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The absolute stability of nonlinear systems 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 the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor Members of 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 i n 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 advanced 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 the 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 . Depar tment o f 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 ABSTRACT This thesis i s i n two parts, both considering the absolute s t a b i l i t y of nonlinear systems. In the f i r s t two chapters the s t a b i l i t y of c e r t a i n classes of nonlinear time i n v a r i a n t systems involving several n o n l i n e a r i t i e s i s considered. A number of graphical methods are given for t e s t i n g the s t a b i l i t y of these systems. The graphical tests are equivalent to a weakened form of the Popov c r i t e r i o n . The t h i r d chapter derives a s t a b i l i t y condition f o r nonlinear systems involving a l i n e a r time-varying gain. The time-varying gain i s assumed to s a t i s f y conditions on i t s magnitude and rate of change. i i "TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v ACKNOWLEDGEMENT '. . v i i 1. 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 S t a b i l i t y 10 2. THE ABSOLUTE STABILITY OF A TIME INVARIANT NONLINEAR FEEDBACK SYSTEM 11 2.1 Introduction. 11 2.2 A Graphical Test of the Absolute S t a b i l i t y of a Series System with N o n l i n e a r i t i e s and I d e n t i c a l Transfer Functions 11 2.2.1 Main Method-Common Popov Line 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 of the Modified Nichols Chart to Obtain Q 19 2.2.4 An Analogue-Computer Technique 23 2.3 The Absolute S t a b i l i t y of the Series Nonlinear System with D i f f e r e n t Transfer Functions 27 2.4 The Absolute S t a b i l i t y of a P a r a l l e l Nonlinear System 30 3. THE ABSOLUTE STABILITY OF A TIME VARYING FEEDBACK SYSTEM WITH MONOTONIC NONLINEARITIES 34 3.1 The Absolute S t a b i l i t y of the Single-Loop Time Varying Nonlinear Feedback System 34 3.2 The Main Results 35 3.2.1 Theorem 1 35 3.2.2 A Speical Case of Theorem 1 36 3.3 Proof of Main Results • 36 3.4 Examples 41 *in-i Page 3.5 The Absolute S t a b i l i t y of a System with Mul t i p l e N o n l i n e a r i t i e s and Time Varying Gains 46 3.6 Theorem 2 49 3.7 Proof of Theorem 2 50 3.8 Example 51 4. CONCLUSIONS 53 APPENDIX 1 55 APPENDIX 2 , 57 APPENDIX 3 58 APPENDIX 4 59 REFERENCES 60 i v LIST OF ILLUSTRATIONS Page 1.1 General time i n v a r i a n t nonlinear feedback system 2 1.2 General time varying nonlinear feedback system 2 1 . 3 Series system with m n o n l i n e a r i t i e s 4 1 .4 P a r a l l e l system with m n o n l i n e a r i t i e s . . . 5 1 .5 Internal feedback system with m n o n l i n e a r i t i e s 5 1 .5 M u l t i - c i r c u i t system with 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 Graphical c r i t e r i o n f o r the time i n v a r i a n t series system with 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. 2.2 Modified Nyquist l o c i and the common Popov l i n e for the series system, o nf \ 32(s+0.25) . _ • ' . N , m = 3 ' G ( s ) = (s+l)(s+ 2)(s+ 4 ) ' F O R E X A M P L E 2 > 1 • • 1 5 2 .3 Modified Nyquist locus for i = l , and the corresponding Popov l i n e f o r the s e r i e s system, m=3, G(s) = (g+i)(s+5)(s+20) ' E x a m P l e 2.2.1. 17 2 .4 Nyquist p l o t of (1+0.788S)G(s), where G(s) = ( s + 1 ) ( S + 5 ) ( s + 2 0 ) ' A N D a graphical t e s t i n g the s t a b i l i t y of the se r i e s system, m=3, f o r Example 2.2.1 17 2 .5 Modified Nyquist locus f o r i = l , and the corresponding Popov l i n e for the series system, m=4, G(s) = , 1 . , ,rw , 9 n \ > for Example Example 2.2.2 \...\ \..".\ 18 2.6 Nyquist p l o t of (1+0.766s)G(s) , where G(s) = (s+]_) (3+5) ( S+20) ' A N ^ a graphical t e s t i n g the s t a b i l i t y of the se r i e s system, m=4, for Example 2 . 2 . 2 18 2.7 Gain-phase pl o t f o r G(s) = ( S + J Q ) ( S+5Q) a n < ^ a r a m : i - l y o r boundaries for the se r i e s system, m=3, f o r Example 2 .3 21 6 30 2 .8 Gain-phase pl o t f o r G(s) = ( S + 2 Q ) ( S+50) A N C ^ A ^am^y °f .boundaries for the series system, m=4, f o r Example 2 .3 22 580 2.9 Gain-phase p l o t for G(s) = ( s + IQ ) ( S +50) a n d a f a m i l y o f boundaries for the se r i e s system, m=5, for Example 2 .3 22 v Page 2.10A Computer program for R-I generator 24 2.10B Computer program for X-Y producer 25 2.11 The l o c i p l o t t e d by analogue computer 26 2.12 Extension of Popov c r i t e r i o n f o r the p a r a l l e l system with m iden- 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 transfer functions 33 3.1 Single loop time varying nonlinear feedback system with zero- input 42 3.2 Bode diagram f o r a compensator l+y(s) = 42 3.3 The p l o t s of G(s) = T 4 T V 7 ^ \ a n d G'(s) = (|^)G(s) for (s+l)(s+2) 4+s Example 3.1 44 3.4 Modified Nyquist p l o t of G(s) = ( s-f]j (S+D) E x a mP-'- e ^.2 47 3.5 General time varying nonlinear feedback system with zero-input.. 47 v i ACKNOWLEDGEMENT I wish to express my sincere gratitude to Dr. M.S. Davies, the supervisor of th i s p r oject, for h i s invaluable guidance, and Dr. E.V. Bohn for reading the manuscript. Grateful acknowledgement i s given to the National Research Council f o r f i n a n c i a l support received under NRC Grant A-4148, and to the University of B r i t i s h Columbia f o r U.B.C. Fellowship awarded since 1968. I would also l i k e to thank Miss Beverly Harasymchuk for typing the manuscript. VH 1 CHAPTER 1 TIME INVARIANT NONLINEAR FEEDBACK SYSTEM AND TIME VARYING NONLINEAR FEEDBACK SYSTEM §1.1 Introduction This thesis considers the absolute s t a b i l i t y of the equilibrium p o s i t i o n , X = 0, of feedback systems defined by X = AX + BY Y = 0(a) (1.1.1) T a = C X, or, i n the time varying case, X = AX + BY Y = 0(a,t) (1.1.2) T a = C X, where X i s an n-vector, Y i s an m-vector, 0 i s an m-vector, A i s an nxn con- T stant matrix, B i s annxm constant matrix, and C i s an mxn constant matrix. In (1.1.1), each element of 0(a) i s a nonlinear function of a alone, so that the system (1) i s time i n v a r i a n t . In (1.1.2), each element of 0(a,t) i s a time varying nonlinear function of both a and t. It i s assumed that the i - t h element can be separated into a nonlinear part 0 . (a . ) and the time i x varying gain k . ( t ) , where 0.(a.,t) = 0.(a.)k.(t). The system (2) i s thus a l i i i i i time varying nonlinear feedback system. The transfer matrix TXs).of the l i n e a r part i s T(s) = C T ( s I - A ) _ 1 B, (1.1.3) where I i s an nxn unit matrix. The feedback system (1) may be depicted as shown i n F i g . 1.1. The forward path consists of an mxm l i n e a r time inva r i a n t matrix r ( s ) and the n o n l i n e a r i t y matrix N.L. In F i g . 1.2, the forward path of the time varying nonlinear feedback system (2) consists of the transfer 2 F i g . 1.1 General time i n v a r i a n t nonlinear feedback system r(t) + <rtt) N N . U <P((Tl k(t) r(s) C(t) F i g . 1.2 General time varying nonlinear feedback system 3 matrix F ( s ) , the mxm diagonal n o n l i n e a r i t y matrix N.L. with elements 0T(a, ), 0 „(a„),. . • 0 (a ) and the mxm diagonal time-varying gain matrix k(t) 1 1 2 2 m m with elements k, (t),k (t) ,. . .k ( t ) . 1 2 m A wide v a r i e t y of systems may be treated by choice of r(s) , some forms of p a r t i c u l a r i n t e r e s t are: (1) A series system. The forward path consists of the s i n g l e - input single-output l i n e a r time inva r i a n t t r a n s f e r functions separated by amnesic n o n l i n e a r i t i e s . This i s shown i n F i g . 1.3. (2) A p a r a l l e l system. The forward path consists of m-parallel branches, each of which has one n o n l i n e a r i t y i n series with one l i n e a r time in v a r i a n t t r a n s f e r function. This i s shown i n F i g . 1.4. (3) An i n t e r n a l feedback system. The forward path consists of m-single nonlinear feedback loops. This i s shown i n F i g . 1.5. (4) A m u l t i - c i r c u i t system. Such systems do not f a l l into the previous classes. Such a system i s shown i n F i g . 1.6. The elements of the n o n l i n e a r i t y matrix are to be considered i n 4 classes. In each i t i s assumed that 0(a) i s a piece-wise continuous, angle valued function of a. (1) A sector n o n l i n e a r i t y ; any function which s a t i s f i e s the condition k. < ^ < k 9, a i 0 l a I 0(a) =0, a = 0 where k^ > 0 and k^ may be p o s i t i v e , negative or zero. (2) A f i r s t and t h i r d quadrant n o n l i n e a r i t y ; this' i s a s p e c i a l case of (1). 0 < < k < oo k > 0 , a — for a l l f i n i t e nonzero values of a. <ryt) <t>,«r,) Q,ls) <t>i(tTi) Gi(s) ( T m f t ) <Pm<F»t F i g . 1.3 Series system with m n o n l i n e a r i t i e s 5 F i g . 1.5 Internal feedback system with m n o n l i n e a r i t i e s 6 F i g . 1.6 M u l t i - c i r c u i t system with m n o n l i n e a r i t i e s (3) A monotonic n o n l i n e a r i t y ; t h i s belongs to a subclass of (2) , i n which i t i s assumed that 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 to a subclass of (3) , i n which i t i s further assumed that 0(-a) = -0(a) for a l l a. Furthermore, the elements of the time varying gain matrix to be considered are assumed p o s i t i v e , bounded and continuous. §1.2 Lyapunov Method Investigations of the s t a b i l i t y of such systems were i n i t i a t e d by Lure who proposed a Lyapunov function of the form: T m a. V(X) = X PX + Z 3 . / 1 0.(z)dz, (1.2.1) i = l 1 0 1 T where P i s an nxn symmetrical p o s i t i v e d e f i n i t e matrix (P = P > 0), a l l 3 . > 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 matrix C TX. (1.2.2) It i s obvious that V(X) i s a p o s i t i v e function since 0 < 0(a)a < Ka 2. (1.2.3) From (1.1.1), the d e r i v a t i v e of (1.2.1) i s V(X) = X T(AP T+PA) +B TPX0(a)'+ X TPB0(a) - 3AX0(a) + BC0(a), (1.2.4) where g i s a diagonal constant matrix with elements 3 n,$ 0 >•••6 • I 2 m From the Lyapunov second method, i n order to f i n d the s u f f i c i e n t condition of absolute s t a b i l t i y , i t i s necessary to determine the conditions under which V(X) i s negative d e f i n i t e except at the n u l l state, X = 0, where V(X) = 0. In [5], i t was shown that the conditions of absolute s t a b i l i t y of a time varying nonlinear system may be found from using the Lyapunov func- t i o n (1.2.1) with a l l 3^ = 0. This method of using Lyapunov functions was further developed by Narendra and Taylor [6] using the modified Lyapunov func- ti o n , v i z . , T m °i V(X,t) = X PX + E g.k.(t) / 0.(z)dz, (1.2.5) i = l 1 1 0 1 where k^(t) i s a time varying gain. In [4,5], i t Is also proved that the s u f f i c i e n t condition under which the Lyapunov function is- v a l i d i s s i m i l a r to the Popov c r i t e r i o n , d i s - cussed below. §1.3 The Popov C r i t e r i o n The s u f f i c i e n t condition of absolute s t a b i l i t y f o r a c o n t r o l l a b l e and observable time i n v a r i a n t system with one n o n l i n e a r i t y s a t i s f y i n g (1.2.3) was established by V.M. Popov [1]. The Popov c r i t e r i o n takes the form Re[(l+qjco) G(jui)] +~ > 0 (1.3.1) for a l l to, where q i s a nonnegative number. A convenient graphical method e x i s t s f o r t e s t i n g (1.3.1). A s u f f i c i e n t condition f o r the absolute s t a b i l i t y of a system with many n o n l i n e a r i t i e s was given by Jury and Lee [2]. The condition requires the Hermitian matrix: 2K _ 1 + H(jo>) + H T(-ju)) (1.3.2) 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. Here K i s a constant diagonal matrix of elements K... ,K , . . .K , a l l of which are p o s i t i v e numbers such that the in e q u a l i t y 1 I m 0 < 0 . (o\ )o\ < K.o. 2 (1.3.3) i i i i i i s s a t i s f i e d for i = l,2,...m, and where H(joi) = (I + J O J Q ) T O ) , (1.3.4) unstable region Y 'Popov A 'me Stable region F i g . 1.7 Popov c r i t e r i o n 10 where Q i s a constant diagonal matrix with elements q 1,q„...q which are 1 2 m nonnegative numbers. This forms a g e n e r a l i z a t i o n of the Popov r e s u l t . For the time varying nonlinear feedback system, the s u f f i c i e n t condition of absolute s t a b i l i t y established by Rozenvasser i s as below, Re G(jco) + ^ > 0 (1.3.5) K for a l l co. It i s but the Popov c r i t e r i o n with q = 0. Other new c r i t e r i a , s i m i l a r to the Popov's, were introduced by Zames' and Falb [8,9], Yakubovich [4], Narendra and Taylor [6], Baker and Desoer [10,11], Bergen and Rault [12], and Anderson [13]. §1.4 L 2 S t a b i l i t y The concept of the s t a b i l i t y has been introduced by Sandberg [14]. It i s c l o s e l y r e l a t e d to asymptotic s t a b i l i t y . L„ i s the space of square integrable, valued functionjon [t ,°°), i t i s assumed that L -is-'a l i n e a r , inner-product, i/formed space; the inner product of x and y i n i s <x,y> = r x ( t ) - y ( t ) d t '< °°, (1.4.1) ' o and the norm of x i s || x || 2 = /<x,x>. Suppose a(x) i s i n L^fO, 0 0], a(t) i s uniformly continuous, and <j(t) i s bounded, then the state.o(t) approaches n u l l state i f the s u f f i c i e n t condi- t i o n of absolute s t a b i l i t y of a time i n v a r i a n t nonlinear system, the Popov c r i t e r i o n , i s s a t i s f i e d . A further L 2 bounded condition was introduced by Zames [8,9]. 11 CHAPTER 2 THE ABSOLUTE STABILITY OF A TIME INVARIANT NONLINEAR FEEDBACK SYSTEM §2.1 Introduction In the previous chapter, i t was mentioned that the nonlinear systems can be considered i n 4 classes according to the form of the t r a n s f e r matrix res). Testing absolute s t a b i l i t y of a single-loop time invariant nonlinear system using the modified Nyquist diagram was f i r s t i n i t i a t e d by Popov, basing the method on h i s c r i t e r i o n . Further developments using a graphical method to test the absolute s t a b i l i t y of a nonlinear system have been furnished by Naumov [15], Meyer and Hsu [16], and Murphy [17]. A graphical method of t e s t i n g the absolute s t a b i l i t y of a time in v a r i a n t s e r i e s system with m-nonlinearities and m-identical l i n e a r transfer functions was introduced by Davies [18]. ' §2.2 A Graphical Test of the Absolute S t a b i l i t y of a Series System with N o n l i n e a r i t i e s and I d e n t i c a l Transfer Functions Consider a ser i e s system with the l i n e a r time invariant transfer function matrix , 0 . 0 v 0 G(s) r(s) = -G(s) 0 o 0 \ \ 0 - G ( s > N \ N \ \ 0 \ \ -G(s) 0 (2.2.1) The input, a . ( t ) , and output, 0^(oO> of the i-th nonlinear element s a t i s f y the i n e q u a l i t y 0 < a . 0 .(a.) < a. i i i l (2.2.2) 12 F i g . 2.1 Graphical c r i t e r i o n for the time i n v a r i a n t series system with 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 within the c l a s s for which a Popov-like s t a b i l i t y c r i t e r i o n has been, established. In applying t h i s r e s u l t , the matrix where Q i s an a r b i t r a r y , semi-positive, diagonal matrix of constants, i s con- sidered. A s u f f i c i e n t condition to e s t a b l i s h the absolute s t a b i l i t y of a nonlinear system i s that the Hermitian matrix (1.3.2) must be p o s i t i v e d e f i n i t e for a l l co. I f a l l nonlinear elements are assumed to have the same upper bound; that i s , = K for i = l,2,...m, then, without loss of g e n e r a l i t y , K may be taken as the i n d e n t i t y matrix. If Q=0, the s t a b i l i t y c r i t e r i o n i s equivalent to the Nyquist p l o t of G(s) l y i n g within a symmetric m-sided polygon. In the subsequent development i t i s not required that Q=0, but rather that a l l elements of Q are equal; that i s , Q=ql, where q i s a p o s i t i v e s c a l a r constant. where r'(jco) i s i d e n t i c a l to r(jco) except that G(jco) has been replaced by G'(jco) = (l+jcoq)G(jco) • Thus i t i s possible to consider the case 0=0 by applying the e a r l i e r r e s u l t s for Q=0 to G'(jco) instead of G(jco) i t s e l f . I f i t can be shown that G'(jco) l i e s within the appropriate polygon, for any p o s i t i v e q, then s t a b i l i t y has been established. H(jui) = (I+jcoQ)r(jco), (2.2.3) I f Q i s r e s t r i c t e d i n t h i s manner, then (I+jcoQ)T(jco) = r'(jco), (2.2.4) §2.2.1 Main Method-Common Popov Line Let |G'(jco)| = *f and G'(jco) = 6. If G' (jco) l i e s within a polygon, then y cos (6 - a) < 1, (2.2.5) where a = 2iiT m and i = 1,2,... m, 14 and where i i s one of m values each corresponding to the m sides of the polygon. '• Now where Thus and Define G'(jco) = (l+jwq)G(» = (l+jo)q)(R+jI), G(ju) = R(u) + jl(oo) Re G' (jcj) = R - jtol = ycosG , Ira G' (jco) = uqR + I = f sine , (R-coql)cosa + (ioqR+I)sina < 1, (Rcosa+Isina) + qco(-Icosa+Rsina) < 1. -X(co) = Rcosa + I s i n a , (2.2.6) (2.2.7A) (2.2.7B) (2.2.8A) (2.2.8B) (2.2.9A) (2.2.9B) Y ( ( j j ) = co(-Icosa + Rsina). (2.2.8B) gives -X + qY < 1. • (2.2.10) From (2.2.9A) and (2.2.9B), the m d i f f e r e n t modified Nyquist l c c i are pl o t t e d each corresponding to one of the m sides of the polygon. To s a t i s f y i n e q u a l i t y (2.2.10), a l l these l o c i must be to the right side of a s t r a i g h t l i n e , the Popov l i n e , which passes the point (-1,0) having slope 1/q. If such a s t r a i g h t l i n e e x i s t s , then the absolute s t a b i l i t y of the system i s established. Example.2.1 Consider a feedback system of the type shown i n F i g . 1.3 (m=3). Let every n o n l i n e a r i t y s a t i s f y the i n e q u a l i t y 0 < 0.a. < a., l l l where i = 1,2,3 and l e t every l i n e a r block be represented by (2.2.11) 15 Fig . 2.2 Modified Nyquist l o c i and the common Popov l i n e for the series system, „, . . 32(s+0.25) • m-3, G(s) = ( s + 1 ) ( s + 2 ) ( s + 4 r ' f ° r E x a m P l e 2- 1' 16 n . , 32K(s+0,25) /, , , ^ l (s+1)(s+2)(s+4) . The m d i f f e r e n t modified Nyquist l o c i and common Popov l i n e are shown i n F i g . 2.2. Then, by s e t t i n g q = 0.23, the absolute s t a b i l i t y condition K < 0.476 (2.2.13) i s obtained. §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 P l o t t i n g the m d i f f e r e n t modified Nyquist l o c i i s tedious i f m >_ 4. A simpler and more d i r e c t approach i s possible i f G(s) i s of the form G(s) = -p (2.2.14) n (S+D . ) j = i 3 or n k n . , (s+N.) G(s) = — V — (2.2.15) P n ( S + D . ) j=i 2 where p > 1+m, N. and D . are r e a l p o s i t i v e constants, and N. > D . for i = j . i 3 1 J I f the tran s f e r function G' (jco) = (l+jioq)G(jo)) having q^ and k^ which are found from the modified Nyquist locus f o r i = 1, s a t i s f i e s the above conditions, then ̂ (1) The phase of G'(jco) i s decreasing as co increases. (2) For some co = co , giving Z.G'(jco ) > and |G'(jco )| < 1, |G'(jco)| c c — m c decreases when co increases f o r co > co . c And k^ i s the maximum value s a t i s f y i n g the absolute s t a b i l i t y conditon. 4 Y(u) 17 J - r 2.3 Modified Nyquist locus f or i = l , and the corresponding Popov l i n e f o r the series system, m=3, G (s ) = ( s + 1 ) ( s ^ s + 2 Q ) ., for Example 2.2.1. 2.4 Nyquist p l o t of (1+0.788$)G(s), where G(s) = 100 , and a (s+l)(s+5)(s+20) graphical testing the s t a b i l i t v 18 a graphical t e s t i n g the s t a b i l i t y of the s e r i e s system, m=4. for Example 2.2.2. - - 19 Example 2.2 (1) Consider m = 3 with every l i n e a r element having a transfer function G(s) = 1 0 ° k (s+l)(s+5)(s+20). (2.2.16) From F i g . 2*3, q± = 0.788, (2.2.17) and k = 1.46. (2.2.18) Setting q = 0.788, k < 1.46 i s the absolute s t a b i l i t y condition since, as shown i n F i g . 2.4, G'(s) s t a i s f i e s the above conditions. (2) Using the same approach the absolute s t a b i l i t y condition of the system with 4 n o n l i n e a r i t i e s i s obtained as: k < 1.234 (2.2.19) by s e t t i n g q = 0.766. The information required for the previous method may be obtained d i r e c t l y from the Nyquist Plot of G(s) by noting that the c r i t i c a l point i s that having phase a^, and the corresponding value of q i s given by tan(a^~3) where tanp i s the slope of G(s) at a= a^. If the transfer function G(s) does not s a t i s f y the s p e c i a l form of (2.2.14) or (2.2.15), the Nyquist locus of G'(s) haivng q^ and k^ found from i n e q u a l i t y (2.2.10) for i = l may be tested by the polygon c r i t e r i o n . I f t h i s f a i l s , the general approach must be adopted. §2.2.3 Use of the Modified Nichols Chart to Obtain 0 The use of the modified Nichols chart to obtain q and to test the absolute s t a b i l i t y of the system with many n o n l i n e a r i t i e s i s also possible. A polygon w i l l be described by some r e l a t i o n s h i p between the log amplitude and the phase of the form 20 M = 20 log, r t sec(- ^ + - - 0)db, (2.2.20) 10 m m > 2ii 2n where - (i-1) < 6 < i , m — — • m and i = 1,2,...m. The stable boundary G' can be represented i n the modified Nichols chart. Let us consider _1 M = M - 20 l o g 1 ( ) ( l + N 2 ) 2 , (2.2.21A) and where j = 0, t a n " 1 N. , (2.2.21B) J and N_. i s an a r b i t r a r y , p o s i t i v e constant. (2.2.21A) and (2.2.21B) give a family of the stable boundaries as shown i n F i g . 2.7, each corresponding to one of the constants N. Note that these curves are a l l of the same form and may e a s i l y be sketched. I f the locus of G(jco) i s sketched and q i s chosen, such that each point co. on the locus of G(jco) on the modified Nichols chart i s beneath the corresponding stable boundary N_. = cô  <J, then the absolute s t a b i l i t y of the system i s established. Also, i f the p o s i t i o n of the point co. i s known, and i s beneath a 2 N. family of stable boundaries N, then the corresponding values of q. = — j = 0,...°°, i s known, so that every point has one corresponding stable q range. Then, any q i n the i n t e r i o r q range i s permitted to be chosen for es t a b l i s h i n g the s t a b i l i t y of the system. This seemingly complex procedure i s , i n f a c t , quite straight-forward. I f a range of q's i s permissible, that value of q i s chosen which gives the greatest possible value of k. Example 2.3 Consider the system i n the Example 2.2 with every l i n e a r element G(s) where 700 F i g . 2.7 Gain-phase pl o t for. G(s) = -,—TTTTCT—, and a family of boundaries for (s+10)(s+50) the s e r i e s system, m=3, for Example 2.3. 22 6db F i g . 2.8 630 Gain-phase p l o t for G(s) = ( s +xo)( s+50) a n d a f a m i l y o f D ° u n d a r i e s f o r the ser i e s system, m=4, for Example 2.3. Fig. 2.9 Gain-phase pl o t for G(s) - 5 8 0 .____ (s+10)(s+50) a n d a family of boundaries f o r e s e r i e s system, m=5, for Example 2.3. 23 G ( S ) = (s +10)(s +20T ( 2 ' 2 - 2 2 ) and m = 3. Comparing the frequency response curve of G(juj) with a family of the stable boundaries as i n F i g . 2.7, the value of q and stable condition are q = 0.067 (2.2.23A) k £ 700 . (2.2.23B) S i m i l a r l y , from F i g . 2.8 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 for m = 4, . (2.2.24) and k £ 580 for m = 5, (2.2.25) are obtained. §2.2.4 An Analogue-Computer Technique This l a s t method makes use of the analogue-computer to test the absolute s t a b i l i t y of a nonlinear system. The computer arrangement, which i s shown i n F i g . 2.10A and F i g . 2.10B, i s divided into two main parts. The f i r s t generates R and I, and depends on the p a r t i c u l a r t r a n s f e r function being considered while the second gives the components of G(jco) which are set to remain unchanged for d i f f e r i n g systems. The e f f e c t of varying q on the G'(jco) locus i s e a s i l y obtained by adjusting a potentiometer. This 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 s c a l i n g . Besides, the analogue-computer set-up becomes more complex with increase i n system order. It i s also noted that to sketch the complete locus of G'(jco) from co = 0 to co = 0 0 by the computer i s impossible because the describing time o M i s proportional to . To improve the accuracy of the output i t i s necessary to use a three- (or two-) stage programming and rescalin'g technique, and also to l i m i t 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 .  2.10B Computer program for X-Y producer  Example 2.4 The system (m=3) has l i n e a r elements G^(s) where G i ( s ) (s+0.5)(s+l) (2.2.26) The c i r c u i t s of the two parts of the analogue computer set-up are shown i n F i g . 2.10A and F i g . 2.10B, and the t y p i c a l resultant l o c i for various values of q shown'in F i g . 2.11 lead to the choice q = 1.5. This therefore permits the choice of k = 0.685. max §2.3 Absolute S t a b i l i t y of the Series Nonlinear System with D i f f e r e n t Transfer Functions Let us consider the case where the i d e n t i c a l l i n e a r transfer functions i n the previous section have been replaced by d i f f e r e n t l i n e a r transfer functions, Now the tran s f e r matrix r(s) may be written / 0 r(s) = o . o -G-^s) 0 - G 2 ( s J \ G (s) N, m 0 I I 0 (2.3.1) -G ^ ( s ) 0 m-1 Let us consider H(>) = (I+jwQ)r(ju)), where I i s a unit matrix and Q i s a diagonal constant matrix with elements q^,q 2,...q^; thus H(s) may be considered as below, / 0 H(s) = G'(s)^ m G^(s) 0 -Gl(s) (2.3.2) 28 where G'(s) = (1+jwq.)G.(s), i=l,2,...m. i 1 1 (2.3.3) Let |Ĝ (jco)I = Y £ and Z G ^ ( J W ) = 6 i; then the Herraitian matrix 21 + H(jco) + H (-ju>) becomes / 2 - Y l e ~ j 6 l 0 Y e j 6m '1 \ 'm -Y 2e 2 r J 6 9 ~ Y 0 R 2 0 I _ j 9 _ _ _ 0 < v e m ^ -j8 , \ -y e m-1 -Y ,e m-1^ 'm-1 2 (2.3.4) and must be p o s i t i v e d e f i n i t e for a l l a> to s a t i s f y the s t a b i l i t y condition. The f i r s t (m-1) p r i n c i p a l minors of A^ are generated by the recurrence r e l a t i o n A. = 2A. l 1-1 "  Y i - 1 A i - 2 > 0, i = 3,4,...m-1, with A^ = 2 and = 4 - Y j _ A = m The l a s t condition i s ri1 •y.e ^1 0 V 9 1 2 'Y 9 e 2 v e 2 m m - Y 2 E 2 e^m 'm Ym-1 \ ie \ 0 — Y _e J m-1 2 'm-1 ! 6 e-J6m-2 > 0, (2.3.5) that i s , 9 9 m m A = 2A , - ( Y , + Y )A „+(-!)( TT Y . ) e ( E 9.) m m-1 'm-1 m m-2 . .. ' i . , l 1=1 i = l m (-E 6.) m • -i 1 + ( - l ) m ( T f Y , ) e 1 = 1 i = l 2. 2, 2 A - , - ( Y / + Y " ) A „ + ( - D m ( Jl Y.)2 cos( E 0.) > 0. m-1 'm-1 'm m-2 . ., ' l . , l 1=1 1=1 m m (2.3.6) If a l l y. < 1 f ° r a H w, i=l,2,...m, then 1 and A 2 =-4-Y^ > 3, 2 2 A 3 ~A 2 = A 2-Y 2 A - L > ^2~^1 = - 2 > 1, 2 A3 > A 2 + 1 = 4 ~ Y 1 + 1 > 4: V A 3 = V Y 3 A 2 > A3" A2 > X ' > A 3+l > 5, 2 A.-A. , = A. ., —Y. , i . , > A. .-A. 0 > 1, l l - l l - l 1-2 l - l i-2 A. > A. ,+1 > i+1, l i - l ' 2 A = A -Y „A > A -A > 1, m-1 m-2 m-2 m-2 m-J m-2 m-3 A T > A „+l > m, m-1 m-2 m A > 2A ,-2A „+(-l) m2cos( E 6.) m m-l m-z . . l i = l m > 2+(-l) m2cos( E 6.) >_0. (2.3.8) 1=1 1 . Therefore, the absolute s t a b i l i t y of a nonlinear system i s assured i f every locus of G^(joo) l i e s within'a c i r c l e of unit radius. Obviously, must be chosen zero so that G'(jw) = G i(jo J) In order to test absolute s t a b i l i t y of nonlinear system i t i s thus necessary to sketch the l o c i of G^(jio) and to observe whether a l l of the l o c i l i e within the unit c i r c l e . 30 As m -> °°, this r e s u l t coincides with the previous r e s u l t i n §2.2, but here the G_^(jaj)'s are not nec e s s a r i l y the same. §2.4 Absolute S t a b i l i t y of the P a r a l l e l Nonlinear System The transfer matrix of the p a r a l l e l nonlinear system shown i n F i g . 1.4 may be written Define and r(s) = , G^s) G (s) G (s) 1 / m G l ( s ) ' j ! i G 1(s) G 2(s) m (2.4.1) then Suppose that G.(s) = G(s), i=l,2,...m, and G'(joo) = (l+jcoq)G(ju)) , ^G|(jw) G 2(jco) G| (ja>) ^G^(jco) G 2 ( j u ) G' (ju>) ̂  m. , i I = G'(jco) 1 1 1 ' (2.4.2) G(jco) = R(co) + jl(co) , R' (jio) = R(co) + ql(w) , (2.4.3A) (2.4.3B) I 1 (ju) = qtoR(w) + I(w) (2.4.3C) Let us suppose the i - t h n o n l i n e a r i t y s a t i s f i e s 0 < a.0.(a ) < a.. 1 x i I (2.2 The Hermitian matrix i s 21 + 2R' ,1 1 1- I I I ^1 1 1/ /2(1+R f) 2R' 2R' \ I 2R' 2(1+R')\ ' | \ 2R* s 2R' 2R' 2(1+R')' (2.4 The s u f f i c i e n t condition of absolute s t a b i l i t y i s that the Hermitian matrix must be p o s i t i v e d e f i n i t e for a l l w; consequently, > 0. Now 2(1+R') 2R' 2R' \ I 2R' 2(1+R'>s I \ \ \ I 2R1- -2R* \ I 2R' \ 2(1+R') 2(1+R') 2R1 2R* -2 2 0, -2 2 \ ° N 1 ! \ \ S i I \ \ • 0 .0 2 32 = 2A. . + 21R' i - l = 2(2A i_ 2 + 2 1 _ 1R') + 21R' = 2 1~ 1A 1 + ( i - l ) 2 V = 2 1(l+iR') > 0. (2.4.5) Substituting (2.4.3B) i n (2.4.5), . R(u>) - q'ajl(u) + j > 0. (2.4.6) Define X(ui) = R(u>), (2.4.7A) and ^ Y(u>) = COI(OJ). (2.4.7B) (2.4.7A) and (2.4.7B) give X(u)) - qY(w) + j > 0. (2.4.8) The condition A = X(u>) - qY(u) + - > 0. (2.4.9) m m implies that A > 0, i = 1,2,...m-1. (2.4.5) Hence,the new Popov line,shown i n F i g . 2.12, passes through the point (- ̂ , 0) with slope —. q 33 2.12 Extension of Popov c r i t e r i o n f o r the p a r a l l e l system with 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 functions 34 Chapter 3 ABSOLUTE STABILITY OF A TIME VARYING FEEDBACK SYSTEM WITH MONOTONIC NONLINEARITIES §3.1 The Absolute S t a b i l i t y of the Single-Loop Time Varying Nonlinear Feedback System In recent years some r e s u l t s concerning the absolute s t a b i l i t y of a single-loop nonlinear system with a time varying gain have been obtained by Rozenvasser [5], Zames [22], Bergen and Rault [12]. The r e s u l t s to be presented here extend t h i s previous work. Let us consider single-loop time varying nonlinear system shown i n F i g . 3.1. In that system, a (t) = St g(t-r) e ( i ) d T f o r t > 0 e o ~~ (3.1.1) = 0 for t < 0 i s the zero-state response of the l i n e a r time i n v a r i a n t part with t r a n s f e r function G(jw) = ^ [ g ( t ) ] . The input il(t) represents the zero-input response of g ( t ) . The complete response of g(t) i s thus c ( t ) = o (t) + n ( t ) . (3.1.2) e The input, a(t) = - c ( t ) , and the output 0(c) of the amnesic n o n l i n e a r i t y N.L. are r e l a t e d i n the following manner: (1) 0 < 00(a) < a 2 for a i 0, 0(0) = 0, 0 ( a ) - 0 ( a ) (2) 0 < a _ a — — 1 f o r CTx * V The block, k ( t ) , represents a l i n e a r time varying gain, thus e(t) = k ( t ) 0 [ a ( t ) ] . The instantaneous value of t h i s gain i s constrained so that (1) K± < k(t) < K 2, where K 2 > K± > 0, (2) bk(t)< k(t) < a k ( t ) , where the number a> 0,and the number b (3.1.3) i s f i n i t e . It i s assumed that the l i n e a r part i s stable, more s p e c i f i c a l l y , (1) g(t) E L 2(0,«0, g(t) e L 1(0, K>) j (2) n ( t ) e ^ ( O . c o ) , (3) n(t) i s d i f f e r e n t i a b l e and f)(t) e L (0,»). Condition (1) above ensures that g(t) i s bounded on (0,°°) and that g(t) 0 as t •+ °°; besides, conditions (2) and (3) ensure that n(t) behaves i n the same manner. Denote n(t) = sup | n ( t ) | , t>0 g(t) = Sup | g ( t ) | . U t>0 The Fourier transforms of g ( t ) , e ( t ) , etc., are denoted by G(jco) E(jco), etc. The notation f| • II denotes norms i n the space L^(0,°°). Thus ||n(t)|l = /~ | n ( t ) | d t . §3.2 • The Main Result The main r e s u l t i s the following theorem. §3.2.1 Theorem 1 Consider the system shown i n F i g . 3.1 to which the assumptions made above apply. Let y ( t ) be any r e a l function such that (1) y(t) = 0 for t < 0, (2) y(t) < 0 ' f o r t > 0 , K l (3) ||y(t)|| < . K2 and l e t q be any nonnegative number. If Re( [l+qjw+Y(jco)] [G(jto)-4-]+aqG(jto')} K2 - l l y ( t ) l l ( f - f ' - 0 ( Q 1 ) K l - K2 36 for a l l to, then (1) Sun |a(t) | < «, t>o (2) a(t) -v 0 as t <», (3) as |ln(t)|| + U n( t) 1 ->- 0, the corresponding a(t) has the property that Sup |o(t) J ->- 0. f >0 §3.2.2 A Special Case of the Theorem It should be noted that i f the time varying gain k(t) i s monotonically non-increasing and a <_ 0 V t >̂  0, then the condition (01) for absolute s t a b i l i t y may be replaced by Re[l+qj<^Y(juO][G(ju))+^H - || y ( t ) II - K2 K l K2 3 > 0 CQ1') for a l l a), 2 K 2 where again, q i s any nonnegative r e a l number. §3.3 Proof of Main Result The body of the proof of Theorem 1 w i l l be given i n a seri e s of appendices; a b r i e f summary i s given below i n t h i s section. Define a(t) for t < T a T ( - ' 0 for t > T, ,(t) ={ and Thus e T ( t ) = k ( t ) 0 [ a T ( t ) ] , a e T ( t ) = g(t-x) e T(x)dT. (3.3.1) a T ( t ) = a (t) for t < T, eT e — and a e T ( t ) e L 1(T,-) 37 Define The notation (x*y)(t) denotes convolution between x(t) and y ( t ) ; (x*y)(t) = x( T ) y ( t - T ) d T . . (3.3.2) a - o + o*y, c = c + c*y. m- • J m J Then T e (t) / ' [aAt) - -~—-]e(t)dt U m K.̂. •= / 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-T) ^ — - ] k ( t ) 0 [ a T ( t ) ] d t . (3.3.4) Now where and R(x) = R 1 ( T ) + R 2 ( T . ) , (3.3.5) R 1 ( t ) = f0 to T ( t-T ) -0 [a T ( t-T ) ] ]k(t )0 [a T ( t ) ]dt , (3.3.6) R 2 ( T ) = /Q[1 - k ^ T ) ]k(t)0[a T(t-x)]0[a T(t)]dt. (3.3.7) Frem Appendix 1. ro K k(t)0[a (t)] R 1 ( T ) l / 0 K [ a T ( t ) K ]k(t)0[a T(t)]dt, (3.3.8) and from Appendix 2, oo 1 1 2 R 2 ( T ) <_ / 0 ( ^ - - K ~ ) e T ( t ) dt. (3.3.9) Thus bo K 2 e T ( t ) R ( T ) ± V K a T ( t ) ~ ~ i ] e x ( t ) d t - (3.3.10) The f i r s t term of the ri g h t side of (3.3.3) i s always p o s i t i v e . Let us now consider the second term of the ri g h t side of (3.3.3) ; 0 t y * ( a " K ^ ] ( T ) D T 38 = / J /Jy(T ) [o- ( t-T) - e ( ^ " T ) ] e ( t ) d T dt co co e ( t - T ) = fQ / Q y ( x ) [ o T ( t - T ) - R ] e T ( t ) d T dt = / 0y(T)R(x)dx. (3.3.11) But, employing (3.3. 10), i t may be shown that fl[y*(o - f - ) ] ( t ) e ( t ) d t 0 > /gyCOdx a T ( t ) - - | ^ - ] e T ( t ) d t (3.3.12) - ' l l y l l / 0 [ K ^ ° T ( t ) " K ^ ~ ] e T ( t ) d t - (3.3.13) Substitu t i o n of (3.3.13) i n (3,3.3) y i e l d s e (t) /S[0»(t) - -YT-^t + W " y" ; o e T ( t ) 2 d t S T 6 T ( T ) > (1 - - ^ l l y l l / i [ a T ( t ) - - | ]e _ ( t ) d t (3.3.14) K l 0 T K2 T and, from the assumptions of the Theorem 1, the r i g h t side of (3.3.14) i s non- negative. Hence, e (t) - - f 2 - ] e ( t ) d t + lly" 'SeT(t)2dt - °- ( 3 - 3 - 1 5 ) Consider the following i n t e g r a l . T E M ( T ) 1 1 I = /o [ _ C Tem ( t ) ~ q°e(t) " ~T~ aqa e(t) + ( f - - £-) ||y|l e ( t ) ] e ( t ) d t = I 1 + I 2 , (3.3.16) where and T e ( t ) i l I = /'[ - a (t) - + ( f - - f - ) ||yll e ( t ) ] e ( t ) d t , (3.3.17) ± u em i^-^. 1̂ 2 I 0 = fl - q[a (t) + aa ( t ) ] e ( t ) d t . (3.3.18) 2 0 e e Now 39 I x = V ° m ( t ) + ^ - ^ ) H y l l e ( t ) ] e ( t ) d t + n m ( t ) e ( t ) d t (3.3.19) givi n g , due to (3.3.15) I > i'l n ( t ) e ( t ) d t . (3.3.20) 1 — 0 m In a s i m i l a r manner, I 2 = /Qq [a(t )+aa(t)]e(t)dt+/Jq[n(t )+an(t)]e(t)dt. (3.3.21) Invoking the r e s u l t of Appendix 3, I 2 L q[k(T)$(T)-k(0)$(0)+/Qq[fi(t)+an(t)']e(t)dt. (3.3.22) R e c a l l that / J f ( t )d t = / 0 f T ( t )d t , where f ^ ( t ) i s the truncated version of f ( t ) to between 0 and T. Hence from (3.3.16), 1 = V ^ e T m ^ " q 6 e T ( t ) " " q a C T e T ( t ) + ( ~ - jp ) ||y I! e T ( t ) ] e T ( t ) d t (3.3.23) and, from the conditions of Theorem 1 and since 1 CO 1 1 = " 27 L,Re{[l+qjco+Y(jw)][G(juO + + aqG(jw) -(^- " ~) llyll }E T ( j u3)E*(j W)du), i t follows that I 1 0. (3.3.24) Since I = I + 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)+qn(t)]e(t)dt. (3.3.25) — U r n Define e = Sup |e(t)1 0 < t < T 40 and invoking the conditions already imposed upon n ( t ) , fi(t) and y ( t ) , the rig h t side of (3.3.25) must be less than the quantity e M [ ( l + llyll +aq) ||n|| +q llfill ] = MeM. (3.3.26) Furthermore, considering the f i r s t term of t h e ' l e f t side of (3.3.25), since 0 i s monotonic, *(t) > |{0[a(t)]} 2. (3.3.27) Using (3.3.26) and (3.3.27) i n (3.3.25) y i e l d s ' | k(T){0[ a(T)]} 2 <_ qk(0)<D(0) + Me^. ' (3.3.28) From (3.3.28), since < 1, 2 ~ • L - e ( T ) 2 £ qk(0)*(0) + MeM. (3.3.29) The i n e q u a l i t y (3.3.29) holds .for any T >_0 and implies K M K M Sup | e ( t ) | ± + [ ( - ~ - ) 2 + 2 K 2 k ( 0 ) $ ( 0 ) ] 1 / 2 . (3.3.30) Furthermore, since a (0) = 0 by (3.3.1), t h i s bound on e(t) tends to zero with II hli + l lf i l l • It remains to be shown that | e ( t ) | + 0 as t ->• °°. Af t e r s u b s t i t u t i n g i n (3.3.14) of(A 3.3) of Appendix 3 and using . (3.3.24) , K (1 - - ^ l l y l l ) / J [ a ( t ) - ^-]e(t ) d t+/Qn m ( t)e ( t)dt -qk(0)$(0) + q/J[fi(t)+an(t)]e(t)dt <. 0. (3.3.31) Thus K (1 - r ^ l l y l l )fl[a(t) - ^ - ] e ( t ) d t < qk(0)*(0) .+ Me . (3.3.32) K^ 0 K 2 — M From which i t follows that T e(t) qk(0)*(0) + Me fQ[o(t) - ^ - ] e ( t ) d t i ^ . (3.3.33) 2 i - / l l y l l K l 41 Since r i g h t side of (3.3.33) i s independent of T, then l e t t i n g l.-> «>, e ( t ) qk(0)*(0) + Me fAo(t) - ^ - ] e ( t ) d t < M 0 L U V W K_ J C ^ ' U I " - K , (3.3.34) 2 l l y l l 1 and t h i s bound on the i n t e g r a l tends to zero with || nil + llnll . However, the bound already placed on e together with the conditions demanded of g(t) require 0" (t) to be bounded and to tend to zero. It may now be shown that the i n t e g r a l of (3.3.34) i s i n f i n i t e unless a (t) + 0 as t co, thus co n t r a d i c t i n g (3.3.34). Therefore i t can be concluded that a(t) -> 0 as t 0 0. Since a (t) i s uniformly continuous and i f Sup | a ( t ) | does not go to zero as |lnll + l l n l l 0, then (3.3.34) does not tend to zero e i t h e r . This i s a c o n t r a d i c t i o n . Hence, as l l n l l + II n II + 0, Sup |a ( t ) | -> 0. The proof i s t>0 therefore completed. §3.4 Examples Example 3.1 In the single-loop nonlinear system with a l i n e a r time varying gain shown i n F i g . 3.1, the l i n e a r part has a transfer function G ( s ) = (s+l)(s+2) ' the input and output of the nonlinear part, s a t i s f y (3.1.2), and the time varying gain i s such that (1) 1 <_ k(t) <_ 1.2, i . e . , K = 1, K = 1.2, (2) bk(t) <_ k(t) <_ ak(t) , where a,b are r e a l numbers such that a > 0, and b i s f i n i t e . Suppose that a i s large enough, q must be chosen zero. Let us assume that y ( t ) i s an exponential function such that 42 rct)=o p" o N.L. eft) Gcs) i | C t 7 c<t) -f 3.1 Single loop time varying nonlinear feedback system with zero-input 4 3 e ' as t > 0 y M - { T 0 as t > 0, 2 where y > 0 , and g > 1 . 2 = — . Then 1 lly(t)H =rQ l - i e ' ^ l dt . Thus the conditions on y(t) are s a t i s f i e d . A f t e r taking the Laplace transform of y ( t ) , 1 + Y(s) = 1 - - 1 = + T S + Y S + 1 3 + YS . ( 3 . 4 . 1 ) • Y The s u f f i c i e n t condition of absolute s t a b i l i t y i s R e { [ C p - l ) + Y . V ] [ G ( j a ) ) +•_! ] _ I ^ | o ( 3 . 4 > 2 ) 3 + Y JW ! • 2 3 1 - 2 — for a l l 03. Let us define G 1(Ja )) = GCjo,) + and From the r e l a t i o n ( 3 . 4 . 2 ) , the locus of G|(joi) must l i e on the r i g h t side of the v e r t i c a l l i n e passing through the point (r~L^~~ > 0 ) • The m u l t i p l i e r of 3 G^(jw) i n G^(jo)) may be considered as a compensator which i s shown on the Bode diagram F i g . 3 . 2 . The function of the compensator i s to improve the c h a r a c t e r i s t i c s of G^(jo3). From the p l o t of G ^ ( j w ) i n F i g . 3 . 3 , i t i s a simple matter to choose the proper values of g and y. From the fac t that the left-most point O J ^ = 2 . 5 • 3 — 1 3 on the locus of G.. (JOJ) l i e s between and — and that g s a t i s f i e d the r e l a t i o n 1 y y li-l 1 -j 1 . , . . ft"Urz _ ir~J < < c the s t a b i l i t y condition 3 K x K 2 K 2 K < 2 3 . 7 5 ( 3 . 4 . 3 ) 44 G= (UJOJXUJCO) Kf =23. 75 F i g . 3.3 The plot s of G(s) = ( 3 + i ) ( g + 2 ) a n d G'<s) = Ĉ Ĵ-)GCs) for Example 3.1. 45 has been found by running a s u i t a b l y w r itten programme on the d i g i t a l computer. It i s noted that the combination 3 = 2 and y = i s not the best one because of the p a r t i c u l a r choice of y ( t ) . That optimum y(t) which gives the best combination of 3 and y m a Y 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 constant, instead of the-G(s) i n the previous problem and l e t 0 < k(t) <_ 1, . and bk(t) <_ R(t) <_ ak(t) . In t h i s case, l e t us suppose y(t) = 0, then the' condition of absolute s t a b i l i t y i s Re{(l+qjio) [G(jco)+l]+aqG(jco)} >_ 0 C3 4.4) for a l l co. Rewritting, R(co) - qcol(co) + 1 + aqR(oj) ^ 0 (3.4.5) ' for a l l co, since G(jco) = R(co) + jl(co) . (3.4.6) Here „ / n 2. K^CD-co ) and Define -K (1+D) oi = ( l + c o 2 ) ( D 2 + . 2 ) ( 3 - 4 ' 7 B ) X(co) = R(co) (3.4.8A) and Y(OJ) = u i y - aR(oj). (3.4.8B) Substituting (3.4.8A) and (3.4.8B) i n (3.4.5), X(OJ) - qY(a)) + 1 >0. (3.4.9) To s a t i s f y i n e q u a l i t y (3.4.9), the locus must be on the r i g h t side of the s t r a i g h t l i n e passing through the point (-1,0) having p o s i t i v e slope —. q From (3.4.8A), (.3.4. 8B). and (3.4.9), i f a <_ l+D- e, where the small number e > 0 i s a r b i t r a r i l y chosen, and q i s then the s t a b i l i t y condition (Ql) i s s a t i s f i e d for any nonnegative r e a l constant k. Besides, the modified Nyquist pl o t i s on the r i g h t side of the Popov l i n e . This i s shown i n F i g . 3.4. If a ^ 0, the s u f f i c i e n t condition of absolute s t a b i l i t y i s R(OJ) - qoKu) + 1 >_ 0. (3.4.10) This i s the Popov c r i t e r i o n and i s s a t i s f i e d for any nonnegative r e a l constant k 1 by choosing q = 1+D" §3.5 Absolute S t a b i l i t y of a System with Many N o n l i n e a r i t i e s and Many Time Varying Gains In the previous secLions, the absoluLe s t a b i l i t y of the system with one n o n l i n e a r i t y and one time varying gain i s established. Now, l e t us consider i system with many n o n l i n e a r i t i e s and many time varying gains. Such a system i s shown i n F i g . 3.5. The input a. and the output 0.(o.) of the i-th n o n l i n e a r i t y l i i are r e l a t e d by the following: (1) 0 < a.0.(a.) < a 2 for a. i 0, 0.(0) = 0, l i i — i l l d0.(a.) (3.5.1) <2> 0<-<t^ l 1 - i and the instantaneous value of the i - t h time varying gain i s constrained so that (1) K, . < k.(t) < K. . , where K„. > K_ . > 0, l i — I — z i 2 i l i (2) b.k(t) < k.(t) < a . k . ( t ) , where the number a > 0 and the number 1 1 1 1 i b. i s f i n i t e , l 47 F i g . 3.5 General time varying nonlinear feedback system with zero-input 48 Besides, ° = fn r(t - T)e(x ) d T e (J — i s the zero state response of the l i n e a r time inv a r i a n t transfer matrix r(jw) = J [F(t) ] , where / G 1 1 ( J O J ) G 1 2 ( j a j ) - G 2 1 ( j u ) G ( j o ) — • r(ju) = G l m ( ^ ^ (3.5.2) I I kGml(U) G^Cjo.) The input vector n_(t) represents the zero-input response of r(t). The complete response of T(t) i s thus 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 matrix r(t) are s t a b l e , more s p e c i f i c a l l y , (1) g ± j ( t ) e L 2(0,-),. g (t) £ ^ ( 0 , 0 0 ) , i=l,2,...m, j = l,2,...m, (2) n.(t) e ^ ( O . o o ) , (3) n i ( t ) i s d i f f e r e n t i a b l e and f^Ct) c L (0,«). Condition (1) above ensures that each element of r(t) i s bounded on (0,<») and that g_̂ .—»nj and conditions (2) and (3) ensure that ^(t) behaves i n the same manner as g ( t ) . i j Denote n = E n. = E Sup |n. ( t ) | i = l i = l t>0 m m m m E E Sup |g ( t ) | . r = E E g.... . . 1=1 j = l J i = l j = l t>0 J The notation 11-11 denotes norms i n the space L ( 0 , c ° ) such that, for example, 49 m H II = 'n E ' n i ( t ) l d t i = l Define and K̂ "*" = diagonal matrix O^-^l ' K] 2 '' ' * " Klm^ = diagonal matrix ( K 21' K22'"''* K2m^' A = diagonal matrix ( a ^ , , . . . . a ^ ) §3.6 Theorem 2 . • Consider the system shown i n F i g . 3.5 to which assumptions given above apply. Let Y(t) = diag{y ( t ) , y (t) . . . .y (t) } be such that each element i s a r e a l function and that (1) y ± ( t ) ' = 0 for t < 0, i = 1,2,...m, (2) y.(t) < 0 for t > 0, l — — K (3) l|y.(t)ll < i K N Let Q be any p o s i t i v e semi-definite constant diagonal matrix. I f there e x i s t s anmxm matrix H(jco) such that H(jco) = [I+JWQ+Y(jaj)] [r(jco)+K21] + AQTCjoj) - II YIt [ K " 1 - K " 1 ] , (02) and then T (1) H(jco) + H (-jco) i s a p o s i t i v e semi-definite Hermitian matrix for a l l co, (2) H*(jw) = H(-jco) (3) Every element' of H(jco) i s a n a l y t i c f o r a l l co, (1) Sup | o.(t) | < «=, and Sup|a.(t)| < i=l,2,...m, t>0 t>0 1 50 (3) (2) o;(t) -> 0 as t -> 0 0, and o\ -> 0 as t -> o o , a s Hull + Ilil II _ > 0> t n e corresponding _g_ has the property Sup|o_(t)| -K), and Sup| C T.(t)| ->0. t>0 t>0 1 §3.7 Proof of Theorem 2 This proof follows the same vein as that of the previous Theorem. The only d i f f e r e n c e here i s that a l l vectors such as e ( t ) , a (t) , n ( t ) , n ( t ) , — —e — — jc(t) , 0^.t) , cr(t) are m-vectors, and a l l matrices such as 0,Y are mxm matrices. A l l formulae and the proofs of which developed i n §3.3 s t i l l hold, except that the proof of (3.3.24) must be performed i n the following manner. where h(t) i s the inverse Fourier transform of H(jco). From the condition of Theorem 2 and Newcomb's r e s u l t [25], From (3.3.23), (3.3.23') = - / ^ e J ( t ) / Q h ( t - T ) e T ( T ) d x dt, I' < 0. (3.3.24') This, however, i s the same as (3.3.24). Hence, by the same argument used i n the proof of Theorem 1, (1) Sup|a(t)| < t>0 CO (2) a(t) + 0 as t -> CO Now, Sup|o.(t)| t>0 < 0 0 i f and only i f every component a.(t) s a t i s f i e s S u p 1°" (t) I < 00. t>0 i S i m i l a r l y , cr(t) -> 0 as t -> ~ i f and only i f every con\ponent ^ ( t ) s a t i s f i e s rj^(t) -> 0 as t -> c o , and Sup|c[(t)| -> 0 i f and only i f every component a. (t) s a t i s f i e s t>0 Sup\o. (t) I -> 0. t>0 1 §3.8 Example Example 3.3 Let us consider the p a r a l l e l system, where m = 3, each branch of which has one n o n l i n e a r i t y and one time varying gain i n series with one l i n e a r time i n v a r i a n t t r a n s f e r function. Here these three parts of each the three branches are i d e n t i c a l to the corresponding ones used i n Example 3.2. Suppose that the matrix Y(t) = 0. Let us consider the matrix H(jw) = (I+ju)Q)(r(ju))+I) + a Qr(joi), (3.8.1) where r(jco) = G(ju) 1 1 IN 1 1 1 1 1 1 ' ' and Q = ql- Rewritting, H(jco) = (l+jcoq+aq)G(jco) 1 1 1 + U+jcoq)!. I l l 1 1 1 Obviously, H*(jco) = H(-jco), and the elements of H(jco) are a l l a n a l y t i c for a l l 60. - Invoking the proof of §2.4, the Hermatian matrix (3.8.2) 52 H(ju) + H T ( - j w ) = Re(l+ju)q+aq)G(joj) + - j . ( 3 . 8 . 3 ) Following the argument used in Example 3 . 2 , i f a <_ 1+D-E. and q i s Y> the s u f f i c i e n t condition Q2 of absolute s t a b i l i t y i s s a t i s f i e d for any p o s i t i v e r e a l constant K. 53 Chapter 4 CONCLUSIONS A graphical method using the Popov l i n e i s possible for a p a r t i c u l a r class of time i n v a r i a n t nonlinear system. The method may be s i m p l i f i e d i n a number of cases. Two a l t e r n a t i v e approaches, one using the Nichols chart, the other the analogue computer, are mentioned b r i e f l y and i l l u s t r a t e d . No simple graphical method e x i s t s to test the absolute s t a b i l i t y of the p a r a l l e l system with many d i f f e r e n t l i n e a r transfer functions, although a graphical method using the Popov l i n e to test the absolute s t a b i l i t y of the p a r a l l e l system with many i d e n t i c a l nonlinear t r a n s f e r functions i s possible. Neither i s there any simple graphical 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 of absolute s t a b i l i t y of a m u l t i - c i r c u i t or an i n t e r n a l feedback system. However, work on the determination of the c r i t e r i o n of absolute s t a b i l i t y f o r any one of the four classes 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 [23] i s i n essence concerned with l o c a t i o n of the optimum combination of matrices Q and K which w i l l define the ̂ boundary of absolute s t a b i l i t y region. I t must be pointed out, however, that with the systems that have been discussed so far i n th i s thesis the graphical method i s so far simpler and less cumbersome i n obtaining the r e q u i s i t e conditions for absolute s t a b i l i t y . In chapter 3, Theorems 1 and 2 provide the s u f f i c i e n t , but not necessary, conditions f o r the absolute s t a b i l i t y of a time varying nonlinear system i n which the nonlinear part must be monotonically nonlinear. Of great importance i n e s t a b l i s h i n g the s u f f i c i e n t condition of absolute s t a b i l i t y i s the appropriate choice of A and y ( t ) . 1 + Y(s) i s i d e n t i c a l to the function describing some RC passive network i f y(t) i s an exponential function. The optimum region of absolute s t a b i l i t y may be found by a d i g i t a l technique. 54 If the time varying gain i s frozen, that i s , k ( t ) = 1, the s u f f i c i e n t condition. (Ql) may be rewritten Re[l+qjw+Y(jui)] [G(jto) > 0. 2 This i s the r e s u l t of Baker and Desoer's [11]. For some classes of time varying nonlinear systems, i t i s possible to use the- graphical method discussed i n chapter 2. APPENDIX 1 From (3.3.6) R ^ O - R ^ T ) = / o ( [ a T ( t ) - 0 [ a T ( t ) ] ] - [ a T ( t - T ) - 0 [ a T ( t - T ) 1 ] } 0[a T ( t ) ] k ( t ) d t Noting that 0(t) and [a(t)-0(t)] are monotonic, and 0 < < 1 " r°2 ' or or (tf 1-a 2)(0 1-0 2) - (01"02)2 > 0, [ ( a 1 - 0 1 ) - ( a 2 - 0 2 ) 1(0̂ 02) > 0, thus 0(t) i s monotonic increasing i n [a(t) - 0(t)]. Let us define A = ([c T(t)-0[a T(t)]-[a T(t - T)-0[a T(t-x)])0[a T(t)], and observe that A >_P(t) - P ( t - r ) , where a T ( t ) P(t) =rQ 0[a T(x)]d[a T ( T)-0[a T(x)]]. From ( A l . l ) , (A1.4) and (A1.5), 00 R 1(0)-R 1(x) >_ / Q [ P ( t ) - P ( t - T ) ] k ( t ) d t and But R 1 ( T ) 1 ^ ( 0 ) + / 0 [ k ( t + T ) - k ( t ) ] P ( t ) d t . V 0 ) -C[0l(t) - ^ t a T ( t ) ] ] k ( t ) 0 [ a T ( t ) ] d t , and,since, P(t) < [ o T ( t ) - 0 [ a T ( t ) ] ] 0 [ a T ( t ) l < t a T ( t ) - - 1 0ta T(t)]] ^ 0 [ a T ( t ) ] , then / ~ [ k ( t + x ) - k ( t ) ] P ( t ) d t <_ (Z2~\) /QP(t)dt K —K J"o [ aT ( t ) - ^ 0 [ a T ( t ) ] ] k ( t ) 0 [ C T T ( t ) ] d t . Substituting (A1.9) and (ALIO) i n (A1.8) y i e l d s 57 APPENDIX 2 From (3.3.7) R 2 ( T ) = /Q[1 - ̂ ^ - ] k ( t ) 0 [ a T ( t - T ) ] 0 [ a T ( t ) ] d t . (A2.1) Therefore, R 2 ( T ) - Qk^O ~ K ^ l l^-tW^tt-!)]] | k ( t ) 0 [ a T ( t ) ] |dt - " K"' l k ( t - r ) 0 [ a T ( t - T ) ] I | k ( t ) 0 [ a T ( t ) ] |dt (A2.2) giving r 2 ( t ) - I ( K " " ^ H y K ( T ~ T ) 0 [ A T ( T ~ T ) ] 2 ) D T + /Q[k(t)0[a T(t)] 2Jdt} (A2.3)' from which (3.3.9) follows. 58 cl e a r l y ^ APPENDIX 3 Since ak(t) > k ( t ) , ak(t) > 0, and a(t)0[a(t)] >_faQ(t) 0[a]da = *(t) > 0, /Jak(t)a(t)0 [a(t ) ]dt >/Jk(t)$(t)dt, (A3.1) J = q/J(d(t)aa(t))k(t)0(t) >q [k(t)4>(t)+k(t)0[a(t)] d(t)]dt - q fl IF t k ( t ) * ( t ) ] d t >_ q[k(T)$(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 ) <_ ak ( t ) <_ 0 and 0 <_ M ^ i £ ( t ) J _ ,< $ - ( t ) f c l e a r l y , fl ^ e ( t ) 2 d t = / J a k ( t ) • ^ t ) 0 [ a ( t ) ] 2 ^ ,T U t ) $ ( t ) d t j ( M < 1 ) J' = q/J[6(t) + ~ - e ( t ) ] e ( t ) d t >_ q /J [k$ (t)+k (t) 0 [a (t) ]d (t) ] dt >_ q[k(T)$(T) - k(0)$(0)] . 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