagrange equation  (A.50),  71 w i t h the a i d of ( 3 . 1 l ) , \ Integrating  c  n  w r i t t e n as  D e  = a V  2  X  g  (3.16)  3  (3.16) gives X  = a V  2  where C~ i s the i n i t i a l d. Subtracting  a  e  X  + C  3  (3.17)  2  c o n d i t i o n of X«~- a V X--.. <ZV e _}U  (3.13) from (3.14) and s o l v i n g f o r X^ y i e l d s X V X„ = -x—2 TT (~4 2 g cos 0 mV  e  B  Substituting  + - - g s i n ©) m s  y  v  (3.18) '  (3.17) and (3.18) i n t o (3.13) and s o l v i n g f o r X  3  results i n 2V(C 3 ~ DV ~ - + mv e 1  X  where X^ =  cos 0 + C s i n ©) 3D — + g s i n © - 2aW m e 2  5  i s a constant,  Euler-Lagrange equation Now  where  a r e s u l t which follows from the  (A.49).  s  letting  (3.19) sin 0  A(©,V) = 2 V(C  =  ;L  B(v!g; !o)  ( - °) 3  m  cos © + C  B(V,h,m,©) =  + ^  2  s i n ©)  (3.2l)  + g s i n S - 2aW  e  2  sin© e  (3.22) and  differentiating  (3.20) gives X . ^ J  5  ^ B^  I t f o l l o w s from (3.1l) and (3.20) that  (3.23)  72 Eliminating  by the a i d of (3.23) gives A B D  =  mV « A  where  OA  B A -  v  . O AQ  m  6A / OA  •  6B v ^ O B 6h 6v  B  v  OB  +  h  /  =  S u b s t i t u t i n g A and B i n t o  i  V  6B '  '  6m"  +  m  ^  6©  +  6  cos O  6B * V  n  6A  P  9  D^ ^ 6B e - — J + vr? nr 0^  n  _6B UB £ g 5© f  K  6A e  Dx  Q  • s  6m  (3.25)  A B  P A  m  6B O  v + s-jn  . V sin 0 Q  cos © A  (3.25) and s o l v i n g f o r (3 r e s u l t s i n the  optimal c o n t r o l v a r i a b l e f o r the v a r i a b l e t h r u s t  subarc  u = (3 1 ~  g sin©  m V  F  e A  E  0  cos  V -  B  BD  g Sill © jyf  m  6B  6v  . AD +  6B  6B O©  6A _ i£ A 6A V cos © ^ OY (3.26)  where T?  *  A A Ve 6B - m- 5v  . 6B A  B  6m  Ve O A m 6v  and  6A 6V 6A  5© 6B  6v 6B 5h 6B  6©  2 Cj cos © + 2 C r-2  mV  e  sin©  2  Cj V s i n © + 2 C  2  V cos ©  + ^ - 2a V s i n © mV e e  g cos © - 2 a V V  g  cos ©  73  dm" =  m  V"' e  By the a i d of equations  (3.17),  (  -  switching f u n c t i o n f  f  s  3  +  —  w  (3.18),  (3.20) and (3.14), the  can be obtained  = C, V cos © + 0 V s i n 0 + a V V s i n © | - ^ g s i n © i £ e D dx> o  _ ADV _ 3AD _ 2mBY 2mB e The  (  U  x  \i^U  optimal c o n t r o l law f o r the four d i f f e r e n t problems  of rocket f l i g h t has been d e r i v e d .  For t h i s c l a s s of optimal  c o n t r o l problems the f u e l consumption has been minimized. ever, the technique  How-  can a l s o be a p p l i e d to problems of maximum  range and minimum f l i g h t time, e t c .  The f o l l o w i n g block diagram  represents the c o n t r o l scheme f o r . a l l four problems. i n each problem three modes of c o n t r o l corresponding b o o s t i n g subarc, the v a r i a b l e t h r u s t subarc  There are to the  and the c o a s t i n g  subarcs (see F i g * 3.3), The  switching time t ^ i s determined when the switching  f u n c t i o n f goes through zero s The  c o n t r o l l e r then operates  time i s reached.  (see (3.2), (3.6), (3.9) and (3.27)) to keep f  = 0 until  the c u t - o f f  I n the problem of z e r o - l i f t f l i g h t * the  i n i t i a l values of the Lagrange m u l t i p l i e r s i n t o the optimal c o n t r o l law.  ,^  a n <  !  enter  The method f o r e v a l u a t i n g the  i n i t i a l values i s d i s c u s s e d i n Chapter 4. 3.3  Analogue Computer Technique f o r the Synthesis of Optimal Controllers The  c o n d i t i o n s f o r optimal  c o n t r o l d e r i v e d i n the l a s t  Impulsive  Boosting  or Maximum Thrust  Optimal •  u(t)  x(t ) o  —  1 x(t)  Mt ) o  Control Law  Performance Criterion  F i g . 3.3  The modes of c o n t r o l f o r optimum rocket f l i g h t 4^  s e c t i o n can be used to synthesize optimal c o n t r o l l e r s . computers  are s u i t a b l e f o r numerical computation.  analogue computers  However,  appear b e t t e r s u i t e d f o r the synthesis of  comparatively simple r e a l - t i m e c o n t r o l l e r s . iterative  Digital  computations of the d i g i t a l  The lengthy  computer  are r e p l a c e d by  r e l a t i v e l y high—speed feedback loops where an e r r o r s i g n a l i s a p p l i e d to a high—gain a m p l i f i e r and the a m p l i f i e r output can be used as the optimal c o n t r o l v a r i a b l e . F i g . 3.4 3.4  The block diagram of  shows t h i s technique.  Analogue Computer Study of the Sounding Rocket Problem The analogue computer  technique d i s c u s s e d i n S e c t i o n  w i l l now be a p p l i e d to the sounding rocket problem, 231-R  analogue computer  computer  A PACE  was used and a schematic diagram of the  program i s i l l u s t r a t e d i n F i g , 3.5.  computed backward  3.3  The problem i s  i n time.  In F i g . 3.5 the e r r o r s i g n a l i s given by the switching function f  = e ( t ) k mg - D ( l +  )  (3.28)  e and the c o n t r o l v a r i a b l e by u ( t ) = -K e ( t ) The reason f o r computing the problem backward  (3.29) i n time  i s that the f i n a l v e l o c i t y , a l t i t u d e , and mass are known. f o r backward  time computation no i t e r a t i o n i s r e q u i r e d f o r  determining the optimal t r a j e c t o r y . The numerical v a l u e s chpsen are the f o l l o w i n g :  Thus  Impulsive  Boosting  or Maximum Thrust  Optimal  u(t)  1  Mt ) o  Control Law  Performance  P  Criterion  P i g . 3.4  Synthesis of optimal c o n t r o l l e r s by means of analogue computers  ON  P i g . 3.5  Analogue computer program f o r the sounding rocket problem  78  h  =  f  4 , 8 8 9 , 5 0 0  ft.  = 1 0 slug V  f  = 0 ft/sec  D = k V e" 2  V  e  =  a  =  K =  ft/sec  5500  k = 10~  a h  4  slug - f t ,  1/22000  ft"  1  100  The r e s u l t i n g state v a r i a b l e s are shown i n F i g . 3 . 6 where T =  - t i s the backward time v a r i a b l e . The f u n c t i o n E ( T )  when  E  (7" ) 2  =  ®*  At  T  h  =  2  V  =  2  i s used to determine the i n s t a n t T > 2  7"  2  the f o l l o w i n g values are obtained:  62,600  ft.  = 5 , 3 1 3 ft/sec = 1 0 slug sec.  X,  =  u  = 0 . 7 2 sl^g/sec.  2  161.3  and the feedback computation of t h r u s t based on E ( T ) = 0 i s introduced by means of a r e l a y . are  At T =  , the f o l l o w i n g values  obtained: =  0  =  2275  m.  =  20.85  slug  1^  =  179.5  sec.  V  1  ±  ft/sec  u^ = 0 . 5 slug/sec At  T=  Tt Q  the i n i t i a l mass i n c l u d i n g f u e l i s  79 4,889,500 4,000,000  2,000,000 "  161.3 lOOe 32,200 20,000-10,000-T sec  0  F i g . 3.6  Experimental r e s u l t s f o r the sounding problem  rocket  80 Vj m  o  ~  l  m  ^V"^ e  e x p  - 31.5 At the  i n s t a n t T=  7", 2  slug  a r e l a y switches the  control  v a r i a b l e u i n t o the  input  of the mass i n t e g r a t o r *  coasting  input  to the mass i n t e g r a t o r i s zero and  subarc the  mass i s constant* and  At the  final altitude  e r r o r s i g n a l e ( T ) i s m^g.  the  with T i t can be to zero.  ^he  2  Since both D and V  r e l a y operates and  v a r i a b l e t h r u s t subarc.  the the  the v e l o c i t y i s zero  seen from (3.28) that the  T= T  At  For  increase  e r r o r s i g n a l decreases the rocket  enters  the  When h = 0, a second r e l a y i s used  to clamp a l l i n t e g r a t o r inputs  at zero, f r e e z i n g the  operation,  (12) Leitmann Section of the  2.3)  and  v  ' has  used the a n a l y t i c a l r e s u l t s  an IBM  sounding rocket  used i n t h i s s e c t i o n .  701  digital  computer f o r the  problem with the His r e s u l t s h  2  = 5,308 f t / s e c  m  2  = 10  was  slug  = 0.74 - 7"  T  same given data as  = 62,576 f t .  2  2  solution  are  V  u  (see  2  slug/sec  = 18.7  sec.  nij = 21 slug m =31.4 slug 0  6  Uj = 0.51 i n general t h i s approach of using compute the  slug/sec, the a n a l y t i c a l r e s u l t to  s o l u t i o n i s not p o s s i b l e , since the a n a l y t i c a l  r e s u l t i s not  obtainable.  general a p p l i c a b i l i t y .  However, the  approach of F i g . 3.4  has  Comparison of the r e s u l t s shows t h a t  the  81 experimental  r e s u l t s f o r the sounding  rocket problem are v e r y  satisfactory. 3.5  Some Other P o s s i b l e Optimal C o n t r o l l e r s In the preceding  s e c t i o n the switching f u n c t i o n given by  (3.28) has been used f o r the s y n t h e s i s of the optimal v a r i a b l e u by an analogue computer.  The switching i n s t a n t 7"  2  s e p a r a t i n g the c o a s t i n g subarc from the i s determined  by f ( T ) = 0.  control  v a r i a b l e t h r u s t subarc  On the v a r i a b l e t h r u s t subarc a  0  feedback loop around a high-gain a m p l i f i e r i s used to s a t i s f y the c o n d i t i o n f o r optimal c o n t r o l which r e q u i r e s t h a t e ( T ) = 0. I t should be noted t h a t the switching f u n c t i o n f (T) i s a S  f u n c t i o n of s t a t e v a r i a b l e s .  In the general case of F i g * 3.1  such a switching f u n c t i o n may not be o b t a i n a b l e .  In t h i s  case  some other means must be used i n order to determine the c o n t r o l v a r i a b l e u f o r the optimal t r a j e c t o r y .  These can be obtained  from the switching f u n c t i o n e  k m K  2  - V  and the f i r s t i n t e g r a l  5  X  e  (3.30)  3  (provided i t e x i s t s , see (A.18)). y  e  3  k C - X  2  V - X ( g + |) - 0 ( X - X 3  5  3  ^  )  (3.31)  Therefore there are three p o s s i b l e f u n c t i o n s which can be used f o r the s y n t h e s i s of c o n t r o l v a r i a b l e u f o r the optimal t r a j e c t o r y by means of a h i g h — g a i n a m p l i f i e r . e  1  e  2  £  3  =  These are  = mg - D ( l + (r-) e  (3.32)  = m X  (3.33)  C - X  5  2  - V  e  V - X  X  3  3  ( g  +  £ )  -f3(X - X 5  3  ^) (3,3.4)  82 A switching f u n c t i o n of the type given by since i t r e s u l t s i n an extremely  (3.32) i s p r e f e r a b l e  simple c o n t r o l l e r .  Otherwise  the Lagrange m u l t i p l i e r s must be computed.  In such a case  and  was  can be used i n the same manner as  used.  I t should  be noted, howevery t h a t e-j = 0 f o r the complete t r a j e c t o r y  and  i s not, therefore> a switching f u n c t i o n even though i t can be used to synthesize the c o n t r o l v a r i a b l e In order to use  u.  (3.33), the Lagrange m u l t i p l i e r s X^  and  must be solved simultaneously w i t h the equations of motion. I t i s of i n t e r e s t to note that X^ and X^ can be obtained by s o l v i n g the two  differential X  3  equations  (see (3.1l) and  (3.12))  = - ^ f  (3.35) e  e I f the f i r s t  i n t e g r a l i s to be used f o r s y n t h e s i z i n g the  control  v a r i a b l e u f o r the optimal t r a j e c t o r y , the complete set of E u l e r Lagrange equations must be s o l v e d . than the case of s o l v i n g equations  This i s much more complicated (3.35) and  (3.36).  83 4.  4.1  THE MODIFIED STEEPEST DESCENT METHOD  Introduction Computational methods f o r the s o l u t i o n of  problems have had  two  d i r e c t approach and  primary d i r e c t i o n s i n the past:  the i n d i r e c t approach.  approach, equations of motion are initial  optimization  c o n t r o l v a r i a b l e and  The  In the d i r e c t  solved by s e l e c t i n g an  then performing an i t e r a t i o n  the c o n t r o l v a r i a b l e so that each new performance f u n c t i o n to be optimized.  i t e r a t i o n improves The  on the  i n d i r e c t approach  i n v o l v e s the development of an i t e r a t i v e technique f o r s o l v i n g the equations of motion and The  the Euler-Lagrange  equations.  d i r e c t approach i s u s u a l l y a s s o c i a t e d w i t h the  method or the method of steepest  descent.  In t h i s chapter a modified described  gradient  steepest  descent method i s  f o r the s o l u t i o n of o p t i m i z a t i o n problems which can  be  programmed on analogue computers. 4.2  Basic The  discussed  Concept of the M o d i f i e d Mayer f o r m u l a t i o n i n Chapter 2.  Steepest Descent Method  of v a r i a t i o n a l problems has  In the case of the four rocket  problems s t u d i e d i n Chapter 3, the optimal  u f o r the optimal and  t r a j e c t o r y may  In g e n e r a l ,  b a s i s of the m o d i f i e d  f o r the optimum value  can  feedback  the c o n t r o l v a r i a b l e  i n v o l v e Lagrange m u l t i p l i e r s  the computation of u becomes much more The  flight  control variable  be determined as a f u n c t i o n of state v a r i a b l e s and c o n t r o l methods can be employed.  been  steepest  complicated. descent i s to  search  of the performance f u n c t i o n by r e p l a c i n g a  84 search i n f u n c t i o n space by a search i n parameter space. g r e a t l y reduces the d i m e n s i o n a l i t y of the problem.  This  The p e r -  formance f u n c t i o n i s considered  as a f u n c t i o n of unknown  terminal c o n d i t i o n s .  state of the system i s determined  The f i n a l  by the s o l u t i o n of the equations of motion and the i n i t i a l values  of the state v a r i a b l e s .  optimal  The c o n t r o l v a r i a b l e f o r the  t r a j e c t o r y i s determined by the state v a r i a b l e s and  Lagrange m u l t i p l i e r s . be considered  The performance f u n c t i o n may, t h e r e f o r e ,  as a f u n c t i o n of the unknown terminal  f o r the s t a t e v a r i a b l e s and Lagrange m u l t i p l i e r s .  conditions In theory, i f  the t e r m i n a l c o n d i t i o n s f o r the state v a r i a b l e s and Lagrange m u l t i p l i e r s are a l l known, the o p t i m i z a t i o n problem can be solved by the method d i s c u s s e d i n S e c t i o n 3.2. In many p r a c t i c a l problems the terminal u s u a l l y not a l l known.  This complicates  c o n t r o l v a r i a b l e u f o r the optimal some of the terminal  c o n d i t i o n s are  the synthesis of the  trajectory.  In such cases  c o n d i t i o n s may be approximately determined  by some means, and then the performance f u n c t i o n i s optimized with respect to the remaining terminal g r a d i e n t method.  This i s the e s s e n t i a l f e a t u r e of the modified  method of steepest Consider  c o n d i t i o n s , using the  descent.  the problem of minimizing  the performance  function P =P(  a i  , .  • . t  (4.1) =  P(t,x) J t  subject to the equations of motion  0  85 x. = f . ( t , x , u ) , j = l , . . . , n . J  (4.2)  J  where x = (x, ,...,x )• u = (u,»... u ), and the f u n c t i o n s 1 n 1 m 7  9  and f . are given f u n c t i o n s 3 Following augmented  P  of t h e i r arguments.,  the theory of c a l c u l u s of v a r i a t i o n s , the  function n (4.3) 3=1  i s formed which s a t i s f i e s the Euler-Lagrange equations ( s ) ~" "—"—- = 0 6x..< ...6x.  dt  j  6F  s  J.  P  » o o )  n» (4.4)  a,  = o,  6u,  j =  Ii. '— I  ;i o e » ^ H i  i  and the t r a n s v e r s a l i t y c o n d i t i o n n  dP + (F -  x  N  V  — — x.; dt + > — 7 — dx. . /Ox. 4-v6x. 3=1 3 3=1 3 — t J  = 0  J  (4.5)  0  S u b s t i t u t i n g the f u n c t i o n F i n t o equations (4.4) and (4.5) gives  n X  3  =  "*  of,  Z i=l  A  (4.6)  X. i 6 T  and J  n  dP -  n X . f . dt + > X - dx ^3 3 ^_—/ 3 3 3=1 3=1 ->t  f 0  (4.7)  0  I f the f u n c t i o n F does not depend on t e x p l i c i t l y , the f i r s t integral exists!  86  (4.8)  = C  that i s  n  "S~^ /  X.f  i  .  3 3  = C  (4.9)  3=1 I t f o l l o w s from the t r a n s v e r s a l i t y c o n d i t i o n that i f e i t h e r t or t  n  i s f r e e the f i r s t The  computational  i n t e g r a l i s equal technique  to zero.  f o r the s o l u t i o n of the  o p t i m i z a t i o n problem i s to solve equations  (4.2) and (4.6)  subject to the c o n d i t i o n s (4.7) and (4.9) so t h a t the perform mance f u n c t i o n P i s a minimum.  Note that the t r a n s v e r s a l i t y  c o n d i t i o n y i e l d s i n f o r m a t i o n about the terminal values A.  I f the f i r s t  i n t e g r a l i s known, i t may give some i n f o r m a t i o n  about the t e r m i n a l values of x and A . terminal values  of the  However, u s u a l l y not a l l  of x are given and not a l l terminal values of  X can be determined by the t r a n s v e r s a l i t y c o n d i t i o n and the first  integral. For a minimum problem having n s t a t e v a r i a b l e s x. the  performance f u n c t i o n P w i l l , i n g e n e r a l , have n unknown parameters a.. 3  I f the f i r s t  i n t e g r a l i s known (provided i t e x i s t s ) ,  (n-1) unknown parameters are independent. the d i m e n s i o n a l i t y a f i r s t  only  In order to reduce  approximation of these  ( n - l ) unknown  parameters may be obtained by computing a subclass of admissible t r a j e c t o r i e s which s a t i s f y the equations  of motion and the  known terminal c o n d i t i o n s of the state v a r i a b l e s . of admissible  The subclass  t r a j e c t o r i e s i s taken to s a t i s f y some, but not  n e c e s s a r i l y a l l , the t e r m i n a l c o n d i t i o n s f o r X, values of x and X f o r the optimal  The i n i t i a l  t r a j e c t o r y can now be determined  87 by the method of steepest descent. In g e n e r a l , a computer program using the modified steepest descent method c o u l d proceed as f o l l o w s .  In order to  s i m p l i f y the d i s c u s s i o n i t i s assumed that more i n i t i a l v a l u e s of the state v a r i a b l e s than f i n a l values are known. (1)  A suitable control u  n  i s s e l e c t e d as a f i r s t  approximation and the equations of motion are solved forward i n time.  I f ^ ( t ^ ) i s known and  x ^ ( t ^ ) i s unknown, an approximation to ^ ( t ^ ) can be obtained by a d j u s t i n g x, ( t . ) u n t i l the f i n a l value of Xj^ takes on the p r e s c r i b e d value X j ^ t ^ K I f both t e r m i n a l values x. ( t . ) and x. (t„) of a k 1 k f state v a r i a b l e x ^ ( t ) are unknown, a f i r s t a p p r o x i mation to ^ ( t ^ ) can be determined by minimizing the performance f u n c t i o n P by the steepest descent method.  The t r a j e c t o r i e s determined i n t h i s manner  form a subclass of admissible t r a j e c t o r i e s . (2)  Y i t h the p r e v i o u s l y determined admissible t r a j e c t o r y the equations of motion and Euler-Lagrange are  equations  simultaneously solved backward i n time.  The  unknown t e r m i n a l values X j ( t ^ ) are adjusted at t = t ^ by i t e r a t i o n u n t i l the p r e s c r i b e d  initial  v a l u e s of the corresponding X• are obtained. first  approximation of i n i t i a l values f o r x and X  has now (3)  A  been determined.  The equations of motion and Euler-Lagrange equations are  simultaneously solved by the feedback  method (see F i g . 1.2)  forward i n time.  t r o l l e r i s i n t r o d u c e d by the feedback  control  The  con-  control  88 technique i s noted.  and the value of the performance f u n c t i o n This subclass of t r a j e c t o r i e s have a  v a r i a b l e t h r u s t subarc and the t h r u s t f o r t h i s subarc (4)  i s determined  by the optimal c o n t r o l law.  The unknown i n i t i a l values of x and A are adjusted according to the m o d i f i e d method of steepest  descent  u n t i l the performance f u n c t i o n i s minimized. 4.3  P o s s i b i l i t y of P r a c t i c a l A p p l i c a t i o n s In p r a c t i c e , there i s o f t e n a need f o r a low cost and  comparatively  simple o n - l i n e method f o r the s o l u t i o n of optimal  c o n t r o l problems. techniques  At the present time many of the computational  e x i s t i n g i n v a r i o u s i n d u s t r i e s o f t e n r e q u i r e the use  of a l a r g e c a p a c i t y general purpose d i g i t a l economical  computer. For  reasons, t h i s may not be acceptable i n many p o s s i b l e  applications.  However, the modified steepest descent method can (4)  be used to r e a l i z e comparatively The  instantaneous  simple o n - l i n e c o n t r o l l e r s .  c o n t r o l p o l i c y i n r e a l time may be obtained  from an analogue computer which operates on a f a s t time The  t r a j e c t o r y i n s t a t e space i s solved by an analogue computer  and a d i g i t a l  computer s t o r e s the data f o r the steepest  adjustment of the unknown parameters. descent method takes account  descent  This m o d i f i e d steepest  of random disturbances since a new  c o n t r o l p o l i c y i s computed f o r each t r a j e c t o r y * 4*4  scale.  (see F i g . 4.1).  Further Investigations The general i d e a of the m o d i f i e d steepest descent method  based on the i n d i r e c t approach of the c a l c u l u s of v a r i a t i o n s  ^(t) u(t)  Process  x'(t) n '  / u(t) Hill-Climbing  a.  Trajectory Computer  Computer (Digital)  P i g , 4*1  a  n  P = P(a ,..,a ) 1  n  (Analogue)  An optimal c o n t r o l l e r f o r a general  process  00  90 seems a very e f f e c t i v e computational method.  The h i g h speed  analogue computer i s p a r t i c u l a r l y s u i t a b l e f o r the determination of t r a j e c t o r i e s and feedback methods can be used to the  control v a r i a b l e .  synthesize  While computational experience w i t h t h i s  method i s l i m i t e d at the present time, i t s p o t e n t i a l as a computational scheme f o r p r a c t i c a l a p p l i c a t i o n s deserves f u r t h e r studie s. I t i s suggested t h a t f u r t h e r i n v e s t i g a t i o n s i n t h i s method should be pursued to f a c i l i t a t e p r a c t i c a l a p p l i c a t i o n s to the following 1.  problems: The a p p l i c a t i o n o f d i g i t a l methods f o r a u t o m a t i c a l l y  h i l l - c l i m b i n g or gradient optimizing  the performance  function. 2.  Hybrid computational methods f o r a u t o m a t i c a l l y  adjusting  the unknown parameters. 3.  The extension of the method to problems of many degrees of freedom.  A l l these problems must be l e f t open f o r f u t u r e  investigations*  91 5.  5.1  FLIGHT SIMULATOR AND  ANALOGUE SIMULATION  Introduction Analogue  computers may be d i v i d e d broadly i n t o d i r e c t  analogues and i n d i r e c t , or f u n c t i o n a l , analogues. of operation  of the d i r e c t analogue computer  The p r i n c i p l e  i s based on a one-  to-one correspondence between the behaviour of the analogue system and that of the p h y s i c a l  system under study.  In the  i n d i r e c t or f u n c t i o n a l analogue computer, the equations which describe  a physical  system are formulated by components, such  as summers, i n t e g r a t o r s , m u l t i p l i e r s , e t c . The f l i g h t  simulator  i s a f u n c t i o n a l analogue  computer  of the e l e c t r o m e c h a n i c a l type and i s i d e a l l y s u i t e d f o r the s o l u t i o n of t r a j e c t o r y problems. f l i g h t problem, a CF-100 f l i g h t  In order to study the simulator  rocket  has been s u i t a b l y  modified. 5.2  Basic  Components of the F l i g h t Simulator  There are f i v e b a s i c  components of the f l i g h t  These are the summer, s e r v o - a m p l i f i e r , detector  and r e l a y .  r e s o l v e r , phase  simulator. sensitive  By means of these components mathematical  operations can be performed.  The summing a m p l i f i e r , or the  summer, c a r r i e s out the a r i t h m e t i c  operations of sign  m u l t i p l i c a t i o n by a constant and summation.  The i n t e g r a t i o n i s  c a r r i e d out by an electromechanical i n t e g r a t o r . c o n s i s t s of a s e r v o - a m p l i f i e r ,  inversion,  This  integrator  a servo-motor and a tachometer.  A gear box i s used to couple the servo-motor to a l i n e a r  92 potentiometer  which converts the s h a f t angle i n t o a v o l t a g e .  Furthermore, the i n t e g r a t o r i s a l s o used to generate and to c a r r y out m u l t i p l i c a t i o n and d i v i s i o n .  The r e s o l v e r  performs t r i g o n o m e t r i c operations i n v o l v i n g the of  coordinates.  functions  transformation  The phase s e n s i t i v e d e t e c t o r i s a device used  to d e t e c t the phase change of an input s i g n a l with r e s p e c t to a reference s i g n a l .  A r e l a y i s energized when the input s i g n a l  changes i t s phase. 5.3  S i m u l a t i o n of the Optimal  Control  Law  This s e c t i o n i s devoted to the s i m u l a t i o n of the  optimal  c o n t r o l law f o r the z e r o - l i f t rocket f l i g h t problem d i s c u s s e d i n Chapter 3.  For the programming of t h i s problem a l a r g e  number of m u l t i p l i e r s and f u n c t i o n generators This cannot be handled  are r e q u i r e d .  by most o r d i n a r y analogue computers since  only a small number of m u l t i p l i e r s and f u n c t i o n generators are normally of  a flight  problem.  available*  The  electromechancial  computing u n i t s  simulator are i d e a l l y s u i t e d f o r t h i s type  In the study of the theory of optimal rocket  of flight,  i t has been shown t h a t the optimal t r a j e c t o r y c o n s i s t s of three subarcs.  A s s o c i a t e d w i t h each subarc  t r o l f o r the c o n t r o l parameter 0. assumed, one  of the subarcs may  i s a mode of con-  I f impulsive boosting i s  be computed a n a l y t i c a l l y .  If  the t h r u s t program c o n s i s t s of maximum t h r u s t , v a r i a b l e t h r u s t and zero t h r u s t , the maximum t h r u s t mode must be i n c l u d e d i n the s i m u l a t o r . thrust  In general there are, t h e r e f o r e , three modes of  control. I t can be seen from the Appendix t h a t the c o n t r o l parameter  93 (3  f  appears i n both equations  (A.46) and (A.48).  The three modes  of t h r u s t c o n t r o l must, t h e r e f o r e , be a p p l i e d to these two equations. The  sequence of the modes i s important.  I t f o l l o w s from  the theory of rocket f l i g h t that the sequence of these modes ares Mode Is  (3 = B , constant t h r u s t , "max Fuel consumption i s at a constant rate and the mass i s a l i n e a r f u n c t i o n of time,  m = m„ - (3 t . 0 max = 0, v a r i a b l e t h r u s t . r  Mode 2l  V The  mass i s constrained.to  s a t i s f y the  v a r i a b l e t h r u s t c o n d i t i o n f o r optimal flight. P = 0, zero t h r u s t .  Mode 3:  The mass i s constant. The  zeros of the f u n c t i o n A  f (m,/0  £ X  V  5  can be used to d e f i n e the three The  - X  3  ^  subarcs  (5.1) (see F i g . 5.1).  switching from Mode 1 to Mode 2 i s performed i n the  simulator by a phase s e n s i t i v e d e t e c t o r and a r e l a y . the r e l a y i s i n the p o s i t i o n f o r maximum t h r u s t . becomes zero, the r e l a y switches  to Mode 2.  In Mode 1,  When f (m.,\)  During Mode 2 the  c o n t r o l parameter 0 i s i m p l i c i t l y c o n s t r a i n e d so that f(m,A) = 0. For Mode 3, the s i g n a l r e p r e s e n t i n g the c o n t r o l parameter 0 i s shorted to ground. 5.4  A n a l y s i s of a Test Problem In order f o r the simulator to perform  satisfactorily,  f(m,X)  P i g . 5.1 various  Three modes of t h r u s t  u n i t s must be c a l i b r a t e d .  control  The c a l i b r a t i o n can be  best performed by s o l v i n g a simple problem of f r e e motion described  by the f o l l o w i n g d i f f e r e n t i a l equations: x = V cos 0 h = V sin 0 V = -g s i n 0 O = - ^ cos 0  The i n i t i a l  conditions  at t = 0 are  x(0) = 0 h(0) = 0 V(0) = V ' o 0(0) = 0  (  5  95 where 0 < 0 < J . q  The s o l u t i o n of t h i s set of d i f f e r e n t i a l equations i s x = V h = V V  2  = V  o  cos 0 t o  o  sin G t 4 o 2 g ° t 2  q  + g t 2  2  2  - 2 g V  tan (| + J ) = tan ( ^ +  O  /Jt  2  sin0  - 2^  o  t  (5.3)  sin 0 t + ^ _ Q  E l i m i n a t i n g the s i n 0 from the second and the t h i r d equations of (5.2) gives  h = -Y V/g I n t e g r a t i n g the above equation y i e l d s V  2  = Y  2  - 2 g h  (5.4)  Since V cannot be zero, i t f o l l o w s from the second equation of (5.2) that s i n 0 must be zero at h . max  Furthermore, because of 7  ( 5 . 4 ) , V i s a minimum when h i s a maximum. From the s o l u t i o n f o r the v e l o c i t y of ( 5 . 3 ) , i t i s seen that V  . = V cos 0 min o o  (5.5) ' s  which i s extremely u s e f u l f o r c a l i b r a t i o n purposes. Another important f a c t i s that the v e l o c i t y i n the xd i r e c t i o n , that i s , x i s always constant.  This gives a good  check f o r the o p e r a t i o n of the s i m u l a t o r . D i f f e r e n t i a t i n g the s o l u t i o n f o r the v e l o c i t y and equating i t to be zero gives  96 V t = — g  sin 0  (5.6)  and a t t h i s i n s t a n t the v e l o c i t y reaches The  i t s minimum.  above equations were used to scale the v o l t a g e s on  the simulator so t h a t f o r the mass used the t r a j e c t o r y a convenient 5.5  range of an xy-recorder.  Experimental  Test of the M o d i f i e d Steepest Descent Method  The b a s i c i d e a f o r the method of modified descent  covered  has been d i s c u s s e d i n Chapter  4.  steepest  I t would e v i d e n t l y  be p r o f i t a b l e to study a p a r t i c u l a r problem which can l e a d to a b e t t e r understanding Consider  of the nature  of the method.  the z e r o - l i f t rocket f l i g h t problem.  The p e r -  formance f u n c t i o n to be minimized i s the f u e l consumption. the i n i t i a l mass m  Q  i s assumed to be given, the problem i s  e q u i v a l e n t to maximizing the f i n a l mass m^. final  If  The i n i t i a l and  c o n d i t i o n s are x(t )  = x  f  h(t ) = h  f  f  f  (5.7)  where m ,  x^ and h^ are given v a l u e s .  Q  The f o l l o w i n g values of  the s t a t e v a r i a b l e s are unknown at the terminal p o i n t s : V„, m „ .  Here m„ i s to be maximized.  The  t r a n s v e r s a l i t y c o n d i t i o n f o r t h i s problem i s  © , 0 q  97 - C dt + Xj dx + X  dh + X  2  3  dV + X  d© + (\ -l)dm  4  = 0  g  t  o (5.8)  The q u a n t i t i e s t , t ^ , Vft  ©»  Q  ^40 XJJ  =  °' ^3f and X ^ The  =  °» *4f  =  0  a  n  > £  d  are f r e e , s o that C = 0,  m  0  ^5f  =  l f  a  n  ^10' ^20' ^30'  d  ^50*  are unknown. first  integral  XjV cos 0 + X V 2  (see(A.55)) i s  s i n 0 - X ( - + g s i n 0) - X 3  4  f  cos © -  8(X  5  V 3  and f o r t = t ^ , B = 0, X ^ 3  tan  Equation  m  = 0 and X ^  = 0.  Hence  ©„ = - ^  r  x  (5.10) 2 f  (5.10) gives a r e l a t i o n between ©^, X-^  the Euler-Lagrange  and X ^ .  equation (A.49) i t i s seen t h a t X^  From  i s a con-  stant f o r the e n t i r e optimal t r a j e c t o r y . For t h i s p a r t i c u l a r problem ©  can not be 90°, as can be  q  seen from the equation of motion (A.47) f o r 9. and the l i f t  i s zero, '  © i s zero i f V  = 90°,  i s not zero, thus the o '  f i n a l p o i n t (x^,h^) cannot be reached.  If 0  cannot be zero, otherwise © w i l l be i n f i n i t e point.  If 0  q  <  90°, then V  at the  initial  Thus an i n i t i a l v e l o c i t y i s e s s e n t i a l which can be  obtained by impulsive b o o s t i n g .  In t h i s case, the  computation  s t a r t s with the v a r i a b l e t h r u s t subarc, since the b o o s t i n g subarc i s very short and may  be n e g l e c t e d .  98  Consider now the case of impulsive  boosting  i s no c o n s t r a i n t on the magnitude of the t h r u s t . time at the end of b o o s t i n g ,  where there L e t t ^ be the  then  t, - t = A t =" 0 1 o x, = x 1 0 h = h 1 o  =0 =0  n  £  V,  0 V  l  ~ o  hi  -  m  m  e  x  x  21 ~  20  A  31 ~  30  X  4  1  51  S  (5.11)  e  io  A  A  (~  p  =  A  X  50  4  = 0  Q  + X  3 0 e^m7-m-) 1 o V  At t = t ^ , the v a r i a b l e t h r u s t subarc s t a r t s , and V X  51  =  X  31mf  <- > 5  12  I f the computation s t a r t s at t = t ^ , the i n i t i a l values f o r the state v a r i a b l e s :  , m^ and 0^ are unknown.  However,  and m^  are r e l a t e d by the r e l a t i o n  m  i  = m  Q  V. exp (- = i ) e  I f the magnitude of the t h r u s t i s constrained  (5.13)  by the  condition  where |3  i s the maximum c o n t r o l parameter, the approximation of  99  (5.11) s t i l l can be a p p l i e d , but the optimal t r a j e c t o r y w i l l with maximum t h r u s t subarc.  Since the i n i t i a l v e l o c i t y V  q  start  i s zero,  some a u x i l i a r y device i s r e q u i r e d to avoid that 0 be i n f i n i t e at the s t a r t .  This can be done by h o l d i n g the rocket on a launcher  w i t h maximum t h r u s t f o r a n e g l i g i b l y short time, and the rocket then s t a r t s w i t h a maximum t h r u s t subarc with an i n i t i a l O  q  l e s s than 90°.  angle  T h i s i s e q u i v a l e n t to the problem of s t a r t i n g  with an i n i t i a l v e l o c i t y  ^ 0 and an i n i t i a l  mass given by  V.  m. £ m  exp (- ^ )  Q  (5.15)  Thus the optimal t r a j e c t o r y s t a r t s w i t h the f o l l o w i n g i n i t i a l conditions: = At a* o  t. - t  i  o x. ^ x = 0 1 o h. * h =0 1 o V. m  i  £0 =  e  2i  *3i *4i 5i  k  In The  x  0  A  10  04.  X  20  Q±  X  30  *li X  V.  m  so  p  (~ (5.16)  40  X  X  50  t h i s case the switching f u n c t i o n may not reach zero at t = t ^ . optimal t r a j e c t o r y must then s t a r t with a maximum t h r u s t  subarc.  When the s w i t c h i n g f u n c t i o n (5.1) i s zero, the t r a j e c t o r y  enters the v a r i a b l e t h r u s t subarc.  The computation s t a r t s a t  100  t = t i. with the i n i t i a l v a l u e s of the state v a r i a b l e s V., 1' and ©^ unknown.  However, "V\ and ^  = m  exp  Q  are r e l a t e d by the  m. l  equation  (- ^ )  (5*17)  For s i m p l i c i t y , the drag f u n c t i o n D used i n the s i m u l a t i o n i s assumed to have the  form  D = D(V,h) = k V * k  To determine  e"  2  (5.18)  a h  r2 1 + ah  a first  approximation  f o r the i n i t i a l  values  of "V\ , HK and ©^, the t r a j e c t o r y i s considered to c o n s i s t of a s u i t a b l e constant t h r u s t subarc or a maximum t h r u s t subarc a zero t h r u s t subarc* (5,17). of  A value  i s s e l e c t e d and  and  computed by  A s u i t a b l e i n i t i a l value ©^ i s chosen and the l e n g t h  the constant t h r u s t subarc v a r i e d so that the f i n a l p o i n t  (x^jhp) i s reached. f o r v a r i o u s ©^,  F i g . 5.2  i l l u s t r a t e s the r e s u l t s obtained  The value of m^  f o r each of these  trajectories  i s noted and the r e s u l t s are p l o t t e d as shown i n F i g . 5*3o In  t h i s manner 0^, V\  A particular  and  are approximately  set of data i s shown i n F i g . 5,4.  on the simulator are i n terms of degrees Since ©^ i s now that A - j ^ and X^f  determined.  A l l quantities  of shaft  rotation.  known at the f i n a l p o i n t , i t f o l l o w s  r e l a t e d by  a r e  A  2f  =  "  c o t  G  f  Note that at the f i n a l p o i n t , A-^  X  l f and X^f  (5*19) a r e  the only unknowns.  101  F i g . 5.3  Determination of approximate i n i t i a l values f o r the state v a r i a b l e s  102  m. 202°+ 201' 200 199  C  u  198°I  •v Fig.  5.4  If  -0. 70° 71° 72° 73°  74° 75°  I  A p a r t i c u l a r s e t of approximate i n i t i a l v a l u e s of the state v a r i a b l e s  is.known,  can be computed by (5.19),  T h e r e f o r e , by  s e l e c t i n g a X-^^t the equations of motion and the Euler-Lagrange equations can be s o l v e d backwards i n time. plier All  X-^f i s v a r i e d u n t i l  The Lagrange m u l t i -  the c o n d i t i o n X^^ = 0 i s s a t i s f i e d .  i n i t i a l v a l u e s are now s p e c i f i e d and i t i s then p o s s i b l e to  compute improved t r a j e c t o r i e s by i n t r o d u c i n g the optimal for  the t r a j e c t o r y and s o l v i n g i t forward i n time.  The f i n a l  m^ i s now considered as a f u n c t i o n of the parameters: A  2i  7  A  3i  T  a  determined of  n  d  0  P"ki  m u m  control mass  9^, A ^ ,  v a l u e s of these parameters can be  by the m o d i f i e d steepest descent method.  The  adjustment  the parameter v a l u e s terminates when m^ reaches a maximum*  This approach  proved f a i r l y  s u c c e s s f u l on the f l i g h t  The numerical r e s u l t i s i i i terms of degrees Since the f l i g h t  simulator.  of ghaft r o t a t i o n .  s i m u l a t o r does not have a high accuracy, no  103 p r e c i s e numerical r e s u l t s have been obtained.  However, a set  of t r a j e c t o r i e s s i m i l a r to F i g . 5.2 c o n s i s t i n g of a maximum t h r u s t subarc* a v a r i a b l e t h r u s t subarc and a zero t h r u s t can be obtained* m_  F i g . 5.5 i l l u s t r a t e s the performance  subarc  function  considered as a f u n c t i o n of the parameter a, .  m^ "  202° +  0  Fig.  a,k opt.  5.5  -*- ak  Optimum performance f u n c t i o n  , the i n i t i a l At the p o i n t a^ = a. opt.  values of the  are a  l  = Q  ±  = 73°  ±  =  a  2  = V  a  3  = m  a  4  = X  a  5  ±  1±  =  50°  = 330° = 168°  X  2 i  = 219°  a  6  = X  3 i  = 253°  a  ?  = X  4 i  = 0°  (This i s known)  For t h i s problem the Lagrange m u l t i p l i e r X^ i s obtained from the first  integral.  Therefore, X^^  i s f i x e d by the f i r s t i n t e g r a l .  104 6.  General  CONCLUSION  optimal c o n t r o l problems formulated by the method  of the c a l c u l u s of v a r i a t i o n s with p a r t i c u l a r emphasis on the problem of Mayer have been s t u d i e d .  S p e c i a l cases of optimal  c o n t r o l can be r e a l i z e d by means of feedback c o n t r o l .  The  Lagrange m u l t i p l i e r s can be e l i m i n a t e d and the c o n t r o l v a r i a b l e f o r the optimal t r a j e c t o r y i s then a f u n c t i o n of the s t a t e variables only.  In t h i s case the optimal c o n t r o l system can be  t r e a t e d as an optimal feedback c o n t r o l system. methods are convenient  Analogue computer  f o r the s o l u t i o n of such problems.  The m o d i f i e d steepest descent method i s s u i t a b l e f o r the s o l u t i o n of c e r t a i n c l a s s e s of optimal c o n t r o l problems. (1)  For v e r y complex problems the d i m e n s i o n a l i t y of the problem can be reduced by u s i n g conventional i t e r a t i v e and g r a d i e n t methods to determine subclasses of admissible  trajectories  i  s a t i s f y i n g some, but not n e c e s s a r i l y a l l , conditions.  of the t e r m i n a l  Thei modified steepest descent method can then  be used to optimize the performance f u n c t i o n which i s con-  ! V  s i d e r e d to be a f u n c t i o n of the remaining  terminal con-  ditions. (2)  Simulator and analogue computer r e s u l t s show that the method i s p r a c t i c a l and can be used to synthesize r e a l time optimal  (3)  controllers.  For complex problems hybrid-computers  are e s s e n t i a l and  are of considerable f u t u r e i n t e r e s t .  This t h e s i s has  d e a l t mainly with the analogue p o r t i o n of the optimal controller.  The o p t i m i z a t i o n of the performance f u n c t i o n has  105 been performed by a manual search.  In an a c t u a l system  the o p t i m i z a t i o n would be performed by a d i g i t a l (see F i g . 4.1). The analogue computer  computer  i s suitable for  high speed t r a j e c t o r y computations while the d i g i t a l is  s u i t a b l e f o r the l o g i c a l operations  computer  i n v o l v e d i n the  o p t i m i z a t i o n of the performance f u n c t i o n .  The r e s u l t s of  the research undertaken show tl\at analogue computers can be used to synthesize  the c o n t r o l v a r i a b l e f o r optimal  t r o l once the c o r r e c t i n i t i a l values w e l l known that d i g i t a l  computers  are known.  con-  It is  can r e a d i l y optimize  a  performance f u n c t i o n P of s e v e r a l v a r i a b l e s by some type of gradient method.  The o p t i m i z a t i o n of P i s used to  determine the c o r r e c t i n i t i a l v a l u e s .  I t can t h e r e f o r e be  concluded that i t i s p o s s i b l e to synthesize trollers means.  optimal  con-  f o r a v a r i e t y of systems by h y b r i d computational  106  REFERENCES  1.  K e l l e y * H.J., "Gradient Theory of Optimal F l i g h t Paths", ARS J o u r n a l 30, 947-954, I960*  2.  Bryson* A*E* , C a r r o l l , F . J . , Mikami K., and. Denham, W.F., "Determination of the L i f t Drag Program that Minimizes Re-entry Heating w i t h A c c e l e r a t i o n or Range C o n s t r a i n t s Using a Steepest Descent Computation Procedure", presented at IAS 29th Annual Meeting, New York, N#T*f Jan*, 23-25, 1961.  3.  Bonn* E*V.,  4.  Bonn, E*V*» "The P r a c t i c a l R e a l i z a t i o n of Optimal C o n t r o l of M u l t i v a r i a b l e Dynamic P r o c e s s e s " , Canadian I n d u s t r i a l Research Conference, C a r l e t o n U n i v e r s i t y , Ottawa, 1964.  5.  Bellman, R.E., "Adaptive Control Processes",: P r i n c e t o n U n i v e r s i t y Press, 1961.  6.  Bellman, R.E. and Dreyfus, S.E.,  " S o l u t i o n of a Class of Optimal Control Problems by a Systematic I t e r a t i v e Technique", Canadian IEEE Convention, Toronto, 1962*  "Applied'Ifynami'Bo'PrQgramming'!, -  P r i n c e t o n U n i v e r s i t y P r e s s , 1962* 7.  Leitmann, G*,  8.  B l i s s , G.A.,  9.  " O p t i m i z a t i o n Techniques", Academic P r e s s , 1962.  "Lectures on the C a l c u l u s of V a r i a t i o n s " , U n i v e r s i t y of Chicago P r e s s , 1946* P o n t r y a g i n , L.S., Boltyansky, V.G., Gambrelidze, R.V. and Mishchenko, E.F., "The Mathematical Theory of Optimal P r o c e s s e s " , I n t e r s c i e n c e P u b l i s h e r s , John V i l e y , 1962.  10.  M i e l e ^ A.,  "General V a r i a t i o n a l Theory of the F l i g h t Paths of Rocket-Power A i r c r a f t , M i s s i l e and S a t e l l i t e C a r r i e r s " , A s t r o n a u t i c a ACTA 4, No* 4, 1958.  11.  T s i e n * H.S* and Evans, R.C., "Optimum Thrust Programming f o r a Sounding Rocket", ARS J o u r n a l 21* No. 5, 1951.  12.  Leitmann, G», "Optimum Thrust Programming f o r H i g h - A l t i t u d e Rockets", Aero/Space Eng., 16, No* 6, 1957*  107 APPENDIX  1.  The Euler-Lagrange Equations f o r Rocket P l i g h t Problems S u b s t i t u t i n g the augmented f u n c t i o n F of (2.10) i n t o  (2.11) y i e l d s the s e t of Euler-Lagrange equations l  X  =  : 2  x  =  X  * 3  X  6  X  +  7  X  U  3 6D . . 64> . d¥ " ^ 6 h 6 6h 7 5h +  X  +  ~ 1 X  c  o  ® ~ 2  s  X  s  i  ^  n  ~~in  +  +  X  4  =  X  (  A  6 OV  l ^  X  s  X  +  ( A  ^" ® ~ 2 ^ n  X  C  0  S  9  +  - X f sin 0 + X g f 6  4  X  5  ~2 ( m  =  +  x n  0  0  =  x  3 6D  V e  6 < > x  ^ 6 x " m V X  +  c  c  +  o  X  s  o  s  V  = 3 -t X  sil  g c  X  +  o  s  L  0  e  7 ( >  x  6Y 7OL  x  6 0L  +  X  - "4 h  S  X  V  > * ~ 4 "mY X  s i n to)  '  ( A  M>  I  N  ( A  W  +  V  X  5 )  *  6 )  60 T (A.7)  y  c  3 )  (A.4)  7  D  *  9  - ) + ~1~( + V  w  + X  0  3  X  m*T  ^.  4  = to-<- 3 h  P  *  3 6D  6V  7  s  0  2 )  cos 0)  - ^  —5  )  QV  L + V B s i n <o  + K  l  u  ( A  X  X =  ,  o  s  «  +  X  6 § ^ +  6* X  73^  6Y (A.8)  108 2.  The V e r t i c a l P l i g h t (The Sounding Rocket) Problem Assume that the t h r u s t d i r e c t i o n i s v e r t i c a l and that  the two a d d i t i o n a l c o n s t r a i n t s are  $ = 0 - 1 Y=  = 0  (A*9)  " •= 0  (A,10)  The equations of motion become <f = h - V = 0  (AU1)  2  B - V p ^  = V + g +  3  = 0  vj> = m + 0 = 0  (A.12) (A.13)  5  The Euler-Lagrange equations are  *2= h A  mSh  = ~ 2 X  5  ( A  ~lof  +  *  1 4 )  (A.15)  = ^ | (V 0 - D)  (A.16)  e  m  The f i r s t  integral i s  XV 2  3.  The H o r i z o n t a l  - X (g +  n  3  - 0(X - A 5  V 3  = C  (A.18)  F l i g h t Problem  I f the f l i g h t path i s assumed to be h o r i z o n t a l and i f the t h r u s t d i r e c t i o n i s p a r a l l e l to V, the a d d i t i o n a l c o n s t r a i n t s are  <£= © = 0  (A.19)  Y= » =  (A. 20)  0.  109 The equations of motion are ^  = x - V = 0  (A*21)  D - V B ^3 = V + vf  = 0  e m  (A.22)  = m + p = 0  5  (A.23)  The Euler-Lagrange equations are Aj = 0 X  3  L  =~h  (A*24) +  ^ O T  = ~ f <V m 0  0  - H  "  D  (^?5) )  +  X  4 \  ( A  - ^  -  (A 27>  = i f <* - S ^  < - «> A  2  -  2 9 )  integral i s  V - * 3 i - K l j - » 4.  2 6 )  m v  5  The f i r s t  *  3  ^  -°  ( A  The A r b i t r a r i l y I n c l i n e d R e c t i l i n e a r P l i g h t Problem I f the f l i g h t path i s r e c t i l i n e a r a t an a r b i t r a r y angle  0 with r e s p e c t to a h o r i z o n t a l plane and i f the t h r u s t i s p a r a l l e l to the f l i g h t path, the a d d i t i o n a l  c o n s t r a i n t s are  <§> = 0 - constant = 0 Y=  direction  «> = 0  „ (A*30)  The equations of motion are ip  = x - V cos 0 = 0  1  KO  0  T  Z  = h - V sin 0 = 0 . D - V p = V + sin 0 + -^-2- = 0 g  (A.31) (A.32) (A.33)  110 ^  (A.34)  = m + 0 = O  5  The Euler-Lagrange equations are  x\ = 0  (A.35)  2=^OT  A  ( A  *  X-j = -X-^ cos © - X X  3 6D  5 = ~l V (  m  "  3 6 )  sin 0  2  + m A  *  (A.37) D )  +  "I  7  ( A  *  3 8 )  m  The f i r s t i n t e g r a l i s X-jV cos © + X  V sin © - X (g  2  3  V ~ A, — ) -> m 5.  The Z e r o - l i f t F l i g h t  = C  s i n © + ^) -  8(X  5  (A*41)  Problem  I f the t h r u s t d i r e c t i o n i s tangent to the f l i g h t : path and i f the l i f t  i s assumed to be zero, the a d d i t i o n a l c o n s t r a i n t s are <£> = L = 0  (A*42)  ^ = <o = 0  (A*43)  The equations of motion are  = x - V cos © = 0  (A.44)  l f = h - V sin Q = 0  (A.45)  f  1  2  Vf  D - V 8 = V + g sin © + jjp^- = 0  3  4^4  =  vf  + f  0  c  o  s  Q  =  (A.46) (A.47)  0  = ra + 6 = 0  5  (A.48)  The Euler-Lagrange equations are X  = 0  x  *  X  3  (A.49)  on  A=-5h  (A.50)  2  * ^m On 6T A- = -X, cos © - X„ s i n © + X  " 4 ^2 X  c  o  s  (A*51)  0  XjV s i n © - X "V cos © + X-jg cos © 2  - X * X  The f i r s t  5  f sin ©  4  (A.52)  X^ =  ~ i  ( V  e  P  "  D  )  ( * > A  53  integral i s  X,V cos © + X V  s i n © - X,(g s i n © + -) - X, # cos © J m 4 V V - P ( X - X -&) = C (A.55) 0  •L  £  5  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Country Views Downloads
China 7 3
United States 3 0
City Views Downloads
Shenzhen 7 3
Ashburn 3 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0302284/manifest

Comment

Related Items