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Aspects of nonlinear system stability. Christensen, Gustav Strom 1966

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ASPECTS OF NONLINEAR SYSTEM STABILITY by GUSTAV ' STROM CHRISTENSEN B.Sc, University of Alberta, 1958 M.A.Sc, University of British Columbia, 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Electrical Engineering We accept this thesis as conforming to the required standard Members of the Department of Electrical Engineering THE UNIVERSITY OF BRITISH COLUMBIA September, 1966 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia^ I agree that the Library s h a l l make i t f r e e l y available f o r reference and study, I further agree that permission f o r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of this thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia Vancouver 8, Canada Date Q e J ^V^- / f ^ The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of GUSTAV STROM CHRISTENSEN B.Sc, University of Alberta, 1958. M.A.Sc, University of B r i t i s h Columbia, i960 MONDAY, OCTOBER 2k AT 3;30 IN ROOM lfl . 8 , HECTOR MacLEOD BUILDING COMMITTEE IN CHARGE Chairman: L. G. James F„ Noakes R. W. Donaldson E. V. Bohn A. C. Soudack C. A. Brockley M.S. Davies . External Examiner: A. R. Bergen Associate Professor University of C a l i f o r n i a Berkeley Research Supervisor: A, C. Soudack ASPECTS OF NONLINEAR SYSTEM STABILITY ABSTRACT This thesis treats system s t a b i l t y from three separate points of view. 1. State Space Analysis 2. Complex Frequency Plane Analysis 3 „ Time Domain Analysis Asymptotic s t a b i l i t y i s considered i n state space. Using state space and the gradient method an expression i s derived for the t o t a l time derivative of the Liapunov function. This expression i s a special case of the general Zubov equation, however, i t does not lend i t s e l f to an e x p l i c i t , exact solution except i n special cases. Global asymptotic s t a b i l i t y and bounded input -bounded output s t a b i l i t y i s considered i n the complex frequency plane. Here a method developed by Sandberg has been applied to some systems the l i n e a r part of which has poles on the imaginary ax i s . The solution of an example of t h i s type v i a the Sandberg method and the Popov method shows that the two methods give e s s e n t i a l l y the same r e s u l t f o r the example considered. Bounded input ~ bounded output s t a b i l i t y i s considered i n the time domain using two separate methods. One, a method developed by Barrett using Volterra series has been extended to cover cases with a nonlinearity of 2nd and kth degree. Two, a method depending on the contraction mapping p r i n c i p l e i s developed and applied to several types of systems. I t i s shown that t h i s method generates the Volterra series found by Barrett's method, and thus we can ac t u a l l y determine a region where, the solution of a given d i f f e r e n t i a l equation can be represented i n the form of a Volterra series. GRADUATE STUDIES F i e l d of Study; E l e c t r i c a l Engineering E l e c t r i c a l Power Systems Network Theory Servomechanisms Analog Computers Numerical Analysis Heat Transfer Electromagnetic Theory Design of E l e c t r i c a l Machinery Nuclear Physics Nonlinear Systems Integral Equations D i g i t a l Computers F, Noakes A. D„ Moore E, V. Bohn W* Dieticker C„ Froese Wm. Wolfe G„ W. Walker J. Szablya G. M, G r i f f i t h s A. C* Soudack E. Macskasy E. V. Bohn GUSTAV STROM CHRISTENSEN. ASPECTS OF NONLINEAR SYSTEM STABILITY. Supervisor• A. Co Soudack. ABSTRACT This thesis treats system s t a b i l i t y from three separate points of view. 1. State Space Analysis 2. Complex Frequency Plane Analysis 3. Time Domain Analysis Asymptotic s t a b i l i t y is considered i n state space. Using state space and the gradient method an expression is derived for the total time derivative of the Liapunov function. This expression is a special case of the general Zubov equation, however, i t does not lend i t s e l f to an e x p l i c i t , exact solution except i n special cases. Global asymptotic s t a b i l i t y and bounded input - bounded output s t a b i l i t y i s considered i n the complex frequency plane. Here a method developed by Sandberg has been applied to some systems the linear part of which has poles on the imaginary axis. The solution of an example of this type v ia the Sandberg method and the Popov method shows that the two methods give essentially the same result for the example considered. Bounded input - bounded output s t a b i l i t y is considered i n the time domain using two separate methods. One, a method developed by Barrett using Volterra series has been extended to cover cases with a nonlinearity of 2nd and 4th degree. Two, a method depending on the contraction mapping principle is developed and applied to several types of systems. It is shown that this method generates the Volterra series found by Barrett 's method, and thus we can actually determine a region where the solution of a given d i f f e r e n t i a l equation can be represented i n the form of a Volterra series. TABLE OF CONTENTS Page L i s t of I l l u s t r a t i o n s v Acknowledgement v i i 1. INTRODUCTION .1 2. STABILITY ANALYSIS USING STATE SPACE 5 2.1 Introduction 5 2.2 Definitions and S t a b i l i t y Theorems 8 2.3 The S t a b i l i t y of Linear, Autonomous Systems 11 2.4 S t a b i l i t y Domains f o r Nonlinear Systems by Zubov's Method 12 2.5 S t a b i l i t y Domains f o r Nonlinear Systems by Gradient Method 13 2.6 Discussion of Results 22 3. STABILITY ANALYSIS USING THE COMPLEX FREQUENCY PLANE 2 3 3.1 Introduction 23 3.2 A Frequency S t a b i l i t y C r i t e r i o n for Nonlinear Systems 25 3.3 Systems with Poles on the Imaginary Axis 35 3.4 The S t a b i l i t y of Some Nonlinear Systems 39 3.5 Discussion of Results 55 4. STABILITY ANALYSIS USING THE TIME DOMAIN 57 4.1 Introduction 57 4.2 S t a b i l i t y v i a Volterra Series 59 4.3 The S t a b i l i t y of Two S p e c i f i c Nonlinear Systems 65 4.4 The S t a b i l i t y of a System with a Nonlinearity of Second Degree ?5 i i i Page 4.5 The S t a b i l i t y of a System with a Nonlinearity of Fourth Degree 80 4.6 Comparison to Desoer's Method 87 4.7 The Contraction Mapping P r i n c i p l e 88 4.8 S t a b i l i t y v i a Contraction Mapping 92 4.9 The S t a b i l i t y of some Nonlinear Systems 96 4.10 The S t a b i l i t y of some Nonlinear Systems with a Simple Pole at the Origin 107 4.11 The V a l i d i t y of Volterra Series Representation 117 4.12 Discussion of Results 122 5. CONCLUSIONS 124 5.1 Summary 124 5.2 Recommendations f o r Future Work 125 APPENDIX A .' 127 REFERENCES 134 i v LIST OF ILLUSTRATIONS Figure Page 2.1 Phase Plane Trajectory 6 2.2 Integration of the Function S 18 3.1 Linear Feedback System with Constant Parameters 23 3.2 Nyquist Plot for F Q(s) = s(1+Q.04s) 2 5 3.3 C r i t i c a l C i r c l e (a > 0) 29 3.4 C r i t i c a l C i r c l e (a < 0) 31 3.5 Bounds on 0 and K 33 3.6 Bounds on 0 and K 34 3.7 Bounds on 0 and K 34 3.8 Block Diagram Representation of (3.47) 39 2 -1 3.9 K(jco) Locus for ^  + x1 - 2x1 x 2 Q e '=o) 4 2 • • • 3.10 K ( » Locus f o r (x + x + 0(x,t) = 0)e=l 44 3.11 K(ju) Locus for Equation (3.74). 48 3.12 S t a b i l i t y Sector for Equation (3.75) 49 t 3.13 W (ju)) Locus for Equation (3.80) 53 3.14 S t a b i l i t y Sector f o r Equation (3.80) 54 4.1 Block Diagram Representation of (4.1) 57 4.2, Graph of Equation (4.18) 63 4.3 Boundary Values of X 66 4.4 |h(t)| versus Time 67 4.5 S t a b i l i t y Sector for Equation (4.27) 70 4.6 S t a b i l i t y Sector f o r Equation (4.49). 74 4.7 Graph of Equation (4.76) 77 4.8 Boundary Values of X (Equation (4.76)) 79 v Figure Page 4.9 80 4.10 83 4.11 86 4.12 87 4.13 92 4.14 93 A . l 128 A.2 130 A. 3 131 ACKNOWLEDGEMENT I wish to express my gratitude to my supervisor Dr. A. C. Soudack and to the head of this department Dr. F. Noakes for encouragement and guidance during the course of th i s study. Further, sincere thanks are given to Drs. H. P. Zeiger, E. V. Bohn and R. W.Donaldson for taking time to l e v e l constructive c r i t i c i s m at th i s project. Also I wish to acknowledge interesting d i s -cussions held with my fellow graduate students concerning the topic treated i n this thesis and subjects related thereto. In p a r t i c u l a r I wish to mention Mr. J . Sutherland i n t h i s connection. Acknowledgement i s g r a t e f u l l y given to the National Research Council for providing assistance for the session 1963-1964 and to the University of B r i t i s h Columbia for awarding U.B.C. graduate fellowships for the sessions 1964-1965 and 1965-1966. v i i 1. INTRODUCTION The s t a b i l i t y of an undriven physical system is determined by i t s behaviour when subjected to external perturbations which displace the system from i t s or iginal rest posit ion. In this thesis a system w i l l be said to be stable i f i t remains close to i t s or iginal rest position for a l l times after the disturbance has ceased, and the system w i l l be said to be asymptotically stable i f i t , i n time, returns to i ts or iginal rest position after the disturbance has ceased. In many cases i t happens that a system exhibits this behaviour only within a limited region around the or iginal rest point, and this region is then called the region of attraction or the s t a b i l i t y region of the given system. The exact location of the boundary of this region is i n most cases very d i f f i c u l t to determine. If no such boundary exists the system is cal led globally or absolutely stable. Also a system may have more than one stable posi t ion, and may indeed come to rest at a point other than the or iginal one i f the applied disturbance is large enough. However, i n general attention is focussed on one rest point. Further, i f the motion of a system does not exhibit any of the characteristics just mentioned, i t i s said to be unstable. In considering the s t a b i l i t y of driven systems we generally use a different concept of s t a b i l i t y and speak of bounded input - bounded output s t a b i l i t y . That i s , we attempt to determine the class of f i n i t e inputs which produce a f i n i t e output. 6* There are other definitions of s t a b i l i t y ; however, these w i l l not be discussed or employed here. * References placed above the l ine of text refer to the bibliography. 2 The modern concept of s t a b i l i t y of physical systems was f i r s t introduced by Lagrange l a t e i n the eighteenth century. He showed that i n order f o r a mechanical system to be stable i t s p otential energy must be a minimum at the singular point. However, th i s applies only to conservative systems which form a very r e s t r i c t e d class of systems i n that the forces acting must be derivable from a scalar potential function. The next major contribution to the theory of s t a b i l i t y of physical systems was made by A.H. Liapunov i n 1892. His doctoral d i s s e r t a t i o n on t h i s subject i s available i n book form under the t i t l e "Probleme General de l a S t a b i l i t e du Mouvement" edited by Princeton University Press. The general contents of th i s d i s s e r t a t i o n are available i n the English literature-'-'^'^. Liapunov considered a class of systems of very general nature, and developed two d i s t i n c t methods for investigation of system s t a b i l i t y . In the l i t e r a t u r e these are generally denoted Liapunov's " f i r s t method" and "second method". The " f i r s t method" consists of a s t a b i l i t y analysis from approximate solutions of the system equations of the perturbed response, obtained by means of a successive approximation procedure. That i s , the " f i r s t method" act u a l l y comprises a l l procedures i n which the e x p l i c i t form of the solutions i s used when represented by i n f i n i t e series 0. The "second method" which i s also often termed the "di r e c t method" gives information about system s t a b i l i t y d i r e c t l y without knowledge of the detailed motion of the system. Here special functions, generally c a l l e d "Liapunov Functions", are formed and u t i l i z e d to investigate the s t a b i l i t y behaviour of the system i n question. I t follows that Lagrange's concept of system s t a b i l i t y belongs to the 3 "second method". As developed by Liapunov, the "second method" constitutes s u f f i c i e n t conditions f o r system s t a b i l i t y ; the inverse problem, that i s , the existence of a Liapunov function f o r a system exhibiting stable motion was not investigated by Liapunov. However, t h i s problem has been successfully solved and found to be true f o r most cases through the ef f o r t s of several Russian s c i e n t i s t s during the past twenty-five years. Probably the most s i g n i f i c a n t developments r e s u l t i n g from these investigations were made by Lure 4 and Zubov5. Lure developed a construc-t i o n procedure for Liapunov functions r e l a t i n g to closed loop control systems, and Zubov devised a constructive proof which shows Liapunov's conditions are both necessary and s u f f i c i e n t to ensure system s t a b i l i t y . While Liapunov's "second method" u t i l i z e s state space analysis and thus leans heavily on the d i f f e r e n t i a l equation approach to investigate s t a b i l i t y there are other methods available f o r th i s purpose. The more w e l l known of these are the "Nyquist Frequency Locus", "Bode Diagrams", and the "Root Locus Method". These s t a b i l i t y c r i t e r i a 7 were developed on th i s continent during the past t h i r t y years and u t i l i z e the complex frequency plane. However, these methods apply only to es s e n t i a l l y l i n e a r systems, although some attempts have been made to extend these methods to nonlinear systems as w e l l 7 . Unfortunately these extensions to nonlinear systems are quite d i f f i c u l t to use. In 1961 V.M. Popov developed a frequency s t a b i l i t y c r i t e r i o n for nonlinear systems 4 which aroused widespread i n t e r e s t , and gave impetus to intensive investigations regarding the generality and lim i t a t i o n s of the method. These investigations are s t i l l continuing. However, the Popov c r i t e r i o n has only been proven to constitute a 4 s u f f i c i e n t condition f o r s t a b i l i t y ^ . On the other hand, Lure's construction procedure has since been shown to be clos e l y connected to the Popov method. In f a c t , i t has been shown t h e o r e t i c a l l y that the Popov condition (see Appendix A) i s necessary and s u f f i c i e n t for the existence of a Liapunov function of the type found by Lure's method^. There are of course many more workers besides those already mentioned who have contributed to the f i e l d of s t a b i l i t y theory, and probably the most complete bibliographies r e l a t i n g to the subject are given i n references 1,4, and 8. In t h i s thesis both state space, frequency plane and time domain methods are u t i l i z e d , and t h e i r respective merits w i l l become evident as various methods are developed, discussed, and employed to investigate the s t a b i l i t y of several systems. 5 2. STABILITY ANALYSIS USING STATE SPACE 2.1 Introduction The most elementary example of state space consists of two perpendicular axes i n a plane. For i l l u s t r a t i v e purposes consider the simple d i f f e r e n t i a l equation • • x^ + x-^  = 0 which represents a conservative system. Let x l = x 2 (2.1) x 2 = -x x (2.2) and divide equation (2.1) by (2.2); we then obtain dx-j HxT x x x 2 L2 *1 which has the solution x-^ 2 + = c^ (2.3) Equation (2.3) may then be represented i n two dimensional state space, that i s , the f a m i l i a r phase plane as shown i n F i g . 2.1. Equation (2.3) i s c a l l e d a phase trajectory and represents a stable motion i n accordance with the d e f i n i t i o n of s t a b i l i t y given i n the introduction. The general procedure for representing a higher order d i f f e r e n t i a l equation, l i n e a r or nonlinear, i n state space i s then just 6 Fig. 2.1 Phase Plane Trajectory an extension of the procedure used above. Thus given the equation - \ S i 1 + ... + g n x 1 = 0 d t n dt n"A we l e t x l = x 2 « x2 " x 3 a §lxn + g2*n-l + ••• + 8n-l x2 + 8^1 In matrix form equation (2.5) becomes x l 0 1 0 . . . 0 x l • = 0 1 * x n 8n 8n-l Sn-2 • ' • 81 x n 7 G e n e r a l l y such a s e t o f equations w i l l be represented by x = Ax = £(x) (2.7) where the non-subscripted v a r i a b l e s x, x and f denote n-dimensional column v e c t o r s , and A denotes an square n by n m a t r i x w i t h constant or v a r i a b l e elements. Further, i f time t does not occur e x p l i c i t l y i n equation (2.7) i t i s c a l l e d autonomous, whereas i f time t does occur e x p l i c i t l y equation (2.7) i s c a l l e d nonautonomous. orthogonal c o o r d i n a t e s , XT, x 2 , x n , i n s t a t e space, and t h a t we can form equations of the type e t c . The s o l u t i o n o f these equations w i l l then give a phase t r a j e c t o r y (provided a s e t o f i n i t i a l c o n d i t i o n s are given) i n the space represented by the n c o o r d i n a t e s . However, g e n e r a l l y these equations cannot be s o l v e d e x p l i c i t l y . The d i s t a n c e between two p o i n t s i n s t a t e space can be d e f i n e d as the e u c l i d i a n norm I t w i l l be noted from equation (2.5) t h a t we now have n |jxj| = ( x x 2 + x 2 2 + ... + x^)h (2.8) where x denotes the transpose o f the column v e c t o r x. 8 In the phase plane, points where dx2 = f 2 ( * i , * 2) _ 0 3xY f i ( X ! , x 2) 0 are called singular points. It should be noted that i f we desire to investigate the s t a b i l i t y of a singular point which i s not located at the or igin we simply translate the coordinates, such that the singular point in question is located at the o r i g i n . 2.2 Definitions and S t a b i l i t y Theorems'^ In mathematical terms the system (2.7) is said to be stable with respect to the solution x = 0 i f , given a small positive number e, there always exists another positive number u , such that any solution of equation (2.7) which i n i t i a l l y sat isf ies |x(t=0)| i u also sat isf ies for a l l t ^ 0 the inequality |x(t):| < e - It w i l l be noted that this assures system s t a b i l i t y i f , b y choosing suff i c ient ly small i n i t i a l conditions, we can guarantee the solution w i l l remain smaller i n magnitude than any predetermined positive number. If i n addition lim x(t) =0 t — - » the system is said to be asymptotically stable with respect to the solution x = 0. A function i s cal led definite i n a domain D containing the origin i f i t has values of only one sign and vanishes only at x ^ = x 2 = . . . = x n = 0. In the l i terature , Liapunov functions are generally denoted by V = V ( X p x n ) . Such a function is called definite (positive or negative) i n a certain domain D : |xjj< H (H > 0 is a constant) i f the sign of V is invariant, and V vanishes only 9 for XT = x 2 = ... = ^ =0 Example V = xTx is a positive definite function. On the other hand, V is called semidefinite i f V = 0 for values other than xn = x 2 = . . . = x n = o. Example V = x x 2 + (x2 + x 3 ) 2 is a semidefinite function since i t vanishes at the point JCXX = 0, x 2 = -x 3 ) . The total time derivative of V = V (xi , . . . ,x n ) is found i n the standard manner v a t L 6H dT 4 - , 1 1=1 1=1 f. by use of equation (2.7). Liapunov's Theorem on Stabi l i ty Given the d i f f e r e n t i a l system (2.7) with the singular point x = 0 located i n a domain D, then i f we can f ind a positive definite function V with a total time derivative V which sat isf ies 0 the system (2.7) is stable i n D with respect to the solution x = 0. For proof see reference 3. Example *1 = X2 10 x. 2 = -x-1 Let V = 1/2 (x x 2 + x22) then V = xnxn + x 0 x = 0 Liapunov's Theorem on Asymptotic S tabi l i ty Given the d i f f e r e n t i a l system (2.7) with the singular point x = 0 located i n a domain D, then i f we can find a positive definite function V which has a total time derivative which is negative definite i n D, then the system (2.7) is asymptotically stable with respect to the solution x = 0. For proof see reference 3. (Note: V may be semidefinite i f the points V = 0 do not form a trajectory.) Example = -x. 1 - x. 2 Let V - + x then + X2 X2 Clearly V< 0 and V> 0 also ensures asymptotic s t a b i l i t y . 11 2.3 The S t a b i l i t y of Linear, Autonomous Systems Since we in this section consider only linear systems i t follows that the domain of s t a b i l i t y wiHl either include a l l points in state space or none at a l l . That i s , the region of asymptotic s t a b i l i t y for such systems is never f i n i t e . Consider the linear system x = Ax (2.9) where the elements of the matrix A, a^j , are constants. For such a system Liapunov used the following procedure to investigate s t a b i l i t y . Let V = - ||x|| 2 (2.10) and V = xTBx (2.11) then from equation (2.11) V = xTBx + xTBx since B is a constant matrix; further", substituting from equation (2.9) V = xTABx + xTBAx (2.12) We then equate the right hand sides of equations (2.10) and (2.12) and obtain A TB + BA = -I (2.13) where I i s the identity matrix. If the eigenvalues of the matrix A a l l have negative real parts we w i l l on solving for the elements of B, b ^ j , i n equations (2.13) f ind that B is a symmetric, positive definite matrix. Positive definiteness is characterized by the fact that a l l the principal minors of B are posi t ive . It therefore follows that 12 Liapunov's Theorem on Asymptotic S t a b i l i t y stated on page 10 is sa t i s f ied 12 for the system (2.9). Gibson gives a pract ical application of this procedure. 2.4 S t a b i l i t y Domains for Nonlinear Systems by Zubov's Method As mentioned in the introduction,Zubov^ developed a constructive method which shows Liapunov's c r i t e r i a to be both necessary and suff ic ient for a given system to be asymptotically stable. His q method was investigated i n detail by Margolis , and the salient features of Zubov's method are the following. Consider the system x =f(x) (2.14) and let D be an open domain i n state space while the closure of D is denoted D. Assume D contains the origin and that x = 0 is a singular point of the system (2.14). Then necessary and suff ic ient conditions for D to be the exact domain of attraction of the equilibrium of (2.14) are the existence of two functions v(x) and 0(x) with the following properties, 1. v(x) is defined and continuous i n D. 2. 0(x) is defined and continuous i n the whole state space. 3. 0(x) is positive definite for a l l x. 4. v(x) is positive definite for x e D, x f 0. 5. 0 < v(x) < 1 6. If y e D - D then lim v(x) = +1; x -> y also lim v(x) = +1 provided that the lat ter l i m i t process can be x* co carried cut for x e.D. 13 n 7. dv = y 5v_ f = . (J) C x ) ( i . v ( x ) ) (2.15) at In cases when equation (2.15) can not be solved e x p l i c i t l y „ (which i t i n general can not) i t may be solved by assuming a series solution of the form V = j=2 k=l £ d j k V _ I P r i * l (2.16) for the two dimensional case. Similar series solutions may be assumed for higher dimensionsj however, i t is evident that the complexity w i l l then increase r a p i d l y 9 , 1 3 • The advantage gained i n using the form (2.16) i s j that recurrence relations arise for the coefficients d ^ . Generally the approximation to the required s t a b i l i t y boundary improves as the number of terms included i n v is increased, however, i t is usually found that the convergence is s low^ ' l^ . j n theory though, as n oo v •> +1 as required by condition 6 stated above. 2.5 S t a b i l i t y Domains for Nonlinear Systems by Gradient Method From the previous section i t i s evident that a direct solution of Zubov's equation is i n most cases d i f f i c u l t . Further, numerical methods are d i f f i c u l t to apply for an arbitrary 0(x) i n (2.15). For this reason a somewhat different approach w i l l be developed here which r e s u l t s i n a particular choice for (f)(x) i n (2.15) The approach taken is t h a t we seek to obtain the time variation of the Liapunov function V ( x 1 , „ . . , x n ) , by using the gradient method 1 0 . Thus we obtain V i n a form which results i n grad V being directed along the system traj ectories. 14 Consider the n-dimensional system x = f(x) (2.17) which has a stable singular point at f(0) = 0 (2.18) Then le t us postulate the existence of a continuous, positive definite function V(x) such that V(0) = 0 (2.19) I t then follows-^ that i n order V has a form which results i n the steepest descent of V along any trajectory we require x^ = - K~— (K = Gain factor) (2.20) In the l i t e r a t u r e ^ K is often required to be a constant; however, here we l e t SV_ = - S ( x i , . . . , x n ) X i Further, using (2.21) we can write c W /5x k x k (2.21) 5v75% = ; (2.22) but dx k ^ \ ^ *m (2.23) Hence i t follows that grad V is directed along the trajectories of (2,17) as required. We can now f i n d an expression for the total time derivative of V, Thus, since 15 we f ind using (2.21) i n (2.24) V = - S ( x 2 2 + . . . + x ^ ) (2.25) S x i 2 or n . n j = 1 J = 1 (2.26) which i s a p a r t i a l d i f f e r e n t i a l equation in V. We can write (2.21) as s _ !_ bV X i &xi (2.27) then using (2.27) i n (2.26) we f ind n n bV • 1 dV e>x, x j = _ • i ^ i z _ j = i 3 x • 2 i " F l " j (2.28) or n . n Y x. dV V • • ^ d x i ^ " 5xT Z _ x ' j = l J 1 j = i 1 (2.29) which i s also a p a r t i a l d i f f e r e n t i a l equation i n V. Here i may have any one value from one to n . We can now compare equation (2.25) to Zubov's equation (2.14), and i t w i l l be observed that i f we le t V = - l n ( l - v) (2.30) then equation (2.25) becomes = (S £ k.2)d - v) (2.31) j-1 16 Therefore i n order that (2.15) and (2.31) are equivalent we require n 2, j=l 0 = (S ^ x / ) (2.32) which requires S to be a positive definite function of the state variables (x^, • ••> ^ . From equation (2.21) i t follows (grad V ) T = (Sx x , Sx 2 , S^) (2.33) and evidently we can require curl grad V = 0 which implies that the matrix formed by dxj is symmetrical. This means that i n the two-dimensional case we can write d*2 d x l C 2 ' 3 4 ) For the two-dimensional case we therefore have the following equations to solve av a/ 2 ' ?-v at = " S ( x l + x 7 2 ) (2.35) X l &q 2 &q - S ^ - bx2) (2.36) On the boundary of the s t a b i l i t y domain D we must have V = 0 and therefore S = 0. This occurs because the s t a b i l i t y boundary of the system (2.17) must necessarily be a trajectory of the system. 17 Therefore i t appears that we only need to solve equation (2.36) for S 14 through the use of the characteristic equations dx dx 2 1 dS x i x 2 &2 <&1 5 * L " <*2 (2.37) i n order to determine the s t a b i l i t y boundary. Further, from equation (2 .35) the characteristic equations for V are dx, dx 1 _ 2 - . dV x l x 2 S ( x x 2 -+ x 2 2 ) (2.38) Comparison of the f i r s t equation of (2.37) and (2.38) respectively indicates that their corresponding trajectories are orthogonal, that i s , dx?. ^ C d x l 3 S = " x 2 (2.39) dx 2 x 2 fe") V = ~ (2-40) 1 x x where the subscripts S and V simply indicate which function the equation refers to. In order to solve equation (2.37) numerically we may start with a set of i n i t i a l values: (S = S 0 , xi = 0, x 2 = 0)* means approximately equal to. 18 and integrate the characteristic equations obtained from (2.37) along a characteristic curve. It w i l l be noted that we know only one point on the solution surface of equation (2.36), and we can therefore only trace out one characteristic curve. Characteristic Curve S t a b i l i t y Boundary S = 0 F i g . 2.2 Integration of the Function S There i s , however, an inherent d i f f i c u l t y i n this approach which is best i l lus t ra ted by an example. Consider the system X l " " h + V X 2 (2.41) x „ = - x„ 2 2 then equations (2.39) and (2.40) become dx2\ HxjJ s -x^ + 2xi z X2 i _ Xi x 2 x 2 (2.42) dx 2 » dxi = ~X2 i ^ 2 V 2 " x l -x 1 + 2 x x ^ 2 (2.43) i n the small, Solution of these two equations i n the small gives respectively 19 + x : 2 ) S = c x c, = constant. = c 2 c-, = constant Therefore using analog or d i g i t a l methods to solve equation (2.36) simply results i n the variables x^ and x 2 remaining close to their i n i t i a l values so that the nonlinear terms i n x-^  and x 2 remain insignificant numerically. The i n i t i a l values were perturbed i n various ways i n an attempt to overcome this d i f f i c u l t y , but to no a v a i l . Therefore this approach was abandoned. becomes evident that S is simply an integrating factor for the equation We know that such a function always exists for the two-dimensional cases, and provided certain conditions are met such a function also exists for higher dimensional cases 1 5 - However, even i n the cases when S can be shown to exist we have no guarantee that S is positive def ini te . For example, for the system (2.41) we may assume a series solution of equation (2.36) of the form If we take a closer look at equations (2.35) and (2.36) i t dx 2 3x7 = £ ( x i . * 2 ) n j+1 j-k+1 k-1 k=l x 1 (2.44) 20 however, when we then attempt to determine the coefficients i t soon becomes evident that the form (2.44) does not sat isfy equation (2.36). Consider again equation (2.33) and make the following assumption OV ^x 2 = S x 2 = F2^x2^ But i n general, for the two dimensional case dV = $L dx, + dx 0 d * i 1 d*2 2 = F 1 (x 1 )dx 1 + F 2 (x 2 )dx 2 therefore V = G ^ ) + G 2 (x 2 ) ( 2 < 4 6 ) which means we can not have cross-products in V. This is of course a severe l imi ta t ion ; however, consider the following example, X l = - x l + 2 x i 2 x 2 (2.47) then from (2.45) x 2 * " x2 s = F l ( x l ) _ F 2 ^ which means x l x 2 curl grad V = 0 The total time derivative of S then becomes 21 dS = & i. + I dt 3x7 1 5x7 2 n 2 S x 1 = - 2xn L 1 x„ (2.48) therefore the characteristic equations are dx.. dx 1 2 dS 1 x l the solution of is dx^ dx^ X l = x 2 x 1 = ^ I X2 + C c = constant (2.50) Then using the value for x^ from equation (2.50) and substituting i n -to equation (2.49) we f ind S = S -c O 2 r x L c 2 SQ = i n i t i a l value of S o r substituting for c from equation (2.50) we obtain 22 It should be noted that S = 0 at = 1 i s the exact 9 13 s t a b i l i t y boundary for system (2.47) ' However, unfortunately this approach cannot be used i n general since: 1. the state equations can in general not be solved e x p l i c i t l y . 2. the Liapunov function V is generally =)= G^(x^) + 2.6 Discussion of Results From the foregoing i t is evident that the equation developed i n section 2.5 for the Liapunov function V, via the method of steepest descent is a special case of Zubov's general equation. However, unfortunately we can not solve equation (2.25) for the general case. The reason for this is that a straight forward solution using the method of characteristics i s usually as d i f f i c u l t as solving the nonlinear equation i t s e l f . Further, we also impose an additional res t r i c t ion on V by requiring i t to have a form which results i n the steepest descent of V along any trajectory. 23 3. STABILITY ANALYSIS USING THE COMPLEX FREQUENCY PLANE 3.1 Introduction s t a b i l i t y c r i te r ion for linear systems. The methods used i n this chapter to investigate the s t a b i l i t y of nonlinear systems are somewhat similar to that developed by Nyquist; however, while his c r i te r ion provides both necessary and suff ic ient conditions for : s t a b i l i t y to exist the methods used here w i l l only provide suff ic ient conditions for a given nonlinear system to be stable. Because o f the re la t ively close correspondence between the linear and nonlinear c r i t e r i a Nyquist's c r i te r ion w i l l be described b r i e f l y . Consider the linear feedback system shown i n F i g . 3.1. F i g . 3.1 Linear Feedback System with Constant Parameters Since the system is l inear with constant parameters i t is governed by a d i f f e r e n t i a l equation of the type (2.4) when the coefficients glf •••» g n are constants. Therefore F 0 (s) represents the Laplace transform o f that type, only i n this case we have a driving force V][(t). Then using standard techniques we f ind It was mentioned ear l ier that Nyquist developed a V 2 (s) v ^ I i L v R(s) FQ(s) V 2 (s) = 1 + F 0 (s) (3.1) 2 4 Since we require the system to be stable (in the sense that i t must have f i n i t e output for every f i n i t e input) the expression F(s) = 1 + F 0 (s) can have no zeroes with positive real parts. This can be shown to be equivalent to the requirement that as we plot the frequency locus F(jio) = 1 + FQ(jaj) for - o o < (jj < oo i n the F(ju) plane then for oi increasing N = Z-P where N = number of encirclements of the origin i n the clockwise direction. P = number of poles of 1 + F 0 (s) (or of F 0 (s)) with positive real parts. Z = number of zeroes of 1 + F 0 (s) with positive real parts. But since we require Z = 0 we obtain N = - P Further, since P for 1 + F Q and P for F Q are equal, and N for 1 + F 0 with respect to the or igin i s equal to N for F 0 with respect to the point (-1,0) we do i n general consider N and P for the open loop transfer function F Q . The Nyquist locus i s usually plotted as shown i n F i g . 3.2 where the function F fs) = 4 ° l J s ( l + 0.04s) was chosen a r b i t r a r i l y for i l l u s t r a t i v e purposes. 25 ImF. F i g . 3.2 Nyquist Plot for F 0 (s) = +\.0*s) 3.2 A Frequency S t a b i l i t y Cri ter ion for Nonlinear Systems S a n d b e r g 1 9 , 2 0 established the following result , "Let k e 6 (a,3), l e t <|>(x,t) e 6 0 (a,3) (see (3.8) to (3.11), and consider the vector nonlinear Volterra integral equation t g(t) = f( t ) + ' / k(t - T ) $ ( f , T )dx 1^0 (3.2) 0 where g e. L 2 N ( 0 , ° ° ) and f e £ ^ (see (3.5)). Then f e L 2 N ( 0 , » ) and there exists a positive constant p, which depends only on k, a, and 6 such that 11*11 =< P ||g|| / ' (3-3) This simply means that the inputs to the nonlinearities (j), are bounded by the function g which is due to driving functions and i n i t i a l conditions. On the other hand, i f the driving functions are zero i t follows ||f|| i s bounded entirely by the i n i t i a l conditions. It is the latter condition which is considered i n this thesis, 26 however, the results obtained here also apply to systems with driving functions. Some definitions are now called for . The set L 2 N (0 ,«> ) i s defined by L 2 N ( 0 , » ) = f e ^ ( 0 , - ) , | f ' f d t < - (3.4) 0 where K^(0,<») is defined as the set of r e a l , measurable N-vector-valued functions of the real variable t , defined on ^ O*"0)' Let y e (0^ «) and define f y ( t ) = f( t ) for t e (0,y) = 0 for t > y for any f e K^(0 ,°°) . We then define the set £ N f e ^ (0 , -0, f e L 2 N (0,~) for 0 < y (3.5) With A being an arbitrary, real measurable N by N matrix valued function of t with elements defined on |o,») le t (p = 1,2) denote 0 dt < n,m =1 , . . . , N (3.6) For an arbitrary f e 1 ^ ( 0 , « ) , le t (j) [ f ( t ) , t denote J\(fpt), ( p 2 ( f 2 , t ) , ())n(fN,t) 1 where (|)^(x,t), (|) (x , t ) , (j).n(x,t) are real valued functions of the real variables x and t for x e ( - 0 0 , 00) and t e ^0,~) such that 27 a) 0 n (O,t ) = 0 t E ( n - l , . . . . N) b ) 0n(x,t) (n = 1, . . . , N ) is a measurable function (3.7) of t whenever x(t) i s measurable. Let a and $ denote real numbers with a ^ 6, then we shall say 0 e 80 ( a , 6) i f and only i f a < 0 n ( x , t ) < 3 (n = 1, N) (3.8) ,-1 for t e Jb,-) and a l l x f 0 Let M denote an arbitrary matrix and M* and M " A the complex conjugate and the inverse respectively of M. Further, le t A (M) denote the largest eigenvalue of (M*M) and let 1^ denote the identity matrix of order N. We shall say k is an element of the set 6 (a ,3 ) i f and only i f k e K j ^ and, with K(s) = J e _ s t k(t) dt 0 and det I N + l (a + 6) K(s) f 0 for <r >. 0 (3.9) (3.10) 7 (3 - a) sup X L \ - O O < (jj < 0 0 J N + |(a + 3) K(ja>) -1 K(ja)) < 1 (3.11) It should be noted carefully that the condition k e means that the s t a b i l i t y c r i t e r i o n does not hold for cases where K(s) has poles on the imaginary axis . Conditions (3.10) and (3.11) can be shown to be sa t i s f ied i f a ^ 0 and j^ K(ju)) + K(jto)*j i s nonnegative definite for a l l a ^ -The preceeding two pages are extracted from reference 19. 28 In this thesis we shal l only consider cases where equation (3.2) has only one component; hence N = 1. In that case (3.10) and (3.11) reduce to the simple form 1 + i ( a + 6 ) K(s) / 0 for (f> 0 (3.12) Li j ( 8 - a ) sup - OO < < oo l4(e+cOK(juO l+|(e+o)K(-ja)) _ < 1 (3.13) Condition (3.12) is noticed to be sa t i s f ied i f K(ju>) does not encircle the point (- ^ + > 0) i n the K(ju>) plane. The second condition (3.13) can be investigated i n the following way. F i r s t take the square of both sides of the inequality (3.13) then l ( g - a ) 2 K 1 + (g+a)Re(K) + ^(g+a)2 | K | < 1 (3.14) and or -ct6 | K l 2 < 1 + (8+a)ReK - i _ < (I + 1) ReK + I K | 2 i f "6 > 0 (3.15) (3.16) or i f a6 > 0 (3.17) 29 We w i l l now consider the inequality (3.15) for some separate cases 1. a = 0 From (3.15) i t follows ReK > 6 2. a > 0 Consider F i g . 3.3 below (3.18) ReK F i g . 3.3 C r i t i c a l Circ le (a > 0) The closed contour shown is a c i r c l e C, of centre P(-d, 0) and radius r Q , where (3.19) d = - \ (i • ±D r - 1 r1 \ r 0 - 2 ( „ + g) Then using the law of cosines rQ2 = |K'| 2 + d 2 - 2K»d cos(180 - 0») (3.21) 30 which upon substitution for d and r 0 from (3.19) and (3.20) becomes - L. = | K'l 2 + (-+h ReK' (3.22) Now consider another point on the locus K, and let K form a triangle with r c > r Q and d as shown. Then r Q 2 < r c 2 = | K | 2 + d 2 - 2dK cos (180 - 0) or r Q 2 < |K 12 + d 2 - 2dK cos (180 - 0) (3.23) which upon substitution for r 0 and d becomes (3.24) therefore,in order to sat isfy the inequality (3.16) the frequency locus K(ju) can not touch or intersect the c r i t i c a l c i r c l e C. Now consider equation (3.22) and let us make i t into an inequality by subtracting a small positive quantity e from the l e f t hand side. Then - 2_ - E < | K | 2 + (I + I)ReK (3.25) and i f we consider a point on the negative real axis we have K = ReK, when upon factorization the inequality (3.25) becomes (K + I +<0 ( K + - -eO > 0 (3.26) (this can be v e r i f i e d by carrying out the indicated multiplication) where <f is a small positive number. Let the point considered be to 31 the l e f t of the point - - , then the f i r s t factor in (3.26) is always negative, and for the inequality sign to hold we therefore require K > - - + 6 . This means K l ies inside the c r i t i c a l c i r c l e C, which has already been shown to be inconsistent with the inequality (3.16). Hence we can conclude the locus of K does not cross the negative real axis to the l e f t of the c i r c l e C, and can therefore not encircle i t . 3. a < 0 Consider F ig . 3.4 below a ImK 1 6 1 ReK a F i g . 3.4 C r i t i c a l Circ le (a< 0) In this case (3.27) d = - r ( - • ? ) 32 then using the law of cosines r Q 2 = | K ' | 2 + d 2 _ 2dK* cos 0' (3.29) which upon substitution from equations (3.27) and (3.28) reduces to - h = I K ' I 2 • ( | 4 ) R e K ' ( 3- 3 0 ) Now consider a c i r c l e of radius r c < r 0 , and let a point on the frequency locus K, form a triangle with d and r c . [ K | 2 + d 2 . 2dK cos 0 + d 2 . 2dReK cos 0 (3.31) which upon substitution for r Q and d becomes -h > l K l 2 + £ + F ) R e K C 3 - 3 2 ) and i t - w i l l be noted (3.32) is identical to (3.17) which must be s a t i s f i e d for a < 0. Therefore the locus of K must l i e entirely inside the c r i t i c a l c i r c l e C, i n this case. It should be observed that condition (3.12) is met i f any one of the three conditions just examined is s a t i s f i e d . The foregoing proofs relating to (3.13) are to the author's knowledge new; however, we do of course only prove an established relation i n this case. It has been pointed out by Professor A.R. Bergen, University of C a l i f o r n i a , that these proofs can also be obtained See F i g . 3.4. Then or 33 by using the trigonometry used in deriving "M - c i r c l e s " . Then to recapitulate, the salient features of this s t a b i l i t y c r i t e r i o n , as used here, are the following: Given a free, nonlinear system (that i s f a system without a driving force) with one nonlinearity, where the transfer function K(s) of the l inear part of the system has (simple) poles only i n the l e f t hand plane, the c r i ter ion provides suff ic ient conditions for the system to be globally asymptotically stable, provided the nonlinearity 0, sa t isf ies 0(x, t) where 8 iL a are real numbers. It w i l l be observed, that here we do not view this c r i t e r i o n as a bounded input - bounded output relationship. Geometrically the conditions enumerated above mean the following: Case 1. a = 0 F i g . 3 . 5 Bounds on 0 and K 34 Case 2. a > 0 F i g . 3.7 Bounds on 0 and K 35 It follows that i f a > 0 and 6 = a the c r i t i c a l c i r c l e degenerates into the c r i t i c a l point (equivalent to the - 1 point) which we must not encircle . Now, i n order to apply this c r i t e r i o n , the obvious thing to do i s simply to plot the K(jw) locus, and then to determine the appropriate c r i t i c a l c i r c l e by inspection o£ the K(ju) locus, using this procedure w© would automatically re ject gysterns th© l i n e a r p a r t o£ which has a transfer function K ( s ) , wi th poles on the imaginary axis. However, i n the following section a simple procedure w i l l be described which shows how this s t a b i l i t y c r i ter ion can be applied to some systems with poles on the imaginary axis. [As stated e a r l i e r , those results are applicable to driven systems as w e l l , provided the inputs are square integrable]. 3.3 Systems with Poles on the Imaginary A x i s 4 Consider the following equation L(x) + 0(x,t) = 0 (3.33) and assume that L is a l inear d i f f e r e n t i a l operator such that has a simple pole at the o r i g i n . Then let us rewrite equation (3,33) as follows: L*(x) - ex + 0(x,t) = 0 (3.34) where L'(x) = L(x) + ex e > 0 (3.35) and le t 0' (x,t) = - ex + 0(x,t) (3.36) 36 then combining equations (3.34) and (3.36) we obtain L* (X) + 0 '(x,t) = 0 (3.37) which can now be treated by using the s t a b i l i t y cr i ter ion described in section 3.2. That i s , we plot K(jco) = rr^r^y* and from this plot we graphically determine a and 8. Then 5 0 J x 1 t O _K 6 ( 3 > 3 8 ) a = x or 0(x,t) - ex a _< ——2—- _< X 1 or a + e J fcll _< 6 + e (3.39) Hence, since we had to choose a numerical value for e i n order to plot K(ju)) we must now use this value for e i n (3.39) to determine the s t a b i l i t y sector for the nonlinear function 0(x,t). Consider an equation of the type L x(x) + L 20(x,t) = 0 (3.40) where and are linear d i f f e r e n t i a l operators , and assume ^ | g y has two conjugate poles on the imaginary axis. Then le t us rewrite equation (3.40) as follows, L x ' (x) + L 2 0*(x,t) = 0 (3.41) where ^ ' ( x ) = L x(x) + L 2(ex) 0 '(x,t) = 0(x,t) - ex 37 thus (3.41) is equivalent to (L x + eL 2) (x) + L 2 [0(x,t) - ex] =0 (3.42) or —i<—— (x) + 0(x,t) - ex = 0 (3.43) L2 We can now choose e such that L x (s) L ^ s ) + eL 2(s) has (simple] poles only i n the l e f t hand plane. Hence we are able L 9 ( ju) to plot the locus K(jto) = z , and from this locus we can \ ( j « ) graphically determine a and 6. Then a < tJ&lL < g (3.44) or 0(x,t) - ex „ a _< ^ ^ _< 3 or a + e ^ 0 ( x ? t : ) =< 8 + e (3.45) The value chosen for e above must now be used i n (3.45) i n order to determine the s t a b i l i t y sector for 0(x,t). It is evident that the previous developments apply to 38 driven systems as w e l l . Consider equation (3.33) and let us apply a driving function y ( t ) . Then (3.33) becomes L(x) + 0(x,t) = y(t) but this equation is equivalent to L'(X) + 0 '(x,t) = y(t) which is obtained by applying the driving function y(t) to (3.37). Hence a forced system with a simple pole at the origin can be treated by Sandberg's s t a b i l i t y c r i ter ion as w e l l . Now consider a system of the type (3.40) becomes Lx(x) + L20(x,t) = y(t) However, this equation is equivalent to b-ljh. (x) + 0( x , t) - ex = f ^ -L2 L2 which also can be treated by Sandberg's s t a b i l i t y c r i t e r i o n , provided YSQ- is square integrable. L2 Comparing (3.36) and (3.43) to (3.7) i t is evident that the nonlinearities given by (3.36) and (3.43) sat isfy (3.7). Hence, for any specif ic value of e equations (3.37) and (3.43) are similar to the equation treated by Sandberg. It therefore follows that Sandberg's s t a b i l i t y c r i ter ion can be applied to some types of undriven and driven systems with poles on the imaginary axis. The previous developments are to the author's knowledge new, and have not appeared i n the li terature before. 39 In the section following the s t a b i l i t y cr i ter ion w i l l be applied to three specif ic examples. 1. A free, nonlinear system with a time-varying nonlinearity. 2. A free, nonlinear system with a simple pole at the o r i g i n . 3. A free, nonlinear system with two simple conjugate poles on the imaginary axis. This example w i l l also be investigated by using the Popov s t a b i l i t y c r i te r ion i n order to compare the results obtained by the two methods. 3.4 The S t a b i l i t y of Some Nonlinear Systems The nonlinear d i f f e r e n t i a l equation , i L(x) + 0(t, a. ~ x) = 0 1 d t 1 can be written P(p)x + 0 [t,Q(p)x] = 0 (3.46) (3.47) where P and Q are l inear , d i f f e r e n t i a l operators when L is a linear d i f f e r e n t i a l operator and a^ = constant (note p = ^-) In block d i n the following way iagram form equation (3.47) may be represented 18 r ( t ) 0 R(p)=0 v 3 ( t ) v i ( t ) P(p)x v 2 ( t ) Q(p) x 0[t,Q(p)x] F i g . 3.8 Block Diagram Representation of (3.47) 40 From F i g . 3.8 we have (3.48) and R(P) " V 3(p) = V x(p) (3.49) or since R(p) = 0 equation (3.49) becomes upon substitution for Vjtp) and Vj^p) from F i g . 3.8 P(p)x + 0 [t,Q(p)x] = 0 (3.50) which is the equation we want to investigate. Equation (3.50) can be put into a different form as follows. We have t v 2 ( t ) = j k 1 ( t - T ) v 1 ( i ) di + g 2 (t) (3.51) 0 where k^(t) is the impulse response corresponding to p ^ j " , and g 2 (t) is an i n i t i a l value function. Therefore t v 2 ( t ) = - J [ T , V 2 ( T ) ] dx + g 2 (t) (3.52) 0 since v x ( t ) = - v 3 ( t ) = " 0 [ t ,v 2 ( t ) ] when we can write equation (3.52) as t g 2 (t) = v 2 ( t ) + [ T , V 2 ( T ) ] di (3.53) 0 41 which is identical in form to equation (3.2) with the exception that g 2 (t) ^ s o n l y dependent on the i n i t i a l conditions since we do not have a driving function in Fig . 3.8. Example 1 ' + ? 2 ~*^ X ^2 x 2 = -x 2 (3.54) this can be rewritten as follows x l + x l " 2 x l 2 x 2 0 e _ t = ^ (3.55) where X 2 Q is the i n i t i a l value of x 2 . Then 0 (x , t ) = - Z x ^ x ^ e " * (3.56) P(p) = p + 1 Q(p) = 1 and using equation (3.48) K(s) = ^-j (3.57) o r K ^ = n u (3-58). We can then plot the frequency locus i n the K ( j i o ) plane, and we obtain t h e graph shown i n Fig . 3.9 on the following page. From F i g . 3.9 i - _ •= 1 + y ( y > 0 small) a therefore 4 2 ImK (3.59) If we le t vi = 0 we must remove the equality signs from (3.59) to satisfy (3.13). In that case (3.59) becomes -1 < -2XjX2Qe < co (3.60) 43 or i f we le t x^ = X2ge t in (3.60) we obtain 0 0 < X 1 X 2  K 7 (3.61) which is a good approximation to the actual s t a b i l i t y region, XyX7 = 1 (see section 2.5). Example 2 x + x + 0(x,t) • 0 (3.62) This equation is of the type (3.33) with L(x) = x + x hence 1 = 1 LTsT s(s + 1) has a pole at the or ig in . Therefore rewriting equation (3.62) i n the form given by (3.37) we obtain x + x + ex + 0 (x,t) = 0 (3.63) where 0'(x,t) = 0(x,t) - ex As mentioned i n section 3.3 we must now choose a value for e. In this case le t e = 1 for the sake of s implic i ty . Then P(p) = p 2 + p + 1 Q(p) = 1 and using equation (3.48) 1 or ( i - U) ) + j u 44 (3.64) Hence we can determine the K(jw) locus shown in F i g . 3.10 below F i g . 3.10 K( joj) Locus for (x + x + 0 (x , t ) =0) e = 1 Then using equation (3.39) we require x = « + 1 _< J&p ^ 3 + 1 but from F i g . 3.10 a = -3 = 0 1 - v hence 0(x,t) i A (3.65) where p > 0 i s small. Hence we obtain quite a conservative bound on the nonlinearity i n this example when we assume e = 1. 45 However, we can also use the following procedure. Using e direct ly i n P(p) without specifying i t s value we have K O ) = ^ (e - w ) + ju> (e - OJ ) loi —Tl —7 ' 2 2 2 (e - ID ) +oj ( e - w ) + u) ( 3 . 6 6 ) then c learly , as e becomes large the locus shown in F ig . 3 . 1 0 w i l l shrink to a point at the origin i n the K(joi) plane. Hence, given a specif ic value of e in ( 3 . 6 6 ) i t is clear we can always pick a and g such that the frequency locus K ( j o j ) l ies entirely inside the c i r c l e whose extremities on the ReK axis are given by - \ and - —. Then p ot - a + c ^ PCx , t ) e + £ ( 3 - 6 7 - ) Further as E + » i t follows.from ( 3 . 6 6 ) that we can let a and 8 -> » i n such a way that the K ( j o j ) locus l ies within the c r i t i c a l c i r c l e . Therefore ( 3 . 6 8 ) 0(x,t) where y ^ > 0 is small and y > 0 is a r b i t r a r i l y large, and y = - a + e y 2 = B + e It follows that i n the l imi t as e -*• <=° , ( 3 . 6 8 ) shows that 0(x,t) 46 may l i e anywhere i n the f i r s t and third quadrant. Example 3 (See reference 4 page 86) Consider the system x l = " c x l + x 2 " ^ x l ^ x 2 " x l + x 3 (3.69) x 3 " ' c x l + b { 3 ( x i ^ ' where c > 0, b > 0 are two constant parameters. We can reduce (3.69) to a single equation through elimination of x 2 and x^. Then we obtain. X l + C X 1 + X l + C X 1 + ' b ( 3 ( x l ^ = 0 (3.70) This equation is of the type (3.40) since from (3.70) L l ( s J (s 2 + l ) (s + c) which has two conjugate poles on the imaginary axis s = + j Therefore we can rewrite equation (3.70) i n the form given by (3.41) s and we obtain x± + (c + e)x 1 + x x + (c - be)x 1 + L 2 0 l (x ] ) = 0 (3.71) where L 7(p) = P 2 - b (3.72) 47 0'Cxp = 0 ( x 1 ) - ex1 (3.73) As mentioned in section 3.3 we must now choose a value for e. In this case we assume c = 2 b = 1 e - 1 therefore equation (3.71) becomes x1 + 3xx + x1 + x1 + L 2 0 , ( x 1 ) = 0 (3.74) and i t follows p (p) = p 5 + y + p + 1 p - 1 Q(P) = 1 when upon using equation (3.48) we f ind K M - 3 g 2 ; 1 ( 3 - 7 5 ) and s + 3s + s + 1 2 K(ju>) — ~ —™ifL_—————™— ^ ^ (1 - 3<0 + j (u -u) From equation (3.76) we can now determine the K(jco) locus shown in F i g . 3.11. We now wish to determine bounds on the nonlinearity 0 (x.p. Clearly we have three choices (as we did i n examples 1 and 2 ) t h a t i s , we can choose a = 0, a <0 or a > 0. However, the simplest choice appears to be a. < 0. Then - I = 2.0 + y (u > 0 small) F i g . 3.11 K(jco) Locus for Equation (3.74) 49 This means that the locus l ies within the i n f i n i t e c i rc le which crosses the real axis at the points (2.0+ u , 0). See F i g . 3.11 Therefore equation (3.45) becomes i n this case 1 + 1 < 0 ( x x ) x. < 0 + 1 or 0.50 1.0 (3.78) Hence, i n order the system (3.69) is asymptotically stable 0(x^) must l i e i n the sector shown i n F i g . 3.12 below. 0 Lower bound excluded F i g . 3.12 Stabi l i ty Sector for Equation (3.74) As mentioned above we had three choices i n con-structing the c r i t i c a l c i r c l e , namely ex = 0, a < 0, or a > 0. In this case we choose a <0 (see F i g . 3.11). Further for a < 0 there clearly exists an i n f i n i t e number of c r i t i c a l c i r c l e s , hence s the one chosen i n F i g . 3.11 is chosen merely for the sake of convenience. In practice, when 0(x^) would be known i n analytical or graphical form, i t would be necessary to investigate several 50 c r i t i c a l c ircles and to vary e in order to determine i f a s t a b i l i t y sector could be found containing 0(x^) entirely . From F i g . 3.11 i t is readily seen that the location of the s t a b i l i t y sector i n the (0,x^) plane depends on which c r i t i c a l c i r c l e is chosen. Further, in reference 4 (see pages 8 and 9) i t is indicated that varying e w i l l result i n a rotation of the s t a b i l i t y sector in the (0,x^) plane. On the other hand, i t is clear that the s t a b i l i t y sector can not include the x^ axis i n this case since this would result i n system (3.69) not being asymptotically stable. However, we shal l now investigate (3.69) by use of the Popov method i n order to show that the two s t a b i l i t y c r i t e r i a give approximately the same result . Example 3 by Popov's Method (See Appendix A) 1 C X , + X •2 " 0 C x l ) X 2 X , + X 3 (3.79) x, 3 cx^ + b0(x^) which can be rewritten as (p 3 + cp + p + c)x, + (p 2 - b)0 (xj = 0 (3.80) Then (3.81) o r 51 when (1 - co )(c + j u ) (3.83) It w i l l be noted that we have a part icular case in that W(p) has poles on the imaginary axis P a 1 3 -To investigate equation (3.80) for s t a b i l i t y - i n - t h e -l imit we must determine d Q + j e Q = lim (p - j) W(p) P + J 2 b Now, le t 1 ( i + j c ) (3.84) 1 + c c = 2 b = 1 as before; then , 1 ( 1 + bs n  d o = 2 ( ~ $ > 0 and the condition for s t a b i l i t y - i n - t h e - l i m i t is s a t i s f i e d . See Appendix A. Using equation (3.83) we can separate W ( j c o ) into i t s real and imaginary parts, and Rew . - <"V»2> • (1 - a )(4 + O then I m W = Cu 2 + B O O (1 - a) )(4 + u ) X = ReW Y = u ImW Y Y _ (to2 + 1) (2 + qto2) A q Y j T (o> - 1)(4 + to ) (g>2 + D(2) 52 4 + u,2 C 3 - 8 5 ^ for q= -2 Hence equation (3.85) states that X - qY ^ 0 therefore X - qY + ^ > 0 (3.86) the "Popov Condition" is sa t i s f ied for ^ = 2 except at the point a) = 0 0 (see F i g , 3.13), however, as mentioned i n reference 4 this is permissible since a s l ight change i n q w i l l sat isfy (3.86) for 0 < to < oo. i t therefore follows system (3.79) is asymptotically stable when the nonlinearity 0(x^) is contained i n the sector shown i n Fig„ 3.14. We can now compare the results obtained by the Sandberg method and the Popov method. For this purpose we compute 54 Fig . 3.14 S t a b i l i t y Sector for Equation (3.80) (u > 0 small) the angular magnitude of each sector i n F i g . 3.12 and 3.14. Hence 6g = t a n " 1 1.0 - tan" 1 0.50 = 1 8 . 9 ° e = tan" 1 0.50 = 2 6 . 6 ° P and i t i s seen that e > e c P s when e = 1. This i s to be expected since a rotation of any sector away from the x^-axis causes this sector to decrease i n angular magnitude^ The values of 6 and 0 found above i l lustrates this s p fact . i 55 No attempt is made here to determine the optimum value of e or the optimum c r i t i c a l c i r c l e , since the primary purpose here is merely to show how we can apply Sandberg's method to some systems whose l i n e a r part has two conjugate poles on the imaginary axis. 3=5 Discussion of Results I t w i l l be noted that we have only treated systems for which Sandberg's vector equation has only one? component i n this chapter. However, i t is evident that a large class of systems belongs to this group. Further, since this chapter is primarily concerned with undriven systems we have used the Sandberg cr i ter ion to investigate such systems for asymptotic s t a b i l i t y . That i s , we have not used the c r i ter ion as an input - output s t a b i l i t y c r i te r ion . From the solution of examples 2 and 3 i t is clear that i t is possible to apply Sandberg's c r i ter ion to some systems with poles on the imaginary axis. Two separate cases are considered. 1. One simple pole at the or igin 2. Two simple, purely complex, conjugate poles. Further, i t i s shown that this development applies to driven systems as w e l l , provided the system inputs are square integrable. AlsOj, Example 1 shows the application of Sandberg's c r i te r ion to an equation i n which the nonlinearity includes time e x p l i c i t l y . It i s evident from the description given of the c r i ter ion and the solution of this example that Sandberg allows / 56 the nonlinearity to have a certain e x p l i c i t , functional dependence on time. Example 3 i l lust ra tes the solution of a system with a nonlinearity which does not depend e x p l i c i t l y on time. This example has been solved by both the Popov and the Sandberg c r i te r ion . From the results obtained i t is evident that both methods i n this case give approximately the same answer. It w i l l be recalled that the Popov criterion! does not apply to systems having nonlinearities containing time e x p l i c i t l y . 57 4. STABILITY ANALYSIS USING THE TIME DOMAIN 4.1 Introduction In this chapter we shal l consider some s p e c i f i c , non-linear systems with arbitrary inputs, and investigate these driven systems for bounded input - bounded output s t a b i l i t y . In order to define our terms consider the following system: LCXj) + 0 (x 1 ) = y(t) ( 4 > 1 ) where L is a l inear , time-invariant, d i f f e r e n t i a l operator, e is a constant, 0(x^) is a nonlinear function of x^, and y(t) is a forcing function. System (4.1) may be represented i n block diagram form as shown i n F i g . 4.1 below, (p = ^ ) yCt) +, e 0 (xx) o 1 x-, (t) L(p) £ 0 F i g . 4.1 Block Diagram Representation of (4.1) It w i l l be observed that F i g . 4.1 is identical to F i g . 3 . 8 , however, while we i n chapter 3 employed the complex frequency plane for s t a b i l i t y analysis we shal l i n this chapter use the time domain instead. Further, i t should be noted that while the previous two chapters dealt mainly with undriven systems, we shal l here consider driven systems, as already mentioned above. 58 One approach taken is the following (Refer to F i g . 4.1). We determine the output x^(t) as a functional of the input y ( t ) , that i s , we f ind x x (t) = F[y(t)] (4.2) In general F[y( t ) ] : i s a Volterra series i n y ( t ) , however, we determine a power series which dominates the Volterra series term by term. We then f ind the region of convergence of the power series which depends on the input y ( t ) . Thus we obtain a bound B, on the input y(t) and consequently a bound A, on the output x^(t). Thus for y(t) <B we have x x (t) < A Hence, we have ensured that system (4.1) exhibits bounded input - bounded output s t a b i l i t y i n a certain region. A second method, which is used here, is to consider the nonlinear system (4.1) as a perturbed version of the following linear system L(x) = y(t) where x^ = x + y Using this procedure we obtain a recursion relation v = A(y ,y) where we can show A to be a contraction operator. We thus determine a region where (4.1) gives a unique bounded output for each bounded input. That i s , we f ind a region where (4.1) exhibits bounded input - bounded output s t a b i l i t y . 59 4 . 2 S tabi l i ty via Volterra Series 23 Barrett developed an interesting i terative method for determining the input - output s t a b i l i t y of the system L(x) + ex3 = y(t) which is of the form (4.1). His procedure w i l l be described in detai l i n the following paragraphs. Consider equation (4.3) where L(p) is a l inear , time-invariant, d i f f e r e n t i a l operator with simple zeroes only i n the l e f t hand plane, and y(t) is an arbitrary driving function, and let us assume the i n i t i a l conditions x(0) = 0 y(O) = 0 (4.4) We can then form the nonlinear integral equation 3 , x(t) + e f h(t - x ) x 3 ( i)dT = f h(t - T)y ( x)dT — on-* —ca-' (4.5) where h(t) is the impulse response corresponding to the linear transfer function T \ \ , It should be noted that we can use the l imits (-°°,°°) L(pJ since x = y = 0 for t< 0 and h(t - T) = 0 for T > t . Equation (4.5) can clearly be considered from the point of view that we endeavour to 3 determine the output x ( t ) , by convolving the input y(t) - ex , with the impulse response of the system h( t ) . This interpretation agrees with F i g . 4.1. ^ We shal l now solve equation (4.5) by i terat ion. For this purpose let us assume as a f i r s t approximation oo x(t) = I h(t • T)y(t) di (4.6) and substitute this into equation (4.5). Then we obtain 60 CO x(t) = h(t - T)y(x)dT - E h(t - x) h(x - T 1)y(T 1)dt 1 (4.7) which upon expansion becomes the second approximation to the true solution of equation (4.5). The last term in (4.7) can be written in the following way oo 3 £ h(t - x) h(t - T 1)y(T 1)dx 1 dx h(t - x) h(x - x. 1)y(x 1)dx 1 |h(x - x 2 )y(x 2 )dx 2 h(x - x 3 )y(x 3 )dx 3 dx Let us now define h 3 ( t - x 1 } t - x 2 , t - x 3) = h(t - x)h(x - x^hfx - x2) h(x - x3) dx then (408) becomes e h(t - x) h(x - T1)y(T1)dr1 dx = (4.8) (4.9) 61 h 3 ( t - x 1, t - T2, t - T 3 ) y ( T 1 ) y ( T 2 ) y ( T 3 ) d T 1 d x 2 d T 3 (4.10) and substituting (4.10) into (4.7) we obtain - co -oo -oo 00 00 00 x(t) = h(t - x ) y ( x ) d x - e / I h 3 y ( T 1 ) y ( T 2 ) y ( T 3 ) d T 1 d T 2 d T 3 (4.11) If we desire higher order approximations we must then substitute equation (4.11) into (4.5). We w i l l then obtain the solution of equation (4.3) i n the form of a Volterra series as follows. x(t) = |h(t - x)y(x)dx - e j j j h ^ t x ^ y ( x ^ y f x ^ d x ^ x ^ + + (4.12) • • • • • • At this point we shal l digress to consider the v a l i d i t y of this series representation. That i s , when does the output x ( t ) , obtained i n this manner represent the solution of the original d i f f e r e n t i a l equation (4.3). When we consider linear forced equations we are always ensured of the existence of a unique steady state solution, however, in general nonlinear d i f f e r e n t i a l equations do not exhibit this characteristic . Here several steady state solutions may obtain, however , some o f these may not exist due to i n s t a b i l i t y . Volterra introduced the functional representation used 22 i n (4.12) and established the concept of an analytic functional . He defined a functional F [y(t,a)] to be analytic i f F[y(t,a)] . is an analytic function of the parameter a, when y(t,a) is an analytic function of t and the parameter a. 62 31 Parente defined an analytic system as a time-invariant, deterministic system whose functional is analytic about zero input at some time t . The reader is referred to reference 22, pages 21 to 26 31 for an exposition of this def ini t ion . Parente also investigated the v a l i d i t y of the Volterra series representation for such systems and found i t to be ,val id provided the following four conditions are s a t i s f i e d . 1. the solution of the given d i f f e r e n t i a l equation exists . 2. the solution of the given d i f f e r e n t i a l equation is unique. 3. the d i f f e r e n t i a l equation must be time-invariant i n the sense that the systems inputs and outputs co-translate i n time. 4. the system functional must be analytic about zero input at some time t , or equivalently the series converges absolutely. We shal l now proceed to estimate the output x(t) from (4.12) for a given input y ( t ) . Thus taking the absolute value of both sides of (4.12) we find 00 f 3 sup |x(t)| ^ sup |y(t)| / |h(t)| dt + |e| sup |y(t>,| 0 + . . . + . . . (4.13) From equation (4.13) i t i s seen that each term i n the Volterra series solution (4.12) is bounded by the corresponding term in the ordinary power series X = 0(Y) - HY + |e| H V + ... (4.14) where sup |y(t)| < Y (4.15) 63 H = / |h(t)| dt 0 and consequently sup |x(t)| < x (4.16) If we consider the algebraic equation .3 (4.17) X - |e| HX J = HY (4.18) and attempt to solve i t by i tera t ion , i t becomes clear that 0(Y) i n (4.14) is a series solution of (4.18). Equation (4.18) may be graphed as shown i n F i g . 4.2 below. In order to determine the values X 2 and Y 2 (see F i g . 4.2)we f i n d , using equation (4.18), Y D(X~, Y ) \ 0 \ 1 / X C ( - X 2 , -Y 2) F ig . 4.2 Graph of Equation (4.18) dY dX = 0 1 - |e| 3HX^  or 64 written X 9 = j— (4.19) 2 (3|e|H)* Y 2 = l e l ^ S H ) 5 / 2 C4.20) For the range |Y| <Y2 the solution o£ (4.18) can be 23 X = 3Y2H s i n ( j s in " 1 y-) (4.21) = 000 or writing this as X = 3Y2H s in U (4.22) we have IT - 1 • -1 Y I s i n y - ( 4 > 2 3 ) We can now expand (4.22) and (4.23) i n power series as f o l l o w s 1 4 , 2 3 . U 3 X = 3Y2H(U - 3- + . . .) (4.24) U = TCTJ  + l ^  + (4- 25) and i t w i l l be observed that we obtain the series 0(Y) i n (4.14) i f we eliminate U from (4.24) and (4.25). Further, (4.24) is convergent for a l l U 1 4 9 and (4.25) i s convergent for Y < Y 2 > therefore (4.24) and thus (4„14) are convergent for Y < Y 2 . Hence we can state the following 23 theorem. Theorem 1. Given the nonlinear system 65 L(x) + ex 3 = y(t) (4.26) where L(p) is a l inear , time-invariant d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane and x = y = 0 for t ^ O then for sup |y(t)| <Y2 we have sup |x(t)| < X = 0(Y) where Y 2 and 0(Y) are given by (4.20) and (4.14) respectively. This theorem states that the system (4.26) exhibits bounded input - bounded output s t a b i l i t y within a certain range for an arbitrary driving function y ( t ) . It w i l l be observed that: we have merely found a bound on the output x ( t ) , i n terms of a bound on the input y ( t ) . That i s , we have shown that the Volterra series (4.12) is convergent; and this sat isf ies condition4 on page 63. F i g . 4.2 is a graph of Y > sup |y(t) | versus X > sup |x(t)|. This graph clearly shows that we have a unique output x( t ) , for a given input y ( t ) , only along the branch COD. When we include points outside this branch we do not have a unique output for a given input i n that to each value of y(t) there may correspond three values of x( t ) . However, the problems of existence, uniqueness and time-invariancy w i l l be treated i n detai l i n section 4.11 and w i l l not be considered further here. 4.3 The S t a b i l i t y of Two Specific Nonlinear Systems Consider F i g . 4.3 shown below. It w i l l be observed that F i g . 4.3 is identical to F i g . 4.2, however, the purpose here is to indicate where the maximum values of X (X^, - X^) are located for |Y| <Y 2 . Form Fig . 4.3 these are at points E and F. Thus |X| < X b for |Y| <Y 2 . 66 Y \ F ( - X b , Y 2 ) D(X 2 ,Y 2 ) l \ I \ 1 \ 1 \ i \ i \ 0 1 X C ( - X 2 , - Y 2 ) W b , - Y 2 J F ig . 4.3 Boundary Values of X In order to i l l u s t r a t e the u t i l i t y of the theory presented i n section 4.2 we w i l l solve two examples. Example 1 x l + x l + x l " x l 3 = y ( t ) ( 4 , 2 7 ) where x(0) = y(0) = 0 Comparison of (4.27) to equation (4.3) shows that with t = + 1 L(p) = p 2 + p + 1 then we f ind (4.28) (4.29) = h(t) (note X = j) 67 and i f we le t a = - X we have b = (1 - \ 2 ) h |h(t)| = i e a t |sin bt| However, we wish to evaluate (4.30) (4.31) H = |h(t)| dt 0 (4.32) and i t w i l l be noted that the graph of the function h(t) has the appearance shown i n F i g . 4.4 |h(t) F ig . 4.4 |h(t)| versus Time From F i g . 4.4 i t is seen that ( 2 k - l ) £ |h(t)| dt = £ 0 k=0 e a t s i n bt dt 2k£ 1 b" k=0 (2k+l), e a t s in bt dt (4.33) 68 but ( 2 > 1 ) £ F at e s in bt dt = — * — 7 ( a s i n D t - bcosbt) a +b 2k; (2k+l) (2k) be a ( 2 k ) £ a 2 + b F aiT/b 7 (1 + e ) (4.34) and (2k+2)| at e s in bt dt = "2— (1 + e b J (2k+l), a + b' (4.35) hence, i f we substitute equations (4.34) and (4.35) into (4.33) we obtain t °° 2ak?-i O / K \ a ( 2 k + l ) £ h(t) I dt - - ^ ( 1 + e ^ / b ) ( 2^  e + e b) ( 4 > 3 f i ) '0 k=0 1 11 l k F . if » a* k=0 It w i l l be recalled the p a r t i a l sum of a geometric series is 1 - r n + 1 S n - V T V - (4.38) and from equation (4.37) air/b 69 but from (4.30) a < 0, therefore r <1 and from (4.38) we find lim sn = j - ! - ( 4 < 3 g ) n -»• 0 0 We can therefore write equation (4.37) as follows air |h(t)| dt - 4 — r 1 * ° aZ+bZ g l (4.40) 0 1 - e It is perhaps best to point out that a much simpler method is 23 available .for • determining H i n this particular case , however, the result obtained here w i l l be used later on i n the particular form (4.40) Numerical evaluation of (4.40) then shows a + b =1 •rra e 5 " = .164 whence H = J |h(t) | dt 0 - 1 + - 1 6 4 1 - .164 = 1.39 (4.41) Then using (4.20) we f ind Y = 2  2 cx^\ n ic\\^-(3 2)(1.39)' = .234 (4.42) 70 We can now determine from equations (4.18), (4.41), and (4.42). Hence 3 X b - 1.39Xb = (1.39)(234) Solution of this equation by t r i a l and error yields X b = .983 which can be compared to X b for (4.27) when i t is undriven 13 x b - 1.0 (4.43) (4.44) (4.45) It is evident that here we can also speak of a s t a b i l i t y sector, although the lat ter is f i n i t e i n this case. See F i g . 4.5 below 0(x) = -x-5 \ " X b Fig . 4.5 S t a b i l i t y Sector for Equation (4.27) Example 2 (Rayleigh's Equation i n reversed time) 3 x, = x 9 + u (- x,) (4.46) x 2 " X l 71 We can rewrite (4.46) i n the following way x 3 • • • • T x i = x 2 " ^ I + w — - at3 or x 3 x1 + y x 1 + x1 - yp - j - = 0 (4.47) which is Rayleigh's equation. This we can i n turn write as 2 x 3 P + ypP + 1 ( x p - y - f - 0 (4.48) and i f we now apply a driving force y ( t ) , we obtain X l 3 L(x x) - p - f = yCt) (4.49) Comparing (4.49) to (4.3) we f ind L(p) = P 2 y P * 1 (4.50) ' " + % We again assume the i n i t i a l conditions x(0) = y(0) = 0 Then X _ 1 ( n T r ) = — ^ ~ T T s in [(1 - X 2 ) t + 6] (4.51) U s J (1 - \ T = h(t) (note X = j) where e - t an" 1 ( 1 " ^ e tan — ( 4 > 5 2 ) If we le t 2 J. b = (1 - x z ) : then and a = -X e a t h(t) = Sg- [sinCbt + 8 ) ] H = j Z r — | s in(bt +e)| dt 0 To evaluate this integral le t bz = bt + 8 then dz = dt and equation (4.55) becomes 8 H = jL e ^ ( e a z |sin bz|dz - J e a z |sinbz|dz) 0 0 However, from equations (4.31), (4.32), and (4.40) we f ind e f -af- a F 1 ^ D / az i . , i , e 1 + e b~ / e l s i n b z l d z = T T ? aT £ a + b 1 - e F Therefore we need only evaluate the integral e/b -ae 1 . T I = e u / e a z I s in bz| dz 73 then To deteremine I e x p l i c i t l y assume v = 0.7 = t a n _yJ Therefore = 1.93 radians = 2.06 radians and B" < TT It therefore follows that we can remove the absolute value sign i n the integral (4.58) and determine I d i rec t ly . Then - a e or e a z s i n bz dz = ^ ^—j (a s in bz - b cos bz) a +b 1 e •ae 1> 0 e E" 0 (4.59) I = a +b (4.60) We can now substitute (4.57) and (4.60) into (4.56), whence o 9 ^ H = e C e „ „ - 1) 1 - e air (4.61) Evaluating this expression numerically there results 74 e = .308 = 1.83 Then using equation (4.20) we obtain Y - 2 ...  Y2 " 37T (4.62) ( 3 ) C 5 ' 4 9 ) = .322 (4.63) We can now determine from equation (4.18) by using (4.62) and (4.63) Thus X b - (0.7) ( i ^ 5 - ) , X b 3 = (1.83) (.322) Solution of (4.92) by t r i a l and error yields X b = 1.77 (4.64) (4.65) In this case the f i n i t e s t a b i l i t y sector has the appearance shown i n F i g . 416 X. - xb ^ ^ ^ ^ ^ X F i g . 4.6 S t a b i l i t y Sector for Equation (4.49) 75 4.4 The S t a b i l i t y of a System with a Nonlinearity of Second Degree In this section Barrett's development (see section 4.2) w i l l be extended to cover systems of the type L(x) + ex 2 = y(t) (4.66) where L(p) i s a l i n e a r , time-invariant, d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane, e i s a constant, and y(t) i s an arbitrary driving function. Inspection of Fig. 4.1 shows that (4.66) can be represented i n block diagram form as w e l l . Let us assume the i n i t i a l conditions x(0) = y(0) = 0 (4.67) We can then form the following nonlinear i n t e g r a l equation from (4.66) x(t) = / h(t - T ) y ( i ) d i - e / h(t - T ) X 2 ( T ) dx (4.68) I t i s clear that (4.68) can be solved by i t e r a t i o n as was equation (4.5). In this case we obtain the Volterra series OO 00 00 x(t) = / h(t - T ) y ( i ) d T -e / / h 2 y ( T 1 ) y ( T 2 ) d T 1 d x 2 •OO *>00 — 00 + ... (4.69) where h 2 ( t - i 1 P t - T 2 ) = J h ( t - T ) h ( T - T 1 ) h ( t - T 2 ) d T (4.70) -00 We can then take the absolute value of both sides of (4.69) whence 76 sup |x(t)| < sup |y(t)| / |h(t)|dt + |e| sup |y(t)| 2 ( J |h(t)|dt) 3 0 0 (4.71) We then observe that each term i n (4.71) is dominated by the corresponding term i n the power series X = 0(Y) = HY + |e| H 3 Y 2 + . . . (4.72) where sup |y(t)| < Y (4.73) H = J |h(t)| dt (4.74) 0 and thus sup |x(t)| < X (4.75) It w i l l now be shown that (4.72) is a series solution of the algebraic equation X - |e| HX2 = HY (4.76) Let X = HY (1st Approximation) then X = HY + | E| HV (2nd Approximation) and X = HY+|e|H3Y2 +2|e| 2HV + | E | 3 H 7 Y 4 + . . . (4.77) and i t is seen that (4.72) is indeed a series solution of (4.76). 77 Consider the graph of equation (4.76) shown i n Fig . 4.7 below. It w i l l be noted that the points A and B are F i g . 4.7 Graph of Equation (4.76) easily determined to correspond to 1 „ 1 A ( X 2 2 lei H Y2 = H ) (4.78) B = B(X = 1 e T H , Y = 0) We must now inquire into the region of convergence of the series (4.72). For this purpose we shal l employ the ratio test. Then th „ | n term (n-1) st term , n - l ^ 2 H2n-1 Y n j n H2n-3 Y n-1 j n-1 < 1 (4.79) where I _^ and I are the integer coefficients of the (n-l)st and the nth term respectively. Let us now assume 1 Y 2 - + - 4 lei H (4.80) (see F ig . 4.7) then (4.107) becomes 78 nth term| 1 I n (n-l)st term| = J • •1 (4.81) We can now investigate the respective values of I n and I n _ ^ I f , in (4 .77)we disregard the factors in |e| and H ( i t w i l l be noted that these factors cancel in (4.79) when (4.80) is substituted into (4.79))then the series (4.77) can be written, X* - ajY + a 2Y 2 + a 3Y 3 + a 4Y 4 + a gY 5 + ... But a l = 1 a 2 = (ax) a 3 = 2 a l a 2 a 4 = ( a 2 ) 2 + 2 a i a 3 a 5 - 2a x a 4 + 2a 2 a 3 etc. and i f we determine the f i r s t few coefficients i t soon becomes evident that n < 1 n - l It therefore follows that the series (4.77) is convergent for 4 |e| H We can then state the following theorem: 79 Theorem 2. Given the nonlinear system L(x) + ex2 - y(t) ' (4.82) where L(p) is a l inear , time-invariant d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane and x = y = 0 for t_< 0 f t then for sup |y(t)| < Y 2 we have sup |x(t)| < X = 0(Y) where Y 2 and 0(Y) are given by (4.80) and (4.72) respectively. This theorem states that the system (4.82) exhibits bounded input - bounded output s t a b i l i t y within a certain range for an arbitrary driving function y ( t ) . In order to test the appl icabi l i ty of the foregoing consider the following system. Example 1. x + x + x - ex2 = y(t) (4.83) From equation (4.41) we have H = 1.39 Consider F i g . 4.8 below. The values of X 9 and Y 7 at point A F i g . 4.8 Boundary Values of X (Equation (4,76)) 80 are given by (4.78). Further, the coordinates of points D and C are given by ( X b 2 , ^ b l » " r e s P e c t i - v e l v ' We desire to determine and X b 2 , the bounds on X within the region of convergence of (4.77). Therefore from (4.80) with e = 1 and substituting for H and Y i n (4.76) we f ind X , = .867 b l X b 2 = - .149 In this case the s t a b i l i t y region for the driven equation (4.79) has the form shown i n F i g . 4.9 below 0(x)=-x* Xb2 X b l 1 X Fig . 4.9 S t a b i l i t y Region for Equation (4.83) 4.5 The S t a b i l i t y of a System with a Nonlinearity of Fourth Degree 81 In this section Barrett 's development (see section 4.2) w i l l be extended to cover systems of the type L(x) + ex 4 = y(t) (4.84) where L(p) is a l inear , time-invariant, d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane, e is a constant, and y(t) is an arbitrary driving function. Inspection of F i g . 4.1 indicates that (4.84) can be represented i n a similar way. Let us assume the i n i t i a l conditions x(0) = y(0) = 0 (4.85) then we can form the following nonlinear integral equation from (4.84) x(t) = | h(t - T)y(x)dx - e I h( t • x)x 4(x)dx (4.86) It is evident that (4.86) can be solved by i terat ion as was equation (4.5). In this case we obtain the Volterra series OO CO o o OO 0 0 x(t) = / h(t - x)y(x)dx / / / / h 4 y ( T l ) y ( T 2 ) — o o — CO — OO — CO y ( T 3 ) y ( x 4 ) d x 1 dx 2 dx 3 dx 4 + ... (4.87) where h 4 ( t - x p t - x 2, t - x 3, t - x 4) = h(t - x)h(x - x x)h(x - x 2)h(x - x 3)h(x - x 4)dx (4.88) 82 We can then take the absolute value of both sides of (4.87), whence 0 0 sup |x(t)| < sup |y(t)| / |h(t)|dt +|e|sup |y(t)| 4 ( / h(t) | dt) 5 +. . . 0 0 ( 4 ' 8 9 ) We then observe that each term i n (4.89) is dominated by the corresponding term i n the power series X = 0(Y) = HY + | E| H V + . . . (4.90) where sup |y(t)| < Y oo H = j |h(t)| dt (4.91) 0 and thus sup |x(t)| < X (4.92) It will*now be shown that (4.90) is a series solution of the algebraic equation X - |e| HX4 = HY (4.93) Let X = HY (1st Approximation) X = HY + | e | H 5 Y 4 (2nd Approximation) and X = HY + |e i H 5 Y 4 +4 |e| 2 H 9 Y 7 +22|e|3 H 1 3 Y 1 0 + . . . (4.94) and i t w i l l be observed that (4.90) is indeed a series solution of (4.93) as was to be shown. Consider the graph of equation (4.93) shown i n F i g . 4.10 below. It w i l l be noted the points A and B can be 83 Y 3 1 1 I J FT(4|e|H)^ / ' \ B 4|e|H 3 \ X F i g . 4.10 Graph of Equation (4.93) determined from equation (4.93) as follows. ^ = 1 - 4|e|HX3 = 0 or when X2 " C4 |e| H ) 3 3 1 , 1 , 1 Y2 " 4 H ( r f n (4.95) (4.96) where X 2 and Y 2 are the coordinates of point A as shown i n F i g . 4.10, Further, for Y = 0 we have 0 (4.97) 84 therefore B = B(X = (Trrrr)3 . Y - o) (4.98) We must now determine the region of convergence of the series (4.94). For this purpose we shal l employ the ratio test. Then nth term (n-l)st term[ T n-1 u4n-3 v3n-2 n E H I T n-2 u4n-7 v 3n V i e H Y < 1 (4.99) where In_^ and I are the integer coefficients of the (n-l)st and the nth term respectively. Let us now assume Y = Y„ = 3 2 " H ( 4 | e | H ) 1 / 3 (4.100) (see F i g . 4.10) then (4.99) becomes nth term | , 27-, " (256"J (n-l)st term| n n-1 < 1 (4.101) At this point we must investigate the respective values of I and I N _ T ' I f» i n (4.94), we disregard the factors i n |e| and H ( i t w i l l be noted that these factors cancel in (4.99) when (4.100) is substituted into (4.99))then the series (4.94) can be written ' 4 7 10 X = a xY + a / + + a 1 Q Y i U + . . . The f i r s t few coefficients are 85 a 7 = 4 a10 " 22 a13 = 140 a16 = 060 a19 = 7084 etc. and i t is evident 27 25o" n n - l < 1 (4.102) It therefore follows that the series (4.94) is convergent for 3 4 H ( 4 | E | H ) I7T (4.103) We can then state the following theorem: Theorem 3. Given the nonlinear system 4 L(x) + ex = y(t) (4.104) where L(p) is a l inear , time-invariant d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane and x = y = 0 for t j= 0 then for sup |y(t) | < we have sup |x(t) | <X = 0(Y) where Y 2 and 0(Y) are given by (4.96) and (4.90) respectively. This theorem states that the system (4.104) exhibits bounded input - bounded output s t a b i l i t y within a certain range for a n arbitrary driving function y ( t ) . In order to test the theory developed in this section theorem 3 w i l l be applied to the following system. 86 Example 1 x + x + x - ex 4 = y(t) (4.105) From equation (4.41) we have H = 1.39 (4.106) Now consider F ig . 4.11 below. The values of X 2 and Y 2 are given by (4.95) and (4.96) respectively. Further, Y A(x 2 ,Y 2 ) Xb2 \ X b l 1 / 0 \ 1 / \ 1 X Y C X b p - Y 2 ) Fig . 4.11 Boundary Values on X (Equation 4.93) the coordinates of points D and C are given by ( X b 2 , - Y 2 ) and ^ X b l ' " ^23 r e s P e c t i v e l y - We desire to determine X ^ and X^ 2 the bounds on X within the region of convergence of (4.94). Hence, from (4.96) with e = 1 3 Y - 7 2 ~ H ( 4 H ) 1 / 3 ( 4 ' 1 0 7 ) and substituting (4.106) into (4.93) we f ind X b l = .590 87 = -.390 (4.108) We can now represent the s t a b i l i t y region i n graphical form as shown i n Fig . 4.12 below F i g . 4.12 S t a b i l i t y Region for Equation (4.105) 4.6 Comparison to Desoer's Method 30 Desoer developed a s t a b i l i t y theorem which consists of the following. He associated to the given nonlinear system S , , a comparison system S, and showed that i f S is stable so is 0(x)=-x 4 x Consider the following nonlinear system t x(t) = z(t) + (4.107) 0 where z(t) is the zero-input response such that z(t) | ^ Z for t ^ 0 and 0 J |0(u,t) | ^ k, u for a l l u and t ^ 0 (*1 > 0) 88 This system Desoer compared to a linear system with impulse response |h(t)| such that 00 J |h(t) | dt = k - z < 0 0 (k = constant) 0 and showed that provided these relations are s a t i s f i e d then the following is true [x(t)| =< k (|y(t)| + Z)) (4.108) Clearly the method used i n the previous sections are of a similar nature. There we found a power series which dominates i t s corresponding Volterra series term by term and then found the region of convergence of that power series . Thus we did indeed determine the s t a b i l i t y of a given system by comparing i t to a system known to be stable. However, i t w i l l be noted that Desoer can treat systems with time-varying non-l inear i t ies which Barrett can not. Further, as has been pointed out before, then Barrett has not shown that his method actually is v a l i d . The close relat ion between Barrett 's method and Desoer's method was drawn to the author's attention by Dr. E .V. Bohn. 24 27 28 4.7 The Contraction Mapping Principle ' ' ' In this section a mathematical i terat ion procedure w i l l be described. This procedure is cal led a contraction mapping because the distance between the images of any two points i n the region considered is always smaller than the distance between the points themselves. In the section following the contraction 89 mapping principle w i l l be used to determine bounded input -bounded output s t a b i l i t y of nonlinear systems. Further, the results thus obtained w i l l be compared to those obtained by Barrett 's method described i n section 4.2. But f i r s t some definitions are called for . Definit ion 1: A real or complex vector space E is called a normed space i f there is given a norm j|x|| for every x i n 24 E such that a) ||x|| ^ 0 b) ||x + y|| < ||x| + ||y|| c) ||xx|| = |x| . ||x|| for every scalar X d) ||x|| = 0 only i f x = 0 Defini t ion 2: A normed vector space E is said to be complete i £ 24,28 II x - x l l 0 11 n m" n,m -+ °° This means that there is an x* i n E such that x -> x* as n + » n that is l|x n - x*|| - 0 n -+ oo Then referring to Definitions 1 and 2, a complete normed vector space is called a Banach space. Thus the space of a l l r e a l , continuous functions of the interval - » <t < » is a Banach space, and i ts norm is always defined as 90 ||x(t) I = sup |x(t) | (4.109) - oo < t < 0 0 Let S be a linear operator i n a Banach space E which maps points of E into i t s e l f . Further, let the impulse response of 6 be h( t ) , then i f the input to Q is Y^(t) the output YQ(t) is found from YQ(t) = h(t - T ) Y . ( t ) d T If we now take absolute values on both sides we f ind 0 0 ||Y0II < ||Y.|| J |h(t)| dt 0 =< ||Vi| h i Therefore the noim of the linear operator g, is 0 0 II31| = J |h(t)| dt (4.109a) 0 since for physical systems we can assume h(t) = 0 for t =< 0. Further, i f 0 0 He l l = j |h(t)| dt < » o i f f o l l o w s 2 4 that 6 is a bounded operator and is therefore 24 continuous . This means $ maps points of E into E i n a cont inuous manne r . 91 Definit ion 3: Let the operator A map the Banach space E into i t s e l f , then this mapping is called a contraction i f for every 24 and %2 i n E we have ||A(x1) - A(x2)|| < K ||x]_ - x2|| (4.110) where K is a constant and 0 < K < 1. This means that the distance between the images of any two points is always smaller than the distance between the points themselves. Further, i f we, i n the Banach space E are given a . sphere of radius ||x - x || ^  a (a = constant) (4.111) and i f ||A(xo) - xo|| < a ( l - K) (4.112) where 0 < K < 1, then the equation A(x) = x (4.113) has a unique solution i n E x = x* (4.114) To prove t h i s , we define a sequence {xn> which is given by x n = A(x n _ 1 ) (n = 1, 2, . . .) (4.115) 24 27 and we assume X q and x^ both l i e within the sphere (4.111) ' Condition (4.112) is called the fixed point condition and means the following. I f , given (4.113), we pick a and K such that A ( X q ) sa t isf ies (4.112) then the sequence x 2 , . . . , x n l ies within the sphere (4.111). 92 Then to recapitulate. I f the mapping, linear or nonlinear, A(x) = x (4.115a) sat isf ies the contraction condition (4.110) then we are guaranteed that a solution of (4.115a) exists . Further, i f i n addition the fixed point condition (4.112) is s a t i s f i e d then we obtain a region where this solution is unique. Hence we obtain the solution of (4.115a) i n the form of a convergent series x* = x + (x, - x ) + . . . + (x„ - x , ) + . . . o v l oJ K n n - l ' 4.8 S t a b i l i t y v i a Contraction Mapping Consider the system L(x) = y(t) (4.116) where L is a l inear , time-invariant d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane and y(t) is an arbitrary, bounded driving function. We can represent (4.116) i n block diagram form as shown i n F i g . 4.13 below/ Clearly we have L'(X) + x = y(t) (4.117) y + L (p)x L ' C P ) X r X F i g . 4.13 System (4.116) 93 when i t follows from (4.116) and (4.117) that L(p) = L'(P) + 1 (4.118) which means F i g . 4.13 represents a stable l inear system with input y(t) and output x ( t ) . Now consider the following nonlinear system L(x x) + e0(x1) = y(t) (4.119) where L is a l inear , time-invariant, d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane, e is a constant, 0(x1) is a nonlinear function of x^, and y(t) is an arbitrary, bounded driving function. System (4.119) can be represented i n block diagram form as shown i n F i g . 4.14 below where N = x^+ 0(x^). Further, i t follows that, as i n F i g . 4.13, we have L(p) = L (p) + 1 (4.120) s L ( p ) x 1 1 x 1 = X + y ) " L'(P) x x + 0 ( x 1 ) N F i g . 4.14 System (4.119) We can now compare (4.116) and (4.119), and i t w i l l be observed that (4.119) is identical to (4.116) i f e =0 . 94 Therefore we shal l i n essence consider system (4.119) as a perturbed version of system (4.116) and assume L and y are identical i n both systems, whereas y (t) A x x ( t ) - x(t) (4.121) as shown in F ig . 4.14. This point of view is closely akin to 17 that taken by Desoer . Now consider (4.119) and let us write that equation i n the form x x = gy - eg 0(xx) (4.122) Here g is a linear operator with impulse response h ( t ) , and from (4.109a) the norm of g is oo II3II = j |h(t)| dt (4.123) 0 Further, i n reference 29 the following theorem is proven. A necessary and suff ic ient condition that a bounded input y ( t ) , to a l inear , time-invariant system gives r ise to a bounded output x(t) is that oo Hell = J |h(t)| dt < « ( 4 _ 1 2 4 ) 0 It w i l l be observed that this constitutes part of Barrett 's theorem. It follows that (4.116) can be written as x = gy .(4.125) 95 Therefore substituting (4.121) and (4.125) into (4.122) we obtain By + y = By - eB 0(By + y) or y = -cB 0(By + y) (4.126) In what follows we s h a l l , for the sake of s implic i ty , use the following notation H I - u I e|| =H (4.127) l|x|| -X llyll -Y It w i l l be observed that (4.126) is equivalent to a nonlinear mapping. Therefore using the terminology of section 4.7 we have A(y ,y) A -eB 0(By +y) (4.128) In order that (4.128) is a contraction mapping, condition (4.110) must be s a t i s f i e d . Therefore we require ||A(y2,y) - A O i ^ y ) ! = || - eB 0(By + y 2 ) + eB 0(3y + y j) || ,<K l ^ - y j where 0 < K <1, or |e| H ||0(6Y + y 2 ) - 0 ( e y + M l ) ||=< K ||w 2 - U x || (4.129) Further, i n order that (4.126) has a unique solution the fixed point condition (4.112) must be s a t i s f i e d . This simply means that for each input y we have only one possible output Xy 96 Consider the case when the nonlinearity 0 is such that 0 _< 0(gy + y) _< k(6y + y) where k > 0 is a constant. Then (4.129) may be written |e| Hk ||y2 .< K ||y2 - y j whence the contraction condition (4.110) reads « e l H k ^ K (4.129a) where 0 < K < 1. We must now determine a sphere (4.111) within which the fixed point condition (4.112) is s a t i s f i e d . Therefore we find from (4.128) " l = IIAC v 0,y|| =< |e| H- ||0.(6y)|| < | e | H2kY Hence, the fixed point condition (4.112) becomes |e| H2kY < (1 - K)a or o ^ |e|H2kY A L ' i 1 - k (4.129b) In this case the contraction condition (4.129a) is clearly d i f f i c u l t to satisfy unless e or H is small (note 0 <K < 1) . However, the fixed point condition (4.129b) is easy to sat isfy for a f i n i t e Y. 4.9 The Stabi l i ty of Some Nonlinear Systems One type of system which can be treated by the method developed i n section 4.8 is the following LCxp + e £ a ^ * = y( t ) n=2 For this system (4.128) becomes k A(y ,y) A - e$ a n ^ y + ^ n=2 97 (4.130) (4.131) and (4.129) becomes k |e| H || Y, an(6y + y 2 ) n - Y a n ( * Y + ^ / n=2 But i n general (By + y 2 ) m - (By + y n ) k I n=2 < K |y 2 - y x | (4.132) 1' / m-1 (By + y 2 ) - (By + y ^ ] £ (By + y 2 ) m " 1 " 1 ( 6 y + M l } 1 i=0 We can therefore write (4.132) as k I e | H | | y 2 - y 1 | | Y n | a n | (HY + U ) n _ 1 ,< K n=2 where we have put m-1 || Y (^ + v 2 ) m " 1 " i ( 8 y || =< m(HY + U) i=0 (4.133) i n 2 " w x l (4.134) m-1 (4.135) Relation (4.135) is j u s t i f i e d on the basis that we have llvjl < IM < y 98 where y is given by (4.121). From (4.134) we can now determine the condition for (4.131) to be a contraction mapping. This condition is seen to be k i n _ 1 - v (4.136) E | H ^ n a n (HY + U) K * < K n=2 where 0 < K < 1. Or equivalently k | e | H ^ n a n (HY + U ) 1 1 " 1 < 1 n=2 (4.137) We must now determine a region within that specified by (4.137) where the fixed point condition (4.112) is s a t i s f i e d . To do this we observe that from (4.131) we obtain A(y 0 ,y ) = y 1 (y 0 - 0) or n (HY) n (4.138) n=2 Therefore combining (4.136) and (4.138) the fixed point condition reads k n (HY)n < U(l - K) (4.139) n=2 where we have put a i n (4.112) equal to U. Refer to equation (4.121). This equation may be written X1 _< X + U (4.140) 99 or using (4.125) we f i n d X x ^ HY + U Therefore (4.137) is equivalent to k (4.141) n X ^ ' 1 < 1 TTTT n=2 (4.142) Further, i f we choose U and K such that (4.137) and (4.139) are sa t i s f ied for the largest possible value of Y which we denote by Y^ then and from (4.141) Y < Y, X l = ^ 1 + U (4.143) (4.144) Thus we have shown that i f (4.142) and (4.143) are s a t i s f i e d then equation (4.131) has a unique solution. This i n essence means that the solution of (4.131) obtained by i terat ion y = y o + (a1 - y Q ) + . . . + ( j i n - y ^ ) + . . . is a convergent series , and since. (4.145) v n = A k n - l ' r t we have y n + l " y n < K n l n - l Therefore, i n the region specified by (4.143) and (4.144) the system (4.130) has a unique output x-^t) , for every input y(t) Hence we have determined a region where (4.130) exhibits bounded input - bounded output s t a b i l i t y . We can now state the following theorem: 1Q0 Theorem 4. Given the nonlinear system k L(Xl) + e Y, = yCt) (4'146) n=2 Where L(p) is a l inear , time-invariant, d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane, and = y = 0 for t =<0 then for sup |y(t) | ,< Y^we have sup |x^(t) | £ HY^ + U where Y^ is determined from (4.137) and (4.139) and X p being single-valued, from (4,144). This theorem states that the system (4.146) has a unique output x^(t), and exhibits bounded input - bounded output s t a b i l i t y within a certain range for an arbitrary driving function y ( t ) . The simplest special case of (4.146) is the system L(x x) + e x ^ = y(t) (4.147) For this system the contraction condition (4.137) reduces to | E | Hm(HY + U ) m _ 1 < 1 (4.148) while the fixed point condition (4.139) reduces to |e| H(HY)m < U(l - K) (4.149) We shal l now consider some speci f ic examples of equation (4.147) Example 1 x1 + x x + x1 + E X ^ = y(t) (4.150) This system has already been investigated by use of Barrett 's procedure (see Example 1 section 4.3). There we found 101 H = 1.39 For (4.149) the nonlinear mapping (4.131) becomes A(ji ,y) = - E g (By + y ) 3 (4.151) the contraction condition (4.148) becomes 3 | E | H (HY + U ) 2 4 K (4.152) and the fixed point condition (4.149) is | E | H (HY)3 < U( l - K) (4.153) The contraction condition (4.152) can be written as follows (on removing the equality sign) or x l < '•TjTfTT3 (4.154) Comparison of (4.19) and (4.154) shows that the right hand sides of the two relations are identical i n both cases. However, i n order to sat isfy theorem 4 we must i n addition ensure that (4.153) is true. Thus, theorem 4, although of a much more general nature than Barrett 's theorem (see theorem 1 section 4,2), imposes s t r i c te r conditions than does Barrett 's method. For the purpose of numerical computations we assume the following K = 0.60 U = 0.10 (4.155) 102 Then from (4.152) we find Y _< 0.20 (4.156) and (4.153) becomes (1.39)(.278) 3 < (0.10)(.40) or .03 _< .04 (4.157) Therefore the fixed point condition (4.153) is s a t i s f i e d . We can now determine the upper bound on the output from (4.144). Thus \ < (1.39)(.20) + 0.10 ^ .38 (4.158) Thus a l l the conditions of theorem 4 are sa t i s f ied within the region specified by (4.156) and (4.158). Using Barrett 's method we obtained (see F i g . 4.2) Y 2 = .234 (4.159) X 2 = .49 (4.160) and i t w i l l be observed that (4.156) and (4.158) provide a good approximation to those given by (4.159) and (4.16b). Refer to F i g . 4.3. It w i l l be observed that along the branch COD we have only one possible output for each input. This range is the one to which theorem 4 applies. It was shown i n section 4.2 that we have bounded input - bounded output s t a b i l i t y for -Y 2 _< Y < Y 2 (4.161) 103 However, i n this interval we have three possible outputs x^(t) for each input y f t ) . This behaviour is not allowed i f we are to satisfy theorem 4 which specifies the output to be unique. Example 2 . . . 2 x l + x l + x l + e X l = (4.162) This system has already been investigated by a different method (see Example 1 section 4.4). We have again H = 1.39 (4.163) For (4.162) the nonlinear mapping (4.131) becomes A(u ,y) = -eg(gy + y ) 2 (4.164) The contraction condition (4.148) becomes 2|e|H (HY + U) <, K (4.165) and the fixed point condition (4.149) is I e I H (HY)2 _< U(l - K) (4.166) On removing the equality sign the contraction condition (4.165) can be written as follows X l < r~f7]~H C4.167) For the purpose of numerical computations we assume the following K = .60 U = 0.075 (4.168) e = -1.0 Then from (4.165) we f ind Y _< .10 (4.169) 104 and (4.166) becomes (1.39)(.45) 2 < (.075)(.40) or .029 _< .030 (4.170) Hence the fixed point condition is sa t i s f ied for the values given by (4.168). We can now determine the corresponding bound on the output x(t) by use of (4.144) and (4.168). Therefore X 1 ^ .22 (4.171) The region specified by (4.169) and (4.171) can now be compared to that obtained i n section 4.4. There we found (see Fig . 4.8) Y ^ Y, = .18 1 (4.172) -.149 j X x _< .36 Clearly the lower bound on X^ has been improved i n (4.171), however, the upper bound on X^, and both upper and lower bounds on Y have decreased i n the present case. This i s in keeping with the fact that the present method imposes somewhat s t r i c te r conditions than does Barrett 's method. The considerations given to the uniqueness of the output i n the previous example are applicable here as w e l l . The present method can only predict s t a b i l i t y along the branch DOA i n F ig . 4.8. It is evident that systems of the type x 1- + x 1 + x1 + ex-j4 = y(t) (4.173) 105 and x l + x l + x l + e X l 5 = y ^ t ) (4.174) are easily treated by the method developed here since they are subcases of (4.147). However, i t i s not considered necessary to solve these examples numerically. A second type of system which can be treated by the method developed in section 4.8 is the following L(x x) + esin x± = y(t) (4.175) where L(p) is a l inear , time invariant, d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane, e is a constant and y(t) is an arbitrary driving function. Clearly (4.175) can at least be approximated by (4.175), however, a simpler approach i s to use the theory developed i n section 4.8 di rec t ly . The nonlinear mapping (4.128) becomes for (4.175) A(y,y) = - e 8 s i n ( B y + y ) (4.176) and (4.129) becomes | e | H || sin(6y + y 2) - sin(6y + y x) || _< K || y 2 - y 1 | (4.177) But ||sin(6y+y 2) - sin(By + y ^  || || s i n By (cos M 2 - cos y ^ ) + cos B y ( s i n y 2 - s i n y ^ ) || ^ || s in By|| ||sin(J - y 2) - s i n ( | - y x) || + || cos ByII ||siny 9 - s i n y , | | ^ 106 ||y2 - y 1 | | ( || sin gy|| + || cos gy|| ) ^ 1.41 ||y2 - y 1|| (4.178) Then substituting (4.178) into (4.177) we obtain 1.41 | e | H ||y2 - y j a< K ||y2 - y 1 | (4.179) Hence in order that (4.176) is a contraction we require K >, 1.41 |e| H (4.180) where 0 < K < 1. We must now ensure that there exists a sphere (4.111) within which the fixed point condition (4.112) is s a t i s f i e d . From (4.176) we f ind ||A(y0,y)|| < |e| ||3|| ||sin6y|| (y Q = 0) (4.181) Therefore the fixed point condition (4.112) becomes. |e| H ^ (1 - K)a (4.182) or |e| H a = 1 - K (4.183) It is evident that (4.183) can be s a t i s f i e d for any 0 <K <1, hence i t follows that the only condition we need to sat isfy i n this case is (4.180). Therefore, for a given H we must constrain e such that (4.180) is true, which of course means that when H is large the perturbation due to the nonlinear term e s in x^, and thus e, must be small. We can now state the following theorem: 107 Theorem 5. Given the nonlinear system LCx.^ + e s i n x± = y(t) (4.184) where L(p) is a l inear , time-invariant, d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane, and x = y = 0 for t ^ 0 then for sup |y(t) | < °° we have sup |x^(t) | < °° provided (2** |e| H) < 1. This theorem states that the system (4.184) exhibits bounded input - bounded output s t a b i l i t y provided (2^ |e| H) < 1. It follows that i f instead of e s i n x^ we had e cos x^ i n (4.175) a similar result would obtain. Further, i t is evident that an equation having a nonlinearity consisting of a combination of trigonometric functions can be treated by a similar procedure. However, i t is not considered necessary to investigate further examples of this type. 4.10 The S t a b i l i t y of Some Nonlinear Systems with a Simple Pole at the Origin . Consider the following system L ' ( X I ) + e0 , (x 1 ) = y(t) (4.185) i where E is a constant, 0 (x^) is a nonlinear function of X p y(t) is an arbitrary driving function and L (p) is a l inear , time-invariant, d i f f e r e n t i a l operator such that L'( P) = pF(p) (4.186) Here F(p) has (simple) zeroes only i n the l e f t hand plane. 108 Clearly (4.185) can not be treated by the method developed i n section 4.8 because (4.124) is not s a t i s f i e d . However, here we can employ the following procedure. Let us rewrite (4.185) as follows i /(Xj) + ax x + e0 , (x 1 ) - ax x = y(t) (4.187) It w i l l be observed we have simply added and subtracted the term ax^, (a > 0) from (4.185). Therefore we can write L(x 1) = L ' C X ^ + ax± (4.188) 0(x x) = 0 , ( x 1 ) - a X l (4.189) It w i l l be noticed that L(p) has zeroes only i n the l e f t hand plane provided we choose a such that i t does not cause L(p) to have purely complex conjugate zeroes. Hence we can write (4.187) as follows L ( X l ) + 0(x x) = y(t) (4.190) which is of the same form as (4.119) and can therefore be treated in a similar manner. In this case the nonlinear mapping (4.128) becomes A(ji ,y) = - ef3 0(By + w) + aB (By + M) (4.191) where B now is a linear operator derived from L(p) , and the corresponding impulse response h( t ) , thus depends on a. The contraction condition (4.129) becomes, i n this case, |E| H|| 0(By + v 2 ) - 0(6y + M X) II + a| | B(By + y 2 ) -109 eCey + y x ) || < K ||u 2 - u x II or | e | H || 0(0y + n 2) - 0($y - y XHI + a H || y 2 - y x || =< K ||y2 - y1|| (4.192) We shal l now treat two systems of the type (4.185). F i r s t consider the system k L ' ( x i ) + e L v i n =y ( t ) ( 4' 1 9 3 ) n=2 and assume L (p) has the form (4.186). We can then write (4.193) as follows k L ( x l ) + z Y, V l " a X l = y ( t ) (4.194) n=2 where L(p) = L'( P) + a (a > 0) and we assume a is such that L(p) only has simple zeroes which are located i n the l e f t hand plane. From (4.194) we f ind the nonlinear mapping (4.191) k A(y ,y) = - e8 Y a n (ey + y ) n + oB (By + y) n = 2 (4.195) We can now compare (4.195) to (4.131) and (4.192) to (4.132). It w i l l be observed that i n both cases these equations are equivalent except for the last term i n (4.192) and (4.195). 110 We can therefore immediately write down the contraction condition for (4.195) by using (4.136). Hence we obtain k | e] H n a n + U ) 1 1 " 1 + a H ^ K (4.196) n=2 where 0 < K < 1. We must now determine a region within that specified by (4.196) where the fixed point condition (4.112) is s a t i s f i e d . In order to do this we f ind from (4.195) A 6 i 0 , y ) = y 1 k 0 = 0 ) k = -ee Y a n ( e y ) n + a$ Cey) n=2 or k u ! =± l e i H Y a n (HY)n + a H 2Y (4.197) n=2 Therefore, combining (4.196) and (4.197) the fixed point condition reads k |e| H Y | a n | (HY)n + a H 2Y <, U(l-K) (4.198) n=2 We shal l now use exactly the same terminology and nomenclature as was used when treating system (4.130) (see section 4,9). We then obtain x 1 _< X + u < HY + U and from (4.196) we f ind on substituting from (4.199) 111 (4.199) n=2 n a n 1 feT T^TH" (4.200) Now let us choose U and K such that (4.196) and (4.197) are sa t i s f ied for the largest possible value of Y. Let us denote this value of Y by Yy Then and from (4.199) Y _< Y1 (4.201) X l = ^ 1 + U (4.202) We can now state the following theorem: Theorem 7. Given the nonlinear system k L'(XX) + E £ a n X ; L n = y(t) (4.203) n=2 where L (p) is a l inear , time-invariant, d i f f e r e n t i a l operator such that L (p) = pF(p) and F(p) has (simple) zeroes only i n the l e f t hand plane, e is a constant, y(t) i s an arbitrary driving function and x^ = y = 0 for t ^ 0 then for sup |y(t)| ^ Y^ we have sup |x^(t)| j HY1 + U where Y^ is determined from (4.196) and (4.198) and X^, being single-valued, from (4.202). 112 The simplest special case of (4.203) is the system L ' C X ^ + e X l m = y(t) (4.204) For this system the contraction condition (4.196) becomes |e| Hm(HY + U ) m _ 1 + a H ^  K (4.205) while the fixed point condition (4.198) reduces to |e| H(HY)m + a H 2Y < U(l - K) (4.206) We shal l now consider one specif ic example of (4.204). Example 1 * L + X l + X l 3 = y ( t ) (4.207) Here we have L'( P) = p(p + 1) (4.208) which has the form (4.186). We now add and subtract a term ax^,(a > 0) from (4.207). Then we obtain . . . , x1 + x 1 + a x x + x 1 - a x 1 = y(t) (4.209) and (4.188) becomes L(p) * L ' ( P ) + a = p 2 + p"+ a (4.210) while (4.189) becomes 0(x x ) = x x 3 - a x x (4.211) For (4.209) the nonlinear mapping (4.195) becomes 113 A(y ,y) = -eB (By + y ) 3 + aB (By + y) (4.212) the contraction condition (4.205) becomes 3 |e| H(HY + U ) 2 + a H _< K (4.213) while the fixed point condition (4.206) becomes |e| H(HY)3 + a H 2Y _< U(l - K) (4.214) We must now choose a value of a and compute H corresponding to (4.210). For this purpose we shal l make use of a formula given i n reference 23. There the following is given. If L(p) = a o p 2 + a l P + a 2 (4.215) and a , a , , a 0 > 0 then o' 1 L i) i f a, > 4a we have J 1 = 0 H - 1 ~ 0 (4.216) 2 i i ) i f a , < 4a we have l = o H = -1 coth(y cot e) (4.217) o where 9 is the angle of the complex roots. Comparison of (4.210) and (4.215) shows that i n this case a o = 1 ' a l = 1 (4.218) 114 Let the f i r s t choice for a be a = 1 (4.219) then we f ind from (4.217) H = 1.39 (4.220) If we now substitute for a and H i n (4.213) we obtain (3)(1.39)(1.39Y + U ) 2 + 1.39 < K (4.221) where 0 <K <1. However, obviously (4.221) can not be sa t i s f ied for any K i n the range specif ied, and we must clearly choose a different value of a than that given by (4.219) Let us now choose a = .30 (4.222) Then from (4.217) we f ind H = 1.0 (4.223) Using these values for a and H i n (4.213) we obtain the contraction condition 3(Y + U) 2 + 0.30 ^ K (4.224) while the fixed point condition (4.214) becomes Y 3 + 0.30Y _< U(l - K) (4.225) In this case we choose U = 0.10 (4.226) K = 0.49 then we f ind from (4.224) Y _< Y x = .15 (4.227) 115 and (4.225) becomes .045 ^ .051 (4.228) hence the fixed point condition is satisfied. From (4.202) we can now determine the bound on the output. Thus X, < .15 + .10 I BO (4.229) < .25 e n Therefore the system (4.207) exhibits bounded input - bounded output s t a b i l i t y within the range specified by (4.227) and (4.229) The second system of type (4.185) which is considered here is the following i / ^ ) + e s in x± = y(t) (4.230) T We assume L (p) has the form (4.186). Hence we can write (4.230) as follows L(x^) + e s in x^ - a x^ = y(t) (4.231) where L(p) = L'( P) + a (a > 0) and we assume a is such that L(p) has simple zeroes which are located i n the l e f t hand plane. From (4.231) we find the nonlinear mapping (4.191) A(y ,y) = - e 6 sin(ey + u) + ae (By + y) (4.232) We can now compare (4.232) to (4.176) and (4.192) to (4.179). It w i l l be observed that i n both cases these equations are equivalent except for the last term in (4.192) and (4.232). 116 We can therefore immediately write down the contraction condition for (4.232) by using (4.179). Hence we obtain 1.41 |e| H Ib^-y-JI + a H 2 l^ - y - J I < K ||u2 - u x || (4.233) or the contraction condition becomes 1.41 |e| H + a H 2 j K (4.234) where 0 < K < 1. We must now ensure that there exists a sphere (4.111) within which the fixed point condition (4.112) is s a t i s f i e d . From (4.232) we f ind l|A(yo,y)|| < | e | | | B | | IIsin By|| +<X | | B|| 2Y (yQ = 0) < |e| H + a H 2Y (4.235) Therefore the fixed point condition (4.112) becomes | e | H + a H 2Y _< (1 - K)a (4.236) or a> H H : a H 2 Y ( 4 > 2 3 7 ) It is evident we can always sat isfy (4.237) for any f i n i t e values of |e| , H, a and Y when 0 <K < 1. Therefore we actually only need to sat isfy (4.234) i n order that (4.232) has a unique solution. We can now state the following theorem: Theorem 8. Given the nonlinear system l / ^ ) + e s i n x x = y(t) (4.238) 117 t where L (p) is a l inear , time-invariant, d i f f e r e n t i a l operator t such that L (p) = pF(p) and F(p) has (simple) zeroes only in the l e f t hand plane, e is a constant, y(t) is an arbitrary driving function, and x^ = y = 0 for t ^ 0 then for sup |y(t) | < °° we have sup |x^(t)| <°° provided (2^ |e|H + a H ) <1. It is evident that substituting cos x^ for s in x^ i n (4.238) would give a similar result . Further, a system having a nonlinearity consisting of a combination of trigonometric functions can presumably be treated i n a similar way, however, such systems w i l l not be considered here. It is evident that the method developed i n section 4.8 is applicable to a wide class of systems. In this thesis we have restricted outselves to systems having nonlinearities i n the form of polynomials and trigonometric functions. However, i t is evident that other types of nonlinearities can be treated by the present method. 4.11 The V a l i d i t y of Volterra Series Representation In section 4.2 i t was pointed out that certain conditions (see page 63) must be sa t i s f ied i n order that we may represent the solution of a given d i f f e r e n t i a l equation i n terms of a Volterra series . In this section we shal l show that these conditions are indeed s a t i s f i e d i n part of the range found by Barrett. This w i l l be done by proving that the method developed v i a contraction mapping i n section 4.8 actually generates the Volterra series found by Barrett. 118 In order to define our terms consider the"following system L C x ^ j * e 0(x 1) = y(t) (4.239) where L(p) is a l inear , time-invariant, d i f f e r e n t i a l operator with (simple) zeroes only i n the l e f t hand plane, e is a constant, 0,'(x.p is a nonlinear function of x^ and y(t) is an arbitrary driving function. We can now write (4.239) as an integral equation i n the form CO o o x x ( t ) = J y(x)h(x - x)dx - eJ 0(x 1(x)h(t - x)dx (4.240) - 0 0 - 0 0 (x^O) = y ( 0 ) = 0 ) where we have convolved y(t) and 0(x^) with the impulse response h(t) corresponding to the linear transfer function • Let us now use the functional operator notation introduced in section 4.8. Thus le t B be the linear operator with impulse response h( t ) . Then (4.240) can be written as follows x ^ t ) = By - eB 0(x 1) (4.241) If we le t 0(x 1) = X-L3 (4.242) we obtain ^ ( t ) = By - eB X j 3 (4.243) 119 which is equivalent to equation (4.5). We shal l now solve (4.243) one, by direct i terat ion (Barrett's method) and two, by the method developed i n section 4 .8 , . Then we shal l show that the two series are identical and thus that the Volterra series representation is v a l i d for (4.239) subject to (4.242). Using direct i terat ion we obtain 1st approximation x^ = By (4.244) 2nd approximation x^ = By - eB (By) (4.245) 3 3 3rd approximation x^ = By - eS [By - eB (By) ] (4.246) etc. If we take absolute values i n (4.245) we f ind l l x ^ t j l < || 31| ||y|| + |e| ||B|| + ||y|| 3 (4.247) where h(t)| dt 0 It w i l l be observed that (4.247) is exactly the f i r s t two terms i n (4.13). Further, i t is readily seen that we i n this manner can generate the i n f i n i t e Volterra series (4.12). Now consider the procedure developed v i a contraction mapping i n section 4.8. There we compared (4.239) to the linear system L(x) = y(t) (4.248) and assumed the input y(t) to be identical i n both'.cases. On 120 the other hand, we defined y (t) A x ^ t ) - x(t) (4.249) that i s , the output of the nonlinear system (4.239) is assumed to be equal to that of the linear system (4.248) plus a quantity y ( t ) , which is due to the presence of the nonlinear term i n (4.239). Let us now again use operator notation. Then (4.248) becomes x = By (4.250) and (4.239) becomes, subject to (4.242), x1 = By - e B x x 3 (4.251) We can now use (4.249) and (4.250) i n (4.251) which then becomes y = - e B (By + y ) 3 (4.252) or A(y ,y) = - e B (By + P ) 3 (4.253) We can then solve (4.263) by i tera t ion . Thus p Q = 0 (4.254) A(y Q ,y) = P x = - e B (By) 3 (4.255) A ^ . y ) = P 2 = - e B [By - e B ( B y ) 3 ] 3 (4.256) etc. Hence we can form a sequence which converges to the true solution given by (4.249). Therefore we have 1st approximation . x^ = x + y Q = By (4.257) 121 x + y x = By - eB (By) 3 (4.258) x + y 2 = By - eB [By - eB (By) 3 ] 3 (4.259) etc. Comparison of (4.257), (4.258) and (4.259) to (4.244), (4.245) and (4.246) shows that the two sequences are ident ica l . However, i n section 4.9 i t was shown that in the range specified by equations (4.148) and (4.149) (note m = 3) the solution of equation (4.253) has the following properties. 1. It exists ((4.253) is a contraction mapping) 2. It is unique ((4.253) sat isf ies the fixed point condition) 3. The series y = y Q + (y 1 - y Q ) + . . . + (y n - y ^ ) + . . . (4,260) converges absolutely. Further, since the solution of (4.253) given by (4.260) is unique, and the system equation (4.239) has constant coefficients we have s a t i s f i e d the requirement of time-invariancy. Thus we have shown that the solution we obtain by the method developed i n section 4.8 is a Volterra series which is identical to that found by Barrett. Further, the series found by the former method sat is f ies a l l the requirements necessary i n order that the Volterra series representation is v a l i d . Therefore Barrett 's method gives v a l i d results i n the range specified by (4.148) and (4.149) (note m = 3). Outside this range we have not v e r i f i e d the Volterra series representation i n this thesis . 2nd approximation x^ = 3rd approximation x, = 122 We have thus developed a relat ively simple method for determining a region i n which the Volterra series represent-ation is v a l i d . Clearly the class of systems which can be treated by this method includes systems of more general nature than that given by (4.239). For example, the following system L(x x) + e^iyf* &202^X1) = yW (4.261) can be investigated by the method developed i n section 4.8. However, the appl icabi l i ty of the method is best investigated for each individual problem. The developments i n sections 4.4, 4.5 and 4.8 to 4.11 are to the author's knowledge new and have not appeared i n the l i terature before. 4.12 Discussion of Results It is evident that the method developed by Barrett requires a large amount of computation for each individual example. Further, the method, i n i t s present form, is of a somewhat restricted nature since we i n each case must determine the region of convergence of a power series which i n most cases is an arduous task. In addition, Barrett only proves convergence of a Volterra series , he does not show the solution he finds is unique, which i t i n fact is not. This feature is discussed i n section 4.2. The method developed i n section 4.8 v i a contraction mapping appears to have several desirable features. The main ones arej one, sat isfying the contraction condition ensures the solution of a given d i f f e r e n t i a l equation exists , and two, satisfying the fixed point condition ensures the solution obtained is unique. Thus we can f ind a range over which a given solution is known to be v a l i d . Further, from the examples solved i t is clear that the method is applicable to a reasonably large class of problems. However, the appl icabi l i ty of the method is best determined for each individual problem. In section 4.11 i t is shown that the method developed i n section 4.8 actually generates the Volterra series we f ind by using Barrett 's method. Thus we have found a method which shows when the solution of a given d i f f e r e n t i a l equation can be represented in the form of a Volterra series . 124 5. CONCLUSIONS 5.1 Summary In this thesis we have treated nonlinear system s t a b i l i t y from three separate points of view. 1. State space analysis. (Undriven systems) 2. Complex frequency plane analysis. (Driven and undriven systems) 3. Time domain analysis. (Driven systems) Here we have also indicated the type of systems a given method is applied to in this thesis. State space analysis has been employed to develop a special form of the general Zubov equation. This special form is designed to determine regions of asymptotic s t a b i l i t y for undriven systems. However, the resulting equations derived for Liapunov's function can not be solved i n general since the constraints we must place on the functions involved can not be s a t i s f i e d except i n special cases. The Sandberg frequency s t a b i l i t y c r i ter ion has been applied to some systems for which the linear part has poles on the imaginary axis. This application is v a l i d for both driven and undriven systems provided the system inputs are square integrable. The examples solved e x p l i c i t l y i n this manner i n this thesis are a l l undriven systems. Therefore, we can only investigate these systems for global asymptotic s t a b i l i t y . Further, the Popov method and the Sandberg method have been compared through the solution of a specif ic undriven system with a nonlinearity not 125 depending e x p l i c i t l y on time. For this particular example i t is found that the two methods give approximately the same answer. However, i t i s emphasized that we have not made a theoretical comparison between the Sandberg method and the Popov method. Using the time domain we have considered bounded input - bounded output s t a b i l i t y by using functional analysis. One, a method developed by Barrett which is based on using a Volterra series solution of a nonlinear d i f f e r e n t i a l equation has been extended to include systems having nonlinearities of the 2nd and 4th degree. This extension rests on the determination of a region of convergence of Volterra series by use of the ratio test. However, i t is emphasized that Barrett has not shown e x p l i c i t l y that his method actually is v a l i d . Two, the contraction mapping principle has been used to derive a general input-output s t a b i l i t y c r i t e r i o n . This c r i te r ion has been derived by comparing a nonlinear system to a linear system and by stipulating that the output of the nonlinear system is a perturbed version of the output of the l inear system. The solution of several examples i l lustrates the u t i l i t y of the c r i t e r i o n . Further, i t is shown that i n using this method to investigate the input-output s t a b i l i t y of a given system we actually generate the Volterra series found by Barrett 's method. Thus we have shown that'the Volterra series solution is a v a l i d representation of a given nonlinear d i f f e r e n t i a l equation i n a certain domain. This means that we have found a region where we can show e x p l i c i t l y that Barrett 's method is v a l i d . 126 5.2 Recommendations for Furture Work The main drawback i n the use of the frequency-s t a b i l i t y c r i t e r i a mentioned i n this thesis is that they can only stipulate that a given nonlinearity must be entirely contained i n a certain sector, Thus these c r i t e r i a refer only to global s t a b i l i t y whereas, local s t a b i l i t y is often the more important. Hence an extension of these c r i t e r i a to local s t a b i l i t y appears desirable. Regarding s t a b i l i t y analysis i n the time domain we have mainly considered systems with nonlinearities i n expl ic i t functional form i n this thesis. It does not seem possible to extend Barrett 's method direc t ly to systems having a nonlinearity of general functional form. However, the s t a b i l i t y theorem based on the contraction mapping principle (see section 4.8) can possibly be extended to cover systems with nonlinearities depending e x p l i c i t l y 30 on time. Desoer developed a method which covers certain systems of that type. There are naturally many more avenues open for research than those mentioned here. S t a b i l i t y analysis offers a rewarding and interesting f i e l d of research, i n part icular v ia functional analysis which has been proven to be a powerful tool i n that f i e l d . APPENDIX A POPOV'S STABILITY CRITERION 4 Consider the system of equations n n dx. 1 dt~ j - l a . .x. + b. 0( IJ 3 I ^  I k=l ckXk> CA..1) ( i = 1, 2 , . . . ,n) where a . . , b . , and c v are constants while 0 is a nonlinear func-tion„ Equation (A.l) is usually written i n the following form n px i - ^ a^x^. + b i 0(a) = 0 (A. 2) j=l n Ci x, k k = 0 k-1 (A. 3) where p = ^ then solving for - a we f ind where D(p) = P " a *21 11 " a12 P -a a n l 22 an2 l l n l2n P " ^ 0 0 0 nn (A. 4 ) or 128 P " a. 11 -a 12 l l n D(p) = *21 P - a 22 l2n (A. 5) a n l *n2 P - a 'nn and K(p) = P - a 11 "21 a n l -a 12 P ' a 22 -a ii2 Let W(p) = J^EJ- then (A. 4) becomes • a l n b l " a2n b 2 p - a b ^ nn n n 0 (A.6) - a = W(p) 0 (A. 7 ) and we may consider W(p) an open loop transfer function with 0 being the input and - 0 ~ the output. In general the nonlinearity 0(a) i s assumed to be confined to the f i r s t and third quadrant as shown i n F i g . A . l below. ka 0(a) F i g . A . l S t a b i l i t y Sector for 0(a) 129 Or i n mathematical form we require 0 < _< k (A.8) The inequality (A.8) is suff ic ient for cases when W(p) does not have poles on the imaginary axis (principal case)y however, when W(p) does have poles on the imaginary axis (particular cases) we substitute for (A.8) the inequality e < 0 i £ i =< k (A.9) where e > 0 is a r b i t r a r i l y small. Further, we define the "modified" frequency response t W ( j u ) ) such that Rew' = ReW Imw' = oalmW (A. 10) Now consider equation (A.4) and le t 0(a) = ha where h can take on any value 0 to k. Then (A. 4) becomes a l inear system with the open loop transfer function hW(p) = h]|g- (A. 11) and we can therefore apply Nyquist's s t a b i l i t y c r i t e r i o n . An equivalent formulation of that c r i te r ion to that l a i d down i n section 3.1, is that the open loop frequency locus intersects the interval (-«=, -1] the same number of times i n the upward and downward directions. However, i n (A.11) the value of h is arbitrary i n the interval [0,k], therefore a necessary and 130 suff ic ient condition that the system (A.l) is stable when (A. 11) is s a t i s f i e d , is that the locus of (A. 11) does not intersect the real axis from - <=° up to and including - p See F i g . A.2 below ImW Forbidden Zone ReW F i g . A.2 S t a b i l i t y Region for Linearized System « - — For part icular cases we do i n addition require that (A.l) be s table - in- the- l imit , i e . , system (A.l) must be stable for suff i c ient ly small h > 0. For nonlinear systems, i . e . , when 0(a) f ha, V.M. Popov proved the following theorem: " If the function W(p) has an index A , where A is the number of characteristic roots .(their m u l t i p l i c i t y must be included) on the imaginary axis of the system ( A . l ) , and i f for some f i n i t e real values of k and q the function G(p) = (1 + qp)W(p) + ^ is a s t r i c t l y positive function of the complex valued argument p , then the system (A.l) is absolutely stable i n the sector [0,k] for the principal case, and i n the sector [e,k] (with e a r b i t r a r i l y small) for the part icular cases". This means Re(l + M)W(ju) + \ > 0 (u ^0 ) (A.12) which is cal led the Popov condition. 131 This can be interpreted geometrically as follows. Let W = X + j Y and on expansion Re(l + j t o q )W ( j u ) = ReW(jco) - qui Im Wfjw) = X - q Y Therefore (A. 12) becomes but X - qY + ^ = 0 is the quation of a straight l ine with slope ^ which passes through the point - ^ on the real axis. This straight l ine is generally called the Popov l i n e . It follows that geometrically the W (jui) locus must be entirely to the right of this l i n e . See F i g . A.3 below. F i g . A.3 Stable Modified Locus To show that the restrictions imposed on the function G(p) sa t isf ies the conditions for s t a b i l i t y - i n - t h e - l i m i t expand W(p) as follows X - qY + | > 0 U i o) (A. 13) Y 132 w ( p ) = f+ Z + w o ( p ) ( A - 1 4 ) 0) 0 where W (p) has poles only i n the l e f t hand plane. The expan-sion (A. 14) can be carried out since, i f G(p) is a positive real function, i t can have only simple poles (with positive residues) on the imaginary axis. The residues of the function G(p) for the poles p = 0 and p = jto are respectively b, (1 + qjcj)u — / = b, 03=0 d + je 0 o •— (p-jco )(l+icoq) / = d -qe to +i (e +qd to ) ^ r J o J M ; / o M o o J o M o o P - 1 0 ) 0)=0) r J o o therefore b± > 0 e o o 2 e d + > 0 0 ^ which shows d > 0 o Hence, since b^ > 0 and dQ> 0 we are assured of s t a b i l i t y - i n - the- l imi t . ( s e e reference 4.) The above description of the Popov S t a b i l i t y Criterion includes only the salient features of this method. For an extensive, detailed treatment the reader is referred to reference 4. REFERENCES 1 3 4 I . Hahn, W., Theory and Application of Liapunov's Direct Method, Prentice Hal l Inc. , Englewood C l i f f s , N . J . , 1963. 2„ La Sal le , J . , and Lefschetz, S„, S tabi l i ty by Liapunov's Direct Method, Academic Press, New York, N . Y . , 1961, 3. Minorsky, N„, Nonlinear Osci l la t ions , D. Van Nostrand Co. Inc. , Princeton, N . J . , 1962. 4. Aizerman, M . A . , and Gantmacher, F . R . , Absolute S tabi l i ty of Regulator Systems, Holden-Day Inc. , San Francisco, T96f. 5. Zubov, V . I . , Mathematical Methods for the Study of Automatic Control Systems, The MacMillan Co . , New York, 1963. 6. Cunningham, W.J . , Introduction to Nonlinear Analysis , McGraw-H i l l Book Co. Inc. , Toronto, 1958. 7o Truxal, J . G . , Automatic Feedback Control System Synthesis, McGraw-Hill Book Co . , Inc. , New York, 1955 8. Kalman, R . E . , and Bertram, J . B . , "Control System Analysis and Design v ia the Second Method of Liapunov", Transactions, American Society of Mechanical Engineers, Series D, V o l . 82, June 1960, pp. 371-393. 9. Margolis, S . G . , Computations of Region of Asymptotic S tabi l i ty for Closed-Loop Systems by'Liapunov's Second Method, Ph.D. Thesis, University of Pittsburgh, 1961. 10. Beckenbach, E . F . , Modern Mathematics for the Engineer, McGraw-Hill Book Co . , Inc . , New York, 1956. II. Rybashov, M . V . , "Analog Solution of Algebraic and Transcendental Equations by the Gradient Method", Automation and  Remote Control, V o l . 22, No. 1, pp. 66-76, 1961. 12. Gibson, J . E . , Nonlinear Automatic Control, McGraw-Hill Book C o . , Inc.,""Toronto, 1963. 15. Rodden, J . J . , "Numerical Applications of Liapunov S t a b i l i t y Theory", Joint Automatic Control Conference, 1964, pp. 261-26T7 14. HiIdebrand, F . B . , Advanced Calculus for Engineers, Prentice-H a l l , Tnc.THewTork, 1954. 15. Margenau, H„, and Murphy, G . M . , The Mathematics of Physics and Chemistry, D. Van Nostrand Co. , Inc. , Princeton, N . j . " , l 9 5 7 . 135 16. Krasovskii, N.N., Stability of Motion, Stanford University Press, Stanford, California, 1963. 17. Cunningham, W.J,, "An Introduction to Lyapunov's Second Method", Transartions on Applications and Industry Ajperican_Innitute of Electrical Engineers, No. 58, 18. Vogt, W.G., "Transient Response from th© Lyapunoy Stability Equation", Joii^.AjrtgM^ic.,Cwitrol Conference, X96S, pp. 23-30* 19. Sandberg, I.W., "Some Results on th© Theory of Physical Systems Governed by Nonlinear Functional Equations", The Bell System Technical Journal, Vol. XLIV, No. 5, 20. Sandberg, I. W., "On the L2"Houndedness of Solutions of Nonlinear Functional Equations", The Bell System Technical Journal, Vol. 43, J u l y' I W , " pp.' 1$81-1600. 21. Brokett, R.W., and WiHems, J. L., "Frequency Domain Stability, Part 1", Joint Automatic Control Conference, pp. 735-741, 1965. ~" r 22. Volterra, V., Theory of Functional and of Integral and Integro-Differential Equations, Dover Publications Inc., New York", 1959. 23* Barrett, J.F., "The Use of Volterra Series to Find Region of Stability of a Nonlinear Differential Equation", Inte^rfiationa Control, March, 1965, Vol. 1, NdrT,"ppT"2(39,-2l6. 24. Kolmogorov, A.N., and Fomin, S.V., Elements of the Theory of Function and Functional Analysis, Vol. 1, Greylock Press, Rochester, N.Y., 1957. 25. B r i l l i a n t , M.B., Theory of the Analysis of Nonlinear Systems, Massachusetts Institute of Technology, Research Laboratory of Electronics, Technical • Report 345, March, 1058. 26. Ceorga, D.A., Continuous Nonlinear Systems, Massachusetts Institute oFTochnoTogy, Research Laboratory of Electronics, Technical Report 355, July, 1959. 27. Des our, CA., "Nonlinear Distortion in Feedback Amplifiers", Transact'! ons on Crrcuit Theory». .Thg .Institute_p_£ .Radio, Engrneofi, voT."TTr9~J N o T T p S c h , 1962, pp. 2-6. ~ 28. Zames, G,, ''Functional Analysis Applied to Nonlinear Feedback Systems," Tranp_<^ions' on.,Circjat.Theory, The Institute of K l ^ ^ ^ ^ ^ ^ ^ e c f r ^ ^ ^ ^ ^ ^ e r s , VcT^wUf-^0 '7"T^.^^ ,September, 1963, pp. ^ 21-404. 136 29. Youla, D . C . , "On the S t a b i l i t y of Linear Systems," Transactions on Circui t Theory, The Institute of  E l e c t r i c a l and Electronics Engineers, V o l . CT-10, No. 2, June, 1963, pp. 276-279. 30. Desoer, C A . , "A S t a b i l i t y Criterion Obtained by a Method of Comparison", Transactions on Automatic Control, The Institute of E l e c t r i c a l and Electronics Engineers, A p r i l , 1965, V o l . AC-10, No. 2, pp. 185-186. 31. Parente, R .B . , Functional Analysis of Systems Characterized by Nonlinear Different ia l Equations, Ph.D."Thesis, Massachusetts Institute of Technology, August 23, 1965. 

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