Technical Report No. 152 5488- 1-T STABILITY OF INTERCONNECTED SYSTEMS by F. N.,,Bailey Approved by: r B.- F. Barton for COOLEY ELE CTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor The National Science Foundation Grant Proposal No. GP-540 Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan July 1964 THE UNIVERSITY OF MICH!GAN ENGINEERING LIBRARY

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PREFACE I would like to acknowledge my indebtedness to my colleagues, both past and present, at the Cooley Electronics Laboratory of the Department of Electrical Engineering, who have contributed immeasurably to my education at The University of Michigan. I am especially grateful to Professors Arch W. Naylor and B. Fred Barton for their continued advice, criticism and encouragement throughout my graduate program. In addition, I wish to thank the members of my doctoral committee, especially Professor Keki B. Irani, for their comments and suggestions throughout the course of this study. I also want to thank Professor Wilfred Kaplan for his comments on the first draft, Mr. Mark J. Damborg for reading the final draft and Miss Ann L. Zorn and Mrs. Lisa Shellman for their invaluable assistance in the preparation of the manuscript. Finally, I want to thank my wife, Joyce, for her patience and for "taking such good care of the ducklings."

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TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv CHAPTER 1. INTRODUCTION 1 CHAPTER 2. BACKGROUND AND SUMMARY 6 2. 1 Historical Background 7 2. 2 A Coordinate Transformation 15 2.3 Lyapunov's Second Method 17 2. 4 Review of Current Literature 22 2. 5 Detailed Summary of Results 32 CHAPTER 3. PRELIMINARY CONCEPTS 42 3. 1 Notation 42 30 2 Composite Systems, Transfer Systems, and Models 44 CHAPTER 4. STABILITY DEFINITIONS AND THEOREMS 49 4. 1 Stability and Boundedness of Solutions of Differential Equations 49 4, 2 Sufficient Conditions for Stability and Boundedness of Solutions of Differential Equations 54 CHAPTER 5. PROPERTIES OF TRANSFER SYSTEMS 65 50 1 Gains for Transfer Systems 65 5. 2 Exponential Stability and Class E 71 CHAPTER 6. STABILITY OF COMPOSITE SYSTEMS 78 60 1 Two Simple Composite Systems 78 60 2 Complex Composite Systems 86 60 3 Examples 90 CHAPTER 7. CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 106 70 1 Conclusions 106 70 2 Suggestions for Future Research 108 APPENDIX. PROOF OF THEOREM 5.2 111 BIBLIOGRAPHY 116

LIST OF ILLUSTRATIONS Page Fig. 1 Stability of an equilibrium point. 10 Fig. 2 Stability of a motion. 12 Fig. 3 A geometric interpretation of LSM when n= 2. 20 Fig. 4 Lur'e's problem of direct control. 28 Fig. 5 Illustration of the concept of a composite system. 33 Fig. 6 A nonlinear characteristic leading to exponential stability- in- the- large. 3 8 Fig. 7 Network for Example 3. 1. 44 Fig. 8 Graph of s[M(U )] versus a for a transfer system in Class G. 68 Fig. 9 Two simple composite systems. 78 Fig. 10 A complex chain. 95 Fig. 11 Composite system for Example 6. 3 98

CHAPTER 1 INTRODUCTION As engineers and scientists have become interested in increasingly complex dynamic systems (physical, biological, economic, etc. ) involving many degrees of freedom and many nonlinear relations, the importance of the so-called "stability problems" has become increasingly evident. At the same time, the area encompassed by the term stability has continually broadened to include a larger and larger group of problems. Of the many reasons for this increasing interest in stability problems, there are two that are of particular significance in engineering applications. The first reason for this rise of interest in stability follows from the older, parochial view of stability where a system is stable if the perturbing effects of small disturbances or parameter variations are also small. This might be called stability in the narrow sense. Certain features of the mathematical modeling techniques commonly used in engineering are responsible for the importance of this narrow concept of stability. The engineer must be continually aware of the approximations implicit in all mathematical models. At best, a useful mathematical model can represent only a few of the features of the system under study. Some features must be neglected and the analyst would like to be certain that these features are indeed negligible; that is, he would like some assurance that his approximations will not lead to erroneous results. Similarily, the designer must choose tolerances that are both reasonably loose to reduce cost and sufficiently tight to insure stability of operation. He

must also insure that small random disturbances do not produce uncontrollable or unstable operation. At this level, both the analyst and the designer are concerned with stability in the narrow sense. These examples lie in what is sometimes called the area of structural stability, an area that has received considerable emphasis in the Soviet literature [Refs. 7, 18, 28]. The second reason for the rise of interest in stability is related to the increasing interest in the "qualitative approach" and the significance of stability problems in the qualitative theory of differential equations [Refs. 34, 35]. One of the full':a>:::enta1 prnoblems in the theory of differential equations is that of describing the properties of the solutions from the form of the equations. This is also a fundamental problem in the applications where differential equations are frequently used as mathematical models. In the simplest cases, the properties of the solutions are obtained by simply integrating (solving) the differential equations. Unfortunately, this procedure is rarely possible in problems of practical interest and one is forced to accept a qualitative description of the solutions as a compromise with the formidable difficulties of exact analysis. Sovre of the central problems of the qualitative theory involve the determination of such properties as asymptotic behavior, boundedness, relationships between neighboring solutions, and effects of disturbances that are not small. These, and similar properties, are frequently classed as stability properties, and are part of what might be called the stability problem in the broad sense [Refs. 4, 6, 41].

One of the effects of this broader view of the stability problem is the enlargement of the scope of the term "stability. " In the narrow view, stability implies an invariance of behavior in the face of perturbations or parameter changes. In this case, a few relatively simple definitions of stability are appropriate for a large class of problems. On the other hand, with the development of the qualitative theory a more pragmatic definition of stability appears [ cf Ref. 16]. There is now a growing tendency to assign the term stable to any desirable behavior (solution) and the term unstable to any undesirable behavior [Ref. 41]. The uninitiated is soon dismayed to find that there are now a considerable number of different stability definitions which appear to be unrelated though there are connections between them [Ref. 6] Once the stability problem is clearly defined, it is necessary to obtain analytic techniques for determining whether the desired stability actually exists. In this area there are many special results, but the most promising general techniques are found in the group of ideas known as Lyapunov's Second (or Direct) Method [Ref. 27]. While originally developed for stability analysis in the narrow sense, Lyapunov's Second Method (abbreviated, LSM) is, in fact, a powerful tool applicable to most of the problems in the broader sense of stability and a wide class of problems in the qualitative theory [Ref. 16]. In its simplest form LSM involves the construction of a generalized metric on the solution space and the analysis of solution behavior in terms of this metric. Once the proper metric has been chosen, the general form of the solutions and

their stability properties are readily determined. This approach is both conceptually satisfying and mathematically rigorous, but in its application there remains one practical difficulty of major importance —the selection of the proper metric (commonly called a "Lyapunov function"). This research attempts to reduce the difficulty of this selection in a particular class of problems; namely, the analysis of the stability of interconnected systems. In the majority of situations, the application of Lyapunovts Second Method (mainly a problem of finding the proper metric) becomes more difficult as the system dimensionality (number of energy storages or degrees of freedom) increases. While many of the current papers on the generation of Lyapunov functions imply that the procedures described will apply to nth -order systems [Refso 13, 183 29, 38], a test on problems of high dimension frequently reveals that only the third- or fourth-order problems worked as examples can be treated with reasonable facility. This limitation, coupled with the fact that many of the higher dimensional systems encountered in practice are actually, or can be considered as, a composite or interconnection of several lower order systems, suggests that one consider a piece-by-piece stability analysis; that is, a separation of the composite system into simpler subsystems to which LSM can be easily applied. The results of this piece-by-piece analysis might then be used to infer stability properties of the original composite systema This thesis presents the results of the application of this piece-by-piece concept of stability analysis. The problem to be considered may be

formally stated as follows: Obtain information about the stability of composite systems from a study of the properties of their subsystems and the topology of their interconnections. The successful solution of this problem leads to a circumvention of the formidable difficulties involved in the direct determination of the stability of high-order nonlinear systems.

CHAPTER 2 BACKGROUND AND SUMMARY This chapter serves the two-fold purpose of (1) putting the stability problem in its proper historical and technical background, and (2) summarizing the new concepts and results to be presented in greater detail in Chapter 3. A limited amount of mathematical notation will be introduced as needed throughout this chapter. However, the emphasis will be on clarity of exposition rather than on careful or rigorous mathematical analysis. For a thorough treatment, the reader is referred to the bibliography and to the following chapters of this thesis. The notation introduced in this chapter is consistent with that used throughout the remainder of this thesis. A detailed discussion of mathematical notation is given in Section 3. 1. Let En be an n-dimensional vector space of vectors x = col[xl,x2, n 1 0.. Xn] with the inner product (xy) = Z xiYi and norm Ixl = (x,x)2. The transpose of a vector is indicated with a prime as x'. In general, x = f(x, t) is a vector differential equation (x and f are n-vectors) with a vector solution x(t;xo, t ) passing through the point x = x at t = to O0 A solution path in En is sometimes called a motion referring to the motion of the point x in this space (state space) as t increases. The solution x(t) O is called the null solution.

2o 1 Historical Background As scientists study increasingly complex systems, the use of mathematical models becomes increasingly necessary. If the model provides an adequate description of the system, it is possible to predict system behavior from manipulation of the model. However, it was noticed long ago that certain forms of behavior predicted by usually valid models were never observed in the actual system. A common class of models for many systems are ordinary differential equations. In a large number of practical problems a state variable description [Ref. 46] of the system under study leads to a vector differential equation model of the form y = f(y,t). (21) A point y in En where fy, t) = 0 for all t is called, for obvious reasons, an equilibrium point of the model and (for a valid model) corresponds to an equilibrium state for the original system. Early'in the study of mathematical models for mechanical systems it was noted that certain equilibrium states predicted by an otherwise valid model were never observed in the physical system being modeled [Refo 7]o A typical example is a small sphere rolling on the outside of a larger, fixed sphere in a uniform downward gravitational field. According to most simple models, there is an equilibrium point at the top of the large sphereo These models predict that if the small sphere is placed at this point, it will remain there~ On the other hand, this equilibrium state is never observed in practice.

Experiences such as this suggested that there must be some additional property associated with those equilibrium points of the model which corresponded to observable equilibrium states. Eventually, it was recognized that this additional property was stability; the observed equilibrium positions were stable and the unobserved equilibrium positions were unstable. It was also recognized that the prediction of these unstable equilibrium states was due to an improper idealization in the mathematical model; i e., certain disturbances were neglected that were, in this cases not negligible. This was the beginning of stability theory in the narrow sense. In the example of the small sphere rolling on the outside of the larger sphere, the equilibrium point at the top of the large sphere is unstable;" that is, small deviations in the initial location or small vibrations neglected in the model produce large deviations in later positionso Since these small deviations or vibrations can never be eliminated in the real world (as opposed to the mathematical world)9 the event of the small sphere remaining at or very close to the top of the large sphere is never observed. (This is actually an example of a strong form of instability sometimes called complete instability [Refo 12] ) Because of these experiences9 some of the first systematic studies of stabllity emphasized stability of equilibrium positions. One common stability definition was what is now called Lyapunov stability or stability in the sense of Lyapunov[Refo 6]. According to this defin ition9 an equilibrium state y in an n-dimensional state space En is

considered stable if any motion y(t) can be kept in an e-neighborhood of the equilibrium y, for all future times simply by starting it in some 6-neighborhood of the equilibrium That is, if for every e > 0 there is a 6 > 0 such that I/y(t0) - ylI < 6 implies ly(t) - yII < e for all t > t (see Figo 1 ). An equilibrium point that is not stable is said to be unstable. This is the well-known ball on a smooth plane concept of stability commonly used even today. Even though it presently bears his name, this definition was actually considered long before Lyapunov's time in connection with the stability of equilibrium points of conservative systems [Ref. 6]. In 17889 Lagrange gave one of the early stability theorems when he proved that for a conservative system an equilibrium position where the potential function has an isolated minimum9 is stable (in the above sense) [Ref. 20]. In practice9 the concept of Lyapunov stability is found to be less important than asymptotic stability. Asymptotic stability requires Lyapunov stability plus convergence to the equilibrium of all solutions starting sufficiently close; that is9 for some 60 > 09 it holds that Ilyo - yil < 6 implies that lim ly(t;yo,to) - = 0. This is the type t-0 0 of stability exhibited by a ball at the bottom of a hemispherical cupo A related problem is the stability of a motion rather than an equilibrium point. An unstable motion predicted by a mathematical model will also be unobservable in the physical system. The same reasoning applies as in the case of the unstable equilibrium point9 and a stable motion is defined as a generalization of the stable equilibrium

Y2 Y2 --- y2o~~~~ Y1 Y1 Fig. 1. Stability of an equilibrium point.

11 point concept. For each a in a set A let y(t;a) define a motion (a curve parameterized by time) in En. A particular motion y(t;a) corresponding to a particular a in A is said to be stable in the sense of Lyapunov if all neighboring motions can be kept in a tube of radius e centered on y(t;a) by starting them at time to in a disc of radius 6 centered on y(t;a). That is, the motion y(t;a) is Lyapunov stable if for every E > 0 there is a 6 > 0 such that lly(to;a) - y(to;a) II < 6 implies Ily(t;a) - y(t;a) II < c for all t > t. In addition, the motion y(t;a) is asymptotically stable if the - O motions sufficiently close at to converge to y(t;a) as t increases. That is, if for some 60 > 0 it holds that Ily(to;a) - y(t;a) II < 60 implies lim Ily(t;a) - y(t;a) II - 0, [Ref. 12]. Once again, a motion that is not t-o 0 stable is termed unstable. In problems where the mathematical models are differential equations, the parameters, a, are usually the initial condition vectors, Yo. Such a situation is illustrated in Fig. 2. In 1892, A. M. Lyapunov showed that all problems of stability of an equilibrium point and stability of motion can be reduced to a single problem of the stability of the null solution (an equilibrium point at the origin) of a special equation called the equation of perturbed motion [Ref. 27]. Moreover, he suggested an ingenious method for solving this particular problem —a method of such versatility that its full potential has not yet been realized. Lyapunov considered only two types of stability: what is presently called Lyapunov stability and asymptotic stability. It is worthwhile to note the similarity between these concepts of stability and the very fundamental mathematical concepts of continuity and convergence.

Y2 - y(t to )7 Y20 - = _ _ /Yo y t Fig. 2. y(t; Yoto) Fig. 2. Stability of a motion.

Stability of an equilibrium point is actually a statement of the continuity at that equilibrium point, of the solutions in the initial condition yo and the uniformity of this continuity with respect to time. Similarily, asymptotic stability includes a statement of the convergence of the solutions to the equilibrium point. The fact that these two definitions of stability (and their variations to be described below) are so successful in such a wide variety of situations is undoubtedly due in part to this close relation to the very fundamental concepts of continuity and convergence. The works of Lyapunov [Refo 27] and some of the older Soviet authors such as Chetayev [Ref. 7] emphasized the narrow viewpoint in stability theory. With the rise of the qualitative approach and the emphasis of the broader viewpoint, new stability definitions began to appear. The Lyapunov stability and asymptotic stability definitions were extended in conformity with the various continuity and convergence concepts of analysis. The earlier results were equi-stability, uniform stability, uniform asymptotic stability, etc. [Refso 169 31]. More recently, the concepts of boundedness [Ref. 45], stability-in-the-large [Ref. 18], regions of asymptotic stability [Refo 21], practical stability [Ref. 22], and ultimate stability [Refo 23], have also been introduced by authors interested in studying particular qualitative properties. 1 Many of these concepts represent attempts to remedy the fact that asymptotic stability in itself states only that there is a nonzero region of attraction (region of initial states for which the perturbed motions will converge to the unperturbed motions). In the majority of practical problems9 an estimate of the size of this region (the

It is remarkable that the basic techniques developed by Lyapunov in 1892 are still applicable to a majority of these new stability concepts introduced in the qualitative approach. The concept of "bounded input gives bounded output" stability introduced by James, Nichols and Phillips [Refo 14] is an example of an apparently different stability involving an input-output relation. Yet even this type of stability can be treated by LSM (see Section 5. 1, below). In summary then, the stability problem as considered here is basically a problem in mathematics with important engineering implicationso It is, in the most general form, a problem of ascertaining certain significant features of the solutions of a given mathematical model. In this thesis the mathematical models are assumed to be ordinary differential equations. The engineer can contribute to this basically mathematical problem by (1) suggesting solution features of particular significance, or (2) suggesting methods of attack motivated by physical experience. This thesis attempts to contribute in the second area. region of asymptotic stability) or an assurance of a region of at least a certain minimum size is required [Refo 22]. One of the most common remedies is to seek asymptotic stability-in-thelarge (abbreviated ASL) where the region of attraction is the whole space~ The practical importance of this property has motivated a considerable number of attempts to obtain conditions guaranteeing asymptotic stability-in-the-large in different types of systems [Refs. 2,3,251.

2. 2 A Coordinate Transformation In his now famous dissertation of 1892 [Ref. 271, Lyapunov introduced a simple method for reducing a broad class of problems in stability of equilibrium points and stability of motion to one basic problem. The procedure is the following. First, consider a differential equation = ~ g(y t) (2 2) and assume that y = y is an equilibrium point; that is, assume that g(0,t) = 0 for all t. Now let x(t) = y(t) - yso that x - - = g(x +,t) - g(,t) g(x + yt) f(x, t). (2 3) The new equation x= f(x,t), (2.4) where f(x, t) is defined in (2. 3), has an equilibrium point at the origin since f(O, t) = g(0 + -, t) = 0 and this equilibrium point will have the same stability properties as the equilibrium point y of (2~ 2). Next, let y(t) be a known solution of (2. 2) and choose x(t) = y(t) - y(t). Now x= y - y = g(x +,t) - gd(,t) f(x, t) (20 5) and the new equation x f(xt), (2. 6) where f(x, t) is now defined in (2. 5), also has an equilibrium point at the origin since f(0, t) = g(0 + y, t) = 0. Moreover, the stability properties of this equilibrium point of (2n 6) indicate the stability properties of the motion y(t,) The same notation is used for the rlght-hand sides of

(2. 4) and (2. 6) to emphasize the fact that both problems have been reduced to the same form: a differential equation with an equilibrium point at the origin, Lyapunov calls y or y(t) the unperturbed motion and (2. 4) or (2. 6) the equation of perturbed motion (it is actually an equation describing the perturbation as a function of time). In this terminology, a neighboring solution to y or y(t) such as y(t) is called a perturbed motion. The transformation introduced by Lyapunov reduces all problems concerning stability of motions or stability of equilibrium points of differential equations to a single problem concerning the stability of the equilibrium point x = 0 (the null solution) of the equation of perturbed motion x = f(x,t) f(O,t) 0. (2. 7) Because of this transformation, a major portion of the current literature on stability theory considers only the problem of stability of the null solution of (2. 7). It is assumed that all other problems can be reduced to this form. For similar reasons, this thesis will consider only the problem of determining the stability of the null solution of an equation in the form of (2, 7). Having made the above simplification, Lyapunov went on to describe methods of solving the problem of stability of the null solution of the equation of perturbed motion. He considered all techniques for solving this stability problem as divided into two classes. Those that require a determination of the solutions of the equation of perturbed

17 motion are put in the first class which includes the techniques of linear approximation, direct integration, series solution, etc. The second class contains only methods which do not require direct determination of the solution but only a determination of its properties from the form of the differential equation (qualitative methods). Lyapunov's first method was then a technique, basically a series solution technique, illustrating the first class. While not without merit, this first method (for a discussion of the first method, see Cesari, Ref. 6), does not appear to have the versatility of his second method, a technique he included as an illustration of the second class. LyapunovTs Second Method (now sometimes called Lyapunov's Direct Method), has, in recent years, proven to be a tool of amazing versatility~ 2. 3 LyapunovIs Second Method In its original form LSM involves the use of a positive definite function (sometimes called a Lyapunov function2) as a generalized metric~ a measure of the distance from the origin of points in the state space. A continuous9 real valued function v(x) on En with continuous first partial derivatives is said to be positive definite [positive semidefinite] if v(O) = 0 and v(x) > 0, [v(x) > o0] for all x,0 O Similarly, This terminology is not uniform Many authors call a positive definite scalar function a Lyapunov function only if its total derivative meets certain requirements. In this thesis, unless otherwise noted, the terms Lyapunov function and posltive-definite functon are interchangable.

18 a continuous, real valued function v(x) on E with continuous first partial derivatives is said to be negative definite [negative semi-definite] if v(O) - 0 and v(x) < 0 [v(x) < 0] for all x A, 0. The application of LSM amounts to the choosing of a particular positive definite function v(x) to establish a generalized metric on the state space and then testing its time derivative along the solution paths of the equation to determine whether the solutions are diverging from or converging to the origin. For example, consider the equation of perturbed motion, x = f(x) (2 8) with f(O) = 0. Choose a positive definite function v(x) and then3 v(x) idt ~. (2.9) i 1 Now along the solutions of (2~ 8) dx. _ = fi(x), (2 10) and thus the time derivative of v along the solutions of (208) is (x) = E av f(x) (2 11) i 1 3 If the right-hand side of (2. 8) is time depende int, it may be necessary to choose a time dependent Lyapunov function v(x, t). The terms positive definite, etco then require slightly different definitions which are given in Section 3o 5o In this case xQ(xt) = -8 fi(xt) + -- i 1

Equation (2. 11) is frequently called the total derivative of v with respect to (2. 8). The important fact is that xv (x) is determined directly from the right-hand side of (2. 8). No reduction or solution is necessary. It can now be shown that if v (x) is negative semi-definite9 then the null solution of (2~ 8) is stable in the sense of Lyapunovo If,v(x) is negative definite, then the null solution is asymptotically stable. An even stronger form of stability can be assured. if v(x) is chosen so that lim v(x) = oo. In Ilx - 0 this case when,v (x) is negative definite the null solution is ASL. These are special cases of the theorems discussed in detail in Section 4.'2 (also see Ref. 12). A geometric interpretation of these results in the case n = 2 is given in Fig. 3. Since v(O) 0 afd iv(,) is econtinuous with v(x) > Q for x A 0, the v constant curves for small values of the constant must be closed curves encircling the origin. The solution path9 coming in from the left (Fig. 3)9 apparently crosses the v(x) = constant curves in such a way that,v (x) will be negative. If v (x) is negative for all x A 09 then the solution must be proceeding into smaller and smaller v constant regions and thus approaching the origin. This suggests asymptotic stability. Similarily, if v is only negative semi-definite, then the solution must be at least staying inside the v O constant contours in which it started. Otherwise, v would at some time be positive. This suggests Lyapunov stabilityo If v(x) has been chosen so that lim v(x) = co, then the v = constant curves are closed curves encirIlx Il- o cling the origin for all values of the constant. In this case9 these curves

20 0'tt X(t2> > > t~~~~c3 (tv t 3h ge~o metric~~ tC3 /~~~~~~~~~~ x(,~ ~~ Vcs )~~~~ (t1 5 t > t > t tz ti 3 c4 ~~~~~~~~~~~ 0 at the origin Fig,3,,A g omeric interpretation of LS W en n=2.

"gradually" fill the entire plane as the constant increases. If v (x) is now negative for all x, 09 solutions starting at any point in the plane must converge to the origin; this is ASL. These simple geometric concepts are really the heart of LSM. More sophisticated procedures are necessary in the application of these concepts to complex problems, but the basic idea of using a generalized metric and studying the derivatives of the generalized distance along the solutions carries through in every case. Once the above example is understood9 the advantages of LSM become obvious. The uses of this technique are limited only by the imagination of the user and his ability to analyze the results. Stability9 location of solution curves9 approximation of solutions and many other problems'in the qualitative theory are all readily visualized as problems in the selection of Lyapunov functions and the studying of their derivatives. Unfortunately, this selection problem is quite difficult and the results obtained are often strongly dependent on the Lyapunov function chosen. While there are many results showing that the necessary Lyapunov functions exist [Ref. 121, there are few results showing how to find them. One of the reasons for this problem can be seen in Figo 30 It is clear that a different choice of Lyapunov function9 say9 where the v - constant curves are circles, would not have resulted in vr being negative definite along the solution path even though it did, in fact9 converge to the origin. This is a common problem that arises because the theorems provide only sufficient conditions and thus negative results are inconclusive0

22 In Fig. 3, as is often the case, it is necessary to choose the Lyapunov function so that the v = constant curves are in some sense similar to the solution paths. Since the difficulty in doing this increases with the order and complexity of the differential equation, the problem of choosing Lyapunov functions is a major stumbling block in the application of LSM to complex high order systems. One of the aims of this thesis is a simplification of this problem for a special class of systems. 2. 4 Review of Current Literature There are many methods for analysis of the basic stability problem, namely, the problem of the stability of the null solution of the equation of perturbed motion. However, many of these procedures are special techniques developed for a special class of problems. For instance, for linear constant coefficient systems there are the eigenvalue location techniques such as the Routh-Hurwitz criteria [Refs. 6, 17, 43]. For general linear systems there are the techniques of functional analysis. For second-order equation there are the phase-plane techniques. When considering only local stability (such as Lyapunov stability), there are a number of results obtained through the use of integral equations. While each of these procedures has its advantages, the problem considered in this thesis, a general study of interconnected linear and nonlinear systems, requires a unified approach to the stability analysis of each "piece. " Moreover, it is important to be able to consider stability in the broad sense (such as asymptotic stability-in-the-large)

23 since local stability is of doubtful value in practical applicationso In view of these requirements, LSM has been selected as the basic tool for the analysis which follows. As noted above, the application of LSM in stability analysis involves the frequently tricky problem of choosing the proper generalized metric or Lyapunov function. Since the results often depend on the particular Lyapunov function chosen9 this choice is frequently a very difficult problem requiring, at present9 a considerable amount of ingenuity9 intuition9 experience or whatever else the analyst can call on for aid. Moreover9 the difficulty of this.choice is, in general9 strongly dependent on the order of the differential equation under analysis with further complication arising when stronger forms of stability are sought. In this thesis asymptotic stability- in-the-large is of major interest. There are a number of rules and procedures for the construction of Lyapunov functions available in the current literatureo Most of these procedures are, in theory, applicable to the construction of Lyapunov functions for systems of arbitrary ordero However, the user of these procedures finds that while the results obtained for second-9 third-, and even fourth-order systems are frequently quite impressive9 there are formidable computational problems encountered when attempting to handle9 say, tenth-order systemso This order limitation on presently available procedures has led to the separation concepts gthe idea of separating a high order system into several lower order systems) to be introduced below. However9 before going on to this new approach,

24 it is worthwhile to emphasize the problems encountered in treating high order systems by considering the currently available procedures in more detail. First, consider the linear, constant-coefficient differential equation. This is one of the few cases where there are clearly defined procedures for problems of any order. The following theorem is due to Lyapunov [Ref. 27]1 Theorem 2. 1~ The equilibrium solution x = 0 of the differential equation x = Ax (2. 12) is asymptotically stable (in-the-large, of course, since the equation is linear) if and only if given any symmetric, positive definite matrix4 Q, there exists a symmetric positive definite matrix P such that Al P + P A -Q o (2~ 13) The proof of sufficiency is obvious if one uses the Lyapunov function v(x) = x'Px so that Vv = 2Px and thus 5 4 A symmetric matrix S is said to be positive definite if the function v(x) = x'Sx is positive definite (see Section 2. 3). 5 [ av avi The vector Vv is defined as Vv = col[ aX o -] It is useful ax1 x in working with Lyapunov functions since (x)- avf (x) = (Vv, f(x))..i 1

25 v(x) = (VvAx) = 2xPAx x'(P A+ AiP) x. (2. 14) The simplicity of this result has challenged a number of authors to find similar results for nonlinear systems. Krasovskii [Ref. 19] gives the following theorem for the general nonlinear differential equation x = f(xt) with f(O t) 0 (2. 15) where f is continuous on En and has uniformly bounded first partial derivatives in every bounded neighborhood of the origin. Theorem 2. 2: Let J(x, t) be the Jacobian matrix fI (x t) J(xt) ax.j (2o 16) of (2. 15) and P be a positive definite symmetric constant matrix. Then the null solution x = 0 of (20 15) is asymptotically stable-in-the-large if J(x, t) P + P J(x, t) = -Q(x, t) (2. 17) and the eigenvalues Xi(x t), (i = I,.., n), of Q(x, t) are greater than some constant y > 0 for all x and to The proof of this theorem in the autonomous case (where f is independent of t) is easily completed with the Lyapunov function v(x) = fI P f [Ref. 12]. In the general case, a more involved argument using the Lyapunov function v(x) = x'Px is required [Refo 13]. Another variation on the linear, constant-coefficient procedure for nonlinear autonomous equations is suggested by Ingwerson [Ref. 13]

26 who chooses a positive definite symmetric matrix C and solves the equation J'(x) P(x) + P(x) J(x) = -C (2. 18) for P(x). This general P(x) is then modified by dropping the dependence on all but the variables x. and x. in the ij term. Then Pij' the ijth term of P(x), is a function only of xi and x.. The components of the gradient vector, are then computed from this new matrix, P(xi, xj), as X1 X2 x av f P l(x xil)dxl + S P 2(xix2)dx2++ S P. (X Xn) dx 1 0 0 0 (2. 19) and v(x) is computed from Vv as a standard line integral. Schultz and Gibson [Refo 38] have proppsed a related procedure they call the Variable Gradient Method. This involves choosing Vv directly in a special form that ensures that it is the gradient of some scalar v(x). The constants in Vv can then be adjusted to make v~ (x) = (Vv, f(x)) negative definiteo Stability information is obtained from a study of v(x) which is again obtained by a line integration on Vv. It is clear that all of these procedures involve formidable difficulties when applied to high order systemso Just the solution of the algebraic problems in (2. 17) and (2. 18) becomes extremely difficult since numerical techniques cannot be used. While Schultz and Gibson avoid this problem, they have another problem in checking that the Vv chosen is a proper gradient of some scalar. Even if this were not a problem, the interpretation and determination of positive definite properties in the resulting Lyapunov functions becomes very difficult for high order systemso

27 A different approach is taken by Zubov [Ref. 47]. For the autonomous system x = f(x) with f(0) = 0, he obtains a Lyapunov function by solving the partial differential equation av(x) f(x) - (x)[1 v(x)] [1 + If(x) 2] 2 0[ (2. 20) i= axo i i1 l 1 The function 0(x) must be positive definite on En but is otherwise arbitrary. In addition, the solution must be such that -1 < v(x) < 0 for all x, 0 (Zubov works with negative definite Lyapunov functions)0 This approach at least suggests a systematic procedure for determining v(x), but, as yet, no one has been able to utilize this fact to significant advantage in higher order systems [Refs. 29, 42]. An important step in the direction of simplifying the use of LSM has recently been developed by Sell [Refo 39]. He notes that in certafi problems it is possible to show that particular types of stability exist if and only if every Lyapunov function in a given class has a corresponding property. This eliminates the problem of choosing a Lyapunov function and replaces it with the (hopefully easier) problem of analyzing the properties of some standard Lyapunov function along the solutions of the equation through a study of Vo Unfortunately, Sell's work deals mainly with local stability problemso It remains to be seen whether similar procedures can be applied to generalized stability concepts such as ASLo

28 Problems specifically related to automatic control have also received a considerable amount of attention. The best known of these are probably the problems of Lur'e [Ref. 25]. Lur'e's problem of direct control is concerned with a system modeled by r x= Ax + pf(a) a (y p b' x (2. 21) where A is a constant matrix, p and b are constant vectors, x is the system state vector and f(a) is a scalar function. In addition, it is assumed that f(O) = 0 and uf(a)> 0 for all a. (2. 22) This may be visualized as a single-input, single-output linear plant with a single nonlinear controller as shown in Fig. 4. Fg 4 Lfix - x-Ax + u' f(a) Af() Fig. 4. Lur'e's problem of direct control.

Lurte considers the so-called absolute stability problem of determining the element of b (with A and p given) so that the null solution of (2. 21) is ASL for all continuous f such that (2. 22) are satisfiedo This problem has received a great deal of attention from both Soviet and western authors [ see discussion in Ref. 12]. A common approach is to use LSM with the Lyapunov function v(x) x' Px + f f(0) d (2 23) where P is a constant, positive definite, symmetric matrix and: is a constant. The major problem in this approach is the determination of definiteness criteria for the scalar functions involvedo For higher order systems, say of order greater than sixth, this becomes very difficult [Refs. 12, 26]. The method can, in theory, be extended in several ways such as the inclusion of several nonlinear elements, but computational problems are then even more difficult [Refo 12]. For the problem with only one nonlinear element, a new approach utilizing the frequency domain behavior of the linear part (see Fig. 4), may simplify some of the computational difficulties [Refo 36]. In another control problem, Aizerman [Refo 1] considers a system modeled by x O Ax + f(x) (2o 24) where A is a constant matrix

30 fl(xj) 0 f(x) = 0 (2. 25) 0 f(0) = 0 and x. is one of the components of x. It is assumed that there is a range of some parameter a, say a < a < 3, such that the linear system obtained from (2. 24) by replacing fl(xj) by axj is asymptotically stable. In his famous conjecture [Ref. 2] Aizerman suggested that the nonlinear system (2. 24) might be ASL for an f (xj) such that ax 2 < xj f (x.) < ox.2 forx# 0. (2. 26) It has since been shown that this is not generally true even for thirdorder systems [Ref. 26]. However, it is still quite important to, ascertain when Aizerman's Conjecture does hold. This problem and its generalizations have attracted a great deal of attention in the control systems literature [Refs. 12, 18, 33] (see also a similar conjecture by Kalman, Ref. 15, and the discussion in Ref. 16), but complete results are currently available -only for second-order systems [Ref. 33]. A few special third- and fourth- order cases have also been considered (see summary in Ref. 12). (In Section 6. 3, the separation concepts developed in this thesis are used to verify a generalization of Aizerman's Conjecture in an nth-order problem with n nonlinear elements. )

31 In the recent mathematics literature a new viewpoint related to LSM has developed using the conceptof'the Auxiliary Equation [Refs. 5, 9, 10, 40]. In using the auxiliary equation to study the nth -order equation (2. 15), one seeks a positive definite function v(x, t) and a scalar function w(v, t) with w(O, t) = 0 such that along the solutions of (2. 15) v(x,t) < w(v(x,t),t). (2. 27) It can then be shown that, with certain reasonable restrictions, the stability properties of the first-order auxiliary equation r w(r,t) (2.28) correspond to the stability properties of the original system. The importance of this approach can be appreciated when it is realized that this reduces the problem of stability of the equilibrium of the n -order system to the problem of stability of the equilibrium of a first-order system. However, once again, there are still formidable problems involved in selecting a v' and an w(v, t) such that the relation v < w(v, t) can be realized and as might be suspected, the difficulty of these problems also increases with the order of the original system. Both Rosen [Ref. 37] and Sell [Ref. 40] have attempted to solve this problem by using the standard Lyapunov function v(x) = Il x 1l (this function does not have continuous derivatives so a modification of some of the original theorems is necessary). This allows Rosen to reduce certain stability problems to nonlinear programming problems.

32 In this thesis auxiliary equations are used, but the separation concepts reduce the problem of finding one suitable auxiliary equation for the entire system to the simpler problem of finding several auxiliary equations for the "pieces" of the system; i. e., a vector auxiliary equation. Thus, the choice of a single Lyapunov function is circumvented by reducing the problem of stability of the n h-order equation to the problem of stability of a system of several simpler first-order equations The form of this vector auxiliary equation depends on the topological interconnection features of the overall system. Apparently such an approach has not been considered in the literature to date. Thus, there are no previous developments of this problem to be reviewed. 2. 5 Detailed Summary of Results This section gives a detailed summary of the results to follow. It is intended first to provide an overall picture of the research for those who are not interested in the finer details and second, to help the prospective reader of the finer details obtain the necessary perspective. References will be omitted throughout this section since they are given in the detailed discussion which follows in Chapter 3. An important concept in the development is that of the composite system. A complex system obtained by interconnecting a set of simpler subsystems which will be called transfer systems because they will normally be input output devices. The composite system may be, say, a servo system and the transfer systems will be its parts: the motors,

33 amplifiers and transducers A basic assumption is that all transfer systems to be studied can be modeled by ordinary differential equations. A general transfer system model is Cx = f(x, t, u(t)) l y(t) = h(x(t),t) (2. 29) where x is the transfer system state vector, u(t) is the vector (or scalar) input and y(t) is the vector (or scalar) output. In a composite system the transfer systems are interconnected so that the outputs of some are the inputs of others. Thus the input, ui(t), for the ith transfer system in some composite system is a sum of the other outputs, say u.(t) = ~ B.ijY(t), (2 30) where the B.'s are constant matrices. An example is shown in the block diagrams of Fig. 5. u(t) y(t) S (a) A transfer system. u1 X3' = xX2 S 1 2 2 x3 ] g u3=x2 (b) A simple composite system made up of three transfer systems, S1, S2, and S3. Fig.. 5. Illustration of the concept of a composite system.

34 Note that this type of interconnection (as given in (2. 30)) implies the usual system theory assumption that the transfer system models are not affected by the interconnection structure. That is, there is no "loading" of one transfer system by another. With transfer systems clearly defined, it is possible to go on to study some of their properties using LSM. Of particular interest is the case where a transfer system has a region in its state space that is ASL. Here, the definitions of Lyapunov stability, asymptotic stability, and asymptotic stability-in-the-large have been generalized to include stability of regions. A region M, which is a subset of E, is said to be stable if for every e > 0 there is a 6 > 0 such that if the distance from Xo = X(to) to M is less that 6, then the distance from x(t;xo, t0) to M will be less than e for all t > t. Asymptotic stability of M requires that M be stable and also that for all x less than some distance 6 > O from 0o~ o0 M, the distance x(t;xo, t ) from M goes to zero as t - oo. For asymptotic stability-in-the-large, this must occur for all 60 taken arbitrarily large. These are basically generalizations of the usual definitions of the stability of the point x = 0. If the set M is put equal to { 0}, the usual definitions result. With the aid of these new stability definitions it is possible to define what will be called a "gain" for transfer systems. However, it is important to realize that this new gain is basically different from the gain considered in the frequency analysis of single-input, single-output transfer systems modeled by linear differential equations with constant

35 coefficients. For such systems, this new gain is similar to the maximum magnitude of the transfer function H(jw). For example, consider the case of a system modeled by the first-order (one pole) linear equation I x -ax + bu(t) y = x where x is a scalar. The transfer function is H(s) b and the maximum magnitude of the gain is the d-c gain b. In this particular case, the gain obtained by estimating the size of stable regions in the state space is also b (see Example 5. 1). In higher order linear a constant-coefficient systems the gain obtained from these state. space techniques will be greater than or equal to the maximum magnitude of H(jw). For other systems (multiple-input, multiple-output, nonlinear, time-varying, etc. ), this gain can best be visualized by considering the ratio A/B where A = the size of the region (the norm of the largest vector in this region) in state space to which all of the solution trajectories converge, B = the size (norm) of the input. The gain is an upper bound on this ratio that is valid for all continuous and bounded inputs. For a transfer system to have a gain of this type, it. must have regions in its state space that are ASL when the input has a certain size (norm) and the size of these regions should depend on the

36 size of the input. Transfer systems having these properties are said to be in Class G and the gain of a transfer system in Class G can be estimated using LSM and the concepts of regions that are ASL. A particular important subclass of Class G is those transfer, systems with models of the form:x = f(x, t) + Du(t) (2. 31) y = Hx (2.32) where H and D are constant matrices and the null solution of the unforced model x = f(x,t) f(0Ot) = 0 (2.33) is exponentially stable-in-the-large (abbreviated ESL). This means that there are positive constants kI and k2 such that for any xo E En the solutions x(t;xo, to) of (2. 33) satisfy the inequality ilx(t;xo, to) ll < k llxol exp { -k 2(t-to)). (2. 34) This subclass of Class G is denoted Class E. Class E is particularly appealing for several reasons. First, Class E includes systems modeled by (2. 31) and (2. 32) when: (a) f(x, t) = Ax where A is a stable constant matrix (all eigenvalues of A have negative real parts), (b) f(x, t) = A(t)x for a large class of variable matrices A(t), (c) f(x, t) is a member of an important class of nonlinear functions. This last statement in case (c) is purposely vague since a precise

37 characterization of the class of nonlinear functions for which (2. 33) is ESL is not available. However, it is easily shown (see Section 5. 2) that all f(x, t) where f(O9 t) = 0 and x'f(x,t) <-clcx112< 0 forx# 0 lead to ESL in (2. 33). The essence of this restriction is best seen in the first-order, time-invariant case where xf(x) < -cx2 < 0 for x 0. Graphically, this means that f(x) lies in the shaded sectors of Fig. 6. In higher dimensional situations the nonlinear characteristic is restricted in a similar fashion. While such characteristics are not completely general, they do represent a class of considerable importance. One obvious requirement is that f(x, t) have a nonzero slope at the origin. This type of nonlinearity is commonly encountered in applications where linear behavior is observed in a small neighborhood of some equilibrium but nonlinear effects become important when larger displacements are encountered. A second appealing feature of Class E is the fact that for any transfer system in Class E there is a Lyapunov function v(x, t) such that c1 lxii 2 < v(xt) < c211 xl (2. 35) and its total derivative with respect to (2. 31) is such that xr < ~av + yiluti K (2.36) where a, y, cl, and c2 are all positive constants.

38 y y =.x) Fig. 6. A nonlinear characteristic leading to exponential stability- in- the- large.

39 This second feature is very important because it opens the door to the interconnection of the auxiliary equations [(2. 36) with inequality replaced by equality] for the transfer systems in the same way that the transfer systems are interconnected in the composite system. To illustrate this interconnection, consider the simple composite system shown in Fig. 5(b). Assume that each of the three transfer systems is in Class E and has a Lyapunov function cilIIxill < vi(xi, t) < ci2 11 xil2 i = 2,3 (2. 37) whose total derivative is Vi < -aev + yilluill2. i= 1,2,3 (2. 38) When the transfer systems are interconnected llu 112 = 11x3112 < 1 31 Ilu 11 2 =lix 12 1 2x 2 _- C11 21 Using (2. 39) in (2. 38) with 1 =c31 2 13~ 2=c Y2' and =3 =- 73,there results a system of differential inequalities, C21 V1 < — alVl + PlV3 v 2 < -ae2v2 + 22Vl

40 replaced by rt's, there results the vector auxiliary system'r ib0 i 30 /33 -a3 r3- (2. 41) or, in vector notation = Ar (2. 42) where A is obviously defined from (2. 41). This is a generalization of the auxiliary equation described in Section 2. 4. Instead of reducing the original system model (the composite system model of order nl+n2+n3 where n. is the order of the ith transfer system) to a first123 1 order auxiliary equation for stability analysis, this procedure reduces it to a third-order system of auxiliary equations (in general, the auxiliary system will have an order equal to the number of transfer systems in the composite system). It is now easy to show that under certain reasonable restrictions the stability properties of the composite system, a high order nonlinear system, are the same as the stability properties of the third order linear system (2. 41). Moreover, it has been necessary to find Lyapunov functions only for the lower order transfer systems. The construction of a single Lyapunov function for the high order composite system as required in all previous applications of LSM has been avoided! This procedure is, of course, generalized to a treatment of a composite system made up of an arbitrary number of transfer systems with

arbitrary interconnections. The only restriction is that each composite system must be in Class E. Several examples illustrating this procedure are discussed near the end of Chapter 6.

CHAPTER 3 PRELIMINARY CONCEPTS With the material in Chapter 2 to provide a background, an analysis of the basic problem can begin. Section 3. 1 deals with notation while Section 3. 2 introduces concepts pertaining to composite systems and their internal structure. 3. 1 Notation The vector notation used is similar to that employed by Hahn [Ref. 12] or Cesari [Ref. 6]. (For an introduction to vector notation used with ordinary differential equations, see Coddington and Levinson, Ref. 8. ) Let En denote the n-dimensional Euclidean space of n vectors, x = col(x1, x2,...,x 9n), where the xi's (i - 1,..., n) are real numbers or real valued functions on the interval T = [0, co) of the real line. The transpose of x is denoted by x' and for all x and y in En the inner product n is defined as (x, y) = x'y = xiyi. The norm of a vector in E is i=1 1 the Euclidean norm Ixl I= (x,x)2 and if P is an m x n matrix of real elements, then IIPII = min {a a xIIx > I[PxIl for allx c En}. A useful metric on En is d(x, y) = lx-y 1l and when limits and continuity are mentioned the implied topology is taken with respect to this metric. For any subset A of En the distance from x to A is d(x, A) = inf d(x, y) and yeA for any e > 0, Se(A) = { x d(x, A)< }. A subset A of En is said to be bounded if there is a finite e > 0 such that A c SE(0). 42

43 The notation A c B means A is a subset of B, while A C B means A is a proper subset of B. If R c En" then Rc is the complement of R in E, R is the closure of R and B(R) is the boundary of R. If M C En and 0 e M, then s[M] = sup llyll is the "size" of M. If M1 and M2 ye M are two subsets of En, containing the zero vector, then M1 is smaller than M2 if s[M1] < s[M2] and the cartesian product M1 x M2 is defined in the usual manner as M x M2 = { (ml, m2) m1 e M1 and m2 M2 }. If f(x) is defined on R c En and B cR, then f(B) I { f(x) x e B }. If v(x) is a positive definite scalar function (defined in Section 4. 2), then Rh = { x v(x) < h}. For any clearly definedt. e T the set [ti, 0o) will be denoted by Ti [i. e., if to e T, then To is the set [to, co)]. The differential equation x = f(x, t) is normally an nth -order vector differential equation with x(t) and f(x, t) denoting n-dimensional vector valued functions defined on T and En x T, respectively. A solution to this differential equation is a function x(t;xo, to) such that to E T. x(t0;x0,t) = xo, and d[x(t;x,t )] = f(x(t;xo,to),t) for all t e T. It is generally assumed that all differential equations satisfy conditions sufficient to guarantee the existence, uniqueness, and continuity of all solutions in t, x0, and t (continuity from the inside is implied at the boundary of any closed region). 6 6 A variety of such sufficient conditions are available in the literature [Refs. 6, 40], but no specific conditions are assumed here. Necessary and sufficient conditions are not available.

44 Finally, let C be the normed linear space of continuous, bounded n-dimensional vector functions on T with norm I x(t) -= sup ix(t) ii. teT Similarily, let C - C such that x e C implies lim x(t) = 0. 0 0 t _ x0 3. 2 Composite Systems, Transfer Systems, and Models As mentioned in the introduction, a composite system is an interconnection of simpler subsystems. The basic building blocks of the composite systems considered in this report are called transfer systems. Def. 3. 1: A transfer system is any input-output device whose terminal variables may be characterized by relations of the form f(x, t, u(t)) (3. 1) y(t) = h(x(t),t) (3. 2) where x(t) is an n-dimensional state vector, u(t) is a pdimensional input vector and y(t) is a q-dimensional output vector. Def. 3. 2: The terminal relations (3. 1) and (3. 2) characterizing the transfer system are called the transfer system model. Example 3. 1: Consider the electric network shown in Fig. 7. II i I c C 4G e=-x=y L _____ _ Fig. 7. Network for Example 3. 1.

45 The dotted box contains a transfer system. When the input u(t) is a current u(t) = i(t), the state x(t) is the voltage across the capacitor x(t) = e(t), and the output v(t) is this same open circuit voltage y(t) = e(t), then the transfer system model is - G + u(t) y(t)= x(t). A composite system can now be defined as an interconnection of transfer systems. Def. 3. 3: Consider a set of m transfer systems, Si i = 1.., m. A composite system is an interconnection of these transfer systems so that for the ith transfer system the (vector) input ui is given as m u. = BijYj + G.u i=,... m th where y] is the (vector output of the jth transfer system, u is an external (vector) input to the composite system and Bij, Gi are constant matrices. (Note that only linear interconnections are allowed. ) The partitioned matrix B'B B Bll 12, im m B= 21 22 B B2m 1 (3. 3) B i < B ~ o i ml m2 mmj _ i k: mm

46 where the submatrices B.., (i, j = 1,., m) are the same as those used in Def. 3. 3, will be termed the composite system interconnection matrix or simply the interconnection matrix since it indicates the type of interconnections present in the composite system. th If the individual transfer systems are n.. -order systems modeled by { x.i =fi(xi, t, ui(t)) =Yi hi(xi' t), with state vectors x col[xi,..,xin] for i = 1,., m, then the composite system will be an n = n th-order system with state ini 1 vector x = col[iXl,...,Xin,x21,...,X2n,...Xml...,xmn ] and the composite system model will be m |1 f l(x1, t, Blj hj(xjt) + G1 u) j=1 m 2 = l -f2(x2, tf, B2j hj(xj,t) + G2 u) = f(x,t,u) m * m Jrfm((Xm, t, Bmhjh(xj t) + Gmu) y(t) = h(x,t), (3. 4) where y(t) is the output of the composite system. It should be pointed out that this sort of interconnection implies the usual system theory assumption [Ref. 44] that the individual transfer system models are not affected by the various types of interconnections; that is, there is no

47 "loading" effect of one system on another. This in itself greatly simplifies the mechanisms through which instability of the composite system can occur. The external input u is included to emphasize the fact that the composite system itself might be a transfer system in a larger composite. If the individual tranfer system models have the form:.i = fi(xit) + D.u. (3.5) y= H. x (3e 6) Yi i for i =1,...,m, where D. and H. are matrices and the interconnections 1 1 are such that u. B.. y + G. u Ui. Bij Yj + Gi then since y. = H. x. D = D. DB.. H. x. + D. G. u i 1. i 1J J J 1 1 and the composite system model takes on the particularly simple form 1 fl(xl't) + C11X1 + C + C13X3 + + Cmx + K1 U 2 f2(x2't) + C 21X1 + C22x2 + C23x3 +.. + m + K u xK f (x mt) + C x + C xC + C x + + C x +K u mm ml 1 im2X2 + Cm3 + mm m m y = h(x,t), (3. 7) where CiD.B..H. and Ki= Di Gi. Equation (3.7) can be further refined to the form

48 x = f(x, t) + Cx + Ku (3. 8) y = h(x,t) (3.9) where x is the composite system state vector, f is a column vector of the fi's and C is the partitioned matrix I I ~ l. I I C C 12 i C13 C'"......, -- -~ t — r -— C 21 22 23 I'" I 2m Cml Cm2 Cm3 Cm (3. 10) and K is the partitioned matrix K2 K 2 K - (3. 11) Since C = D. Bi.H the C matrix defined in (3. 10) will serve the same purpose as the B matrix of (3. 3) in indicating the type of interconnections present in the composite system.

CHAPTER 4 STABILITY DEFINITIONS AND THEOREMS Precise definitions of the various types of stability to be used throughout this thesis will now be given. Sufficient conditions for the existence of these various types of stability are then obtained using LSM. The stability definitions are given in terms of regions in the state space. This generalization of the usual definitions will be useful in Chapter 5. 4. 1 Stability and Boundedness of Solutions of Differential Equations The stability and boundedness definitions to be given refer to solutions of the ordinary differential equations which are models for the systems (transfer systems and composite systems) of interest in this report. 7 The definitions given below are a generalization of the standard stability definitions [Ref. 12] which characterize the behavior of solutions of the differential equation in the neighborhood of the null solution x = 0. (The behavior in the vicinity of any fixed solution can be reduced to this problem by a change of variables. See Section 2. 3). The generalization is the characterization of solution behavior in the neighborhood of a fixed set M or set of solutions rather than a fixed equilibrium point x = 0 or a fixed solution. This generalization of the No attempt is made to give a complete list of commonly used stability definitions and the omission of specific definitions does not imply that the present results cannot be extended to these forms. Those chosen are representative of the stability concepts found valuable in a majority of applications. 49

50 stability definitions (and the associated stability theorems) leads to the introduction of several new concepts in Section 5 and appears to have possibilities of further applications not considered in this thesis. When M = { 0 }, these definitions reduce to the familiar forms for the behavior in the neighborhood of (stability of) the equilibrium solution, x = O [Ref. 12]. Consider the solutions of the vector differential equation = f(x t) x(to) x (4. 1) where f(O, T) - 0. This equation can be considered as, a model for either an unforced system (u(t) - O) or a forced system with a fixed u(t) that is included in f(x, t). Def. 4. 1: The solution x(t;xo, to) of (4. 1) is bounded on a bounded set M if x(T;xo, to) C M. Def, 4. 2: A bounded set M in the state space of (4. 1) is stable if for every E > 0 there exists a 6 > 0 such that x(To;S,(M), to) cs (M). 8 C The following theorem shows that Def. 4. 2 implies Def. 4. 1. Theorem 4. 1: If a bounded set M in the state space of (4. 1) is stable, then, the solutions of (4. 1), starting on M, are bounded on M. Recall that x(To;S6(M), to) is the set of all points x(t;xo, to) for (t,xo) E T x S (M) or; in other words, the set of all solution paths x(t;xo, to) for t> t obtained with x e S(M). 00 - o (M6'

51 Proof: For any (x to)e M x T, let dM(Xo,to) = sup [ d(x(t;x t ), M)]. te T o If dM(M, T) - 0, the Lemma is proved. If dM(x, to) / 0 for some (xI, t') e M x T then let e dM(x-, t'). From the stability of M there is then a 6 > 0 such that xo e S6(M) implies dM(Xo, to)< E. But xV e M implies x' ES (M) and thus d (xItI) < = o o 6M oO d(xo, t). Since d > 0 this implies that dM(x, t) - 0 on M x T. Thus, the solutions starting on M are bounded on M. Def. 4. 3 A solution x(t;xo, t ) of (4. 1) approaches a set M if lim d(x(t;x,t o)M) = 0. t-oo o0 If all the solutions starting in some S6(M) approach M, then the set M might be called quasi- asymptotically stable. This property in itself does not insure that the solutions are uniformly bounded. Def. 4. 4: A bounded set M in the state space of (4. 1) is asymptotically stable (AS) if it is stable and if there is some y > 0 such that every solution x(t;xo, t ) of (4. 1) with xo e S (M) approaches M; that is lim d(x(t;xo, to), M) = 0 for every t- oo x e S (M). O' The above definitions consider behavior in an arbitrarily small neighborhood of M. For applications this is frequently unsatisfactory and neighborhoods of reasonable size must be considered. One approach to this problem is to consider stability-in-the-large.

52 Def. 4. 5: A bounded set M in the state space of (4. 1) is asymptotically stable-in-the-large (ASL) if it is stable and for every (x,to) E x T, lim d(x(t;x t ),M) = 0. t-*o A stronger form of ASL is ultimate boundedness [Ref. 45]. Def. 4. 6: The solutions of (4. 1) are ultimately bounded (UB) on a bounded set M if for every (Xo, t ) e En x T there is a T > to such that t > T implies that x(t;xo, t ) e M. If M = { 0}, then the above definitions (with the exception of Def. 4. 6) reduce to the usual definitions [Refs. 12, 16 or Section 2 above] for stability of the equilibrium solution, x = 0. Def. 4. 7: The equilibrium solution x = 0 of (4. 1) is said to be stable, asymptotically stable, or asymptotically stable-in-thelarge if the set M = { 0 } is stable, asymptotically stable, or asymptotically stable-in-the-large, respectively. Two further implications of ASL are shown by the following theorems. Theorem 4. 2: If a set M in the state space of (4. 1) is ASL, then the solutions of (4. 1) are bounded for t > to; that is, there is a constant b(x,to) such that Ilx(t;xo, to) < b(xo, t ) for all t>t o - o

Proof: Given any E > 0 there is aT > t such that t > T implies 0 d(x(t;xo, to), M) < eo Thus, for all t > T, the solutions are bounded in S (M)o Then choose any fixed 1 > T. On the closed interval [to T1], x(t;xO to) must be bounded because it is a continuous function of t (continuous on the left at t )o Thus, the solutions are bounded for all t > t 0- Theorem 4. 3: Let M be a bounded set in the state space of (40 1) containing the origin. Then, for any (xo, t) e En x T, IIx(t;xo,to) l < d(x(t;xo, to) M) + s[M] and, in addition, if M is ASL (a) lim d(x(t;xo t ),M) = 0 for all (xo to) E x To (0 0 l 0 t — o (b) d(x(t;xo to) M) = O for all (t, xo, to) e To x O x T. (c) d(x(t;x, to), M) is continuous in t, xo, and to at all points in T x En x To 0 Proof: The bound on the norm of x(t;xo, to) is a restatement of the triangle inequality while prperty' (a)- follows immediately from the definition of ASLo Property (b) holds because x = 0 is an equilibrium point that is contained in M. Property (c) follows from the continuity of d(x, M) in x and the continuity of x(t;xo, to) in t, xo, and t 0 00 0~~~~~o

54 4. 2 Sufficient Conditions for Stability and Boundedness of Solutions of Differential Equations In this section, sufficient conditions for the several types of stability and boundedness defined in Section 4. 1 will be obtained through the application of LSM. Two different approaches will be employed to obtain two sets of sufficient conditions for each of the definitions in Section 4o 1. The first approach is essentially a straightforward application of the original procedures attributed to Lyapunov [Refs. 12, 27] where sign definite (or semi-definite) functions (Lyapunov functions) are used to establish solution behavior in terms of a generalized metric on the state space. Let R c Se(O) for some e> O with O e Ro Defo 4. 8: A real valued function v(x), defined in En, is said to be positive definite [positive semi-definite] on RC (recall that Rc is the complement of R in En) if x Rc implies that v(x)> 0 [v(x)> 0] and v(O) = 0. Def. 4. 9. A real valued function v(x, t) defined on En x T is said to be positive definite on R x T if v(O, T) = 0 and there is a function w(x), that is positive definite on R and such that v(x, t) > w(x) on RC x T. Note that the behavior of v(x), v(x, t) or w(x) inside R is immaterial.

55 Defo 4. 10: A real valued function v(x, t) defined on En x T is said to be positive semi-definite on RC x T if v(x, t) > 0 on R x T and v(O, T) = 0. Defo 4. 11: The functions v(x) and v(x, t) are said to be negative definite [negative semi-definite] if -v(x) and -v(x, t) are positive definite [positive semi-definite]. If R = { 0}, then replace Rc in the above definitions by Eno For instance, a real valued function v(x), defined in En, is said to be positive definite on En if x En implies that v(x)> O for x # O and v(O) = 0o These are the usual definitions of sign definite functions [Ref. 12]o In their domain of definition, all of the definite and semi-definite functions defined above are assumed to be continuous and to have continuous first partial derivatives with respect to all arguments. In all cases where v(x, t) is a Lyapunov function, v (x, t) denotes the total derivative of v(x, t) with respect to the differential equation under consideration (see Section 2. 3). Theorem 4. 4: Let v(x) be positive definite on E If Rh (as defined in Section 3. 1) is a bounded subset of En and v (x, t) is negative semi-definite on Rh x T, then all solutions of (4. 1) starting on Rh are bounded on Rho Proof: With (Xo, to) Rh x T, assume that for some t2 > to — c — h the solution x(t2;xo, to) E R (note that R ) Since the solutions are assumed continuous in t and v(x) is continuous in x, there is a time tl where to < t1 ~ t2 such that x(tl) e B(Rh)o

Since x(t2) e Rh, then v(x(t2)) > h - v(x(tl)) in a region where iT(x, t) < 0o This contradiction shows that the solutions remain in Rh for all t > t Theorem 4. 5: Under the hypothesis of Theorem 4, 4, the set Rh is stable. Proof: Given any e > 0, choose a y > O such that Rh+y CSE(Rh). This is always possible due to the continuity of v(x) in xo Then choose 6 > O so that S6(Rh) c Rh+y Now, if xo e S6(Rh) C Rh+y, Pthe solution x(T;S5(Rh), t) is bounded on Rh+y C S (Rh) by Theorem 4. 4. Thus, Rh is stable. Theorem 4. 6: Let v(x) be positive definite on En and assume that Rh is a bounded set. If fv (x, t) is negative definite on Rh x T and negative semi-definite on B(Rh) x T, then every bounded solution of (4. 1) approaches Rho Proof: If (xo, to) e Rh x T. then by Theorem 4. 4, x(t;x o t ) is bounded on Rh and Defo 4. 3 is satisfied trivially. If x(t;x, to) h 00 O is any other bounded solution, then there is a constant P such that IIx(t;x,to)ll < P and a closed, bounded set D where s[D]> P such that D D Rh and x(t;xo, t ) is bounded on Do Choose 61> 0 such that Rh 6 C D and x e (D-R ) Since the region h+6 0 h+R 61 (D-Rh+61) c Rh is closed and bounded (and hence compact), (x, t) < -w(x) < 0 (w(x) is positive definite on E ) takes on a maximum value, say -a < O therein. Alongthesolutionx(t;xo, to) 00

in this region, t v(x) v(X) + f dt < v(x) -a(t-to). t 0 Since v(x)> O for all x 4 0, the solution can remain in this region for only a finite length of time, Since the solution is bounded on D, it must then approach Rho Thus, given any 6 > 0, there is a Tl > to such that t > T1 implies that x(t;xo, to) e Rh+60 It is now only necessary to show that this last statement implies that d(x(t), Rh) becomes arbitrarily small. Given any e > 0, choose 6 > O so that Rh+6 C S (Rh)o Then, for any E > 0 there is a T > t such that t> implies x(t;x,' to) e Rh+ C S(Rh)o Thus, t > T implies that d(x(t;xo, to), Rh)< e, and the solution approaches Rho Theorem 4. 7: Let v(x) be positive definite on En and R be h a bounded subset of En. If H (x, t) is negative definite on Rh x T, and negative semi-definite on B(Rh) x T, then the set h h Rh is asymptotically stable (AS)o Proof: Stability follows from Theorem 4. 5o Since Rh is stable, the solutions may be bounded in some S (Rh) by choosing x0 in some S (Rh)o Then choose D as some closed bounded set such that S (Rh) c D All solutions with x e Ss(Rh) are then bounded on the closed bounded set D D Rho The remaining hypotheses of Theorem 40 6 are then satisfied so all solutions starting in SA(Rh) approach Rh and this set is AS.

58 Theorem 4. 8: Let v(x) be positive definite on En, If v (x, t) is negative definite on Rh x T, negative semi-definite on B(Rh) x T, and lim v(x) = co, then the set Rh is ASLo Proof: The set Rh is stable by Theorem 4. 5o Given any (xo, to) e Enx T, there is a scalar x > h1> v(xo) such that Rh is bounded, I 0 1 Rh D Rh, x ce Rh (the selection of such an hl is always possible 1 1 because lim v(x) = co), and the solution x(t;xo, t) of (4. 1) is IIxIl- - co bounded on Rh by Theorem 4. 4. Since Rh is a closed bounded 1 1 set in which the solution x(t;xo, to) is bounded, it follows from Theorem 4. 6 that this solution approaches Rho Moreover, this is true for every solution x(t;xo, to) with (Xo, to) e E x T. Theorem 4. 9: Let v(x) be positive definite on En with c lim v(x) = c. If xv (x, t) is negative definite on Rh x T, then the solutions x(t;En, T) of (4. 1) are ultimately bounded (UB) on Rho Proof: If (Xo, t ) c Rh x T, then by Theorem 4. 4, the solution x(t;xo, to) is bounded on Rh. For every (Xo, t) e R x T there is a scalar hI > v(xo) such that R is bounded Rh R and the I 0 h h h, solution x(t;x, t ) is bounded on Rh by Theorem 4. 4. Thus, the 1 solution is bounded and lies initially in the closed region -h Rh) C Rh where v(xt) < -w(x) < O. In this region the c1tu fC continuous function w(x) must take on a maximum value -a < O,

59 and along the solution in this region t ) = v(x0) ( + f v dt < V(XO)-a(t-to) t o Since v(x) is positive for x A 0, the solution cannot remain in this region for more than a finite length of time. Thus, there is some c to such thatt > T implies x(t;x,t t)c (Rh - Rh) = Rh C U Rho Since the solution is bounded in Rh it must enter Rh for t> T. As noted above, it then remains in Rh for all t > T. The second approach to this problem of obtaining sufficient conditions for stability and boundedness follows Conti [Refso 5, 9] who views the positive definite functions as dependent variables in a firstorder auxiliary equation (Brauer calls this a comparison equation). For example, let v(x, t) be a positive definite function on En x T and vx be the total derivative of v with respect to (4. 1). If there is a function w(v, t) such that along the solutions of (4. 1) v < (v t) (4~ 2) then with quite mild restrictions it can be shown that (4. 1) has the same stability and boundedness properties as the first-order (scalar) differential equation r= w(r, t) (40 3) when ro r(t) =v(x(to)) vo This reduces the problem of determining the stability of an nth-order system to that of determining the- stability of a first-order system —a very significant simplification0

60 (However, it is still necessary and frequently difficult, to choose the proper Lyapunov function to obtain (4~ 2).) An important tool in this approach is the following lemma: Lemma 4o 1: Let o(r, t) be a real valued continuous function on T x T with co(0, T) > 0 and satisfying conditions sufficient9 to insure that for all (ro, t ) e T x T the first-order differential equation r = w(r, t) has a unique solution r(t;r, t0) that is continuous in t, ro, and t Let v(t;vo, to) be a real valued function on T x T x T such that (a) v(t;vo, t) is continuous in t, vo, and t (b) v(t0;vo,t0 ) v0 (c) for any (vo, to) T x T, the function v(t;v o to) A dv satisfies the differential inequality dt < w(v, t) for all t E To0 Then, for any (vo, t ) and (r, t ) in T x T with 0<v v0 < r it follows that r(t;r,to)> 0 and r(t;ro t ) > v(t;vo, to) for all t c T Proof: Since w(0, T) > 0, no solution r(t;r t ) of (4. 3) with r > 0 can cross the r = 0 axis into the region where r is negativeo Thus, r > 0 implies that r(t;r,t ) > 0 for all t e T See footnote 6, page 43~

61 The fact that v(t;v, to) < r(t;r, to) follows from a point-bypoint consideration of the implications of the facts that v < r and v (t) < i (t) whenever v = r. To see this assume that v < r but v(t2;v to) > r(t2;ro, to) for some t2 > t. Since both v and r are continuous functions of time, there must be some tl 1 to such that v(t1;vo, to) = r(t;ro, to) and v(t;vo, to)> r(t;ro, to) for t > t. But this means that v (t1) > I (t1) which is a contradiction. Thus, v(t;vo, t) < r(t;ro, t) for all t T0 Using Lemma 4. 1, one may obtain the following theorem: En Theorem 4. 10: Let v(x) be positive definite on E and Rh be a bounded set. Assume that along the solutions of (4. 1), v(x(t)) satisfies the inequality vx < co(v, t) with co(v, t) satisfying the hypotheses of Lemma 4. 1. Then (a) the solutions of (4. 1) will be bounded on Rh if the solutions of (4. 3) are bounded on the set Ph { r 0 < r < h} (b) the set Rh in the state space of (4.0 1) will be stable, AS, or ASL if the set Ph = {r0 o < r < h} is stable, AS, or ASL in the state space of (4. 3). Proof: If the solutions of (4. 3) are bounded on the set Ph for r(t ) E Ph' it follows by Lemma 4. 1 and the positive definiteness of v that O < v(x(t)) K h and thus x(t) c Rh { xj v(x) < h}.

62 Similarily, if the set Ph is stable, then for every er > 0 there is a 6r > O such that r(t ) < h + 6r implies r(t;ro, t) < h+ er r o - r 000 - r Now, given any E> 0, choose er so that h+E C S(Rh) and r 6 > O so that xo E S(Rh) insures that v(xo) < h + 6r Then v(x(t)) < r(t;ro, t ) < h + er for all t e To and therefore, x(T;xo, to) c S (Rh)Q The proof of AS and ASL follows in an equivalent fashiono Example 4. 1. Consider a transfer system modeled by the first-order equation x = -f(x) + bu(t) (40 4) where b > 0, u(t) is some fixed input function in the normed linear space C, 0 < cx2 < Xf(x) < oo for x 4 O where c is a positive constant and f(O) = 00 Choose a positive definite function v(x) = x2o Then = -xf(x) + bxu(t)< -cx2 + bxu(t) = -cx2 [- bu(t or v < -cx2 1 - cx (4 5) b2 Now let h = ll 2 Then if 2c2 c bllullc Rh = {x| v(x) < h} = {x Ixlj < c it is clear from (4o 5) that xv is negative definite on Rx T and negative semi-definite on B(Rh) x To Thus, the hypotheses of Theorem 40 8 are satisfied and the set Rh is ASLo

63 This example is particularly interesting because the same region Rh is ASL for all u(t) e C that have the same norm llulic This suggests that for such systems a "gain" could be meaningfully defined relating the size of Rh to the norm lJ u lic. In the case considered, the size of Rh is b jull so the gain would be b o This concept of gain is explored further in Section 5. Lo Example 4. 2: Again consider (4. 4), but now note that (see Lemma 50 1) < C 2 bz U2 or v < -cv + i llc Thus, the auxiliary equation is = -cr + lull 2c c and for this system the region 0 < r < lull is ASL since - 2c2 c r(t) = r0t) ba 2cU-c(t-t2) 2 r(t) Lrt llul e c(tt)+i lu2 Thus, by Theorem 4o 10, the region Rh xv(x) b u where h xlv(x) - llul where h _ [llull2 is ASL in the state space of (4L.-. But 2c2 c V(X) i 2 b i i

64 which is the same result as Example 4. 1o In conclusion, note that at the expense of considerable complication of the proofs, all of the theorems of this section may be stated with much weaker hypotheses. For example, relaxation of requirements of uniqueness of the solutions of (4. 1) is considered by Conti [Refo 9] while relaxation of requirements of continuity in the Lyapunov functions is considered by Massera [Ref. 31]. Other generalizations are also possible, but the added complication of the proofs does not appear to be necessary for the present purposes and is therefore omitted.

CHAPTER 5 PROPERTIES OF TRANSFER SYSTEMS With certain restrictions, the stability definitions and theorems of Chapter 4 can be applied to the study of transfer systems. This study leads to the definition of special classes of transfer systems which are important in the development of a stability theory to follow. 5. 1 Gains for Transfer Systems The stable regions discussed in Chapter 4 offer a method for characterizing certain important features of transfer systems. However, this problem is complicated by the fact that transfer systems are inputoutput devices and the characterization must include the relation between input and output. In the discussion of stability of regions, a fixed differential equation was assumed and fixed regions were considered. If u(t) may be one of a class of input functions in the differential equation = f(x, t, u(t)), (5. 1) then the existence and location of regions M exhibiting some of the stability properties described in Section 4. 1, will depend on the specific input. In general, when (5. 1) exhibits stable regions they must be denoted M(u(t)) to indicate this dependence on the input. For example, in a transfer system modeled by (5. 1) it may happen that for each u(t) in some set U there are corresponding regions M(u(t)) that are ASLo There are several interesting special cases of this situation 65

66 Defl 5. 1: A transfer system modeled by (5~ 1) is said to be in Class C if u(t) e C implies 0 that x(t;x, t, u(t)) e C for all, O0 (x to) E x T. Thus, for a system in Class C a continuous bounded input gives a continuous bounded output. This is similar to the bounded input implies bounded output stability defined in Ref. 14. Def. 5. 2: A transfer system modeled by (5. 1) is said to be in Class C if u(t) e C implies x(t;xo, to) e C for all (x, to) e En x T. A situation of particular interest occurs when there is one fixed set in the state space that is ASL for all u(t) in some set UO For instance, suppose that for every u(t) in a set Ua C C there is one bounded set M(U ) containing the origin that is ASL. [Since any set N D M(U ) will also be ASL, let M(Ua) be the smallest of all those that are ASL for u(t) e U ]. Then, by Theorem 4. 3, for each u(t) e Usa lx(t;xo, t0, u(t)) l < d(x(t;xo, to, u(t)), M(U)) + s[ M(Ua)] (5. 2) In order that the right-hand side of (5. 2) be independent of u(t), except through Ua, assume that d(x(t;xo, to, u(t)), M(Ua)) can be bounded above by a scalar function y(t, xo, to) that is independent of u(t) and approaches O as t increases (the distance d(x, M) has this last property for each 10 Here C refers to the normed linear space of continuous, bounded functions on To See Section 30 1o

67 fixed u(t) —see' Theorem 4o 3). The inequality (5. 2) then becomes lJx(t;xo, to u(t))ll < y(t,o,t) + s[M(U )] (5~ 3) The solutions x(t;x, to, u(t)) now approach a region M(Ua) whose size is determined by u(t) through Uao This suggests that for certain classes of systems it is possible to define a "gain" relating the size of the input to the size of this region approached by the output. The following class of transfer systems has the necessary properties. Defo 50 3: Let y(s, x, t) be defined on T x En x T and continuous n in s, x, and t at all points in T x E x T with y(T, O, T) = O and lim y(s, En, T) = 0. A transfer system modeled by (5~ 1) S'- o00 is said to be in Class G if (a) for every u(t) e C there is a bounded region M(u) containing the origin that is ASLo (b) u(t) 0 implies M(u) = { O}. (c) there exists a function y(s, x, t) with the above properties such that for al.: u(t) e C and all (x, t ) e En x T d(x(t;Xo, tu(t)), M(:)) y(a, o t)o It is clear from the statement of Defo 5. 3 that Class G is a subclass of Class CO Now if UU = {u(t)l u(t) e C and Ilullc = a }, for transfer systems in Class G, a plot of s[M(Ua)] versus a appears as shown in Fig. 8.

68 y y = s[ M(Ua)1 0 0 a Fig. 8. Graph of s[M(Ua)J versus a for a system in Class G. The fact that this curve goes through the origin follows from part (b) of Def. 5. 3. Clearly then, there are constant /3 such that 3 a> s[ M(u)]. This leads to a definition of what will be called the gain of a system in Class G. Def. 5. 4: The gain 71 of a transfer system in Class G is d6fined as;? = inf{ I|: a > s[M(Ua)] for all u(t) C } Theorem 5. 1: For every transfer system in Class G there is a gain 71 and a function y(t;xo, to) such that for all u(t) e C 1Ix(t;xo,tou(t))lI < y(t,xo,to) + 7/11uulc * (5. 4) Proof: This follows immediately from Defs. 5. 3 and 5. 4 and Theorem 4. 3.

69 Note that Class G and the definition of gain represent, to a certain extent, attempts to extend single-input, single-output, linear system concepts to a wider class of transfer systems. The term y(t, xo, to) in (5. 4) is similar to the transient response of a linear system while the term 11 ullc corresponds to the steady state solution. The gain 71 is then a constant relating the size of the steady state response to the size (here the norm Hl ulc) of the input. The following examples illustrate these ideas. Example 5. 1: Consider a single-input, single-output, first-order linear transfer system modeled by i x = -ax + bu(t) x(to) x x (5 5) where a and b are positive constants. The solution is -a(t-t ) t -a(t-z) x(t;xo,tou(t)) = e x0 + e bu(T) dr t o giving -a(t-t ) t -a(t-r) or llx(t;xo, to, u(t)) IlI < Ie e x to + b + llc (5u6) To see if this system is in Class G, check Def. 5. 3. For part (a), it is clear from (5, 6) that for every u(t) e C the bounded region M(u(t)) =

70 { x lxll < b Ilullc } is ASL and contains the origin. Since the null solution M { 0 } is ASL when u(t) = 0, part (b) is satisfied. For part (c) note that d(x(t;xo, t, u(t)), M(u(t))= (t;x t (t)) - u < ie ja(tto)x which has all the required properties of afunction y(s, x, t). Thus, the transfer system modeled by ( 5. 5) is in Class G. If Ua = { u(t) u(t) e C and lullc = a }, then it is clear from (5. 6) that s[M(Ua)] < b a and a so the gain of this system is less than or equal to -. (This is a typical case where only an upper bound on the gain can be obtained ) Note that this is just the maximum gain of the transfer function H(s) = (s) which occurs at zero frequency (s = 0)o Example 5. 2: Consider the general linear, time-invariant system modeled by A x + B u(t) x(t) = x y Hx (5. 7) where A, B, and H are constant matrices and it is assumed that all of the eigenvalues of A have negative real partso The well-known solution to (5. 7) is [Ref. 8] A(t-t ) t A(t-r) x(t;xo,tou(t)) = e x + f e Bu(r) dr. (50 8) t

Because all the eigenvalues of A have negative real parts, there are positive constants 13 and y such that IIleAt l < p eYt (see Refo 6 or use the Gronwall- Bellman lemma, Ref. 4). Thus, -~,(t-t ) t -,(t-z) IIx(t;x, t, u(t))lI < pe o'ix t + lie IBI dTr Ilulli or IIx(t;xo t u(t)l < /e -Yt-t x II +,IBI IIUIl Reasoning as in Example 5. 1 shows that this system is in Class G, and the set { xj llXII < a Bll1 a } is ASL for all u(t) e Ua = { u(t) t) t) e C and 1lullc = a }. Thus, M(U) C { x lxli < I-Bi a} and the gain 7r is less than or equal to - Il Blli 50 2 Exponential Stability and Class E The above discussion has led to the definition of a class of transfer systems (Class G) having special properties that are important in the study of composite systemso In this section, an important subclass of Class G is defined and it is shown that Class G is "large enough" to contain many systems of practical significanceo The systems in this subclass of Class G are related by the property of exponential stability-in-the- largeo

72 Def. 5. 5: The null solution of the equation x f(x,t) f(O, T) 0 (5. 9) is said to be exponentially stable-in-the-large (ESL) if there are two positive constants a and 3 such that -a(t-t ) lx(t;xo,to) lI < ilx ii e 0 for all (xo, to) E x T. Def. 5. 6: A vector function f(x, t) on En x T is said to be in Class B (denoted f e B) if, for all (x, t) e En x T, f(x, t) is continuous in x and t and has continuous first partial derivaaf. tives with respect to xl x2, I xn such that I i < L (L is aconstant and i, j = 0 0 0, n). Theorem 5. 2: Assume that f(x, t) e B in (5. 9). Then (5. 9) is ESL if and only if there is a positive definite function v(x, t) such that (a) cl Ix12 <K V(X,t) < c2 11x112 (b) v (x,t) < -c31lx 112 (c) llVv l < c4 Ilxli I where cl, c2, c3, and c4 are positive constants. If (5. 9) is autonomous, then v can be chosen independent of t. Proof: See Appendix or Ref. 18, p. 59. Using this theorem, it is possible to show that an nth -order system modeled by - f(x,t) + Du(t) (5.10)

73 is in Class G if x = f(x, t) is ESL and f(x, t) e B. Lemma 5. 1: If a> 0, then for all z T _~t2 a 2 b2 az2 +bz < a z + 2a (5.11) Proof: 2 a 2 b2 -az + bz < - 2 z + b2 2a 2a b2 ba 2 b2 2 2a 2 2aa 2 2a iff- a z + bz - -2a Z - 2z 2-a ( 2 z 2a ) - 2a (az-b) <- a 2 + b2 which is obviously true. Theorem 5. 3: If x = f(x, t) is ESL and f(x, t) e B, then a transfer system modeled by (5. 10) is in Class G with gain 7 ( < (c4i) J I IDII where cl, c2, c3, and c4 are the constants occuring in Theorem 5. 2. Proof: Since x = f(x, t) is ESL and f(x, t) e B, there is a positive definite function v(x, t) satisfying (a), (b), and (c) of Theorem 5. 2. The total derivative of this function with respect to (5. 10) is then v - l (x, t) + Vv' Du < -C3 xI 11 2 + c41lx li Di1l 1u(t)il

74 where v 1 is the total derivative of v(x, t) with respect to (5. 9). Use of Lemma 5. 1 yields c C 2 1D 11 2 v <-2 1 Ixl 2 + 1l u(t) 11 2 or C ~ c2 11D 11 2 C3 c4 v<- v + 11 u(t)112 (5.12) -_-2c2 2c3 This is a differential inequality in v and by Lemma 4. 1, C3 c3 -- (t-t2) C 3 2c 2 o c2 IIDll2 t 2c 2 v(t) < v(t0) e 2 + 2 f e 2 llu(T)1 dT. 23 t o (5. 13) If u(t) E C with norm [lull, then C3 2c (t-t) c C2 lDll2 v(t) < v(to) e 2 + 2 lullc2 -- 0 2 C C3 and from (a) of Theorem 5. 2 C3 C2 2c (t-to) c2clIDII2 Ix112 < 2 x(t )2e 2 + 2 4 2 IuC 1 1C3 or C3 - 4 l )(to ) c2 - c4

75 This result can now be compared with the requirements of Def. 5. 3 to show that the system is in Class G. Requirements (a) and (b) are satisfied since the region M(u) = {x lx ll < ( ) 4 DlI Ilulc c3 contains the origin, is ASL for every u(t) c C, and reduces to { 0) when u(t) 0-. Requirement (c) is satisfied since d(x(t;xo, t, u(t)), M(u)) = [I ix(t;xo t, u(t)) I- IIDII lull u c3 /C\- 4c2 (t-t0) < ( XIX(to)l[ e LC1) and this last term on the right has all the required properties of a function y(s, x, t). Thus, all of the requirements of Def. 5. 3 are satisfied and this system is in Class G. Moreover, from Def. 5. 4, this system has a gain 7 < _c IDi Def. 5.7: A system modeled by (5. 10) will be said to be in Class E if the unforced model [(5. 10) with u(t) = 0] is ESL and f(x, t) e B. The gain estimate determined in Theorem 5. 3 depends on the constants c1, c1, c3 and c4 and thus on the particular Lyapunov function

76 chosen. This is a reoccurrence of the old problem of choosing the proper metric. The significance of this problem in the present context will become more apparent in Chapter 6. From Theorem 5. 3 it is clear that Class E is a subclass of Class G. In addition, it can be shown that Class E (and therefore Class G) contains many systems of practical importance. This follows from the fact that many important equations are ESL. Theorem 5. 4. The following ordinary differential equations are ESL: (a) The linear constant coefficient equation: = Ax when A is stable (i. e., all of its eigenvalues have negative real parts); (b) The linear time-varying equation i A(t) x where A(t) is continuous and bounded and x: A(t) + u(t) is in Class C; (c) The equation x: = f(x, t) where f(O, T) = 0 and x'f(x, t) < -c3x 12 < O for all (x, t) En xT with x 0. Proof: (a) is a well-known result [Ref. 12], (b) see Ref. 32, p. 518, and (c) see Example 5. 3 below. The list given in Theorem 5. 4 is far from exhaustive. A more detailed description of Class E and Class G should be the subject of future study. The following example shows a nonlinear equation that is ESL.

Example 5. 3. Consider the n th-order equation x = f(x,t) f(O,T) = 0 where x'f(x,t) < -c311x1 2 < 0 for x A 0. Let v(x) = llxI 2 so that v (x,t) -x'f(x) < -c3 11 x II = -2c3v Then, by Lemma 4. 1, - 2c3(t-t ) v(t) < v(to) e and thus, -c3(t-t ) Ilx(t;xo to)l I < Ilxol I e showing that this equation is ESL.

CHAPTER 6 STABILITY OF COMPOSITE SYSTEMS Using concepts developed in the previous chapters, one can now develop an approach to the basic problem, the stability of composite systems. 6. 1 Two Simple Composite Systems The tools developed earlier will first be applied in the analysis of the stability of the two simple composite systems shown in Fig. 9. a) Simple chain —P x. u 0) will be considered. x x y m1 m M S m- 1!k+2 I0k+ 1 S b) Simple closed loop —PL Th olwn emswilb fmjriprtnei hsscin j~ _ S... x

79 Lemma 6. 1: Let x(t;xo, to) be a solution of the differential 11 inequality" x: < Ax (6.1) with x(t;x to) - x and let y(t;yo, to) be a solution of the differential equation = Ay. (6. 2) If all the elements aij (i, j = 1,..., n) of A are nonnegative, and x0 = y, thenx(t;xo t ) < y(t;yo, t) for all t T Proof: First note that the solution to (6. 2) is y(t;Yo, to) = A(t-t ) e y. The inequality (6.1) can be rewritten as = Ax - p(t) where p(t) is a vector whose elements are nonnegative functions on T; i. e., p(t) >0. Then, A(t-to) A(t-T) x(t;xo, to)= e oeA - p(T)dr But the integral on the right represents a nonnegative vector because each element of eA(t-T) is nonnegative as long as t > T. Then, since x = yo, Throughout this section, the notation x < y where x and y are nvectors means that xi < yi for i = 1,..., n.

80 x(t;xo, to) = y(t;yo, t) - g(t) where g(t) denotes the integral and hence g(t) > 0. Therefore, x(t;xo, t) < y(t;yo to) for all t e T Lemma 6. 2: Let B be a matrix with negative diagonal elements and nonnegative off-diagonal elements. If x(t;xo, t0) and y(t;y, to) are solutions of K < Bx (6. 3) y = By (6.4) and x = y, then x(t;xo, to) < y(t;Yo, to) for allt e To Proof: Let -d be the smallest of the diagonal elements of B. +dt +dt Application of the transformations v = e x and w = e y changes (6. 3) and (6. 4) to v < (B+dI)v, v = (B + dI) w, and B + dI has all nonnegative elements. Then by Lemma 6. 1, since v = wo, it follows that v(t;v, t ) < w(t;wo, to) and thus x(t;x,to) < y(t;y,to) for all t c T e 12 2 The author is indebted to Dr. J. K. Hale for this proof of Lemma 6. 2.

81 Lemma 6. 3: The null solution of the system of differential equations = -alxl + blXn x2 -a2X2 + b2Xl xn = -a x + b x n n n n n-1 with ai and b real, ai > 0 and b.i > O, (i = 1,..,n), is ASL if and only if n b. n < 1. a. i=1 1 Proof: The characteristic equation for this system is n n (X + a.) - Ib. = 0 i=1 i-1 1 Now replace bl with ibl1 and consider the root locus problem [Ref. 43] for Az > 0. If pi = 0, there are n negative real roots, X = -a.. As,i increases, the largest root (one closest to the origin) moves to the right along the real axis reaching the origin when n n II a. - M I b. O i=1 i=1 1 It is only necessary to ascertain that no complex root has crossed the imaginary axis for a smaller value of A. When bl is replaced by pbl, the characteristic equation can be solved for p giving

82 n (X+ai) M = n b. i- 1 1 as the value of Mi corresponding to each point on the root locus. At the origin X = 0 and n a. ll'= while if the root locus crosses the imaginary axis at X = X1 j O the value of AM at this point is n Ix1+ai n a. b. >1 b1 Thus, the point where the locus crosses the imaginary axis with the smallest value of A is at the origin and the root moving along the real axis must be first to cross as p. increases. If =- 1, the necessary and sufficient condition for stability becomes n n n b. II a. - I b. > 0 or I < 1 1 i1 i1 a. It is now possible to consider the stability of the simple chain Pc c of Fig. 9(a). It is intuitively obvious that an interconnection of individual stable systems will be stable because of the "weak" interconnections involved (no loading assumed, see Section 3. 2). The following theorem gives a simple proof of this fact.

83 Theorem 6. 1: Consider the simple chain Pc with u = u1 = 0 and ui = xi for i = 2,..., m. If the individual transfer systems Si, (i = 1,..., m), are in Class E, then the null solution of the composite system is ASL. Proof: If the individual transfer systems are in Class E, then for the ith transfer system there is an inequality [see (5. 12)] < -a.iV + iluil where c~3 ci4~ 2ID 11 2 a = i3 > Oi4 1 >0. ai 2E > ~' Yi 2c i 2ci3 When the interconnections are made, u1 - 0 and Yi iuI= lx ill2 < c. vii for i = 2,3,., m With =, (i 2,..., m), the resulting system of differential inequalities becomes 1 < -a1V1 v2 < -a2v2 + 22V1 n < -mvm + mvm-1 (6. 5) where v. > 0. Since = 0, the null solution of the system of equations (6. 5) with inequalities replaced by equalities (a system of auxiliary equations) is ASL (Lemma 6. 3). Then by Lemma 6. 2,

84 the system (6. 5) is also ASL which means that each vi approaches 0 Since lx. 112 < k1.,i' the equilibrium solution x = 0 of the 01 Sc- cil 1 composite system is therefore ASL. A more impressive result is obtained for the simple closed loop PL' In this case, the stability of the individual systems is not sufficient. The additional requirement is that the loop gain estimated from the chosen Lyapunov functions be less than unity —again an intuitively plausible requirement. Theorem 6. 2: Consider the simple closed loop PI with u = 0 (no external input) and ul = xm, and ui = xi 1for i = 2,...,m. If the individual transfer systems Si, are in Class E with gain estimates ti (i = 1,..., m), then the null solution of the composite system will be ASL if II tji < 1. i= 1 Proof: As in the proof of Theorem 6. 1, there is a system of differential inequalities V 1 < -alvl + 1vn v2 < -a2v2 + P2Vl i < -a v + i v (6. 6) m - mm m m-1 where 1 2c > 0 i= 1,...,m

85 c 2lD I[ 2 c14 ID1 > 1 2C13cml C4 2Di 11 2 =2c c > i = 2,...,m ci3 i-l 1 where the v. > 0, for i = 1,..., m. The null solution of the composite system will be ASL if the system of auxiliary equations [(6. 6) with inequalities replaced by equalities] is ASL. But, by Lemma 6. 3, this will occur if m p. II -- < 1 a. < Now hi Ci2Ci4 Cil 2 a.i Ci-1, 1 i 2 1, (where Ci_1 1 - cml when i = 1) and so the requirement that m m. ii r7 < 1 or II <1 i=l i=1 a thereby giving ASL. The possibility of requiring only that the individual transfer systems be in Class G is suggested by the following theorem. Theorem 6. 3: If the transfer systems in Theorem 6. 2 are only required to be in Class G with II ri < 1, then the null solution of the composite system PL is stable.

86 Proof: This follows easily from Theorem 5. 4 and the fact that the loop gain is less than one. It seems reasonable to expect that further study of Class G will allow an extension of this result to ASL or a clarification of the relation between Classes G and E. 6. 2 Complex Composite Systems In Section 3. 2, it was noted that when the transfer systems of a composite system are individually modeled by | i = fi(xi't) + Diui(t), (6.7) lYi = Hxi i (6. 8) for i = 1,.., m, then the composite system modeled has the particularly simple form x = f(x,t) + Cx + Ku (6. 9) y = h(x, t). (6. 10) Since transfer systems in Class E have models of the form (6. 7), (6. 8), this situation is of particular interest in this analysis. Since Cij = DiBi jHj, (see Section 3. 2), the C matrix of (6. 9) serves the same purpose as the B matrix of (3. 3) in indicating the topology of the interconnections in the composite system. For example, in the case of the simple chain, Pc, the C matrix has the partitioned form

87 [jll- ~ I1 I._ _lJ8.to _. 0 0 CC o I o I... o o _ __.. I7-. —(- - [- I | O O ~ ( ~' l Cm mm-1 0O and for the simple closed loop, PL the C matrix has the form C - 0 0._o O... I = L ~ | C 32! o W, o SL I 4 I 0 0 0 L... t O O t- - - _... - 01 0 0. l ~ i ~ ~... m,m l li J of nonzero elements (submatrices) in the C matrix. It is now possible to give sufficient conditions for ASL of the x O solution of a composite system with arbirary interconnections; that is, a system where C has an arbitrary number of nonzero elements. It will, however, be assumed that Cii = O so that there is no direct feedback around any individual transfer system. Theorem 6. 4: Let P be a general composite system made up of m transfer systems Si of order ni - 1,...,.m) and modeled by (6. 9) with Civ i O. Assume that each transfer system Si

88 is in Class E and has a Lyapunov function vi(xit) satisfying the bounds listed in Theorem 5. 2 with coefficients cil, ci2, ci3, ci4 (all coefficients are positive). Consider the m th-order linear system of auxiliary equations r Ar (6.11) where A is an m x m matrix of elements Ci3 for i j 2ci2 ij CA42 lCi l 2j1 for i ~ j 2ci3 c.1 with C..i an element (submatrix) of C. The equilibrium solition x 0 O of the unforced composite system model [(6. 9) witl u - 0] th will be ASL if the equilibrium solution r 0- of the m -order linear auxiliary system (6. 11) is ASL. Proof: The ith transfer system can be modeled by 1 = 1 fi(xi + Du(t) + Diui(6. 12) = Hii (6. 13) and due to the interconnections u. = i Bijyj. (6.14) Since this ith transfer system is in Class E, there is a

89 Lyapunov function vi(x, t) such that i-< -ci3 lxi112 + ci4 lxi1l IID.ul (6. 15) Now a substitution (6. 13) and (6. 14) into (6. 15) and application of Lemma 5. 1 gives c2 2 v K - c3lx 11i2 + i4 ( D.B H x 1 CO ~1+ 2c ii j#.i which, by Holder's inequality, shows that C. 2 m m,~ < _c I[xii2 + ci4 IID. B..H.Il2 Z [[x IIx2 i - 2 i3 1 + 2ci3 j=l i 1J i j-1 j/i jti Now a reintroduction of the inequalities cil Ixii2 < vi(xi, t) < ci2 C xi112 ii =,...,m and a use of the relation C= D.B. H. yields J 1J J C C 2 n m m v. < cVi 2c+ 4 1ij1 c1 2ci2 i 2ci3 j=1 j1 C jAi The resulting system is c13 v 1 +; 1 <- - c2 1 2 m3 = ICj\ rnv.\j'.'c.'. 1c ( 2C12 V1 f \2C13 m - vJ K Cm4 (Z j- l j.

90 or v <Av, (6. 16) where v col[vl,v2,...,vm] and < means each component is <. Now by Lemma 6. 2, v(t) < r(t). Thus, x (it;xot ll < —- v(t) < r(t) i= 1,...,m and so ASL of the solution r 0 O of (6. 11) implies ASL of the solution x 0- of the composite system. This result does not have the intuitive appeal found in the results of Theorems 6. 1 and 6. 2. That is, there is nothing like a loop gain to suggest that the resulting stability criteria is reasonable. However, this is not surprising since the same difficulty is encountered even in a linear composite system with arbitrary interconnections. Theorems 6. 1 and 6. 2 are now corollaries to Theorem 6. 4 when the C matrix has the form C or CL' c L 6. 3 Examples As noted in the introduction, there is no known previous work that has expressed the viewpoint suggested in this report. On the other hand, many previous results and some interesting new results can be obtained using the techniques developed above. A few examples are given in this section.

Example 6. 1: Aizerman's Conjecture As noted in several previous examples (Examples 4. 1, 4. 2, and 5. 3), a transfer system modeled by the first-order equation x = -f(x) + bu(t), where 0 < cx2 < xf(x) < o0 for x, 0, f(0) = 0 and b > 0 is in Class E and has a gain 7 < b. Now consider a simple closed loop of n such systems modeled by the first-order (scalar) equations i = -fi(xi)+ biui(t) 0 < c.x.2 < xifi(xi) < 0o (6. 17) for i = 1, 2,.., n. Here it is required that for each of the system models c. = max axil < fi(xi) for allx. c E1} (6.18) so that an accurate gain estimate is obtained. The individual gains are b. 1 then -- and the loop will be ASL if n b. n 1 < 1 i 1 Aizerman's Conjecture (see Section 2. 4), in its most general form, implies the following problemo Consider an n th-order equation: = f(x) f(O) = 0 (6. 19) and the related parametric family of linear equations = J(x) y (6.20)

92 [af.i where J(x) is the Jacobian Matrix i/]. When does the asymptotic stability of (6. 20) for all values of x c En imply asymptotic stability-inthe-large of the null solution of (6. 19)? This question has received a considerable amount of attention in the control and applied mathematics literature [Refs. 2, 12, 15, 16, 18, 33] The best results to date consider only fourth-order systems In the following example, Theorem 6. 2 is applied in obtaining an answer for an nth -order system. In the composite system which is a loop of the subsystems modeled by (6. 17), the composite system model corresponding to (6. 19) is 1 -fl(Xl) + blxn 2 -f2(x2) + b2X' xn -fn(xn) + b x n n-1, and the related linear equations corresponding to (6. 20) are afl(xl) 0 0... 0 b ax af2(x2) b 0... 0 0 2 ax2 af (xn 0 0 0 b - n ax n The stability of the linear system depends on the roots of the characteristic equation

93 n afi(x i)n 1 n ax - X + (-1)n l Ib. = 0 i i i=1 or n )f. n *I (+ X. ) - IIb. = 0. (6. 21) i=1 ax i=1 1 This equation will have LHP roots if and only if (see Lemma 6. 3): af.(xi) ax. i > 0 ( i =,.., n) for allx En (6. 22) and n afi(xi) n H1 ax. H b. > O for allx E. (6. 23) i=1 ax1 i=l 1 NOw (6. 18) and (6. 22) together imply that for each i = 1,..., n 1 af.(xi) there is ax.i' E E such that = Ci. > 0 and thus, 1 ax. 1 n af.(xi) n n I > H c. > O forallx E E ax. - i 1 1=1 n af.(x!) n H ax. - H ci > 0 for x' = (XI, * * * Xh). i=1 1 i=1 Equation (6. 23) implies that n afi(xi) n n I > bi > O forallx E, i=1 ax. 1

94 and thus (6. 18), (6. 22), and (6. 23) together imply that n n II c. > II b. =1 1 i= 1 or n b. 12 I c < 1d i. That is, the stability of the family of linear equations (6. 20) implies that the estimated loop gain of the composite system (6. 19) is less than unity. By Theorem 6. 2, the composite system is ASL and Aizerman's Conjecture holds. This is apparently the first verification of Aizerman's Conjecture th for an n -order system. Moreover, (6. 22) suggests a root locus approach to the stability of this nonlinear system. To see this replace bl with IZbl, where Mi > 0, and write n n i-1 afi(xi) where gi(xi) ax. Normal root locus procedures can then be used on (6. 24) if one notes that the pole positions depend on x (also note that this root locus provides only a sufficient condition for stability). This is still a formidable task if all the fi.s are nonlinear. However, if only a few of the fi's are nonlinear, a useful stability criterionr for' (6. 19) is obtained.

95 Example 6. 2: A Complex Composite System In a recent paper, Markus and Yamabe [Ref. 30] have considered th the n -order system of equations = f(x) (6. 25) where af. (a) 0 (x) = O for j > i and for all x E, J af. (b) (x) < for i 1,...,n and for all x E x. (c) f(x) = 0 if and only if x =. (6. 26) They have shown that the null solution, x = 0, of the system of equations (6. 25) is ASL under the restrictions provided in (6. 26). When (6. 25) is viewed as an interconnection of first-order transfer systems, this result is quite reasonable. The condition af. = 0 for j > i shows that the composite system is basically a chain J with inputs to each transfer system coming only from the preceding system as shown in Fig. 10. x x 1 2 33 " 2 x3 n nFig. 10. A complex chain.

96 The stability of each of the transfer systems is ensured by the second and third assumptions. Since there are no closed loops, one would expect this very simple stability criteria. While it is not presently possible to prove this theorem as stated using the techniques suggested in this paper, a proof of a slightly weaker version is given below. Moreover, the validity of the general result suggests that the viewpoint adopted of the present paper can be extended to a broader class of problems. Theorem 6. 5: If each of the first-order equations in (6. 25) is of the form 1i b xI + bi2x2 +. + b. + fi(x i. (6. 27) where b10- 0, then the null solution of the (6. 25) system will be ASL under the restrictions af. 1 af. (b) ax (x) ~ -c. < 0 for i 1,..., n and for all XCEn (c) f(x) 0 if and only if x =. (6. 28) Proof: Although Theorem 6. 4 applies in this case, a step-by-step construction of an auxiliary system will be used to illustrate the procedure. For the ith subsystem take the Lyapunov function

97 vi(xi) =2 xi2 sothat xbui + + x ib- 1Ui, + Xifi(x i = 1... where u.. are the inputs to ith subsystems (u.. = x. when the inter1] 1] ]J connections are made) and u10 - O. Let b - max bij and note i, j that restrictions (b) and (c) imply that xifi(xi) < -c.x.2. Thus.i < -c.X.2 + b(Ui + " +Ui, i-) xi i = 1,...,n or Ci 2 b2(uil+ * + ui' il) Q1 < i i 2x 2c. when Lemma 5. 1 is used. Now note that (uil+ i.i < nuil +.. +nu' i21 so that 2 i < i 2 nb i3 i -C3V3 + -- -w-=, nb nb V.. < -cv += v. +. vn nn c 1 c n (6.29) n n

98 It is obvious that the corresponding system of auxiliary equations (equations obtained when the inequalities are replaced by equalities) has a null solution that is ASL. Thus, the null solution of (6. 2) is 12 ASL by Lemma 6. 2. Since vi(xi) = - xi, the null solution of the system of equation (6. 27) will also be ASL. Example 6. 3: Another Complex Composite System Consider the ninth-order composite system shown in Fig. 11. u! U2 Y2 FCepY3 u3C i Uysu Fig. 11. Composite system for Example 6.3. The individual transfer systems are assumed to have the following models: S1: Linear, constant-coefficient, third-order x1 = A1xl + D1Ul(t), Y1= H1 x, (6. 30) where -2 0 0 A1 S -3 -1 S O O -5 (6.31)

99 and S is any nonsingular 3 x 3 matrix. S2: Linear, variable coefficient, second-order x 2 = A2(t) x2 + D2 U2(t) Y2 = H2X2' (6.32) where 0 a(t) A2(t) =: (6. 33) - 1 - 2a(t) and a(t) is a continuous real valued function. In addition, it is assumed that a (t) exists for all t T, < d a- (t) <, (6. 34) dt and.5 < a-l(t) < 1. (6.35) S3: Nonlinear, first-order x3 f3(x3) + D3 u3(t)' Y3 = H3x3 (6.36) where f3(x3) = -x3 - 2 sin 2 x3. (6.37) S4: Nonlinear, third-order fx4 = f4(x4) + D4u4(t), =4- H4X4, (6.38) Y4& H4x4'

100 where f4(0) 0. The problem here is to determine a value of a positive constant k that will insure that the composite system is ASL if x4f4(x4) < -k lix4112 < 0 for x4# O. (6. 39) The interconnections suggested in Fig. 11 are assumed to be U1 4 Y4 U2 Yl + B23 Y3 U3 =Y2 U4 - Y2 and the resulting interconnection matrix B [see (3. 3)] has the partitioned form 0 0i 1 IoIB lo B =- -- - (6.40) _ I [ 0 [ 0 o0 I 0 O0 where I is an identity matrix. Because of the form of the models for S1 through S4 it is possible to describe these interconnections with the matrix C whose partitioned elements are C.i = D. Bij H. In this case, C has the partitioned form

' 3 101,I, OC14 c 21 0 C 0 o C = ---- 23 --- (6.41) 0 C32 I 0 1421 and the composite system model is x = f(x,t) + Cx, (6. 42) where x = col [X11, x12, x13, x21, x22, x31, x41, x42, x43] is the composite system state vector. When written out in detail, the composite system model has the form 11 12 13 = 11 + a 12X2 + a 13X3 + c14x41 + C14x42 + C14X43 21 22 23 12 21 11 a2212 23x13 + C1441 + c14x42 + 1443 31 32 33 13 = 31Xll +11 a32x12 + a33x13 + c14x41 + C14x42 + c1443 11 12 13 11 x 21 a(t)x22 + c21 x11 + c21x12 + C21x13 + C23x31 21 22 23 21 -x21- 2a(t)x22 + 11 + 21X12+ C21X13 +23 31 e (x11 12 31= f3(x31) + c32x21 c32x22 11 12 x 41 f1(x41'x42 x43) + c42x21 + c42x22 21 22 X42 = f42(X41'x42,x43) + c42x21 +c42x22, 31 32 43 = f43(x41' x42'x43) + c42x21 C42x22'

102 mn t nth th where the c's are the mn elements of Cij the ij submatrix in the partition of C. A straightforward approach to this problem would involve choosing a Lyapunov function involving the 9 state variables, evaluating its derivative with respect to (6. 43), and studying this derivative to obtain conditions under which it is negative definite. Clearly, this approach would involve the solution of some very difficult problems. On the other hand, the techniques developed in Section 6. 2 provide a method for obtaining a solution to this problem rather quickly. The first step is to find Lyapunov functions of the type described in Theorem 5. 2 for each of the transfer systems. This will, of course, be possible if and only if these transfer systems are in Class E. S1' Since this is constant-coefficient linear system, standard techniques can be used (see Theorem 2. 1) to obtain the Lyapunov function v(xl) = x1 Px1 where 1 0 0 P = (S- 1) 2 0 S-1 0 0 3 which has a derivative v(xl) = -xt'Qx1 where Q (S 1)' 12 0 S1

103 For this Lyapunov function the following inequalities are satisfied x1 l 2 < Vl(X1) < 3IIx 11 2 vl(xl) < -4 1x1 12 IIvvl(Xl)ll < 211PIl IIxllI = 6llxlII S2: In this case, choose v2(x2, t) x2' P(t) x2 where 2+a- (t) 1 P(t) Then v 2(x2, t) = -x2 Q(t) x2 where 02 Q(t) = 4 For this Lyapunov function the inequalities are 511,x2 112 < v2(2' t) < 3. 5 11x2 11 2 2(x2, t) < -2I2 x 2 IlVv2(x2, t) l < 71lx2 11 S3S Since this model if first-order, x3 = x31. Choose 12 v3(x3) =- X 31 and then = x3f) The inequalities are now

104 1 2 1 2 x31 v3(x3) < x31 v3(x3) <.56x31,2 IiVv3(x3)[ 1 ~Ix31 i = Ix311 oS~~~~~~~~~ 1 S4: In this case, choose v4(x4) -= x4t x4 and note that x4 f(x4) < -kll x4 11 2 Then x4 112 v4(x4) < -2 1x411 v4(x4) < -kIx4 112 4 -' 4 4 1l Vv4(x4)11 < 1lx4 1i With Lyapunov functions chosen for each of the transfer systems, it is possible to apply Theorem 6. 4. The system of auxiliary equations is, in this case, the fourth-order system =1 = -arl + ~1r4, 2 - a2r2 + 2r1 + y2r3 r3 =- a3r + P3r2 r 4 _ - a4r4 + 34r2. (6. 44 Here, 2 1 l = 3 a2 - a3 = 56 a4 = k and

105 i= 18 clC14 l 49 II+ ]lC ) /2 29= 2(1C21II + 11C2311) 2 /3 = 5I]C32II 2 [[C4 I] 4 114211 ~2 = 49 ( 1C2111 + C2311), where Cij are submatrices of C. (The numerical values of the above constants are obtained from the bounds on the Lyapunov functions chosen for the individual transfer systems. See Theorem 6. 4 and Eq. 6. 11) According to Theorem 6. 4, the null solution of the composite system (Fig. 11) will be ASL if the null solution of the system of auxiliary equations is ASL. The latter will occur if and only if the roots of thee characteristic equation [-a4(k)-X] [-al-X] [(-a1-X)(-a3-X)-7y23] + 134(k)[(-a3-X)P12] 0 (6.45) all have negative real parts. [Here the dependence of a4 and 34 on k is indicated as a4(k) and 14(k). ] Thus the original problem has been reduced to a much simpler problem that can be solved by root locus or numerical techniques.

CHAPTER 7 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 7, 1 Conclusions Lyapunov's Second Method is a powerful tool applicable to a broad class of stability problems regardless of order, complexity, or existence of nonlinearitieso Unfortunately, the full promise of this important tool has not been realized because of formidable problems encountered in the actual application of LSM to high-order, nonlinear systems. These problems are encountered at basically two points: the selection of a suitable Lyapunov function and the determination of suitable properties in the total derivative of this Lyapunov function. It is noted that the difficulty of these two problems is strongly dependent on the order of the system under study. Thus, low-order (third, fourth, even fifth) problems have received a considerable amount of attention while high-order problems are still inaccessible. These same order dependent limitations are present in the several general techniques that have been proposed for constructing Lyapunov functions (Krasovskii, Lur'e, etc. )o While most of these techniques are also theoretically applicable to problems of arbitrary order, practical limitations due to algebraic complexity, etco, grow rapidly with the order of the systems being treated. With these problems in mind, this thesis has described a circumvention of these two points of difficulty in the application of LSM by noting that many high-order systems are actually, or effectively, an interconnection of lower order systems for which both points of difficulty hold 106

107 less significance. The recognition of this interconnection structure has led to the definition of a composite system as an interconnection of simpler systems, termed transfer systems, and to a study of the properties of these transfer systems with the aid of LSM and the associated concept of the auxiliary equation. These transfer systems are input-output systems and, by using LSM, it has been possible to define a broad class of transfer systems that have what is called a "gain" relating the size of state space regions that are ASL to the size of the input. It has also been found that for composite systems made up of transfer systems belonging to a reasonably broad class, the auxiliary equations for the individual transfer systems can be interconnected following the interconnections existing in the given composite system. There results a system of auxiliary equations whose solutions have the same stability properties as those of the given composite system. Moreover, in the construction of this system of auxiliary equations it has been necessary to find Lyapunov functions only for the lower order transfer systemso The construction of one Lyapunov function for the high-order composite system has actually been avoided. This system of auxiliary equations gives a simple method of determining asymptotic stability-in-the-large for a class of high-order composite systems. As might be suspected, this simplification of the original problem is not obtained without some attendant sacrifice. The main disadvantage of this approach and a fault common to most attempts to obtain general sufficient conditions for stability is that it is sometimes overly restrictive

108 (overly sufficient) due to a failure to make best use of available information about the detailed structure of the system under analysis. This fine structure is "washed out" at points where matrix norms or absolute values are usedo On the other hands the introduction of transfer systems and interconnection inforn.ation into stability analysis has increased the class of problems to which L,~M haIlas practical. application. This is ind.cate by the proof of thle special case of Aizerman's Conjecture and other examples given in Section 6. 3o 7, 2 Suggestions for Future Research During the course of the investigation reported here, numerous areas for further investigation have become evident, A few of the more promising are mentioned briefly below. First is the general concept of vector Lyapunov functions and their application. The proofs of Chapter 6 actually involve the development and use of vector Lyapunov functions whose components are scalar Lyapunov functions for the individual transi'r s stems. An obvious question is whether these vector Lyapu-io-v i unctions have other applications, A first trial might be along the lines that scalar Lyapunov functions have proved valuable; for example, optimization problems. The vector Lyapunov functions may also be of value as an aid to finding single scalar Lyapunov functions when such are more desirable, A second area involves a direct extension of the above approach to alarger classof composite systems, For example, it is important to be

109 able to handle composite systems where one or more of the transfer systems is unstable when isolated from the composite system. This is the well-known problem of feedback stabilization which has been considered by other authors using conventional techniques [Refso 25, 37]. Another important extension would be a strengthening of Theorem 6. 2, to obtain stability criteria for composite systems where the loop gain is greater than unity. The theory of linear, constant-coefficient systems suggests that this might be an important class of problemso Both of these extensions will probably require the inclusion of a larger amount of interconnection information since the "polarity" of the individual interconnections will now become important. A third area for future research involves a study of the importance of Class G and Class E and the relation between them. For example, Class G appears to be larger than Class Eo Thus, there may be theorems similar to Theorems 6o 1, 60 2, and 60 4 that can be developed for systems in Class G. There may be even broader classes than E or G for which the transfer systems have auxiliary equations of the general o form r = w(r, t, u(t)). The extension of the basic approach used in Chapter 6 to such broader classes would then depend on an extension of Lemma 60 2 to cover the differential inequalities that were encountered. Finally, the special Lyapunov function v(x) = lix II leads to a useful auxiliary equation that has been employed by Sell [Refo 44], Rosen [Refo 37] and others, Using this approach, Rosen has reduced certain stability problems to nonlinear programming problems amenable to numerical

solution. It would be of interest to consider an extension of this type of procedure to a study of composite systems.

APPENDIX PROOF OF THEOREM 5.2 Consider the nth-order vector differential equation x = f(x,t) f(O0, T) - 0 (A. 1) where f(x, t) is in Class B (Defo 5. 6). The following two lemmas will be needed in the proof of the theorem. The first is due to Krasovskii (Ref. 18) o afi Lemma A. 1 If f(x,t) is in Class B with ax. < L (i,j=, o o.,n) and x(t;xo, t ) is a solution of (A. 1), then -nL(t- t ) x(t;x o, t0) 11 > Ix11 e 0 (A. 2) afO Proof: Since fc(i B with < L it follows easily that If.i(xt) I L 1Ixll Now note that n d 2 = X Z x i (Ao3) dt i=1 Since Ix. f.(x,t)l < Ixi LIlxll < Ll1x 21 II - I then Xi x = xi f.(x,t) > -Ix. (xt)l > -LI0x112 and thus

ddt xlx112 > -2 S Llx112 = -2n LIx112 i=1 Then, as in the proof of Lemma 4. 1, -2nL(t- t ) lx(t;Xoto)l 2 > I Xo 11 2 e and (A. 2) follows immediately. The second lemma, due to Nemytskii and Stepanov (Ref. 34), shows the continuity of the solutions of (A. 1) in the initial state. Lemma A. 2: If x(t;xo, to) is a solution to (A. 1), then axj (t;x, to) nL(t- to) <n e (A. 4) ax. fori=l, oaa, n; j=l, o o, nandt>to(xio is theith cortiponent of x). Proof: See Ref. 34, po 140 It is now possible to prove Theorem 5. 2 which is restated below. Theorem A. 1: The null solution of (A. 1) is ESL if and only if there is a positive definite function v(x, t) such that (a) cll xl0 2 < v(x, t) < c2 I1x I 2 (b) ( t)< ~ 2t Il x (c) 11lVI v cJ Ix 1,

where c1, c2, c3, and c4 are positive constants. Proof: (1) Sufficiency: if there is a positive definite function satisfying requirements (a), (b), and (c) above, then c3 and by Lemma 6. 1, c3 e (t- to) C2 v < v e -- O Thus c3 c (t- to) t1Ie2 c o l x(t;xoto)12 < -x 102 e or c3 Ilx(t;Xo, t ) Il < I1x x1 e which shows that the null solution of (Ao 1) is ESLo (2) Necessity: If there are positive constants a and i such that for all (Xo, to)e En x To -a(t- t ) Ilx(t;x0,t0)l <) /3IQx0 IDe (Ao 5)

then let T = I Qn 3 2 and choose t+T v(x, t) = fS Ix(O;x, t) 1 2 dO. (A. 6) t Now substituting (A. 5) in (Ao 6) gives v(x, t) < f 32 llxl 2 e2a( t) dO = c2 11x11 t which proves the right inequality in (a). To prove the left inequality substitute (A. 2) in (A. 6) giving v(x,t) > f III l e2nL(t)dO = cl1 1xa t To prove (b), v(x, t) is differentiated with respect to t along the solutions of (A. 1)o Thus, when x = x(t;xo to) dv(x(t;xo, t ), t) d t dt t L 1I0x(a;x(t;xo, to), t1 2 dO t + llx(t+, x(t;xo, to) t) [i 2 - I x(t;x(t;xot) t) t10 2 t a t a x(O; x(t;xo0, t),tl d (A 7) But x(O; x(t+ At; x, t), t+ At) = x(O;x(t;xo, to), t) so the derivative inside the integral in the last term on the right in (A. 7) is zero.

From (A. 5) and the fact that T = 1 fn /1'2 it is clear that IIx(t+r; x(t;x, t ),t) 2 < 32 IIx(t;x, to)II 2 e- 2a(t+T-t) 1- Ix(t;x to) 11 2 Thus, at the point x(t;xo, to), dv 1 2 _ dt < 2 IIx(t;xo to)0l - I2 x(t;x 1 lx(t;x t )112 which proves (b). Now to prove (c) note that a v(x,t) t+T n axi(o;x, t) ax~. 2 f xi(O;x, t) dO (A 8) axj t i=1 axj Substitution of (Ao 5) and (A. 4) in (Ao 8) gives av(x, t) e t+T(nL - a)( - Iax. I < S 2n2 3 1xl e(nL a)(- t dO kllx II j t Now n'2 ~VvIll2 = S ax I < nk2llxll2 so proving ( < cncluding the proof of the heorem proving (c) and concluding the proof of the theoremO

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