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The absolute stability of nonlinear systems Chang, Te-Lung 1970

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THE ABSOLUTE STABILITY OF NONLINEAR SYSTEMS  by  TE-LUNG CHANG B.A.Sc. i n E.E., Cheng Kung U n i v e r s i t y , 1966  A THESIS SUBMITTED IN PARTIAL FULFILMENT  OF THE  REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED  SCIENCE  i n t h e Department o f Electrical  We a c c e p t  this  Engineering  t h e s i s as conforming t o the  required  standard  Research S u p e r v i s o r Members o f Committee  Acting  Head o f Department  Members o f the Department of E l e c t r i c a l THE UNIVERSITY  Engineering  OF BRITISH COLUMBIA  F e b r u a r y , 1970  In  presenting  this  an a d v a n c e d  degree  the L i b r a r y  shall  I  further  for  scholarly  by h i s of  agree  this  written  thesis at  the U n i v e r s i t y  make i t  tha  freely  permission  for  It  of  Columbia,  British  by  for  gain  of Columbia  shall  the  requirements  reference copying  of  I agree and this  that  not  copying  or  for that  study. thesis  t h e Head o f my D e p a r t m e n t  is understood  financial  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  of  for extensive  permission.  Department  fulfilment  available  p u r p o s e s may be g r a n t e d  representatives. thesis  in p a r t i a l  or  publication  be a l l o w e d w i t h o u t  my  ABSTRACT  T h i s t h e s i s i s i n two of n o n l i n e a r systems.  p a r t s , both  In the f i r s t  two  c o n s i d e r i n g the a b s o l u t e  chapters  the s t a b i l i t y of  stability  certain  c l a s s e s o f n o n l i n e a r time i n v a r i a n t systems i n v o l v i n g s e v e r a l n o n l i n e a r i t i e s i s considered.  A number of g r a p h i c a l methods are g i v e n f o r t e s t i n g  s t a b i l i t y o f these systems.  The  form of the Popov c r i t e r i o n .  The  g r a p h i c a l t e s t s are e q u i v a l e n t to a weakened third  chapter  derives a s t a b i l i t y condition  f o r n o n l i n e a r systems i n v o l v i n g a l i n e a r t i m e - v a r y i n g g a i n i s assumed to s a t i s f y  the  gain.  c o n d i t i o n s on i t s magnitude and  ii  The  time-varying  r a t e o f change.  "TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS  v  ACKNOWLEDGEMENT 1.  2.  v i i  TIME INVARIANT NONLINEAR FEEDBACK SYSTEMS AND TIME VARYING NONLINEAR FEEDBACK'SYSTEMS  1  1.1  Introduction  1  1.2  Lyapunov Methods  7  1.3  The Popov C r i t e r i o n  8  1.4  L  2  Stability  10  THE ABSOLUTE STABILITY OF A TIME INVARIANT NONLINEAR FEEDBACK SYSTEM  11  2.1  Introduction.  11  2.2  A G r a p h i c a l T e s t o f the A b s o l u t e S t a b i l i t y  2.3 2.4 3.  '. .  o f a S e r i e s System  w i t h N o n l i n e a r i t i e s and I d e n t i c a l T r a n s f e r F u n c t i o n s  11  2.2.1  Main Method-Common Popov L i n e  13  2.2.2  S i m p l i f i c a t i o n i n P a r t i c u l a r Cases  16  2.2.3  Use o f the M o d i f i e d N i c h o l s Chart t o O b t a i n Q  19  2.2.4  An Analogue-Computer Technique  23  The A b s o l u t e S t a b i l i t y o f the S e r i e s N o n l i n e a r System w i t h D i f f e r e n t Transfer Functions The A b s o l u t e S t a b i l i t y o f a P a r a l l e l N o n l i n e a r System  THE ABSOLUTE STABILITY OF A TIME VARYING FEEDBACK SYSTEM WITH MONOTONIC NONLINEARITIES 3.1  3.2  27 30  34  The A b s o l u t e S t a b i l i t y o f the S i n g l e - L o o p Time V a r y i n g N o n l i n e a r Feedback System  34  The Main R e s u l t s  35  3.2.1  Theorem 1  35  3.2.2  A S p e i c a l Case o f Theorem 1  36  3.3  P r o o f o f Main R e s u l t s  3.4  Examples  •  36 41  *in-i  Page 3.5  4.  The A b s o l u t e S t a b i l i t y o f a System w i t h M u l t i p l e  Nonlinearities  and Time V a r y i n g Gains  46  3.6  Theorem 2  49  3.7  P r o o f o f Theorem 2  50  3.8  Example  51  CONCLUSIONS  53  APPENDIX 1  55  APPENDIX 2  ,  57  APPENDIX 3  58  APPENDIX 4  59  REFERENCES  60  iv  LIST OF  ILLUSTRATIONS Page  1.1  G e n e r a l time i n v a r i a n t  1.2  G e n e r a l time v a r y i n g n o n l i n e a r feedback system  2  1.3  Series  4  1.4  P a r a l l e l system w i t h m n o n l i n e a r i t i e s . . .  5  1.5  Internal  5  1.5  M u l t i - c i r c u i t system w i t h m n o n l i n e a r i t i e s  6  1.7  Popov c r i t e r i o n  9  2.1  G r a p h i c a l c r i t e r i o n f o r the time i n v a r i a n t i d e n t i c a l l i n e a r t r a n s f e r functions.  2.2  M o d i f i e d N y q u i s t l o c i and t h e common Popov l i n e f o r t h e s e r i e s o nf \ 32(s+0.25) . _ • ' . ' (s+l)(s+2)(s+4)' • •  m  2.3  =  3  (  2  system w i t h m n o n l i n e a r i t i e s  s  feedback system w i t h m n o n l i n e a r i t i e s  ) =  F  O  R  E  X  A  M  P  L  E  system w i t h 5  series  2  >  system, N  1  1  series  system, m=3,  testing  '  G(s) = (g+i)(s+5)(s+20)  N y q u i s t p l o t o f (1+0.788S)G(s), where G(s) = ( a graphical  the s t a b i l i t y  s + 1  of the s e r i e s  E  x  a  m  P l  e  2.2.1.  )( +5)(s+20)' S  A  N  system, m=3, f o r 17  M o d i f i e d N y q u i s t l o c u s f o r i = l , and t h e c o r r e s p o n d i n g Popov l i n e f o r the s e r i e s Example 2.2.2  2.6  system, m=4,  Gain-phase p l o t  Gain-phase p l o t f o r the s e r i e s  2.9  1  . , , r w , \ > f o r Example \...\ \..".\ 9 n  S  s  f o r the s e r i e s 2.8  G(s) = ,  N y q u i s t p l o t o f (1+0.766s)G(s) , where G(s) = ( +]_) ( 3 + 5 ) ( +20) ' a graphical testing Example 2 . 2 . 2  2.7  17  D  Example 2.2.1 2.5  ,  5  M o d i f i e d N y q u i s t l o c u s f o r i = l , and t h e c o r r e s p o n d i n g Popov l i n e f o r the  2.4  G  n o n l i n e a r feedback system  Gain-phase p l o t f o r the s e r i e s  the s t a b i l i t y  o f the s e r i e s  A  18 N  ^  system, m=4, f o r 18  f o r G(s) = ( system, m=3, f o r G(s) ( =  S +  system, m=4,  S +  J Q ) ( +5Q)  ^  a  r a m : i  -ly  o  boundaries  r  f o r Example 2 . 3 6 30 2 Q ) ( +50) S  A N C  ^  21 A  ^ ^y  °f .boundaries  am  f o r Example 2 . 3  580 f o r G(s) = ( I Q ) ( + 5 0 ) s +  system, m=5,  a n <  S  a  n  d  S  f o r Example 2 . 3  v  22 a  f  a  m  i  l  y  o  f  boundaries 22  Page 2.10A  Computer program f o r R-I  generator  24  2.10B  Computer program f o r X-Y  producer  25  2.11  The  2.12  E x t e n s i o n of Popov c r i t e r i o n f o r the p a r a l l e l system w i t h m i d e n t i c a l n o n l i n e a r i t i e s and m l i n e a r t r a n s f e r f u n c t i o n s  33  S i n g l e loop time v a r y i n g n o n l i n e a r feedback system w i t h input  42  3.1  l o c i p l o t t e d by analogue computer  26  3.2  Bode diagram f o r a compensator l + y ( s ) =  3.3  The  p l o t s o f G(s) = T 4 T V 7 ^ \ (s+l)(s+2) Example 3.1  3.4  Modified Nyquist  3.5  General  G'(s)  a n d  42 =  p l o t of G(s) = ( - f ] j ( +D) s  S  (|^)G(s) for 4+s  Exam  P-'-  time v a r y i n g n o n l i n e a r feedback system w i t h  vi  zero-  e  ^.2  zero-input..  44 47 47  ACKNOWLEDGEMENT  I wish to express my s i n c e r e g r a t i t u d e t o Dr. M.S. D a v i e s , t h e s u p e r v i s o r o f t h i s p r o j e c t , f o r h i s i n v a l u a b l e guidance, and Dr. E.V. Bohn for reading  the manuscript.  G r a t e f u l acknowledgement i s g i v e n  N a t i o n a l Research C o u n c i l f o r f i n a n c i a l support  to t h e  r e c e i v e d under NRC Grant  A-4148, and to t h e U n i v e r s i t y o f B r i t i s h Columbia f o r U.B.C.  Fellowship  awarded s i n c e 1968.  Harasymchuk  for typing  I would a l s o l i k e to thank Miss B e v e r l y  the manuscript.  VH  1  CHAPTER 1  TIME INVARIANT NONLINEAR FEEDBACK SYSTEM  AND  TIME VARYING NONLINEAR FEEDBACK SYSTEM  §1.1  Introduction T h i s t h e s i s c o n s i d e r s the a b s o l u t e s t a b i l i t y o f the e q u i l i b r i u m  p o s i t i o n , X = 0, of feedback systems X = AX +  d e f i n e d by  BY  Y = 0(a)  (1.1.1)  T a = C X, o r , i n the time v a r y i n g X = AX +  case,  BY  Y = 0(a,t)  (1.1.2)  T a = C X, where X i s an n - v e c t o r , Y i s an m-vector,  0 i s an m-vector, A i s an nxn  con-  T s t a n t m a t r i x , B i s annxm c o n s t a n t m a t r i x , and C In (1.1.1), each element o f 0(a) so t h a t the system  (1) i s time i n v a r i a n t .  i s an mxn  constant matrix.  i s a nonlinear function of a alone, In (1.1.2), each element  i s a time v a r y i n g n o n l i n e a r f u n c t i o n o f both a and t .  of 0(a,t)  I t i s assumed t h a t  the i - t h element  can be s e p a r a t e d i n t o a n o n l i n e a r p a r t 0 . (a . ) and the time i x v a r y i n g g a i n k . ( t ) , where 0.(a.,t) = 0 . ( a . ) k . ( t ) . The system (2) i s thus a l i i i i i time v a r y i n g n o n l i n e a r feedback system. The t r a n s f e r m a t r i x TXs).of  the l i n e a r p a r t i s  T(s) = C ( s I - A ) T  where I i s an nxn u n i t m a t r i x . shown i n F i g . 1.1.  _ 1  B,  (1.1.3)  The feedback system  (1) may  The forward path c o n s i s t s o f an mxm  be d e p i c t e d as  l i n e a r time  invariant  m a t r i x r ( s ) and the n o n l i n e a r i t y m a t r i x N.L.  In F i g . 1.2,  the forward path  of the time v a r y i n g n o n l i n e a r feedback system  (2) c o n s i s t s of the t r a n s f e r  2  Fig.  1.1  r(t)  G e n e r a l time i n v a r i a n t n o n l i n e a r  <rtt)  N  N.U  <P((Tl  k(t)  feedback  system  r(s)  +  F i g . 1.2  G e n e r a l time v a r y i n g n o n l i n e a r  feedback  system  C(t)  3  m a t r i x F ( s ) , the mxm  diagonal nonlinearity  0 T ( a , ) , 0 „ ( a „ ) , . . • 0 (a ) and the mxm 1 1 2 2 mm  m a t r i x N.L. w i t h elements  diagonal time-varying gain matrix k ( t )  w i t h elements k, ( t ) , k ( t ) ,. . .k ( t ) . 1 2 m A wide v a r i e t y  o f systems may be t r e a t e d  by c h o i c e o f r ( s ) , some  forms o f p a r t i c u l a r i n t e r e s t a r e : (1) input  A series  single-output  system.  l i n e a r time i n v a r i a n t  amnesic n o n l i n e a r i t i e s . (2)  The forward path c o n s i s t s  This  transfer  A p a r a l l e l system.  transfer (3)  function.  The forward path c o n s i s t s  This  m - s i n g l e n o n l i n e a r feedback l o o p s .  The 4 classes.  This  A m u l t i - c i r c u i t system.  previous classes.  of m - p a r a l l e l  i n s e r i e s w i t h one l i n e a r time  The forward path c o n s i s t s of  i s shown i n F i g . 1.5. Such systems  do not f a l l  i n t o the  Such a system i s shown i n F i g . 1.6.  elements o f the n o n l i n e a r i t y m a t r i x a r e t o be c o n s i d e r e d i n  In each i t i s assumed that  valued function (1)  s e p a r a t e d by  i s shown i n F i g . 1.4.  An i n t e r n a l feedback system.  (4)  functions  i s shown i n F i g . 1.3.  b r a n c h e s , each o f which has one n o n l i n e a r i t y invariant  o f the s i n g l e -  0(a) i s a p i e c e - w i s e continuous, angle  o f a. A sector  nonlinearity;  condition  k.l  any f u n c t i o n which s a t i s f i e s the  9  a i 0  0(a) = 0 ,  a = 0  < ^ < a  k ,  I  where k^ > 0 and k^ may be p o s i t i v e , n e g a t i v e o r z e r o . (2)  A first  and t h i r d quadrant n o n l i n e a r i t y ;  case o f ( 1 ) . 0 <  a  < k < oo —  f o r a l l f i n i t e nonzero v a l u e s o f a.  k > 0 ,  this' i s a s p e c i a l  <ryt)  <t>,«r,)  Q,ls)  F i g . 1.3  <t>i(tTi)  Gi(s)  S e r i e s system w i t h m  nonlinearities  ( T m f t )  <Pm<F»t  5  Fig.  1.5  Internal  feedback system w i t h m  nonlinearities  6  Fig.  1.6  Multi-circuit  system w i t h m  nonlinearities  (3)  A monotonic n o n l i n e a r i t y ; t h i s belongs  to a s u b c l a s s of  (2) , i n which i t i s assumed t h a t  d|M>o. do (4)  A monotonic odd n o n l i n e a r i t y ; t h i s belongs  (3) , i n which i t i s f u r t h e r assumed t h a t 0(-a) = -0(a) Furthermore,  the elements  f o r a l l a.  of the time v a r y i n g g a i n m a t r i x to be  c o n s i d e r e d are assumed p o s i t i v e , bounded and §1.2  to a s u b c l a s s of  continuous.  Lyapunov Method I n v e s t i g a t i o n s o f the s t a b i l i t y of such systems were i n i t i a t e d  Lure who  proposed  a Lyapunov f u n c t i o n of the T  V(X) = X PX +  m a. Z 3. / i=l 0 1  1  where P i s an nxn all  3.  by  form:  0.(z)dz,  (1.2.1)  1  symmetrical p o s i t i v e d e f i n i t e m a t r i x  (P = P  T  > 0),  > 0 and the upper l i m i t s of the i n t e g r a l terms are the elements  of  the m a t r i x C X.  (1.2.2)  T  I t i s obvious  t h a t V(X) 0 <  i s a positive function since  0(a)a  < Ka .  (1.2.3)  2  From (1.1.1), the d e r i v a t i v e of (1.2.1) i s V(X) = X (AP +PA) +B PX0(a)'+ X PB0(a) T  T  -  T  T  3AX0(a) + BC0(a),  where g i s a d i a g o n a l c o n s t a n t m a t r i x w i t h elements  (1.2.4) 3 , $ >•••6 • I 2 m n  0  From the Lyapunov second method, i n o r d e r to f i n d  the  c o n d i t i o n of a b s o l u t e s t a b i l t i y , i t i s n e c e s s a r y to determine under which V(X) V(X)  = 0.  sufficient  the c o n d i t i o n s  i s n e g a t i v e d e f i n i t e except at the n u l l s t a t e , X = 0, where  In  [ 5 ] , i t was shown that  the c o n d i t i o n s of a b s o l u t e  of a time v a r y i n g n o n l i n e a r system tion  (1.2.1) w i t h a l l 3^ = 0.  may be found  stability  from u s i n g the Lyapunov f u n c -  T h i s method o f u s i n g Lyapunov f u n c t i o n s was  f u r t h e r developed by Narendra  and T a y l o r [6] u s i n g the m o d i f i e d Lyapunov f u n c -  tion , viz. , T V(X,t) = X PX +  °i g.k.(t) /  m  E i=l  1  0.(z)dz,  0  1  (1.2.5)  1  where k ^ ( t ) i s a time v a r y i n g g a i n . In  [ 4 , 5 ] , i t Is a l s o proved  which the Lyapunov f u n c t i o n is- v a l i d  that the s u f f i c i e n t  c o n d i t i o n under  i s s i m i l a r to the Popov c r i t e r i o n ,  dis-  cussed below.  §1.3  The Popov  The  Criterion  sufficient  condition of absolute s t a b i l i t y  fora controllable  and o b s e r v a b l e time i n v a r i a n t system w i t h one n o n l i n e a r i t y s a t i s f y i n g was e s t a b l i s h e d by V.M. Popov  [1].  The Popov c r i t e r i o n  Re[(l+qjco) G(jui)] +~ f o r a l l to, where q i s a nonnegative  takes the form  > 0  (1.3.1)  number.  A convenient g r a p h i c a l method e x i s t s f o r t e s t i n g A sufficient  (1.2.3)  c o n d i t i o n f o r the a b s o l u t e s t a b i l i t y  many n o n l i n e a r i t i e s was g i v e n by J u r y and Lee [ 2 ] .  (1.3.1). o f a system  with  The c o n d i t i o n r e q u i r e s the  Hermitian matrix: 2K  _ 1  + H(jo>) + H (-ju))  to be p o s i t i v e d e f i n i t e f o r a l l OJ. elements  Here K i s a c o n s t a n t d i a g o n a l m a t r i x o f  K... ,K , . . .K , a l l o f which a r e p o s i t i v e numbers such 1 I m 0 < 0.(o\)o\ i  is satisfied  (1.3.2)  T  i  i  < K.o. i  2  i  that  the i n e q u a l i t y  (1.3.3)  f o r i = l,2,...m, and where H(joi) = (I +  JOJQ) T O ) ,  (1.3.4)  Y  'Popov'me A  unstable region  F i g . 1.7  Popov  Stable region  criterion  10  where Q i s a c o n s t a n t  diagonal matrix  with  n o n n e g a t i v e numbers.  T h i s forms a g e n e r a l i z a t i o n of the Popov  For the time v a r y i n g n o n l i n e a r c o n d i t i o n of absolute  stability  elements q ,q„...q 1 2 m  which  1  feedback system, the  are  result.  sufficient  e s t a b l i s h e d by Rozenvasser i s as below,  Re G(jco) +  ^ > 0  (1.3.5)  K  f o r a l l co.  I t i s but  Other new and  Falb  criteria,  [8,9], Yakubovich  [10,11], Bergen and §1.4  L  2  the Popov c r i t e r i o n w i t h q =  Rault  similar  to the Popov's, were i n t r o d u c e d by Zames'  [ 4 ] , Narendra and [12], and  0.  Anderson  Taylor  [ 6 ] , Baker and  Desoer  [13].  Stability  The  concept of the  s t a b i l i t y has  I t i s c l o s e l y r e l a t e d to asymptotic  been i n t r o d u c e d by  and y i n  -is-'a l i n e a r ,  i n n e r - p r o d u c t , i/formed  functionjon  [t ,°°), i t i s  space; the i n n e r product  the norm of x i s Suppose a(x)  || x ||  2  (1.4.1)  = /<x,x>.  i s i n L^fO, ],  a(t) i s uniformly  0 0  continuous,  and  i s bounded, then the s t a t e . o ( t ) approaches n u l l s t a t e i f the s u f f i c i e n t t i o n of absolute criterion, Zames  of x  is <x,y> = r x ( t ) - y ( t ) d t '< °°, ' o  and  [14].  stability.  L„ i s the space o f square i n t e g r a b l e , v a l u e d assumed t h a t L  Sandberg  stability  is satisfied.  [8,9].  <j(t) condi-  of a time i n v a r i a n t n o n l i n e a r system, the Popov A further L  2  bounded c o n d i t i o n was  introduced  by  11 CHAPTER 2  THE  ABSOLUTE STABILITY OF A TIME INVARIANT NONLINEAR  FEEDBACK SYSTEM  §2.1  Introduction In the p r e v i o u s  can be c o n s i d e r e d  chapter,  i t was  mentioned that the n o n l i n e a r systems  i n 4 c l a s s e s a c c o r d i n g to the form of the t r a n s f e r m a t r i x  res). T e s t i n g a b s o l u t e s t a b i l i t y of a s i n g l e - l o o p time i n v a r i a n t n o n l i n e a r system u s i n g the m o d i f i e d Nyquist the method on h i s c r i t e r i o n .  diagram was  first  i n i t i a t e d by Popov, b a s i n g  F u r t h e r developments u s i n g a g r a p h i c a l method  to t e s t the a b s o l u t e s t a b i l i t y of a n o n l i n e a r system have been f u r n i s h e d by Naumov [15], Meyer and Hsu  [16], and Murphy  [17].  A g r a p h i c a l method of t e s t i n g the a b s o l u t e s t a b i l i t y of a time i n v a r i a n t s e r i e s system w i t h m - n o n l i n e a r i t i e s and m - i d e n t i c a l l i n e a r t r a n s f e r f u n c t i o n s was  §2.2  i n t r o d u c e d by Davies  [18].  A G r a p h i c a l Test of the A b s o l u t e N o n l i n e a r i t i e s and  Consider function  '  S t a b i l i t y of a S e r i e s System w i t h  Identical Transfer  a s e r i e s system w i t h  Functions  the l i n e a r  time i n v a r i a n t t r a n s f e r  matrix , 0  .o 0  -G(s)  r(s) =  0  00  v  \  G(s)  \  0 (2.2.1)  -G(s> N  \  N  \  \  0  \ \ -G(s) 0  The  i n p u t , a . ( t ) , and  output,  0 ^ ( o O > of the i-th n o n l i n e a r element s a t i s f y  the  inequality  0 < a . 0 . ( a . ) < a. i i i  l  (2.2.2)  12  Fig.  2.1  G r a p h i c a l c r i t e r i o n f o r the time i n v a r i a n t s e r i e s system w i t h 5 i d e a l n o n l i n e a r i t i e s and 5 i d e n t i c a l l i n e a r t r a n s f e r functions.  Such a system f a l l s w i t h i n has  been, e s t a b l i s h e d .  the c l a s s f o r which a P o p o v - l i k e  In a p p l y i n g  H(jui) =  stability  criterion  t h i s r e s u l t , the m a t r i x  (2.2.3)  (I+jcoQ)r(jco),  where Q i s an a r b i t r a r y , s e m i - p o s i t i v e ,  d i a g o n a l m a t r i x of c o n s t a n t s ,  sidered.  A s u f f i c i e n t c o n d i t i o n to e s t a b l i s h the a b s o l u t e  nonlinear  system i s t h a t the H e r m i t i a n m a t r i x  i s con-  s t a b i l i t y of a  (1.3.2) must be p o s i t i v e d e f i n i t e  f o r a l l co. If a l l nonlinear that i s ,  elements are assumed to have the same upper bound;  = K f o r i = l,2,...m, then, w i t h o u t l o s s of g e n e r a l i t y , K may  taken as the i n d e n t i t y m a t r i x . to the N y q u i s t p l o t of G(s)  I f Q=0,  equivalent  l y i n g w i t h i n a symmetric m-sided p o l y g o n .  subsequent development i t i s not of Q are e q u a l ;  the s t a b i l i t y c r i t e r i o n i s  required  t h a t Q=0,  but  be  In  the  r a t h e r t h a t a l l elements  t h a t i s , Q=ql, where q i s a p o s i t i v e s c a l a r  constant.  I f Q i s r e s t r i c t e d i n t h i s manner, then (2.2.4)  (I+jcoQ)T(jco) = r'(jco), where r'(jco) i s i d e n t i c a l to r(jco) except t h a t G(jco) has G'(jco) =  (l+jcoq)G(jco) •  Thus i t i s p o s s i b l e to c o n s i d e r  the e a r l i e r r e s u l t s f o r Q=0 t h a t G'(jco) l i e s w i t h i n s t a b i l i t y has  §2.2.1  been r e p l a c e d the case 0=0  to G'(jco) i n s t e a d of G(jco) i t s e l f .  the a p p r o p r i a t e  p o l y g o n , f o r any  by  by applying  I f i t can be  p o s i t i v e q,  shown  then  been e s t a b l i s h e d .  Main Method-Common Popov L i n e  Let  |G'(jco)| = * f and  G'(jco) = 6.  I f G' (jco) l i e s w i t h i n a polygon,  then y cos where and  a =  2iiT m  i = 1,2,...m,  (6 - a)  <  1,  (2.2.5)  14  and where i i s one o f m v a l u e s each c o r r e s p o n d i n g to the m s i d e s of the polygon. '•  Now (2.2.6)  G'(jco) = ( l + j w q ) G ( » = (l+jo)q)(R+jI), where G ( j u ) = R(u) + jl(oo) Thus Re G' (jcj) = R - jtol = ycosG ,  (2.2.7A)  Ira G' (jco) = uqR + I = f sine ,  (2.2.7B)  (R-coql)cosa + (ioqR+I)sina < 1,  (2.2.8A)  and  (Rcosa+Isina)  + qco(-Icosa+Rsina)  (2.2.8B)  < 1.  Define -X(co) = Rcosa + Y((jj)  (2.2.9A)  Isina,  (2.2.9B)  = co(-Icosa + R s i n a ) .  (2.2.8B) g i v e s -X + qY < 1. From  (2.2.9A) and (2.2.9B),  •  (2.2.10)  the m d i f f e r e n t m o d i f i e d N y q u i s t l c c i a r e  p l o t t e d each c o r r e s p o n d i n g to one o f the m s i d e s o f the p o l y g o n . inequality  (2.2.10), a l l these l o c i must be t o the r i g h t  the Popov l i n e , which passes  o f the system  satisfy  side of a s t r a i g h t  the p o i n t (-1,0) h a v i n g s l o p e 1/q.  l i n e e x i s t s , then the a b s o l u t e s t a b i l i t y  To  line,  I f such a s t r a i g h t  i s established.  Example.2.1 C o n s i d e r a feedback L e t every n o n l i n e a r i t y  satisfy  system  o f the type shown i n F i g . 1.3  the i n e q u a l i t y  0 < 0.a. < a., l  where i = 1,2,3  (m=3).  l  l  and l e t every l i n e a r b l o c k be r e p r e s e n t e d by  (2.2.11)  15  Fig.  2.2  Modified Nyquist • m-3,  „, . . G(s) =  loci  and the common Popov l i n e  32(s+0.25) (  s  +  1  )  (  s  +  2)(  s  +  4  r  ' ° f  r E  x  a  m  P  l  e  2- ' 1  f o r the s e r i e s  system,  16  . ,  n  32K(s+0,25) (s+1)(s+2)(s+4) .  l  The m d i f f e r e n t m o d i f i e d N y q u i s t shown i n F i g . 2.2.  /, ,  l o c i and  Then, by s e t t i n g q = 0.23,  common Popov l i n e  are  the a b s o l u t e s t a b i l i t y c o n d i t i o n  K < 0.476 is  ,^  (2.2.13)  obtained.  §2.2.2  Simplification  Plotting  in Particular  Cases  the m d i f f e r e n t m o d i f i e d N y q u i s t  A s i m p l e r and more d i r e c t  l o c i i s t e d i o u s i f m >_ 4.  approach i s p o s s i b l e i f G(s)  i s of the  G(s) = -p  form  (2.2.14)  n  (S+D.)  j=i  3  or n  k .n , G(s)  = — V  P n  j=i where p > N. i  (s+N.) —  (2.2.15)  (S+D.) 2  1+m,  and D . are r e a l p o s i t i v e c o n s t a n t s , and N. 3 1  I f the t r a n s f e r f u n c t i o n G' (jco) = a r e found from the m o d i f i e d N y q u i s t  > D. J  for i = j .  (l+jioq)G(jo)) h a v i n g q^ and k^ which  l o c u s f o r i = 1, s a t i s f i e s  the above c o n d i t i o n s ,  then ^ (1)  For some co = co , g i v i n g Z.G'(jco ) > and |G'(jco )| < 1, |G'(jco)| c c — m c d e c r e a s e s when co i n c r e a s e s f o r co > co . c k^ i s the maximum v a l u e s a t i s f y i n g the a b s o l u t e s t a b i l i t y c o n d i t o n . (2)  And  The phase of G'(jco) i s d e c r e a s i n g as co i n c r e a s e s .  17  4 Y(u)  J  2.3  M o d i f i e d N y q u i s t l o c u s f o r i = l , and the c o r r e s p o n d i n g Popov l i n e s e r i e s system, m=3,  2.4  -r  G(s)  =  (  s  +  1  ) (  s  ^  s  +  2  Q  )  ., f o r Example 2.2.1.  N y q u i s t p l o t of (1+0.788$)G(s), where G(s) = graphical testing  the s t a b i l i t v  f o r the  100 , and a (s+l)(s+5)(s+20)  18  a g r a p h i c a l t e s t i n g the s t a b i l i t y Example 2.2.2. --  of  the  series  system, m=4.  for  19  Example  2.2  (1)  C o n s i d e r m = 3 with every l i n e a r element h a v i n g a t r a n s f e r  G(s) = ( s + l ) ( s +°5 ) ( s + 2 0 ) . 1  0  function  (2.2.16)  k  From F i g . 2*3, q  ±  = 0.788,  (2.2.17)  = 1.46.  (2.2.18)  and k  S e t t i n g q = 0.788, k < 1.46 as shown i n F i g . 2.4, (2)  G'(s)  i s the a b s o l u t e s t a b i l i t y  staisfies  U s i n g the same approach  condition  since,  the above c o n d i t i o n s . the a b s o l u t e s t a b i l i t y  c o n d i t i o n of the  system w i t h 4 n o n l i n e a r i t i e s i s o b t a i n e d a s : k < 1.234  (2.2.19)  by s e t t i n g q = 0.766. The i n f o r m a t i o n r e q u i r e d f o r the p r e v i o u s method may directly  from the N y q u i s t P l o t o f G(s) by n o t i n g t h a t the c r i t i c a l p o i n t i s  t h a t h a v i n g phase a^, and  the c o r r e s p o n d i n g v a l u e o f q i s g i v e n by  where tanp i s the s l o p e o f G(s) at a=  (2.2.14) or (2.2.15), inequality  the s p e c i a l form o f  the N y q u i s t l o c u s o f G'(s) h a i v n g q^ and k^ found  (2.2.10) f o r i = l may  the g e n e r a l approach  §2.2.3  tan(a^~3)  a^.  I f the t r a n s f e r f u n c t i o n G(s) does not s a t i s f y  fails,  be o b t a i n e d  be t e s t e d by the polygon  must be  criterion.  from  If this  adopted.  Use o f the M o d i f i e d N i c h o l s Chart to O b t a i n 0  The use of the m o d i f i e d N i c h o l s c h a r t to o b t a i n q and absolute s t a b i l i t y  to t e s t  o f the system with many n o n l i n e a r i t i e s i s a l s o  the  possible.  A polygon w i l l be d e s c r i b e d by some r e l a t i o n s h i p between the l o g amplitude  and  the phase of the  form  20 M = 20 l o g , sec(- ^ 10 2ii 2n (i-1) < 6 < i,  + - - 0)db, m m  r t  > where  m and  —  (2.2.20)  — • m  i = 1,2,...m. The  chart.  s t a b l e boundary G' can be r e p r e s e n t e d  L e t us  i n the m o d i f i e d  Nichols  consider M  = M - 20 l o g  _1 (l+N ) , 2  1 ( )  (2.2.21A)  2  and tan"  1  N. , J  (2.2.21B)  where j = 0, and  N_. i s an a r b i t r a r y , p o s i t i v e c o n s t a n t .  (2.2.21A) and (2.2.21B) g i v e a  f a m i l y o f the s t a b l e b o u n d a r i e s as shown i n F i g . 2.7, each c o r r e s p o n d i n g to one  of the constants  N.  Note t h a t these curves a r e a l l o f the same form and  may e a s i l y be s k e t c h e d . If  the l o c u s o f G(jco) i s sketched and q i s chosen, such t h a t  p o i n t co. on the l o c u s o f G(jco) on the m o d i f i e d  Nichols  each  c h a r t i s beneath the  c o r r e s p o n d i n g s t a b l e boundary N_. = co^ <J, then the a b s o l u t e  stability  o f the  system i s e s t a b l i s h e d . A l s o , i f the p o s i t i o n o f the p o i n t co. i s known, and i s beneath a N. f a m i l y o f s t a b l e b o u n d a r i e s N, then the c o r r e s p o n d i n g v a l u e s o f q. = — 2  j = 0,...°°, i s known, so t h a t every p o i n t has one c o r r e s p o n d i n g s t a b l e q range.  Then, any q i n the i n t e r i o r q range i s p e r m i t t e d  establishing is,  the s t a b i l i t y  o f the system.  i n fact, quite straight-forward.  value  o f q i s chosen which g i v e s  t o be chosen f o r  T h i s seemingly complex procedure  I f a range o f q's i s p e r m i s s i b l e ,  the g r e a t e s t  p o s s i b l e value  that  o f k.  Example 2.3 C o n s i d e r the system i n the Example 2.2 w i t h every l i n e a r element G(s) where  F i g . 2.7  Gain-phase p l o t the  series  700 and a f a m i l y (s+10)(s+50) f o r Example 2.3.  f o r . G(s) =  system, m=3,  -,—TTTTCT—,  of boundaries f o r  22  6db  F i g . 2.8  Gain-phase p l o t series  Fig.  2.9  630  f o r G(s) = ( x o ) ( + 5 0 ) s+  system, m=4,  Gain-phase p l o t  a  series  d  a  f  a  m  i  l  y  o  f  D  °  u n d  aries  f o r the  f o r Example 2.3.  f o r G(s) -  .____ (s+10)(s+50) f o r Example 2.3. 5  8  0  a  e  n  s  system, m=5,  n  d  a  f a m i l y o f boundaries f o r  23  G ( S )  (s 10)(s 20T  =  +  and m = 3.  ( 2  +  Comparing the frequency response  of the s t a b l e boundaries  as i n F i g . 2.7,  ' 2  2 2 )  curve of G(juj) w i t h a f a m i l y  the v a l u e of q and s t a b l e  condition  are q = 0.067  (2.2.23A)  k £ 700 . S i m i l a r l y , from F i g . 2.8  (2.2.23B)  and F i g . 2.9,  by s e t t i n g q = 0.067, the  following absolute s t a b i l i t y conditions k £ 630  f o r m = 4,  k £ 580  f o r m = 5,  .  (2.2.24)  and (2.2.25)  are o b t a i n e d .  §2.2.4  An Analogue-Computer  Technique  T h i s l a s t method makes use of the analogue-computer to t e s t  the  a b s o l u t e s t a b i l i t y of a n o n l i n e a r system. The  computer arrangement, which i s shown i n F i g . 2.10A  is divided into the p a r t i c u l a r  two main p a r t s .  The  first  generates R and I , and depends on  t r a n s f e r f u n c t i o n b e i n g c o n s i d e r e d w h i l e the second  components o f G(jco) which are s e t to remain The  and F i g . 2.10B,  g i v e s the  unchanged f o r d i f f e r i n g  systems.  e f f e c t o f v a r y i n g q on the G'(jco) l o c u s i s e a s i l y o b t a i n e d by a d j u s t i n g  a potentiometer. T h i s method, however, s u f f e r s from d i f f i c u l t i e s  i n amplitude  scaling.  B e s i d e s , the analogue-computer set-up becomes more complex w i t h i n c r e a s e i n system  order. I t i s a l s o noted  co = 0 to o  co = M  00  l o c u s o f G'(jco) from  by the computer i s i m p o s s i b l e because the d e s c r i b i n g  i s p r o p o r t i o n a l to  .  time  To improve the a c c u r a c y of the output i t i s n e c e s s a r y  to use a t h r e e - (or two-) to l i m i t  t h a t to s k e t c h the complete  stage programming and rescalin'g t e c h n i q u e , and  the to range which i s dependent on the computer, c h a r a c t e r i s t i c s .  also  2.10B  Computer program f o r X-Y p r o d u c e r  Example  2.4 The  system  (m=3)  G  The i n F i g . 2.10A  §2.3  (  s  l i n e a r elements G^(s)  where  (2.2.26)  (s+0.5)(s+l)  )  c i r c u i t s o f the two  p a r t s o f the analogue computer set-up  and  the t y p i c a l r e s u l t a n t l o c i  F i g . 2.10B, and  of q shown'in F i g . 2.11 choice of k  i  has  l e a d to the c h o i c e q = 1.5.  are shown  f o r various values  This therefore permits  the  = 0.685.  max  Absolute  S t a b i l i t y o f the S e r i e s N o n l i n e a r System w i t h D i f f e r e n t  Transfer  Functions  L e t us c o n s i d e r the case where the i d e n t i c a l l i n e a r t r a n s f e r f u n c t i o n s i n the p r e v i o u s s e c t i o n have been r e p l a c e d by d i f f e r e n t Now  the t r a n s f e r m a t r i x r ( s ) may /  r(s)  0  o.  -G-^s)  0  =  linear  transfer functions,  be w r i t t e n  o  G  m  (s) N,  0 I  -G (sJ\  (2.3.1)  I  2  0  -G L e t us  ^(s) m-1  0  consider  H ( > ) = (I+jwQ)r(ju)), where I i s a u n i t m a t r i x and Q i s a d i a g o n a l c o n s t a n t m a t r i x w i t h q^,q ,...q^; thus H(s) may 2  be c o n s i d e r e d as below, G'(s)^ m  / 0 H(s)  =  G^(s)  elements  0 (2.3.2) -Gl(s)  28 where G'(s) = i Let  (1+jwq.)G.(s), i=l,2,...m. 1  (2.3.3)  1  |G^(jco)I  and  = Y£  ZG^(JW) =  6 ; then the Herraitian m a t r i x i  21 + H(jco) + H (-ju>) becomes  -  / 2  e~ '1  Y  j  l  6  l  0  e m 'm j 6  Y  \  -Y e  2  2  rJ 9 ~Y 2 6  (2.3.4)  0  R  0  ^ -y  \ I < v  _ j ___0 m  and  ,e 'm-1  m-1^  -Y  9  e  must be p o s i t i v e d e f i n i t e f o r a l l a> to s a t i s f y  A. = 2A. 1-1 " i - 1 l Y  w i t h A^ = 2 and  condition.  A  i-2  > 0,  i = 3,4,...m-1,  Y  condition i s  •y.e  A = m  the s t a b i l i t y  = 4 - j_  The l a s t  V  2  (m-1) p r i n c i p a l minors o f A^ a r e generated by the r e c u r r e n c e  The f i r s t relation  -j8 , e m-1  ri^1 1  9 1  e^m 'm  0  2  -Y  2  E  >  2  0, (2.3.5)  'Y  e  !  2  9  6 e- m-2 J6  m-1  Y  v e m that  2  m  \  0  —  ie  \  _ e m-1 'm-1 J  Y  2  is , 9 9 A = 2A , - ( Y , + Y )A „ + ( - ! ) ( TT . ) e ( E 9.) m m-1 'm-1 m m-2 . .. ' i . , l 1=1 i=l m (-E 6.) m • -i +(-l) (Tf , ) e i=l m  m  Y  1  m  1 = 1  Y  m m 2. 2, 2A-,-( /+Y")A „ + ( - D ( Jl . ) 2 c o s ( E 0.) > 0. m-1 'm-1 'm m-2 . ., ' l . , l m  Y  Y  1=1  1=1  (2.3.6)  I f a l l y. < 1 f °  H  r a  1  A ~A 3  2  A  2  2 = A -Y A-  L  2  2  w,  i=l,2,...m,  then  =-4-Y^ > 3,  > ^ ~^1  2  =  2  - 2 > 1,  2 A  V  A  3  3  >  A  2  V  =  +  1  =  4  ~Y +  >  1  1  Y  3  A  2  >  A  4:  3" 2 > ' A  X  > A + l > 5, 3  A.-A. , = A. ., —Y. l l - l l - l  2 , i . , > A. .-A. > 1, 1-2 l - l i-2 0  A. > A. ,+1 > i + 1 , l i - l '  2 = A -Y „A > A -A > 1, m-2 m-2 m-2 m-J m-2 m-3  A m-1  A  T  m-1  > A  „+l > m, m-2  and A  m > 2A ,-2A „+(-l) 2cos( E 6.) m m-l m-z . . l i=l m  > 2+(-l) 2cos( m  m E 6.) >_0. 1=1 .  (2.3.8)  1  Therefore,  the a b s o l u t e s t a b i l i t y  o f a n o n l i n e a r system i s assured  of G^(joo) l i e s w i t h i n ' a c i r c l e o f u n i t  radius.  Obviously,  i f every  locus  must be chosen  zero so t h a t G'(jw) = G ( j o ) i  In o r d e r sketch unit  to t e s t a b s o l u t e s t a b i l i t y  J  o f n o n l i n e a r system i t i s thus n e c e s s a r y to  the l o c i o f G^(jio) and t o observe whether a l l o f the l o c i l i e w i t h i n the  circle.  30  As m -> °°, t h i s but here  §2.4  result  c o i n c i d e s w i t h the p r e v i o u s r e s u l t  the G_^(jaj)'s a r e not n e c e s s a r i l y  Absolute  i n §2.2,  the same.  S t a b i l i t y o f the P a r a l l e l N o n l i n e a r  System  The t r a n s f e r m a t r i x o f the p a r a l l e l n o n l i n e a r system shown i n F i g . 1.4 may be w r i t t e n , G^s) 1 r(s) =  G  G (s) /  '  s)  !  i  G (s)  G (s)  l (  1  G (s) m  j  (2.4.1)  m  2  Suppose t h a t G.(s) = G ( s ) , i=l,2,...m, and G'(joo) = (l+jcoq)G(ju)) , then ^G|(jw)  G (jco)  G' (ju>) ^ m.  2  G| (ja>)  ^G^(jco)  ,i  G (ju) 2  I  (2.4.2)  = G'(jco)  1  1  1 '  Define G(jco) = R(co) + j l ( c o ) ,  (2.4.3A)  R' (jio) = R(co) +  (2.4.3B)  ql(w) ,  and I  1  ( j u ) = qtoR(w) + I ( w )  (2.4.3C)  L e t us suppose the i - t h n o n l i n e a r i t y  satisfies  0 < a.0.(a ) < a..  x  1  (2.2  I  i  The H e r m i t i a n m a t r i x i s ,1  1  I  21 + 2R'  I  ^1  /2(1+R )  1-  I  1  1/  2R'  f  2R'  \  2R'  I  2(1+R')\  '| (2.4 \  s  2R'  2R*  2R'  The s u f f i c i e n t c o n d i t i o n  2(1+R')'  of absolute s t a b i l i t y  i s that the H e r m i t i a n m a t r i x  must be p o s i t i v e d e f i n i t e f o r a l l w; c o n s e q u e n t l y , 2(1+R')  2R'  2R'  \  2R'  I  2(1+R'>s  \  > 0.  I  \  \  I \ I 2R'  \  -2R*  2R 1  2(1+R') 2R  1  -2 2  2(1+R')  -2  \  2  1  °N  0,  ! \ \ 2R*  I  0  \  .0  i  S  \ • 2  Now  32 = 2A. . + 2 R' i-l 1  = 2(2A _ i  = 2 ~ A 1  1  1  2  + 2 +  1_1  R ' ) + 2 R' 1  (i-l)2V  = 2 ( l + i R ' ) > 0.  (2.4.5)  1  Substituting  (2.4.3B) i n (2.4.5),  .  R(u>) - q'ajl(u) + j > 0.  (2.4.6)  Define X(ui) = R(u>), and  (2.4.7A)  ^ Y(u>) = COI(OJ).  (2.4.7B)  (2.4.7A) and (2.4.7B) g i v e  X(u)) - qY(w) + j  > 0.  (2.4.8)  The c o n d i t i o n A = X(u>) - qY(u) + - > 0. m m implies  (2.4.9)  that A  > 0,  i = 1,2,...m-1.  Hence,the new Popov line,shown i n F i g . 2.12, passes through the p o i n t with slope —. q  (2.4.5) (- ^ , 0)  33  2.12  E x t e n s i o n o f Popov c r i t e r i o n f o r the p a r a l l e l system w i t h m nonlinearities and m l i n e a r t r a n s f e r f u n c t i o n s  identical  34 Chapter 3  ABSOLUTE STABILITY OF A TIME VARYING  FEEDBACK  SYSTEM WITH MONOTONIC NONLINEARITIES  §3.1  The A b s o l u t e  S t a b i l i t y o f the Single-Loop  Time V a r y i n g  Nonlinear  Feedback System  In r e c e n t years  some r e s u l t s c o n c e r n i n g  the a b s o l u t e s t a b i l i t y o f a  s i n g l e - l o o p n o n l i n e a r system w i t h a time v a r y i n g g a i n have been o b t a i n e d by Rozenvasser  [ 5 ] , Zames  [22], Bergen and R a u l t  here extend  t h i s p r e v i o u s work.  [ 1 2 ] . The r e s u l t s t o be p r e s e n t e d  L e t us c o n s i d e r s i n g l e - l o o p time v a r y i n g n o n l i n e a r system shown i n F i g . 3.1. In t h a t system, a (t) = S g(t-r) e ( i ) d T e o  for t > 0 ~~  t  (3.1.1) = 0  for t < 0  i s the z e r o - s t a t e response o f the l i n e a r f u n c t i o n G(jw) = ^ [ g ( t ) ] . g(t).  time i n v a r i a n t p a r t w i t h t r a n s f e r  The i n p u t i l ( t ) r e p r e s e n t s  the z e r o - i n p u t response o f  The complete response o f g ( t ) i s thus c(t) = o (t)+ n(t). e  The  (3.1.2)  0 ( c ) o f the amnesic n o n l i n e a r i t y N.L.  i n p u t , a ( t ) = - c ( t ) , and the output  are r e l a t e d i n the f o l l o w i n g manner: (1) 0 < 00(a) < a f o r a i 0, 0(0) = 0, 2  (2)  0  <  (3.1.3)  0(a) - 0(a) a  _ a  — — 1  x* V  f o r CT  The b l o c k , k ( t ) , r e p r e s e n t s a l i n e a r time v a r y i n g g a i n ,  thus  e(t) = k(t)0[a(t)]. The  instantaneous  v a l u e o f t h i s g a i n i s c o n s t r a i n e d so t h a t  (1)  K  (2)  bk(t)<  ±  < k(t)  < K , where 2  K  2  > K  ±  > 0,  k ( t ) < a k ( t ) , where the number a> 0,and the number b  is  finite.  I t i s assumed that (1)  g(t)  (2)  n  (3) Condition  the l i n e a r p a r t  (t)  E  e  L (0,«0, 2  i s s t a b l e , more s p e c i f i c a l l y ,  g(t) e L (0, >) 1  K  j  ^(O.co),  n(t) i s d i f f e r e n t i a b l e  and  f)(t)  e  L  (0,»).  (1) above ensures t h a t g ( t ) i s bounded on  as t •+ °°; b e s i d e s , c o n d i t i o n s  (2) and  (0,°°) and that g ( t )  (3) ensure that n ( t ) behaves  0  i n the same  manner. Denote n(t)  g(t) U  = sup t>0  | n(t)|,  = Sup t>0  |g(t)|.  The F o u r i e r t r a n s f o r m s of g ( t ) , e ( t ) , e t c . , are denoted by G(jco) E(jco), e t c .  The n o t a t i o n  f| • II denotes norms i n the space L^(0,°°).  ||n(t)|l  §3.2 • The Main  = /~ | ( t ) | d t . n  Result  The main r e s u l t §3.2.1  i s the f o l l o w i n g  theorem.  Theorem 1  C o n s i d e r the system shown i n F i g . 3.1 above a p p l y .  L e t y ( t ) be any r e a l f u n c t i o n such  (1)  y(t) =0  (2)  y(t) <0  (3)  ||y(t)|| <  for '  t <  for  to which the assumptions made that  0,  t>0,  l  K  .  Thus  K  2  and l e t q be any nonnegative number.  If  Re( [l+qjw+Y(jco)] [G(jto)-4-]+aqG(jto')} 2 K  -lly(t)ll  ( f lK  - f ' 2 K  0  (  Q  1  )  36 for  a l l to, then (1)  Sun |a(t) | < «,  t>o (2)  a ( t ) -v 0  (3)  as  It  <»,  that Sup |o(t) J ->- 0. f >0  s h o u l d be noted  that i f the time v a r y i n g g a i n k ( t ) i s m o n o t o n i c a l l y  and a <_ 0 V t >^ 0, then  the c o n d i t i o n (01) f o r a b s o l u t e  stability  be r e p l a c e d by  - || y ( t ) II  Re[l+qj<^Y(juO][G(ju))+^H K  a l l a),  2 K  2  K  l  K  2  > 0  3  for  ->- 0, the corresponding a ( t ) has the  A S p e c i a l Case o f the Theorem  non-increasing may  t  |ln(t)|| + U n( t) 11  property  §3.2.2  as  CQ1')  2  where a g a i n , q i s any nonnegative r e a l number.  §3.3  P r o o f o f Main R e s u l t The body o f the p r o o f o f Theorem  appendices;  1 w i l l be g i v e n i n a  s e r i e s of  a b r i e f summary i s g i v e n below i n t h i s s e c t i o n .  Define a(t) ={ ' 0  a ,(t) (T  e (t)  for  t < T  for  t > T,  = k(t)0[a (t)],  T  T  and a  e T  g(t-x) e (x)dT.  (t) =  T  Thus a ( t ) = a (t) eT e T  for  t < T, —  and a  e T  (t)  e  L (T,-) 1  (3.3.1)  37  The n o t a t i o n  ( x * y ) ( t ) denotes c o n v o l u t i o n between x ( t ) and y ( t ) ;  (x*y)(t) =  x( )y(t-T)dT.  .  T  (3.3.2)  Define a - o + o*y, m- •  c m  J  = c + c*y. J  Then e (t) -~—-]e(t)dt K.^.  T  / ' [aAt) U m  -  •= / J [ a ( t ) - ^ - ] e ( t ) d t  + /J[y*(a - | - ) ( t ) e ( t ) d t .  (3.3.3)  Define k(t-x)0[a (t-T.)] T  R(x) •= / J [ a ( t - T )  ]k(t)0[a (t)]dt.  ^ — -  T  T  (3.3.4)  Now R(x) = R ( T ) + R ( T . ) , 1  (3.3.5)  2  where R  1  ( t )  to (t-T)-0[a (t-T)]]k(t)0[a (t)]dt,  0  f  =  T  T  T  (3.3.6)  and R ( T ) = /Q[1 -  ^  k  2  T  ]k(t)0[a (t-x)]0[a (t)]dt.  )  T  T  (3.3.7)  Frem Appendix 1. ro  R (T) l / 1  and  0  K  k(t)0[a ( t ) ]  K  T  [ a  K  ( t )  ]k(t)0[a (t)]dt, T  (3.3.8)  from Appendix 2, oo 1 1 2 <_ / ( ^ - - K ~ ) e ( t ) d t .  R (T) 2  0  (3.3.9)  T  Thus bo  The f i r s t  K  V  R(T) ±  e  2  K  term o f the r i g h t  a  T  (  t  )  0  t  y  *  t  )  ]  e x  (t)  d t  -  s i d e o f (3.3.3) i s always p o s i t i v e .  c o n s i d e r the second term o f the r i g h t  ;  (  T  ~~ i  (  a  " K ^  ]  (  s i d e o f (3.3.3)  T  )  D  T  (3.3.10) L e t us now  38  = / J /Jy(T)[o-(t-T) -  co  = f  e (  co  T )  ]e(t)d  dt  T  e(t- ) ] e ( t ) d dt T  /Qy(x)[o (t- ) -  Q  ^"  T  T  R  T  T  = / y(T)R(x)dx.  (3.3.11)  0  But,  employing  (3.3. 10), i t may be shown that - f-)](t)e(t)dt  fl[y*(o 0  >  /gyCOdx  -  '  l l y l l /  a (t) - -|^-]e (t)dt T  0 K^ °T [  (3.3.12)  T  "  ( t )  K^~  ] e  T  ( t ) d t  -  (3.3.13)  S u b s t i t u t i o n o f (3.3.13) i n (3,3.3) y i e l d s  S» -  /  [0 (t)  e (t)  W  +  -YT-^t  S T > (1 - - ^ l l y l l / [ a ( t )  6  T  and,  l  0  y  T  (  T  T  K  ;  o T e  ( t ) 2 d t  )  - -|  i  K  ""  ]e_(t)dt  2  (3.3.14)  T  from the assumptions o f the Theorem 1, the r i g h t s i d e o f (3.3.14) i s non-  negative.  Hence, e (t) - - f  2  -  ]  e  (  t  )  d  t  lly  +  " 'S T e  (t)2dt  - °-  ( 3  - 3  1 5 )  C o n s i d e r the f o l l o w i n g i n t e g r a l . T  E  = I  (  T  1  )  M  I = /o  1  [ _ C T  em  ( t )  ~ °e q  (t)  " ~T~  1  a q a ( t ) + ( f - - £-) ||y|l e ( t ) ] e ( t ) d t e  + I ,  (3.3.16)  2  where I  e  T  = / ' [ - a (t) ± u em  ( t )  i  l  + ( f - - f - ) ||yll e ( t ) ] e ( t ) d t , i^-^. 1^2  (3.3.17)  and I  2 0  = fl - q[a ( t ) + aa ( t ) ] e ( t ) d t . 0 e e  (3.3.18)  39  Now I  = V ° m  x  +  (  t  )  ^-^)Hylle(t)]e(t)d  +  t  n (t)e(t)dt  (3.3.19)  m  g i v i n g , due to  (3.3.15) > i'l n ( t ) e ( t ) d t . 1 — 0 m  (3.3.20)  = /Qq[a(t)+aa(t)]e(t)dt+/Jq[n(t)+an(t)]e(t)dt.  (3.3.21)  I In a s i m i l a r manner, I  2  I n v o k i n g the r e s u l t I Recall  o f Appendix 3,  L q[k(T)$(T)-k(0)$(0)+/Qq[fi(t)+an(t)']e(t)dt.  2  (3.3.22)  that /Jf(t)dt = /  f (t)dt,  0  T  where f ^ ( t ) i s the t r u n c a t e d v e r s i o n o f f ( t ) t o between 0 and T.  Hence from  (3.3.16), 1  = V ^ e T m ^  +  "  q 6  eT  T  eT  ( t )  (3.3.23)  1 and s i n c e 1  = " 27 L,Re{[l+qjco+Y(jw)][G(juO +  -(^- " ~) llyll follows  q a C T  T  C O  1  it  "  ( ~ - j p ) ||y I! e ( t ) ] e ( t ) d t  and, from t h e c o n d i t i o n s o f Theorem  1  "  ( t )  + aqG(jw)  }E (ju3)E*(j )du), T  W  that I 1 0.  Since I = I  •  (3.3.24)  + I , from Appendix 3 and a f t e r s u b s t i t u t i n g (3.3.20) i n (3.3.24), ?  q[k(T)*(T)-k(0)*(0)]  < - /*[aqn(t)+n ( t ) + q n ( t ) ] e ( t ) d t . — U r n  Define  e  = Sup 0 < t <T  |e(t)1  (3.3.25)  40  and  invoking  the c o n d i t i o n s a l r e a d y  imposed upon n ( t ) , fi(t) and y ( t ) , the  r i g h t s i d e o f (3.3.25) must be l e s s than the q u a n t i t y e [ ( l + llyll +aq) ||n|| +q llfill ] = Me . M  (3.3.26)  M  Furthermore, c o n s i d e r i n g  the f i r s t  term o f t h e ' l e f t  s i d e o f (3.3.25), s i n c e 0  i s monotonic, *(t) Using  (3.3.27)  2  (3.3.26) and (3.3.27) i n (3.3.25) y i e l d s '  | From  > |{0[a(t)]} .  k(T){0[ (T)]}  2  a  (3.3.28), s i n c e  <_ qk(0)<D(0) + Me^.  '  (3.3.28)  < 1, 2  ~  • L - e ( T ) £ q k ( 0 ) * ( 0 ) + Me .  (3.3.29)  2  M  The i n e q u a l i t y (3.3.29) h o l d s .for any T >_0 and i m p l i e s KM KM Sup  |e(t)| ±  + [(-~-)  2  + 2K k(0)$(0)]  1 / 2  2  .  (3.3.30)  Furthermore, s i n c e a (0) = 0 by (3.3.1), t h i s bound on e ( t ) tends t o zero w i t h  II hli + llfill • It in  remains to be shown that  | e ( t ) | + 0 as t ->• °°.  After substituting  (3.3.14) o f ( A 3.3) o f Appendix 3 and u s i n g . (3.3.24) , K (1 - - ^ l l y l l ) / J [ a ( t ) -  ^-]e(t)dt+/Qn (t)e(t)dt m  - q k ( 0 ) $ ( 0 ) + q/J[fi(t)+an(t)]e(t)dt  <. 0.  (3.3.31)  Thus K (1 - r ^ l l y l l K^  )fl[a(t) 0  From which i t f o l l o w s T f [o(t) Q  - ^ - ] e ( t ) d t < q k ( 0 ) * ( 0 ) .+ Me . K — M  (3.3.32)  2  that  e(t) qk(0)*(0) + Me - ^-]e(t)dt i ^ 2  i - / l l y l l l K  .  (3.3.33)  41 Since  right  s i d e of (3.3.33) i s independent of T, then l e t t i n g l.-> «>,  fAo(t) 0 L  U  V  W  e  (  t  q k ( 0 ) * ( 0 ) + Me  )  - ^-]e(t)dt < K_ ^' " J C  M  K  U I  ,  (3.3.34)  llyll  2  1 and  t h i s bound on the i n t e g r a l tends to zero w i t h However,  the bound a l r e a d y  placed  || n i l +  llnll .  on e together  w i t h the c o n d i t i o n s  demanded of g ( t ) r e q u i r e 0" ( t ) to be bounded and to tend to zero. be shown t h a t the i n t e g r a l o f (3.3.34) i s i n f i n i t e u n l e s s thus c o n t r a d i c t i n g t  00  as  |lnll  a (t) i s uniformly  +  llnll  a contradiction.  therefore  §3.4  Therefore  i t can be concluded t h a t  co,  a ( t ) -> 0 as  0, then  Hence, as  continuous and i f Sup  | a ( t ) | does not go to  (3.3.34) does not tend to zero  llnll  +  II n II + 0, Sup t>0  either.  | a ( t ) | -> 0.  This i s  The p r o o f i s  completed.  Examples  Example  3.1 In the s i n g l e - l o o p n o n l i n e a r  system w i t h a l i n e a r time v a r y i n g  shown i n F i g . 3.1, the l i n e a r p a r t has a t r a n s f e r G  the i n p u t gain  a ( t ) + 0 as t  now  . Since  zero  (3.3.34).  I t may  (  s  )  =  function  (s+l)(s+2) '  and output o f the n o n l i n e a r  i s such  gain  part, s a t i s f y  (3.1.2), and the time  varying  that  (1)  1  (2)  b k ( t ) <_ k ( t ) <_ a k ( t ) , where a,b are r e a l numbers  <_  and  k ( t ) <_ 1.2,  i.e. , K  = 1, K  = 1.2, such t h a t a > 0,  b is finite.  Suppose t h a t a i s l a r g e enough, q must be chosen z e r o . t h a t y ( t ) i s an e x p o n e n t i a l  f u n c t i o n such  that  L e t us assume  42  rct)=o p"  3.1  o S i n g l e loop  i|Ct7  N.L.  time v a r y i n g n o n l i n e a r  eft)  Gcs)  c<t)  -f  feedback system w i t h z e r o - i n p u t  43  e  y M - {  '  t >  as  t > 0,  0  T  0 2  where y >  as  0 , and g > 1 . 2 = — .  Then  1  =r  lly(t)H Thus the c o n d i t i o n s  l - i e ' ^ l  Q  dt  on y ( t ) are s a t i s f i e d .  . A f t e r t a k i n g the L a p l a c e  transform  of y ( t ) , 1  + Y(s) =  1  -  -  =  1  Y  +  S  T  +  + 1  3  +  S  YS  (3.4.1)  .  •  Y The s u f f i c i e n t c o n d i t i o n o f a b s o l u t e R e { [  Cp-l) + Y.V 3  for a l l  stability is  +•_!  ] [ G ( j a ) )  + YJW  ] _I  !•  ^ |  3  2  1-  o  ( 3  .  4 > 2 )  —  2  03. L e t us d e f i n e  G (Ja ) = GCjo,) + 1  )  and  (3.4.2),  From the r e l a t i o n  the v e r t i c a l l i n e p a s s i n g  the l o c u s o f G|(joi) must l i e on the r i g h t through the p o i n t  (r~ ^~~ L  > 0)  •  s i d e of  The m u l t i p l i e r  of  3  G^(jw) i n G^(jo)) may be c o n s i d e r e d diagram F i g . 3 . 2 . of G^(jo3).  The f u n c t i o n o f the compensator i s to improve the  From the p l o t  the p r o p e r v a l u e s  as a compensator which i s shown on the Bode  o f G ^ ( j ) i n F i g . 3 . 3 , i t i s a simple matter to choose w  o f g and y.  From the f a c t t h a t  on the l o c u s o f G.. (JOJ) l i e s between  • 3—1  1  li-l  ft"Urz 3 K x  _  1  -j  ir~J K 2  < < c  1  K  characteristics  ., .  y  the l e f t - m o s t  point O J ^ = 2 . 5  3  and — and t h a t g s a t i s f i e d the r e l a t i o n y  .  the s t a b i l i t y c o n d i t i o n 2  K  <  23.75  (3.4.3)  44  G=  (UJOJXUJCO)  K =23. f  Fig.  3.3  The p l o t s o f G(s) = ( + i ) ( g 3  Example 3.1.  + 2  )  a  n  d  75  '< ) = C^^J-)GCs)  G  s  for  45 has been found by r u n n i n g a s u i t a b l y w r i t t e n programme on the d i g i t a l computer. It  i s noted  t h a t the combination  one because o f the p a r t i c u l a r the b e s t combination  3=2  choice of y ( t ) .  o f 3 and y  m a  Y  and y  =  i s not the b e s t  That optimum y ( t ) which g i v e s  be determined  by a d i g i t a l  computer  technique.  Example 3.2 Consider G(s)  K  (s+1)(s+D)'  where D i s any p o s i t i v e r e a l c o n s t a n t , i n s t e a d o f the-G(s)  i n the p r e v i o u s  problem and l e t 0 < k(t)  <_  1,  . and b k ( t ) <_ R ( t ) <_ a k ( t ) . In  this  case, l e t us suppose y ( t ) = 0, then the' c o n d i t i o n o f a b s o l u t e  stability  is Re{(l+qjio) [G(jco)+l]+aqG(jco)} for  >_ 0  a l l co. R e w r i t t i n g , R(co) - qcol(co) + 1 + aqR(oj) ^ 0  for  C3 4.4)  (3.4.5) '  a l l co, s i n c e G(jco) = R(co) + jl(co) .  Here  (3.4.6)  „ 2. K^CD-co ) /  and  n  -K (1+D) oi =  (l co )(D 2  +  2  . ) 2  +  ( 3  - ' 4  7 B )  Define X(co) = R(co)  (3.4.8A)  and Y(OJ) = u i y Substituting  (3.4.8A) and  - aR(oj).  (3.4.8B) i n  (3.4.5),  X(OJ) - qY(a)) + 1 To s a t i s f y  >0.  (3.4.9)  i n e q u a l i t y ( 3 . 4 . 9 ) , the l o c u s must be on the r i g h t s i d e of  the s t r a i g h t l i n e p a s s i n g  From (3.4.8A),  through the p o i n t  (.3.4. 8B). and  i s s a t i s f i e d f o r any  (-1,0) having  e  q is  then the s t a b i l i t y  nonnegative r e a l constant  i s on the r i g h t s i d e of the Popov l i n e . I f a ^ 0,  the s u f f i c i e n t  p o s i t i v e s l o p e —. q  ( 3 . 4 . 9 ) , i f a <_ l + D - , where the  number e > 0 i s a r b i t r a r i l y chosen, and  plot  (3.4.8B)  k.  Besides,  condition  the m o d i f i e d  T h i s i s shown i n F i g .  c o n d i t i o n of a b s o l u t e  by c h o o s i n g  §3.5  q =  Absolute Varying  Nyquist  3.4.  (3.4.10)  i s s a t i s f i e d f o r any nonnegative r e a l constant  k  1 1+D"  S t a b i l i t y of a System w i t h Many N o n l i n e a r i t i e s and Many Time Gains  In the p r e v i o u s one  (Ql)  stability is  R(OJ) - q o K u ) + 1 >_ 0. T h i s i s the Popov c r i t e r i o n and  small  n o n l i n e a r i t y and  one  s e c L i o n s , the absoluLe s t a b i l i t y  of the system w i t h  time v a r y i n g g a i n i s e s t a b l i s h e d .  Now,  system w i t h many n o n l i n e a r i t i e s and many time v a r y i n g g a i n s . shown i n F i g . 3.5.  i n p u t a. and l the f o l l o w i n g :  are r e l a t e d by (1)  The  0 < a.0.(a.) < a l i i — i  2  the output  for  a. i 0, l  l e t us c o n s i d e r i  Such a system i s  0.(o.) of the i-th n o n l i n e a r i t y i i  0.(0) l  =  0,  d0.(a.)  (3.5.1)  < > -<t^ l 2  0<  1  i  and  the i n s t a n t a n e o u s (1) (2)  v a l u e of the i - t h time v a r y i n g g a i n i s c o n s t r a i n e d so  K, . < k . ( t ) < K. . , where K„. > K_ . > 0, l i — I — z i 2 i l i b . k ( t ) < k . ( t ) < a . k . ( t ) , where the number a 1  b. i s l  1  1  finite,  1  i  > 0 and  that  the number  47  Fig.  3.5  General  time v a r y i n g n o n l i n e a r  feedback system w i t h  zero-input  48 Besides,  °e is  the zero  r(jw)  =  f  n (J r ( t - T ) e—( x ) d  T  s t a t e response of the l i n e a r time i n v a r i a n t t r a n s f e r m a t r i x  = J [ F ( t ) ] , where G  /  1 1  G  (JOJ)  2 1  G  (ju)  1 2  (jaj)-  G  G  l m  (  ^ ^  (jo)—• (3.5.2)  r(ju) =  I  I  kG (U)  G^Cjo.)  ml  The i n p u t v e c t o r  n_(t) r e p r e s e n t s  response o f T(t)  i s thus  response o f r ( t ) .  the z e r o - i n p u t  The  complete  c ( t ) = o ^ t ) + _n(t). I t i s assumed t h a t . a l l elements of the l i n e a r t r a n s f e r m a t r i x  r(t)  are s t a b l e , more s p e c i f i c a l l y ,  Condition and t h a t  g  (2)  n.(t)  (3)  n ( t ) i s d i f f e r e n t i a b l e and f^Ct)  ± j  (t)  e L (0,-),. g  (t) £ ^ ( 0 , 0 0 ) ,  (1)  2  e  i=l,2,...m,  j = l,2,...m,  ^(O.oo),  c L  i  (1) above ensures that each element o f r(t)  g_^.—»n and c o n d i t i o n s j  (2) and  (0,«). i s bounded on  (3) ensure that ^(t)  behaves  (0,<»)  i n the same  manner as g  (t). ij Denote n  =  m  r  =  E 1=1  The n o t a t i o n for  example,  n.  E  =  E  Sup |n. ( t ) | t>0  i=l  i=l  m  m  m  E  E  E  g....  j=l  J  11-11  .  Sup  i = l j = l t>0  .  |g  (t)| .  J  denotes norms i n the space L ( 0 , c ° )  such  that,  49 m H II = 'n  E  ' i n  i=l  (  t  l  )  d t  Define O^-^l ' ] 2 '' ' * " l m ^  K^"*" = d i a g o n a l m a t r i x  K  K  = d i a g o n a l m a t r i x ( 1' 22'"''* 2m^' K  K  K  2  and (a^,,....a^)  A = diagonal matrix  §3.6  Theorem 2 .  •  C o n s i d e r the system above a p p l y .  shown i n F i g . 3.5 to which assumptions  L e t Y ( t ) = diag{y  given  ( t ) , y ( t ) . . . .y ( t ) } be such t h a t each  element  i s a r e a l f u n c t i o n and t h a t (1)  y ( t ) ' = 0 f o r t < 0,  (2)  y . ( t ) < 0 f o r t > 0,  ±  l  —  i = 1,2,...m,  —  K  (3)  l|y.(t)ll < i  K  N  L e t Q be any p o s i t i v e s e m i - d e f i n i t e c o n s t a n t d i a g o n a l m a t r i x .  If  t h e r e e x i s t s anmxm m a t r i x H(jco) such t h a t H(jco) = [I+JWQ+Y(jaj)] [r(jco)+K ] + AQTCjoj) 1  2  - II YIt  [ K " - K " 1  1  (02)  ] ,  and T (1)  H(jco) + H (-jco) i s a p o s i t i v e s e m i - d e f i n i t e H e r m i t i a n m a t r i x f o r a l l co,  (2)  H*(jw) = H(-jco)  (3)  Every element' o f H(jco) i s a n a l y t i c f o r a l l co,  (1)  Sup | o.(t) | < «=, and S u p | a . ( t ) | < t>0 t>0  then  1  i=l,2,...m,  50 (2)  o;(t) -> 0 as t -> , and o\ -> 0 as t ->  (3)  a s  00  Hull  Ilil II  +  _>  0>  t  n  e  oo,  corresponding  _g_ has the p r o p e r t y  Sup|o_(t)| -K), and S u p | . ( t ) | ->0. t>0 t>0 C T  1  §3.7  Proof  o f Theorem 2  This proof  f o l l o w s the same v e i n as t h a t o f the p r e v i o u s  Theorem.  The o n l y d i f f e r e n c e h e r e i s t h a t a l l v e c t o r s such as e ( t ) , a ( t ) , n ( t ) , n ( t ) , — —e — — jc(t) , 0^.t) , cr(t) a r e m - v e c t o r s , and a l l m a t r i c e s All that  formulae and the p r o o f s o f which developed  such as 0,Y a r e mxm  i n §3.3 s t i l l  hold,  matrices.  except  t h e p r o o f o f (3.3.24) must be performed i n the f o l l o w i n g manner. From  (3.3.23),  (3.3.23')  = -/^eJ(t)/Qh(t-T)e (T)dx dt, T  where h ( t ) i s the i n v e r s e F o u r i e r t r a n s f o r m Theorem 2 and Newcomb's  o f H(jco).  From the c o n d i t i o n o f  r e s u l t [25], (3.3.24')  I ' < 0. T h i s , however, i s the same as (3.3.24). Hence, by the same argument used i n the p r o o f o f Theorem 1, (1) S u p | a ( t ) | < t>0 (2) a ( t ) + 0  Now,  Sup|o.(t)| < t>0  00  CO  as  t ->  i f and o n l y  CO  i f every  component a . ( t ) s a t i s f i e s  S u  p  t>0  1°" i  (t)  I<. 00  S i m i l a r l y , cr(t) -> 0 as t -> ~ i f and o n l y i f every con\ponent ^ ( t ) s a t i s f i e s rj^(t) -> 0 and Sup|c[(t)| -> 0 i f  as  t -> c o ,  and o n l y i f every component a. ( t ) s a t i s f i e s  t>0 Sup\o. ( t ) I > 0.  t>0 §3.8  1  Example  Example  3.3 L e t us c o n s i d e r the p a r a l l e l system, where m = 3,  each branch o f  which has one n o n l i n e a r i t y and one time v a r y i n g g a i n i n s e r i e s w i t h one l i n e a r time i n v a r i a n t branches  transfer function.  Here these t h r e e p a r t s o f each the t h r e e  are i d e n t i c a l t o the c o r r e s p o n d i n g ones used i n Example 3.2.  t h a t the m a t r i x Y ( t ) = 0. H(jw)  Suppose  L e t us c o n s i d e r the m a t r i x  = (I+ju)Q)(r(ju))+I)  +  (3.8.1)  a Qr(joi),  where  1 1 r(jco) = G ( j u )  IN  1 1 1 111''  and  Q = qlRewritting,  1 1 1 H(jco) = (l+jcoq+aq)G(jco)  I  l  l  +  U+jcoq)!.  (3.8.2)  1 1 1 O b v i o u s l y , H*(jco) = H(-jco), and the elements all  60.  o f H(jco) are a l l a n a l y t i c f o r  -  Invoking the p r o o f o f §2.4, the Hermatian  matrix  52  (3.8.3)  H(ju) + H ( - j ) = Re(l+ju)q+aq)G(joj) + - j . T  w  F o l l o w i n g the argument used i s Y>  the s u f f i c i e n t  positive real  i n Example 3 . 2 ,  i f a <_ 1+D-E. and q  c o n d i t i o n Q2 of a b s o l u t e s t a b i l i t y i s s a t i s f i e d  c o n s t a n t K.  f o r any  53  Chapter 4  CONCLUSIONS  A g r a p h i c a l method u s i n g the Popov l i n e i s p o s s i b l e f o r a c l a s s o f time i n v a r i a n t n o n l i n e a r system.  particular  The method may be s i m p l i f i e d  ina  number o f cases. Two a l t e r n a t i v e approaches, one u s i n g the N i c h o l s other  c h a r t , the  the analogue computer, a r e mentioned b r i e f l y and i l l u s t r a t e d . No simple g r a p h i c a l method e x i s t s  of the p a r a l l e l system w i t h many d i f f e r e n t  to t e s t the a b s o l u t e linear  a g r a p h i c a l method u s i n g the Popov l i n e t o t e s t  stability  t r a n s f e r f u n c t i o n s , although  the a b s o l u t e  stability  o f the  p a r a l l e l system w i t h many i d e n t i c a l n o n l i n e a r t r a n s f e r f u n c t i o n s i s p o s s i b l e . N e i t h e r i s t h e r e any simple  g r a p h i c a l method a v a i l a b l e to e s t a b l i s h  the c r i t e r i o n o f a b s o l u t e s t a b i l i t y o f a m u l t i - c i r c u i t system.  However, work on the d e t e r m i n a t i o n  or an i n t e r n a l  feedback  o f the c r i t e r i o n o f a b s o l u t e  stability  f o r any one o f the f o u r c l a s s e s mentioned i n §1.1 by d i g i t a l technique i s underway.  The d i g i t a l  technique  of t h e optimum combination of a b s o l u t e s t a b i l i t y  [23] i s i n essence concerned w i t h  of matrices  region.  Q and K which w i l l d e f i n e the ^boundary  I t must be p o i n t e d o u t , however, that w i t h the  systems t h a t have been d i s c u s s e d so f a r i n t h i s i s so f a r s i m p l e r and l e s s cumbersome for absolute  location  t h e s i s the g r a p h i c a l method  i n o b t a i n i n g the r e q u i s i t e  conditions  stability.  In chapter  3, Theorems 1 and 2 p r o v i d e  conditions f o r the absolute s t a b i l i t y  the s u f f i c i e n t , but not n e c e s s a r y ,  o f a time v a r y i n g n o n l i n e a r system i n  which the n o n l i n e a r p a r t must be m o n o t o n i c a l l y  nonlinear.  Of g r e a t importance i n e s t a b l i s h i n g the sufficient  c o n d i t i o n of absolute s t a b i l i t y  and y ( t ) .  1 + Y(s) i s i d e n t i c a l  to the f u n c t i o n d e s c r i b i n g some RC p a s s i v e  network i f y ( t ) i s an e x p o n e n t i a l f u n c t i o n . s t a b i l i t y may be found  i s the a p p r o p r i a t e c h o i c e o f A  by a d i g i t a l  The optimum r e g i o n o f a b s o l u t e  technique.  54  I f the time v a r y i n g condition. (Ql) may  be  g a i n i s f r o z e n , t h a t i s , k ( t ) = 1, the s u f f i c i e n t  rewritten  Re[l+qjw+Y(jui)] [G(jto)  > 0. 2  This  i s the r e s u l t  o f Baker and Desoer's  [11].  For some c l a s s e s o f time v a r y i n g n o n l i n e a r to use the- g r a p h i c a l method d i s c u s s e d  i n c h a p t e r 2.  systems,  i t i s possible  APPENDIX 1  From R^O-R^T)  (3.3.6) = / ([a (t)-0[a (t)]] - [a (t- )-0[a (t- ) 1 ] } o  T  T  T  T  T  T  0[a (t)]k(t)dt T  N o t i n g t h a t 0(t) and [a(t)-0(t)] a r e monotonic, and 0 <  < 1  "r°2  '  or (tf -a )(0 -0 ) 1  2  1  2  (0 "0 ) 1  2  2  > 0,  or [(a -0 )-(a -0 ) 1  1  2  2  1(0^02) >  0,  thus 0(t) i s monotonic i n c r e a s i n g i n [ a ( t ) - 0 ( t ) ] . L e t us d e f i n e A=  ([c (t)-0[a (t)]-[a (t-T)-0[a (t-x)])0[a (t)], T  T  T  T  T  and observe t h a t A >_P(t) - P ( t - r ) , where a (t) T  0[a (x)]d[a (T)-0[a (x)]].  P ( t ) =r  Q  From ( A l . l ) ,  T  T  T  (A1.4) and (A1.5), 00  R ( 0 ) - R ( x ) >_ / [ P ( t ) - P ( t - T ) ] k ( t ) d t 1  1  Q  and R (T) 1  But  1 ^ ( 0 ) + / [k(t+ )-k(t)]P(t)dt. 0  T  V  -C l  [0 (t)  0  )  -  ^ta (t)]]k(t)0[a (t)]dt, T  T  and,since, P(t)  <  [o (t)-0[a (t)]]0[a (t)l T  T  < ta (t) - -  T  1  T  0ta (t)]] ^  0[a (t)],  T  T  then /~[k(t+ )-k(t)]P(t)dt x  <_ (Z2~\)  /QP(t)dt  K —K J"o T [ a  Substituting  ( t )  - ^  0[a (t)]]k(t)0[ T  (A1.9) and ( A L I O ) i n (A1.8) y i e l d s  C T T  (t)]dt.  57 APPENDIX 2  From  (3.3.7)  R (T)  = /Q[1 - ^ ^ - ] k ( t ) 0 [ a ( t - T ) ] 0 [ a ( t ) ] d t .  2  T  T  (A2.1)  Therefore,  R  2  ( T )  -  Qk^O  ~ K ^ l l^-tW^tt-!)]]  | k ( t ) 0 [ a ( t ) ] |dt T  " K " ' l ( t - r ) 0 [ a ( t - ) ] I | k ( t ) 0 [ a ( t ) ] |dt  -  k  T  T  T  (A2.2)  giving r  2  (  t  )  - I  (  K ""  ^ H y  K  (  T  ~  T  )  + /Q[k(t)0[a (t)] Jdt} 2  T  from which (3.3.9) f o l l o w s .  0  [  A  T  (  T  ~  T  )  ]  2  )  D  T  (A2.3)'  58 APPENDIX 3  Since  a k ( t ) > k ( t ) , a k ( t ) > 0, and  >_f  a (t)  a(t)0[a(t)]  Q  0[a]da = *(t) > 0,  clearly^ / J a k ( t ) a ( t ) 0 [ a ( t ) ] d t >/Jk(t)$(t)dt,  J = q/J(d(t)aa(t))k(t)0(t) >q  -  q  f  l  IF  (A3.1)  [k(t)4>(t)+k(t)0[a(t)] d(t)]dt  t ( )*( )]dt k  t  t  >_ q[k(T)$(T) - k ( 0 ) * ( 0 ) ] .  (A3.2)  Therefore, I  2  = i + q /J[f|(t)+an(t)]e(t)dt ^ q [ k ( T ) $ ( T ) - k ( 0 ) $ ( 0 ) + /^[f|(t)+an(t)]e(t)dt.  (A3.3)  59 APPENDIX 4  2 Since k ( t ) <_ a k ( t ) <_ 0 and 0 <_ M ^ i £ ( t ) J _  fl ^  e(t) dt 2  = / J ak(t) • ^ t ) 0 [ a ( t ) ]  J' = q/J[6(t) + ~ - e ( t ) ] e ( t ) d t  2  ^ ,T  U  t  )  $  (  t  )  d  t  ,< - ( ) c l e a r l y , $  j  t  f  (  M  <  1  )  >_ q / J [k$ (t)+k (t) 0 [a (t) ]d (t) ] dt  >_ q[k(T)$(T) - k(0)$(0)] .  (A4.2)  Therefore, 1\ £ q [ k ( T ) $ ( T ) - k(0)$(0) +  fi(t)e(t)dt].  ' (A4.3)  60  REFERENCES  1.  V.M. Popov, "The A b s o l u t e S t a b i l i t y of N o n l i n e a r A u t o m a t i c - C o n t r o l System", Avtomatika i Telemekhanika, V o l . 22, No. 8, pp. 857-875, August, 1961.  2.  E . I . J u r y and B.W. Lee, "The A b s o l u t e S t a b i l i t y o f Systems w i t h Many N o n l i n e a r i t i e s " , Avtomatika i Telemekhanika, V o l . 26, No. 6, pp. 945965, J u l y 1965.  3.  V.A. Y a k u b o v i c h , "The S o l u t i o n of Some M a t r i x I n e q u a l i t i e s O c c u r r i n g i n the Theory o f Automatic C o n t r o l " , D o k l . AN SSSR, V o l . 143, No. 6, 1962.  4.  V..A. Y a k u b o v i c h , "The M a t r i x - I n e q u a l i t y Method i n t h e Theory o f the S t a b i l i t y o f N o n l i n e a r C o n t r o l Systems", Avtomatika i Telemekhanika, V o l . 25, No. 7, pp. 1017-1029, J u l y , 1964.  5.  E.N. Rozenvasser, "The A b s o l u t e S t a b i l i t y o f N o n l i n e a r Systems", i Telemekhanika, V o l . 24, No. 3, pp. 304-313, March, 1963.  6.  Kumpati S. Narendra and James H. T a y l o r , "Lyapunov F u n c t i o n s f o r Nonl i n e a r Time-Varying Systems", I n f o r m a t i o n and C o n t r o l , V o l . 12, pp. 378-393, 1968.  7.  Kumpati S. Narendra and Yo-sung Cho, " S t a b i l i t y A n a l y s i s o f N o n l i n e a r and Time-Varying D i s c r e t e Systems", SIAM J . 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Hsu, "The E v a l u a t i o n of Frequency Response S t a b i l i t y C r i t e r i a f o r N o n l i n e a r System v i a L o g a r i t h m i c Gain-Phase P l o t s " , Proc. 1968 JACC Conf. pp. 791-811.  17.  Gordon J . Murphy, "A Frequency-Domain S t a b i l i t y Chart f o r N o n l i n e a r Feedback Systems", IEEE Trans. Automatic C o n t r o l V o l . AC-12, pp. 740-743, December, 1967.  18.  M.S. Davies., " S t a b i l i t y of a C l a s s o f N o n l i n e a r Systems", E l e c t r o n i c L e t t e r s , V o l . 4, No. 5, pp. 322, J u l y , 1968.  19.  T.L. Chang and M.S. D a v i e s , " S t a b i l i t y of a C l a s s of N o n l i n e a r Systems", I n t . J . C o n t r o l , V o l . 10, No. 2, 1970.  20.  N.M. Trukhan, " S i n g l e - L o o p Systems which are A b s o l u t e S t a b l e i n the Hurwitz S e c t o r " , Avtomatika i Telemekhanika, V o l . 27, No. 11, pp. 5-8, November, 1966.  21.  N.G. Meadows, "New Analog-Computer Technique f o r Automatic Frequency-Response Locus P l o t t i n g " , Proc. IEE, V o l . 114, No..2, December, 1967.  22.  G.  23.  A.G. Dewey, "Frequency Domain S t a b i l i t y C r i t e r i a f o r N o n l i n e a r M u l t i V a r i a b l e Systems", I n t . J . C o n t r o l , V o l . 5, No. 1, pp. 77-84, 1967.  24.  T.L. Chang and M.S. D a v i e s , "The A b s o l u t e S t a b i l i t y o f N o n l i n e a r Systems I n v o l v i n g a Time-Varying Gain", Proc. 1969 A l l e r t o n Conf. pp. 721-729.  25.  R.W.  Zames, " N o n l i n e a r Time-Varying Feedback Systems-Conditions f o r LooBoundedness D e r i v e d U s i n g Conic O p e r a t o r s on E x p o n e n t i a l l y Weighted Spaces", P r o c . 1965 A l l e r t o n Conf. pp. 460-471.  Newcomb, " L i n e a r M u l t i p o r t  Synthesis", McGraw-Hill,  New  York,  1966.  

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