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On the absolute stability of nonlinear control systems Chen, Cheng-Fu 1970

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ON THE ABSOLUTE STABILITY OF NONLINEAR CONTROL SYSTEMS by .CHENG - FU CHEN B.Sc. i n E.E., Shizuoka U n i v e r s i t y , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in. the Department of E l e c t r i c a l Engineering We accept t h i s thesis as confoimii\g to the required standard Research Supervisor .... Members of the Committee Acting Head of Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA February, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Depa r tment ABSTRACT S u f f i c i e n t conditions for the absolute s t a b i l i t y of a class of non-l i n e a r sampled-data systems are derived using the techniques of system trans-formations due to Aizerman and Gantmacher. These c r i t e r i a are based on d i f f e r e n t forms used to approximate the area under the nonlinear c h a r a c t e r i s t i c . I t i s also shown that the s t a b i l i t y c r i t e r i o n can be improved by relaxing a r e s t r i c -t i o n on the slope of nonlinear element. In multiple n o n l i n e a r i l y continuous systems, by the a p p l i c a t i o n of numerical techniques, i t was shown that some improvement over previous s t a b i l i t y bounds can be made. TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS i v ACKNOWLEDGEMENT v 1. INTRODUCTION . 1 2. THE STABILITY OF NONLINEAR SAMPLED-DATA SYSTEMS 3 2.1 Description of System 3 2.2 S u f f i c i e n t Condition for Absolute S t a b i l i t y 5 2.3 Ap p l i c a t i o n of Theorem 1 12 3. THE STABILITY OF NONLINEAR SAMPLED-DATA SYSTEMS WITH A MONOTONE NONLINEARITY 16 3.1 Descr i p t i o n of System 16 3.2 S u f f i c i e n t Condition f o r Absolute S t a b i l i t y 16 3.3 Example 22 4. THE STABILITY OF NONLINEAR CONTINUOUS SYSTEMS WITH SEVERAL NONLINEARITIES 24 4.1 Formulation of the Problem 24 4.2 Statement of Theorem 24 4.3 S p e c i f i c Structures.. 26 4.4 Ap p l i c a t i o n of Theorem III 26 5. CONCLUSIONS 34 APPENDIX A 35 APPENDIX B 36 REFERENCES 38 i i i LIST OF ILLUSTRATIONS Figure Page 2.1 System S 4 2.2 Nonlinear Function 4 2.3 x" - y" Plane . 14 2.4 k - k" Plane 14 m 4.1 Nonlinear M u l t i - V a r i a b l e Feedback System 25 4.2 Two - Variable Cascade System 25 4.3 n - Variable P a r a l l e l System 27 4.4 n - Variable Series System 27 4.5 Flow-Chart of the Mul t i p l e Nonlinear System 30 i v ACKNOWLEDGEMENT I would l i k e to express my gratitude to Dr. M.S. Davies for h i s invaluable guidance during the preparation of t h i s thesis and to Dr. Y.N. Yu for reading the manuscript. In addition, I wish to acknowledge the f i n a n c i a l support of the National Research Council. I would also l i k e to thank my grandfather for h i s constant encourage-ment, and Miss Beverly Harasymchuk for typing the manuscript. 1 1. INTRODUCTION In recent years much attention has been given to the consideration of the absolute s t a b i l i t y of nonlinear control systems. The most f r u i t f u l approaches to the d e r i v a t i o n of s t a b i l i t y c r i t e r i a have been Liapunov's Second Method and Popov's frequency domain method. Before the p u b l i c a t i o n of V.M. Popov's paper [1] the i n v e s t i g a t i o n of nonlinear system was usually based on the Liapunov function of Lurie type [2] -consisting of a quadratic form plus an i n t e g r a l of the n o n l i n e a r i t y . This second method was extended to d i s c r e t e systems by Kalman and Bertram [3]. Although the method y i e l d s s u f f i c i e n t conditions for system s t a b i l i t y and i s general i n a p p l i c a t i o n , nevertheless i t i s s e n s i t i v e to the choice of sui t a b l e functions. In 1961, Popov [1] developed an e n t i r e l y new approach to the c l a s s i c a l problem of absolute s t a b i l i t y and obtained a number of very powerful r e s u l t s . One of the main advantages of Popov's method i s that a s t a b i l i t y c r i t e r i o n i s obtained i n terms of the frequency response of the l i n e a r part of the system, s i m i l a r i n many respects to the Nyquist s t a b i l i t y c r i t e r i o n f o r l i n e a r systems. Kalman [4] obtained an equivalent r e s u l t and showed that the frequency s t a b i l i t y c r i t e r i c v derived through Popov's approach i s a necessary and s u f f i c i e n t condition for the existance of a Liapunov function of the Lurie type. Recently, Tsypkin [5] and Jury and Lee [6], [7] used the Z-transform method to extend Popov's method to a class of sampled-data systems with a sing l e n o n l i n e a r i t y . Szego and Pearson [8] obtained e s s e n t i a l l y the same r e s u l t using the Liapunov method. Jury and Lee [9] then extended t h e i r r e s u l t s [6], [7] further to mu l t i v a r i a b l e systems, and Anderson [10] has used the Liapunov method for m u ltivariable continuous systems. In chapter I I , two s u f f i c i e n t conditions for the absolute s t a b i l i t y of nonlinear sampled-data systems are'derived. Although these c r i t e r i a are i n d e n t i c a l 2 to the r e s u l t s of Jury and Lee [6], [7], the d e r i v a t i o n , using the technique of system transformations of Aizerman and Gantmacher [11], i s more straightforward. Further, an example of the a p p l i c a t i o n of the theorem, when compared with previous work, [6], [8], i s more general. In chapter I I I , the constraints on the slope of the n o n l i n e a r i t y of Theorem I are relaxed. The theorem for absolute s t a b i l i t y of nonlinear sampled-data systems i s proved using the same method applied i n chapter I I . Several examples i l l u s t r a t i n g the best choice of the free parameters, q , and maximum gains, k , of the m u l t i - v a r i a b l e nonlinear continuous systems are considered i n chapter IV. In the general case, since the evaluation of the p r i n c i p a l minors of c r i t e r i a matrices i s very complex, some s i m p l i f i c a t i o n i s necessary i n order to reduce the number of v a r i a b l e parameters. 2. THE STABILITY OF NONLINEAR SAMPLED-DATA SYSTEMS 2.. 1 Description of System Consider the s i n g l e i n p u t - s i n g l e output sampled-data feedback system S shown i n Figure 2.1. It consists of a memoryless n o n l i n e a r i t y N and a l i n e a r , time-invariant plant G subject to the following conditions: (N.l) The nonlinear function 0(a) i s assumed to be piecewise continuous and to s a t i s f y the conditions: 0(0) = 0, 0 <e ^ 0(a)/a <_ k - e y • .. y o 0 (2.1) and -k' <_ 60/da <_ k". (2.2) The output of N i s u(n) - 0[a(n)]. (2.3) Eq. (2.1) r e s t r i c t s the nonlinear function to l i e i n a sector [e, k-e] with e a r b i t r a r i l y small as shown i n Figure 2.2. Eq. (2.2).bounds the slope of the nonlinear function. (G) The system at the nth sampling instant i s described by n a(n) = r(n) - n(n) - £ g ( n - i ) u ( i ) . ( ? i=0 K ' Here g(n) i s the sampled impulse response and n( n) i s the zero-input response of the l i n e a r plant G. The input r(n) i s assumed to tend to zero as n->c° and to be bound In addition, the following conditions are imposed: (G.l) For any i n i t i a l s t ate, n(n) and i t s f i r s t backward difference Vn(n). are bounded, are elements of I-^O,00) and so go to zero with time. (G.2) g(n) and i t s f i r s t backward differ e n c e Vg(n) are bounded, are elements of L (0,°°) and so go to zero with time. It i s assumed further that for some a > 1 T, a ng(n) < co . (2.5) n=0 Figure 2.2 Nonlinear Function. 5 Thus, G(z), the Z-transform of g(n) , i s a n a l y t i c in \z\>a , and has poles within c i r c l e |z| < a, i . e . the l i n e a r plant: i s s t r i c t l y stable. According to the notation of Jury and Lee [7], a nonlinear sampled-data (NSD) system S s a t i s f y i n g these assumptions for s p e c i f i c nonnegative k, k', k" i s r e ferred to as an NSD system S of (0,k; - k 1 , k " ) . In the following section, s u f f i c i e n t conditions are derived for the absolute s t a b i l i t y of the equilibrium point c(n) ='a(n) a u(n) = 0. The zero equilibrium point of the system described above i s said to be absolutely stable i f , for any 0[o(n)] s a t i s f y i n g Eqs. (2.1) and (2.2), the zero s o l u t i o n i s g l o b a l l y asymptotically stable. That i s lim c(n) = lim o(n) = lim u(n) = 0. (2.6) 2.2 S u f f i c i e n t Condition for Absolute S t a b i l i t y  Theorem I The NSD system S of (0,k: -<»,k") i s absolutely stable i f there e x i s t s a nonnegative number q such that ReH (z) = Re{[l+(z-l)q]G(z)} - ^ | ( z - l ) G ( z ) | 2 + ~ >_ 6 > 0 (OJ for a l l |z| = 1. Co r o ll a r y I The NSD system of (0,k; -k',«) i s absolutely stable i f there e x i s t s a nonnegative q such that ReH'(z) = R e { [ l + ^ ^ - q]G(z)} - (z-l)G(z) | 2 + \ > 6 > 0 (Q') for a l l |z| = 1. Note: If q=0, (Q) and (Q') reduce to the Tsypkin c r i t e r i o n [5] ReC(z) + r > 6 > 0 (P) k ~ for a l l |z| = 1, where 6 i s a r b i t r a r i l y small. Proof of Theorem I Rewriting the system equation (2.4) equivalent:!}/ as n o(n) = x(n) - E g ( n - i ) u ( i ) (2.7) i=0 where x(n) = r(n) - n(n) . (2.8) From Eq. (2.7), the f i r s t backward difference of a(n) i s n • - Va(n) = Vx(n) - E V g ( n - i ) u ( i ) . (2.9) i=0 Truncate the v a r i a b l e u(n) at any p o s i t i v e integer N, then define the following a u x i l i a r y functions such that u(n), 0 < n < N u N(n) = j (2.10) 0, otherwise n a (n) = x(n) - E g(n-i)u ( i ) , (2.11) i=0 Va (n) = Vx(n) - E Vg(n-i)u ( i ) . (2.12) . i=0 N Therefore, because of the truncation, u^(n) are elements of L^(0,m). In addition, by assumption (Gl) , x(n) and Vx(n) c l e a r l y belong to L (0, r o) , and by (G2), g(n) and Vg(n) belong to L^(0, O T). Then, making use of the fact that the convolution of and sequences i s an sequence, o^(n) and Vo^(n) belong to L 2(N,«). Let us introduce the notation f 1 ( n - l ) = a N ( n - l ) f qVo^n) - k ' ^ C n - l ) , (2.13) f 2 ( n - l ) = x(n-l) + qVx(n). (2.14) It i s obvious that f^ and f 2 belong to L^iO,<*>) . This guarantees the existance of Z-transform denoted by F^(z) and F 2 ( z ) . Bv s u b s t i t u t i n g Eqs. (2.11) and (2.12) into Eq. (2.13) and using Eq. (2. we obtain • n f (n-1) = f 9 ( n - l ) - E [ g ( n - i - l ) + q Vg („-i) ] u (i) - k _ 1 u (n-1). (2.15) 1 2 . = 0 N N Consider the summation 00 E f 1 ( n - l ) u N ( n - l ) a 2 ( n _ 1 ) (2.16) n=l where a. > 1. Then, from Eq. (2.16), CO co E f 1 ( n - l ) u N ( n - l ) a 2 ( n " 1 ) = E f 2 ( n - l ) u N ( n - l ) 2 ( n _ 1 ) ri=l n=l 00 - E u N ( n - l ) a ( n _ 1 ) { E [g (n-i-1) + • q Vg ( n - i ) ] (i ) } a ( n _ 1 ) n=l i=0 - k" 1 z u ^ n - D c ? ^ . (2.17) n=l It i s now to be noted that ^ r , .,2 2(n-l) N r , .,2 2 (n-1) °° r . ...2 2 (n-1) .„ . • E [Vo(n)l a ~ E [Va N(n)] a <_ E [Va N(n)] a (2.18) n=l n=l n=l and, by using Eq. (2.].2) we obtain N 0 0 n k"q „ r„ , u 2 2 (n-1) k"q r , , . ... ,..,2 2 (n-1) ... . -y 1- E [V0(n)J a ~~2^ l V x ( n ) " E Vg(n-x;u (x) ] a . v2.j.9) n=l n=l i=0 Adding the qu a n t i t i e s on both sides of the i n e q u a l i t y (2.19) to those of Eq. (2.17), the following key i n e q u a l i t y i s established: oo N v ' c r u r i\ 2(n-1) k"q r , . , 2 2(n-1) E f (n-l)xv N(n-l)a + "2 t V a ( n ) ] a n=l n=l >_ E f 2 ( n - l ) u N ( n - l ) a 2 ( n _ 1 ) n=l 00 1*1 - E u N ( n - l ) a ( n _ 1 ) { E [g(n-i-1) + q V g ( n - i ) ] u N ( i ) } a ( n _ 1 ) n=l i=0 CO -1 „ 2 2(n-1) -k E u (n-1)a n=l + Z l'Vx(n) - E. Vg(n-:i.)u ( i ) ] 2 a 2 ( a _ L ) . (2.20) n = l i=0 A p p l y i n g a m o d i f i e d form of P a r s e v a l ' s theorem (Appendix A) to the r i g h t -hand s i d e ( r . h . s . ) of i n e q u a l i t y (2.20) and c o l l e c t i n g l i k e t e r m s , the r i g h t - h a n d s i d e o f Eq. (2.20) can be e x p r e s s e d a s : r . h. s . = -z—r 0 2TTJ J c { [ l + ( Z l - D q ] G ( Z ] ) - ^3- | ( z 1 - l ) G ( z ] ) | 2 + k 1 ) | U N ( Z l ) i 2 Z _ 1 d z + 2 K £ [F *(z,) - k"q|z -1|2 X*(z.)G(z 1)]U. 1(z.)'z ^ z 77 j J c 2 1 1 1 1 1 1 N 1 A 77 j JC + T — V | ( z 1 - l ) X ( z 1 ) | ' z "dz, (2.21) where z = e x p ( j u ) , z^ = -^HSEil^l. a n t j - j n g e n e r a l [12] f o r any x (n) w i t h Z - t r a n s f o r m X ( z x ) , Z.[x(n)a n] = X ( Z l ) , (2.22) and t h e a s t e r i s k (*) i s used to i n d i c a t e t h e c o n j u g a t e o f a complex f u n c t i o n . I t i s c o n v e n i e n t t o d e f i n e t h e f o l l o w i n g f u n c t i o n s H 1 ( z 1 ) = [ 1 + ( z 1 - l ) q ] G ( z 1 ) - | ( z 1 - l ) G ( z 1 ) I 2 + k" 1, (2.23) F 4 * ( Z l ) = F 2 - ( z ] ) - k"q|z - l | ^ X * ( z 1 ) G ( z 1 ) t (2.24) N o t h i n g t h a t t h e r i g h t - h a n d s i d e must be a r e a l q u a n t i t y , a f t e r s u b s t i t u t i n g z^ = e^°Va and z "'"dz = jdu) i n Eq. ( 2 . 2 1 ) , i t can be shown t h a t r . h . s . =-T— f T r R e l l ( e j W / a ) | n T ( e 3 U / c t ) I 2dw 2 7 T J_rr 1 TT , 1 f F* (e^V«)lI(ejW/a0du + " - ^ J U / - M T 477 , / 4 N 4TT -77 «/-T7 F, ( e J " / a ) U * ( e J U J / a ) d c o 4 N -77 where 7^ (- l ) X ( e ] U / a ) /dai (2.25) 4TT ,/ I a • 1 | t F / ^ ( e j a , / a ) U x . ( e j w / a ) + F, ( e j w / a ) U *(e j u )/a) ] = Re[F * ( e > / a ) U . ^ e ^ / a ) ] . (2.26) 2 4- N 4 N 4 N If t hen ReH ( e J W / a ) > 6 > 0, for -v < co < ir, (2.27> .1. — a, — — 7 Re{[l + ( Z l - l ) q ] G ( Z l ) } - ^  | ( z ^ l ) G ( Z ] ) | 2 + k" 1 >_ 8 . (0) Since s a t i s f a c t i o n of (Q) i s assumed, hence (Q) implies (Q ) f ° r s u f f i c i e n t l y small a > 1 [13]. Now completing the.square i n the right-hand side of Eq. (2.25), we obtain f71 . 1 / 9 . F ( e J A 7 A ) 2 r.h.s. = -(2v) 1 J |[ReH 1(e : , 0 J/a)] X U ^ e ^ / a ) + — 1 dco J ~ V 2 [ R e H l ( e j w / a ) ] 1 / 2 ATTIF. (e^ /a) | . ,, p-n j co . 0 + 1 [ l_4 d u + f ^ | | ( — - l ) X ( e J 0 7 A ) | 2 dco. (2.28) 8 ^ J-rr ReH^e ^ / a ) 4 T r J . i r a Removing the f i r s t integration, which i s always negative, from the right-hand side and making use of the condition (2.27) yields ir r.h.s. < 1 V | F / ( e ^ / a ) | 2 d U + ^ (- l ) X ( e J W / a ) |" dco. (2.29) From Appendix B, Eq. (2.29) can be expressed as: r.h.s. < E -^-'{[xCn-l) + qVx(n)] 2 + [ V x ( n ) ] 2 } a 2 D . ( 2 . 3 0 ) n=l a It i s obvious that the right-hand side of Eq. (2.30) does not depend on N, but depends q u a d r a t i c a l l y on the i n i t i a l conditions and tends to zero with the i n i t i a l condition tending to zero. Now returning to the left-hand side (l.h.s.) of Eq. (2.20) and s u b s t i t u t i n g the expression for f (n-1), we obtain CO . l . h . s . = E [ o N ( n - l ) - k ~ 1 u > J ( n - l ) ] u N ( n - l ) a 2 ( n _ 1 ) n=l CO N + q £ n f \ r i \ 2 ( n " l ) , k"q r - , , 2 2 (n-1) ,„ s n Vo„(n)u (n-l)a . + — E [Va(n)j a , (2.31) n=l N N I 1 n=l 10 o r N i u v n 0 [ a ( n - - l ) ] , ( ,. r , .-, 2 ( n - 1 ) l . h . s . = E [1 - , 7 J a ( n - 1 ) 01 a ( n - 1 ) J a n=l N ,, N , _ v + q {E 0 [ a ( n - l ) ] V a ( n ) + E [Va(n) ] } c c ° " ; . (2.32) n = l " n = l The f i r s t sum i n Eq. (2.32) i s al w a y s p o s i t i v e , t h a t i s E (1 - -J P in~K} ]) a (n-1) 0 [ a (n-1) ] > 0, f o r a l l N > 0. (2.33) n k a ( n - l ) — n = l d0 ,, F o r t h e second and t h i r d sums, because o f the c o n s t r a i n t - r o < -3— < k' , — da — the f o l l o w i n g a r e a i n e q u a l i t y , due t o J u r y and Lee [ 6 ] , a p p l i e s . l c" o (*a Cn) 0 [ a ( n - l ) ] V a ( n ) + j - [Vo(n)] >_ I 0 ( a ) d a , f o r a l l n > 0, (2.34) J o ( n - 1 ) and, s i n c e q >_ 0, i t f o l l o w s t h a t N k"a N ' 2 q E 0 [ a ( n - l ) ] V a ( n ) + Z [ V a ( n ) j >_ q [$ (a [N] ) - * ( a [ 0 ] ) ] >_-q$(a[0]) (2.35) n = l n = l where a * ( a ) = 0 ( a ) d a . ^ 0 T h e r e f o r e , by u s i n g t h e i n e q u a l i t y (2.35) i n ( 2 . 3 2 ) , i t . f o l l o w s t h a t l . h . s . > E [1 - K £ ^ - m ] a ( n _ 1 ) 0 [ a ( n _ 1 ) ] c t 2 ( n - l ) _ ^ $ ( a [ ( ) ] ) a 2 ( n - l ) ( 2 i 3 6 ) n=l F i n a l l y , r e t u r n i n g a g a i n t o Eq. (2.20) and u s i n g t h e Eqs. (2.30) and (2.36) t h e i n e q u a l i t y i s r e a r r a n g e d t o y i e l d I { 1 _ ^ ( n - U l j ( n _ 1 ) g [ ( n _ 1 } ] . k o ( n - l ) n = l <c E { y f - t x C n - l ) + q y x ( n ) ] 2 + ^ r T x ( n ) ] 2 } -k[ * (a [ 0 ]) . (2.37) — 46a 2. n=l • o r N {1 _ K^-llli] a ( n - l ) 0 [ a ( n - l ) ] < C. (2.38) E k a ( n - l ) — n = l 11 Here the constant C, which denotes the right-hand side of the i n e q u a l i t y (2.37), depenc only on the i n i t i a l values and. tends to zero together with them. Note t h i s con-stant i s obviously independent of N. Consequently, the i n e q u a l i t y (2.38) i s v a l i d for a l l N, and implies that t h i s sum i s uniformly bounded for a l l N [6], Therefore lim o ( n - l ) 0 ( n - l ) [ l - f ^ i p r ] = 0. (2.39) n->"» ka(n-l) From Eq. (2.1) , [1 - r^T-Hrl > °> for a l l a ± 0. (2.40) ka(n-l) Hence lim a(n) = 0, and lim 0[o(n)] = 0, (2.41) n -y °° n -> 0 0 and the theorem i s proved. Proof of the Corollary For the NSD system S of (0 ,k,-k' ,°°) , the constraints of nonlinear element are s a t i s f i e d by the conditions (2.1) and 1 i d0(a) , 0. -k •= , < 0 0 (2 .42) ~ da —. Since q > 0, these conditions ensure the existance of the following i n e q u a l i t y area [7], that i s ^ ,a(n) q0[a(n)]Va(n) + — • ^ [ V a ( n ) ] 2 >_ q 1 0(a) da, V n l ° , (2.43) ^ a ( n - l ) The a u x i l i a r y functions f^ and f^, f i r s t defined i n Eqs. (2.13) and (2.14), are replaced by f x ( n ) = a N(n) + qAa^n) - k ^ u ^ n ) (2.44) f (n) = x(n) + qVx(n). . (2.45) Now, following the steps of the d e r i v a t i o n given i n Theorem I, i t can be shown that i f the condition, ReH'(e J°7cO > ^ > 0, for -TT <_ w <_ ir (2.46) 12 i s s a t i s f i e d the following i n e q u a l i t y , f i r s t shown i n Eq. (2.37), follows. I { l _ M £ M ] o ( n ) j [ o ( „ ) ] n=0 k o ( n ) 1 E {~- [x(n) + qVx(n)] Z + - ^ I Vx (n) ] ~ } + <$>(a(0)) (2.47) n=0 a or N E i l - f ^ Y ^ - ] a ( n ) 0 [ a ( n ) ] < C \ (2.48) n=0 k a ( n ) Here the constant C', j u s t l i k e the constant C, depends only on the i n i t i a l values and tends to zero together with them. This completes the proof of the c o r o l l a r y . 2 . 3 A p p l i c a t i o n of Theorem I The examples given i n [6] and [8] a l l s a t i s f y the constraint Re(z-l)G(z) > 0, for |z| - 1 (2.49) and the maximum k", denoted by k" , i s j u s t equal to the minimum p o s i t i v e value m of -x" given by: x " = -Re(z-l)G(z) | ( z - l ) G ( z ) | 2 ' Furthermore, the in e q u a l i t y (Q) does depend on the choice of q and no simple gra-p h i c a l procedure has yet been developed for q 4 0. In t h i s t h e s i s , the case of q 4 0 i s considered and the following step by step method for determining maximum k of k" i s suggested. F i r s t rewrite the ineq u a l i t y (Q) as 2 [ R e G ( z ) ± 4 ^ - - q [ - R e J ( z - l ) G ( z ) ] + k I I ] > Q j ( 2 _ 5 0 ) | ( z - l ) G ( z ) | Z | ( z - l ) G ( z ) r and l e t »• = 2(ReG(z) + k" 1)  7 " | ( z - l ) G ( z ) | 2 ' ( 2 ' 5 1 ) x" = z M l z i ^ k l . ( 2 . 5 2 ) | ( z - l ) G ( z ) p 13 Then Eq. (2.50) can be e x p r e s s e d as y" - q(x"+k") >_ 0. (2.53) But the e q u a t i o n y" - q(x"+k") = 0 (2.54) i s t he e q u a t i o n o f a s t r a i g h t l i n e w i t h s l o p e q, p a s s i n g t h r o u g h the p o i n t -k" on t h e r e a l a x i s o f x"-y" p l a n e shown i n F i g u r e 2.3. T h i s s t r a i g h t l i n e d i v i d e s t h e p l a n e i n t o two h a l f p l a n e s . The i n e q u a l i t y (2.53) i s s a t i s f i e d i f a r e a l p o s i t i v e q may be found such t h a t t h e p l o t o f x"+ j y ' 1 as a f u n c t i o n o f ui(0 t o X ^ i n t h e x"-y" p l a n e , l i e s e n t i r e l y t o t h e l e f t o f t h i s s t r a i g h t l i n e . From F i g u r e 2.3 ? f o r y" = 0 a t to = w^ , Eq. (2.54) becomes k" = - x " (2.55) Hence k „ = 2 [ R e ( z - l ) G ( z ) ] m ~ | ( z - l ) G ( z ) | 2 z-.e (2.56) I t i s c l e a r t h a t k" may be found from t h e x"-y" p l a n e f o r d i f f e r e n t m J r . v a l u e s o f k s t a r t i n g w i t h the v a l u e a t q=0. Then, i f k" i s g i v e n , i t i s easy t o f i n d t h e l a r g e s t p e r m i s s i b l e k from k-k" c u r v e , Example 1. The open l o o p t r a n s f e r f u n c t i o n o f t h e l i n e a r p a r t o f system S i s G ( S ) = T^ITITTS-TOTST: • ( 2 - 5 7 ) C o n s i d e r t h e s a m p l i n g p e r i o d T = 1 second, and t a k e t h e Z - t r a n s f o r m o f G(s) G(z) = 5[ ^Ti - Z - , 3 J . >:V, ( 2- 5 8) z-e ' z-e S u b s t i t u t i n g z = e^' i n t o Eq. ( 2 . 5 4 ) . Then s e t t i n g q = 0 i n i n e q u a l i t y ( 0 ) , one can f i n d M i n . R e G ( e j t 0 ) = -3.44. (2.59) 14 F i g u r e 2 . 4 k" P lane , m 15 Therefore k = 0.29. (2.60) Now, according to the procedure mentioned above, the r e l a t i o n between k and k" , i s shown i n Table 1, and i s plo t t e d i n Figure 2.4. Table 1 k k" m 0.3 0.837 0.5 0.809 0.7 0.805 0.8 0.804 0.81 0.804 From Figure 2.4, the minimum p o s i t i v e value of k" that s a t i s f i e s the m constraint k_ < k" i s equal to 0.804. Hence, the system S i s absolutely stable i f k < k" < 0.804. (2.61) 3. THE STABILITY- OF NONLINEAR SAMPLE1)-DATA SYSTEMS WITH A MONOTONE NONLINEARITY 3.1 D e s c r i p t i o n o f System In t h i s c h a p t e r the s t a b i l i t y o f NSD systems v ; i t h a monotone i n c r e a s i n g n o n l i n e a r f u n c t i o n w i l l be c o n s i d e r e d . The system i s s i m i l a r t o t h a t t r e a t e d i n C h a p t e r I I . (N) The n o n l i n e a r f u n c t i o n 0 [ o ( n ) ] , i s assumed t o be a p i e c e w i s e c o n -t i n u o u s and a m o n t o n i c a l l y i n c r e a s i n g r e l a t i o n , and i s d e s c r i b e d by t h e e q u a t i o n s ( 2 . 1 ) , (2.3) and 0 'toJ - 0[o-J 0 < — — < k f o r a , f a „ (3.1) - ° r ° 2 - ' 1 2 (G) By s e t t i n g t he i n p u t r = 0, the system i s d e s c r i b e d by n a(n) = -n(n) - E g ( n - i ) u ( i ) , (3.2) i=0 o r u s i n g t h e symbol * t o denote t h e c o n v o l u t i o n , we o b t a i n a(n) = -n(n) - ( g * u ) ( n ) . (3.3) Here n and g s a t i s f y t h e c o n d i t i o n s ( G l ) and (G2) o f C h a p t e r I I . D e n o t i n g = SUP | n ( n ) | , g M = STIP |g(n)|, (3.4) n >_ 0 . n >_ 0 and u s i n g || • j| to denote L^ norms and || • ^ to denote norms i n d i v i d u a l l y , II y J = E |y |, ||r,||? = ( E p 2 ( n ) ) 1 / 2 (3.5) n=0 n=0 3.2 S u f f i c i e n t C o n d i t i o n f o r A b s o l u t e S t a b i l i t y  Theorem I I . C o n s i d e r a system s a t i s f y i n g t he above c o n d i t i o n s . L e t y ( n ) be any r e a l - v a l u e d f u n c t i o n such t h a t ( i ) y ( n ) = 0 f o r n < 0, ( i i ) y ( n ) < 0 f o r n > 0, ( 3 , 6 ) ( H i ) | y |j < I-17 I f , fen: some q > 0, Rell 2(z) = Re{[l+(z-l)q+Y(z)]G(z)} - ^  | ( z - l ) G ( z ) | 2 + Re [ l + Y ( z ) ] k _ 1 > 8 > 0, for a l l |z| = 1. (3.7) then (i ) sup | a(n) | < oo, (3.8) n >_ 0 ( i i ) l im a ( n ) = 0, (3.9) n -> oo ( i i i ) as || n H2 + || Vn ||9 -> 0, the corresponding a (11) has the property that sup |a(n) [ ->0. n > 0 Lemma. Let x(n) and f[x(n)] be functions i n L ( - 0 0 , 0 0 ) . - If f[x(n)] i s monotonically increasing, then f o r a l l . i , CO CO E x ( n - i ) f [x(n)] < E x ( n ) f [ x ( n ) ] . (3.10) n=-oo n=-co I f , i n add i t i o n , f[x(n)] i s odd (f(x) = - f ( - x ) ) , then the in e q u a l i t y above holds with absolute value taken on the l e f t sum. The proof of th i s lemma i s obtained with s l i g h t modification from that contained i n [14]. This important r e s u l t i s used i n the proof of Theorem I I . Proof of Theorem II For any p o s i t i v e integer N > 0, the following a u x i l i a r y functions are defined: (u (n) 0 < n < N u N(n) = j (3.11) 10, otherwise u N(n) =-0[a N(n)], (3.12) n a „(n) = I g(n-i)u ( i ) , (3.13) eN . _ N i=0 n Aa„(n) = E Vg(n-i)u..(l) . (3.14) eN . r. N i=0 Thus, o „(n) and Va M ( n ) b e l o n g to L 0(N,«). eN eN I L e t a = o + o• " V, u = u + u*y (3.15) m ' m and, i n g e n e r a l , g i v e n any f u n c t i o n x, we d e f i n e [15] x - x + x*y m • Then, l e t us i n t r o d u c e the n o t a t i o n f l ( n - 1 } = o^'U + q V ° N ( n ) - k ~ l u ^ n - 1 ) ' ° - 1 6 ) f„(n~l) = - n (n-1) - qVn(n). (3.17) I m Because a N ( n ) = - n(n) - a ^ n ) , (3.18) t h e r e f o r e , f 1 ( n - l ) = f 2 ( n - l ) - [ a e N m ( n - l ) + q V a ^ n ) + k 1 u m N ( n - l ) ] , (3.19) or Z f 1 ( n - l ) u N ( n - l ) = E f 2 ( n - l ) u N ( n - l ) n=0 n=0 CO - E [a „ (n-l)H-qVa . T ( n ) + k - 1 u ..(n-1) ]u (n-1) . (3.20) _ eNm eN mN N n=0 I t i s 'low t o be n o t e d t h a t 00 CO Z [ V o N ( n ) ] 2 = ^ E [Va (n) + V n ( n ) ] 2 . (3.21) n=0 n=0 M a k i n g use of Eqs. (3.20) and ( 3 . 2 1 ) , we o b t a i n CO CO E f ^ n - D u ^ n - 1 ) + ±f Z [ V a N ( n ) ] 2 n=0 ' n=0 CO o • = E f 0 ( n - l ) u M ( n - l ) - E [a M (n-l)+qVa „ ( n ) + k - 1 u (n-1) ] u M ( n - 1 ) „ 2 N _ eNm eN mN N n=0 n=0 CO + V 1 5 : ^ + V n ( n ) ] 2 (3.22) 2 n = 0 eN(n) Then, a p p l y i n g P a r s e v a l ' s theorem [18] t o t h e r i g h t - h a n d s i d e ( r . h . s . ) o f Eq. (3.22) and c o l l e c t i n g l i k e t e r m s , the r . h . s . o f Eq. (3.22) can be e x p r e s s e d a s : .19 r.h.s. = $ {[l+(z-l)q + Y(z)]G(z) - ^ |(z-l)G(z) + [1+Y(z)]k 1 } | u N ( z ) | 2 z ]"d z + - ~ j> [ F 2 ( z ]-) - kq|z-l| 2X(z 1)G(z)]U N.(z)z 1 dz + | ~ J |(z-l)X(z) | 2 z " 1 dz (3.23) c where X(z) = -Z[n(n)]. Now, define H 2(z) = tl+(z-l)q+Y(z)]G(z) - (z-l)G( ?) | 2 + [1+Y(z)]k \ (3.24) F*(z) = F 2 ( z _ 1 ) - k q | z - l | 2 X ( z _ 1 ) G ( - z ) . (3.25) Since the imaginary part of the right-hand side must be equal to zero, and by the condition of the theorem, ReH (z) >_ 6. Then, following the steps of the d e r i v a t i o n given i n Theorem I (from Eq. (2.25) to (2.30)), i t can be shown that CO r.h.s. < Z {yrtti (n-1) + qVn(n)] 2 + |~[7n. (n) ] 2 }. (3.26) n=0 Now, returning to the left-hand side (l.h.s.) of Eq. (3.22) and s u b s t i t u -t i n g the expression f o r f ^ ( n - l ) , we have -] l.h.s. = E [ o m N ( n - l ) + qVa N(n) - k (n-1) ] U"N (n-1) n=0 , N + f 3 - E [ V a ( n ) ] \ (3.27) n=0 Consider °° u N(n) En[ 0mN ( n ) ~ - J T — ] u N < n ) n=0 °° u. (n) co u = M o N ( n ) - - | ]u N(n) + E [yMoy - ^ ) ] ( n ) u ( n ) . (3.28) n=0 n=0 ' 'v ' 20 D e f i n e u ( n - i ) R ( i ) = I [a,.(n-i) . -]u..(n) (3.29) i-l k N • n=0 S i n c e 0(a) i s a m o n o f c o n i c a l l v i n c r e a s i n g f u n c t i o n , by lemma, R ( i ) <_ R(0) . (3.30) C o n s i d e r i n g the l a s t sum i n Eq. (3.28) co u c o u ( n - i ) V y M 0 N " . l T ) ) ( n ) u N ( n ) = E ( y ( i ) [ a N ( n - i ) - H u ^ n ) n=0 n=0 i=0 oo co u ( n - i ) = E y ( i ) { 2 [ a N ( n - i ) - - ^ - L u ^ n ) } z=0 n=0 i f 0 y ( i ) R ( i ) . . (3-31) S i n c e y <_ 0, s u b s t i t u t i n g Eq. (3.30). i n t o Eq. (3.31) y i e l d s 00 E y ( i ) R ( i ) >_ -R(0) || y || (3.32) i=0 L e t i = 0 . Then s u b s t i t u t i n g Eqs. (3.29) and (3.32) i n t o eq. (3.28) g i v e s u (n) E [ a m N ( n ) ~ "IT ]Vn)>. [ l . - | | y | | ] R ( 0 ) . (3.33) n=0 1 F o r |jy|| <_ 1, t h e r e i s a f i n i t e b (N) >_ ( 1 . - || y || ) > 0, such t h a t N u (n) E [a (n) - ] u ( n ) = b ( N ) R ( 0 ) > 0. (3.34) A m k — n=0 Remember t h a t u (n) = 0[a ( n ) ] . By u s i n g Eq. (3.34) i n Eq. ( 3 . 2 7 ) , i t f o l l o w s t h a t CO ' CO l . h . s . I 0[a ( n - l ) ] V a (n) + y q- E [ V a , , ( n ) ] 2 (3.35) n=0 n=0 d0 A l s o , . because o f t h e c o n s t r a i n t 0 < 5 k , t h e i n e q u a l i t y (2.35) can be used i n Eq. ( 3 . 3 5 ) . Hence l . h . s . >_qS[c(N)] - q<J[a(0)]. (3.36) F i n a l l y , r e t u r n i n g a g a i n to Eq. (3.22) and u s i n g the Eqs. (3.26) and (3.36) and r e a r r a n g i n g t h e i n e q u a l i t y on b o t h s i d e s y i e l d s q * [ a ( N ) ] - q<I'|'a(0)] < E' { y - f n (n-1) + q y n ( n ) ] Z -I- ^ i V n C n ) ] 2 } . (3.37) - n = 0 46 m L e t t i n g CO N ( II 5 1 12' II V n II 2) = E n { f ; [ n (n-D + qVn(n)] 2 + ^ ( v n ( n ) ] 2 } (3.38) n=0 4o 111 z *• and a d d i n g t o b o t h s i d e s of t h e i n e q u a l i t y (3.37) t h e p o s i t i v e q u a n t i t y a<i>[a(0)], a l s o , f o r c o n v e n i e n c e , w r i t i n g H f o r M( jj n j l^, IIVMI^- ' ^ c l " (3.37) becomes • . q * [ o ( N ) ] <_ M + q * [ c r ( 0 ) ] . . (3.39) S i n c e 0 [ a ( n ) ] i s m o n o t o n i c * [ a ( n ) ] > [ 0 ( a ( n ) ) ] 2 / 2k, (3.40) hence u(N) j 2 <_ M q - 1 + * [ a ( 0 ) ] . (3.41) I t i s c l e a r t h a t t h e r i g h t - h a n d s i d e o f i n e q u a l i t y (3.41) i s i n d e p e n d e n t of N, 2 thus |u(N)| i s u n i f o r m l y bounded f o r a l l N. T h e r e f o r e SUP | u ( n ) | <_ {2K[Ma + $ ( a ( 0 ) ) ]} (3.42) n >_ 0 and SUP ^ [ u ( n ) | tends t o z e r o as || n | j 2 + || Vq | [ 2 -> 0. Nov;, l e t us show t h a t o(n) -> 0 as n ->• °°. Prom Eqs. ( 3 . 2 7 ) , (3.34) and (3.37)., we have N r _n b(N) £ ( o ( n - l ) - ^ f ^ - ) u ( n - l ) + q $ ( a ( N ) ) - a $ ( a ( 0 ) ) < M (3.43) n=u Thus N . . _ i • Z [a (n-1) - ^ g^-Mn-1) <_ [H + q* ( a ( 0 ) ) ] (1 - || y || ) \ (3.44) n=0 S i n c e t h e r i g h t - h a n d s i d e o f Eq. (3.44) i s o b v i o u s l y i n d e p e n d e n t o f N, and tends to z e r o w i t h t o g e t h e r jj n ||2 + |J Vq ||2 _ > 0. T h i s p r o p e r t y i m p l i e s t h a t the sum o f Eq. (3.44) i s u n i f o r m l y bounded f o r a l l N. T h e r e f o r e , u s i n g the same arguments as i n Theorem I one can show t h a t lim a(n) = 0 . (3.45) n -> c o I n a d d i t i o n , s i n c e u(n) -> 0 o n l y i f a (n) -> 0, i n e q u a l i t y (3.42) i m p l i e s t h a t 2 2 SUP „ I o (n) I < »and tends to zero as |j n IL + I !! ~y °. n > u i i II / n ii 2 This completes the proof of Theorem I I . 3.3 Example Example 2. Consider the system and transfer function of example 1. By taking q =0, the i n e q u a l i t y (3.7) becomes Re[l + Y(z)][G(z) + k _ 1 ] >_0. (3.46) or [1 + ReY(z)]ReG(z) - ImY(z)ImG(z) + [1 + R e Y ( z ) ] k _ 1 >_ 0. (3.47) Assuming 1 + ReY(z) > 0, (3.48) then, d i v i d i n g 1 + ReY(z) on both sides of i n e q u a l i t y (3.47) y i e l d s ReG(z) = ImY(z)ImG(z)/[l + ReY(z)] + k" 1 >_ 0. (3.49) Let ReG'(z) = ReG(z) - ImY(z)ImG(z)/[1 + ReY(z)]. (3.50) Hence ReG' (z) + k 1 >_ 0. (3.51) 1 *~ cn Suppose y(n) = - -^ e with a choice of c and d to s a t i s f y CO ||y(n)|| = E |- i e " C n | < 1. (3.52) n=0 The Z-transform of y(n) i s Y(z) = Z _r (3.53) d(z - e ) * By choosing c = 1, d = 1.7 and s u b s t i t u t i n g z = e^^ into Eq. (3.53), i t can be shown that Min [1 + ReY(e- 1 w)] > 0. (3.54) Then, making use of Eqs. (2.58) and (3.53) i n Eq. (3.47), we f i n d • Min ReG'(ej'°) = -0.8075 • (3.55) at co - 0. 8 rad. . Therefore k £ 1.237. (3.56) Due to the compensation of ImY(z)ImG(z)/[l+ReY(z)] i n Eq. (3.49), the value of k i s larger than k = 0.29, the value found from Theorem I. In t h i s example, q was taken zero i n order to reduce the complexity of c a l c u l a t i o n . 24 4. THE STABILITY OF NONLINEAR CONTINUOUS SYSTEMS WITH SEVERAL NONLINEARITIES 4 . ]. Formulation of the Problem The system under consideration i s shown i n Figure 4.1, where _r, _a, _u and c are n-vectors. N i s a time-invariant memoryless nonlinearity_ The i t h components of i t s input, o \ ( t ) , and output, 0^(o\(t)) are assumed to be characterized as follows. • 0 < 0.o. < k.a. 2, for a. 4 0, (4.1) - l i — i i I 0.(0) = 0, ( i = 1,2,.. ,n), (4.2) where i s the i t h element of diagonal matrix k. G i s a linear-time i n v a r i a n t sub-system, the r e l a t i o n of i t s input, _0(t) and output, c_(t), i s described by the equation: c(t) = n(t) - f g(t - - T ) 0 ( T)d T, (4.3) o where g(t) i s the nxn impulse response matrix of G, and n(t) i s n-vector and i s the zero-input response of G. It i s assumed that the following conditions are s a t i s f i e d : (Gl) For a l l i n i t i a l states, the zero input response (a) n^Ct) i s bounded on (0,c°), (b) n 1 ( t ) , p j L ( t ) e L2(0,c°) and (c) n ^ t ) -> 0 as c -> co. (G2) Impulse response of G (a) g (t>e L^(0,°°) and (b) g _ (t) ->- 0 as t 0 0. In the f i n i t e - d i m e n s i o n a l case, (Gl) and (G2) are implied by the require-ment that G(s), the Laplace transform of g ( t ) , has no s i n g u l a r i t y i n the r i g h t -hand plane or on the r e a l frequency axis except for a s i n g l e pole at the o r i g i n . 4.2 Statement of Theorem  Theorem III Let k be a r e a l diagonal matrix with p o s i t i v e elements. If the system under consideration s a t i s f i e s the above assumptions and there e x i s t s a r e a l diagonal matrix q, such that Figure 4.1 Nonlinear M u l t i - v a r i a b l e Feedback Sy 11 7-c 21 + £ r. Figure 4.2 Two-Variable Cascade System. • P (co) = W(ju) + W*(jio) > 0. V 'o, (4.4) then the system i s absolutely stable. VJ* i s the complex conjugate transpose of W and W(JOJ) = [I + j oic! ] G (j O.0 + k _ 1 . (4.5) The proof of the theorem was given by Jury and Lee [9], using the Popov approach, and by Anderson [10], using the Liapunov approach. 4 . 3 S p e c i f i c Structures Let us consider the general case of transfer matrix G(s) as G(s) = [g ( s ) ] , ( i , j = l , n ) , ( 4 . 6 ) for which the corresponding system (with n = 2 ) shown i n Figure 4.2 i s a cascade connection. However, when G (s) = [ G l n ( s ) ] s • ( i = 2,n; j = l , n ) , ( 4 . 7 ) the system also implies a p a r a l l e l connection as shown in Figure 4 . 3 . In addition, one serie s case i s considered f o r which G(s) 0 0 0 G m ( s ) - G 2 1 ( S ) o - G 3 2 ( s ) I 0 -G . (s) 0 ,n n-1 ( 4 . 8 ) The corresponding system i s shown i n Figure 4 . 4 . 4 . 4 Application of Theorem I I I In t h i s section, the system shown i n Figures 4 . 2 and 4 . 4 w i l l be con-sidered as example. For the convenience of future use, the G(jw) matrix appearing i n the s t a b i l i t y c r i t e r i o n takes the general form as shown i n Eq. ( 4 . 6 ) . h N l G 11 Figure 4 .3 n - Variable P a r a l l e l System. • N l G b 2 1 °2 Ficure 4.4 n - Variable Series System. ( l . + j u q ^ G ^ (jaO+l^ • (l.+;iuq 1)G (jai) (l.+.iojq-j ) G ] n ( j w) (l.+jwq 2)G 7 ] (jto) I (l.+jtoq )G (jco) n n i (l.+jwq2)G22(j(o)-»-k2-(l.+jtoq )G (ju) _ n n/ (l.+jtoq )G (ju)+k n nn n 1 (4 .9) then by the theorem, a s u f f i c i e n t condition for absolute s t a b i l i t y of the system i s that the following c r i t e r i a matrix " W 1 1(ja))+W 1 1(-ja )) W l 2(jto)+W 2 l(-1a)) --- w ] n ( j w ) + w n ] Hw) W 2 1(ja))+W 1 0(-jio) W 0 0(ja))+W„(-ja)) - -- W^(jaj)+W^ 9(-jaj) 12 22 22 2n n2 W (ju))+W l n(-jaj) W (jco)+W (-ico) nn nn (4.10) be p o s i t i v e d e f i n i t e for a l l to. It i s seen that even for the very simple system, the polynomials produced by the p r i n c i p a l minors of Eq. (4.10) are far too complex for manual evaluation. Even with q. ( i = l,2,...,n) given there i s no known general method for exact f i n d i n g the largest range on the n o n l i n e a r i t y gains ( i = l,2,...,n). Our main purpose i s to f i n d the best q_. with the largest gains k^ by applying two geometric techniques, namely ( i ) Gradient Method ( i i ) Projected Gradient Method [16]. The c a l c u l a t i o n s are made by a d i g i t a l computer, on the basis of the flow-chart shown in Figure 4.5. The o u t l i n e of flow-chart i s as follows (1) The c o e f f i c i e n t s o f transfer matrix C(s) and the i n i t i a l values of a. and k. are read, ' i i. (2) Frequency i s changed from zero u n t i l the minimum p o s i t i v e values of the p r i n c i p a l minors of the c r i t e r i o n matrix P(w) with the corresponding frequency co , are found, m (3) Frequency i s fixed at co , and k. i s increased along the. normal m i gradient d i r e c t i o n u n t i l one of the minors f a l l s into the p o s i t i v e constraint range e_^ . (4) Procedure (2) i s repeated again. (5) If one of the minors i s negative at some frequency, then k i s decreased and the procedures of (2) to (4) are repeated u n t i l one of the minimum minors converging i n the range e , a l l other minimum minors are p o s i t i v e (6) In the general case, q_^  w i l l be increased, and the procedures of (2) - (5) are repeated u n t i l the maximum values of k. are found. (7) In the serie s case, k_. and q w i l l be changed i n the projected gradient d i r e c t i o n on the basis of co . m (8) Procedures of (2) and (7) are repeated u n t i l k. cannot be increas The symbols used i n the flow-chart are explained as follows: N l : the number of n o n l i n e a r i t y . M: the order of numerator of G(s) ; N: the order of denominator of G(s) . a. . , : the c o e f f i c i e n t s of numerator of G(s): i»J»k b. . „: the c o e f f i c i e n t s of denomiiiator of G(s). dk: the increment of k^ i n the normal gradient d i r e c t i o n , de: the converge factor NJ xk .: the c o e f f i c i e n t s of function T = X xk .q . n i ' . _Tri n i n i m=NO e.: the constraint range of A., i 3- ' . . • a..: the increment of projected gradient d i r e c t i o n . JJl P..: the elements of c r i t e r i o n matrix P(s). A_^ : the re a l p r i n c i p a l minors of. P(s). DELX..: projected gradient d i r e c t i o n . .13 MOTE : i , j= (1 ,NL) , j j = (1 ,NJ) , ni= (NO ,NJ) , k= (0 ,M) , ?,= (0 ,N) . 30 Figure 4.5 Flow-Chart of the Multi p l e Nonlinear System. Read NL,ND,M,N NONL+1 ; NJ=2NL Read a. . , , b.. . i , l , k i , J , % V k i dk, de, xk ., e. , a. . n i l n 12 s = cmplx (0. , to) G..(s) = a. . i i j i,3>k ies f 72 Find min. A. at to=to k5 = -1 1 110 Gradient method k. = k. + dk-Vf x i k3 = 1 Check A.>0 l 1 / *o for a l l to. to = 0 k3 = 2 k5 = 1 Yes No pp2 •-P i 3 (l+sq.)G..(s) i 13 = (l-sq.)G. . (-s) 3 13 ppl + pp2 13 2k. +(l+sq.)G..(s)+(l-sq.)G..(-s) i " i 13 1 13 No 55 = -1 k3 = 2 Decrease a . 3 a. = 0:. / 2 r x Go to 300 = 1 L Find new min .to k. = k. - dk-vf. 1 1 Go to 9 Decrease i n i t i a l k. k. = k./2 ' 1 3. Go to 9 31 110 Yes S e r i e s c a s e o f t h e system tO = LO m G e n e r a l c a s e of t h e system WRITE q.. , k. - i X to m q . = a . + a. n i "X X L e t q . n i •- k. , x . . = a . . i J J "JJ F i n d G. . ( s ) i l J s = m P. . ( s ) i j 1 j w J s = m A i Aq. . = 1,1 Aq . . •1,111 9A./9q . i "J = SA./^k. i J F i n d t h e d i r e c t i o n o f p r o j e c t e d gradient" G = Aq i , J J NJ (G G ' ) . . = • E Aq . . . Aq . . . x x i , j ± ' l . J J J , J J MAGD = G G ' x x GI. . = [(G G ') . . i> J " " XX , J J J J ,m NL x > 2 NL E Aq .. E GI. .Aq. j = 1 4 J , J J j = 1 i , j J,nx ( I X X i ; j .niVMAGD .DELX. NJ E xk . .D 2 2 ni=N0 n l g i i J J ,m WRITE DELX.. to J J J M T" 300 Go t o 300 k3 q -1 X . J J J J qo.. = q . . J J J J = a + a..DELX. . 'JJ J J J J Yes Stop Go t o 10 32 Example 1 = 2. Consider a series system as shewn i n F i g . 4.4 with 1 ' „ , s s + 1 G 2 1 ( s ) = C (s) = -12 K ' (s + 2) (s + 3) (4.11) The l i n e a r i z e d system, that i s , t h e system with the nonlinear elements replaced by l i n e a r elements with gains k ( i = 1,2)^is stable for k > -30. A.G. Dewey [17] applied the theorem i n section 4.2, and. found that = q^ = 0.19, the nonlinear system i s stable with k k < 2390. Substituting Eq. (4.11) into Eq. (4.10), i t follows that the nonlinear system i s stable i f (i ) k 1 > 0, ( i i ) 4(kJ.c 2) (.1-1-j uq ]) (1+j to) (1-j toq 2) (2+jto) (3+jto) (5-jco) > o, for a l l to. It i s c l e a r that at to' = 0, the i n e q u a l i t y becomes 4(k ]_k 2) -1 900 > 0. (4.12) (4.13) or k k < 3600. (4.14) Thus the best that we can expect from i n e q u a l i t y (4.12) as the values of q and q2 are varied i s Eq. (4.14). By using of both the normal and pro-jected gradient method, values of q = 0.20, q 2 = 0.166 are found, for which the system i s absolutely stable with k^k < 3599. Example 2 Consider a three-variable general system as shown i n F i g . 4.2, where the matrix transfer function of the l i n e a r part i s G(s) = 1 1 1 s + 1 s + 4 s + 6 1 1 _1_ s + 9 s + .2 s + ~8 1 1 1 s + 10 s + 12 s -1- 3 (4.15) 33 It can be shown that the l i n e a r i z e d system i s stable for a l l p o s i t i v e k ( i = 1,2,3). A.G. Dewey [17] has used the theorem shown i n section 4.2, for i a choice of [q] = 0, he establishes s t a b i l i t y f o r the nonlinear system with k l - k2 = h - 10• By using of the normal gradient method i n this example, it-was found that the nonlinear system i s stable with "18, for [q] = 0 k = k 2 = k ? =< (4.16) .323. for q = q 2 = q^ = 0.2 Although, the r e s u l t s are much better than the previous work, since the minimum values of p r i n c i p a l minors of c r i t e r i a matrix (4.10) do not simul-taneously converge to zero, i t i s assumed that q^ = q^ = q^, so s i m p l i f y i n g the computation. Indeed, only the highest p r i n c i p a l minor converges to zero. 34 5-. CONCLUSIONS S u f f i c i e n t conditions for the absolute s t a b i l i t y of nonlinear sampled-data systems have been derived. The method used i s the technique of system transformation of Aizerman and Gantmacher, adapted to sampled-data system. The c r i t e r i o n i n Theorem I was expressed i n terms of the frequency transfer function of the l i n e a r elements and the bounds on the gain and on the slopes of the nonlinear elements. Corollary I was expresses i n s i m i l a r form except that the slope of nonlinear element was bounded from below. These c r i t e r i a were based on d i f f e r e n t forms used to approximate the area under the nonlinear c h a r a c t e r i s t i c . A simple graphical method f or t e s t i n g s t a b i l i t y of the case q ^ 0 was suggested. Theorem II relaxed the r e s t r i c t i o n on the slope of nonlinear element and introduced a a u x i l i a r y function y(n) which may be used to increase the maximum gain k. However, some d i f f i c u l t i e s occurred i n choosing the function y(n). In the multiple nonlinear continuous systems, by the a p p l i c a t i o n of numerical techniques, i t was shown that some improvement over previous s t a b i l i t y bounds can be made. But, i n the general case, the computational d i f f i c u l t i e s are s i g n i f i c a n t . For th i s reason, s i m p l i f i e d assumptions, such as taking q^ = q^ = q^, are often necessary to s i m p l i f y the conputation. The extension of the nonlinear sampled-data system c r i t e r i a given i n theorems I and II to systems in v o l v i n g time-varying gain i s desirable. Such an extension has been made for the continuous case. APPENDIX A The Z-transform of f 2 ( n ) i s F 2 ( z ) = E f 2 ( n ) z n=0 By inverse i n t e g r a l ^ ] f , N (n-1), 2^ J ] > ^ z ) z d -f 2 ( n ) The Z-transform of f„(n)a* gives "2 By inverse i n t e g r a l , i Z[f (n)a n] = F(z/a) = F ( Z ] ) . t p 2 ( n ) a n ] = ^ J F ( z / a ) z ( n - 1 ) d z . Therefore y 0 0 °° n 1 P* (n—1) E [ f 2 ( n ) a n ] [ u N ( n ) a 1 1 ] = Z ^ N (n) a* ^ - j J F ( z / a ) z n dz F(z/a) [ E u„(n)(az) n]z ] J n=0 N 1 |) F ( z / a ) U N ( l / a z ) z Xdz. 2-TTJ C Let Thus z = z/a. E [ f 2 ( n ) a " ] [u N(n)a ] =^77 ^ F C z ^ U / ^ ) n=0 _ „ C This i s the modified Parseval's theorem 36 APPENDIX B Inequality (2.29) i s repeated for convenience r.h.s. [ " | F , ( e j w / a ) | 2 d U + ^ 8 fro ] '4 1 a v ~7r By modified Parseval's theorem (2 - f I f - - D X ( e J t 7 a ) | - dco. IT J 1 a ' —IT (B.l) TT 1 ) \ |x.(e j a >/a) |2d(o = S 2 2(n-l) 0 n x (n-1)a , n=0 (B.2) lco (27T - 1 ) \ j ( - ^ - - l)X ( e : 1 % . ) | 2 d c o = i [ 7 x ( n ) ] 2 a 2 ( n _ 1 ) -IT n=0 Replacing i n the f i r s t i n t e g r a l i n (B.l) by the expression i n Eq. (2.24), t h e r e f o r e } |F.(e J W/«) 4 |F 2 ( e "j W / a ) + q k " | ( e j % ) - l | 2 X ( e " j ( 0 / a ) G ( e j w / a ) |2dw. Since (B.3) f 2 ( n ) a ( n 1 } = [x(n-l) + q V x ( n ) ] a ( n 1 } ; (B.4) then, by modified Parseval's theorem F 2(e 3 a J/a) = [1 + q ( l - ) ]X(e J a 7 a ) (B-5) and 1 co F,(e JU7a) = [1 + q ( l - ^ —)] X(e J a J/ a) . (B.6) Therefore (B.3) may be rewritten as |F.-(e J t7a)| dto ' 4 1 7r . o-Tw |[l+q(l •ir ^ ) 1 X ( e - j ° 7 a ) j 2 |1 -a 1 CO o k " q | ( ^ — - l)| 2G(e- 1 W/a) i c o 1 + q ( l - - — ) a du. (B.7) Since G(z^) i s a n a l y t i c i n the domain a > 1 for a s u f f i c i e n t l y small, therefore sup k"q l(e-1(7a) - irG(e- 1 B/q) e j t 0 l + q ( l - — ) . < '1 < ro. (B.8) 37 because 12 , , 12 hence i r!F 4(e J W/a)|"du ) 1 E ^ |[1 + q ( l - ^ — ) ]X(e J t°/a) | ^ T  s: < 2TTE E [x(n-l) + q V x ( n ) ] 2 a 2 ( n 1 ) . (B.9) n=0 Making use of (B.2) and (B.9), i n e q u a l i t y (B.l) becomes E r f , . M 1 2 , k"q_ r n , ^ 2 , 2 (n-1) ^ r — [x(n-l) - f qvx(n)J -i ^ i n . , / - „ \ i L„. a which i s i n e q u a l i t y used i n Eq. (2.30) r.h.s. < E { y j - [x(n-l) + Vx(n)T + ^ [ 7 x ( n ) ] }a l n J J , (B.10) n=0 38 REFERENCES 1. Popov, V.M., " 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 o f A u t o m a t i c C o n t r o l " , A u t o m a t i o n and Remote C o n t r o l , 22, pp. 857-875, A u g u s t , 1961. 2. L u r i e , A . I . , "Some N o n l i n e a r Problems i n t h e Theory o f A u t o m a t i c C o n t r o l " , T r a n s l a t i o n : Her M a j e s t y ' s S t a t i o n e r y O f f i c e , London, 1957. 3. 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I . , "Theory and A p p l i c a t i o n o f t h e Z-Transform Method," New Y o r k : W i l e y , 1964. 13. B e r g e n , A.R. and R a u l t , A . J . , " A b s o l u t e I n p u t - O u t p u t S t a b i l i t y o f Feedback System w i t h S i n g l e T i m e - V a r y i n g G a i n , " J o u r n a l o f the F r a n k l i n I n s t i t u t e , V o l . 286, No. 4, pp. 280-294, 1968. 14. F l a b , P.L., and Zames, G., "On C r o s s - C o r r e l a t i o n and the P o s i t i v i t y o f C e r t a i n N o n l i n e a r O p e r a t o r s , " IEEE T r a n s . Auto. C o n t r o l , AC-12; pp. 219-221, 1967. 15. B a k e r , R.A. and Deso e r , C.A., " A s y m p t o t i c S t a b i l i t y i n the L a r g e o f a C l a s s o f S i n g l e - L o o p Feedback Systems," SIAM J . C o n t r o l , V o l . 6, No. 1, pp. 1-8, 1968 16. Hadley, G., " N o n l i n e a r and Dynamic. Programming," A d d i s o n Wesley, 1964. 39 17. Dewey, A.G., "Frequency Domain S t a b i l i t y C r i t e r i a f o r Nonlinear M u l t i - v a r i a b l e System," Int. J. Control, Vol. 5, No. 1, pp. 77-84, 1967. 18. Lir i d o r f f , D.F., "Theory of Sampled-Data Control Systems," Wiley, 1965. 

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