# ON THE CONSTRUCTION OF LIAPUNOV FUNCTIONS by .

ON THE CONSTRUCTION OF LIAPUNOV FUNCTIONS FOR THIRD ORDER CONTROL SYSTEMS WITH LIMIT CYCLES by . Michael John Wozny A Dissertation Submitted to the Faculty of the DEPARTMENT OF ELECTRICAL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 19 6 5 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE I hereby recommend that this dissertation prepared under my direction by entitled Michael John Wozny_______________________________ On The Construction of Liapunov Functions for_____ Third Order Control Systems with Limit Cycles be accepted as fulfilling the dissertation requirement of the degree of _________________________________________________________ After inspection of the dissertation, the following members of the Final Examination Committee concur in its approval and recommend its acceptance:* 4 (/m utf0 5 ^ H, 17 *This approval and acceptance is contingent on the candidate's adequate performance and defense of this dissertation at the final oral examination. The inclusion of this sheet bound into the library copy of the dissertation is evidence of satisfactory performance at the final examination. PLEASE NOTE: Not original copy. Pages tend to "curl" throughout. Filmed in the best possible way. University Microfilms, Inc. STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library, Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made, Requests for permission for extended quotation from or reproduction of this manu script in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED /C ACKNOWLEDGMENTS The author wishes to thank Professor Donald G 0 Schultz for his guidance» continued Interest, and encouragement shown In the preparation of this thesis. Thanks are also due to the author’s wife, Nancy, for her support throughout this study and for her excellent typing of the manuscript0 This research was supported by the National Science Foundation under Grant GP-2237° ill TABLE OF CONTENTS Page ooooeooooeoooooot LIST OF ILLUSTRATIONS vl ABSTRACT O O O O O O O O O O O O © O O O 0 O 6 © O O O < ,Vili CHAPTER Is I © I I n INTRODUCTION AND ORGANIZATION t s i 'o c L u o l y l o z i © © © © © © © © © © , c1 © © © © © © t 1 ©2 Organization of the Work © © © © © © © © . © , , I ©3 Notation © © © @ © © © © © © © @ © @ © © © © < , CHAPTER 2s 2 ©I 2©2 2©3 2.4 2©5 CHAPTER 3s 3 ©1 3©2 3©3 3.4 , . 6 7 , 8 <, 20 <, 34 CONSTRUCTION OF LIAPUNOV FUNCTIONS FOR SECOND ORDER LIMIT CYCLE PROBLEMS © © © © < LIAPUNOV FUNCTIONS FOR THIRD ORDER LIMIT CYCLE PROBLEMS © © OOOOQOOQOOeOOQt c o < SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH, Summary o o o o e e o e o o o o o Suggestions for Future Research iv o e o o © o © 47 47 48 55 65 84 Introduction o e o o o e o o o o o o o o o o Previous Work © © Reduction of Third Order Problem e o © o o Liapunov Formulation of the General Problem CHAPTER 5s 5©1 5.2 6 Introduction © © © © ©* © © © © © © © © © © © < Estimation of the Region of Asymptotic Stability © . © , © © . © © . . « « Application of the Polncare-Bendixson Theorem , Liapunov Functions Definite with Respect to the Limit Cycle e o e o o e o e e c CHAPTER 4s 4.1 4.2 4.3 4.4 . GENERAL BACKGROUND © Introduction © © © © © © © © © © © © © © © © ( State Space Representation © © . © © © © . © Singularities and Limit Cycles in Nonlinear Systems .......... . . . » . © , Stability and the Second Method of Liapunov © Methods of Constructing Liapunov Functions © 1 3 4 o o e o e © c 84 85 94 11.9 134 134 137 v TABLE OF CONTENTS - - Continued Page APPENDIX A al A»2 As SECOND ORDER SYSTEMS WHICH EXHIBIT LIMIT CYCLES EXPRESSIBLE IN CLOSED FORM „ . 139 Introduction o o o e o o o o o o o © © © © © © 139 A Synthesis Method © « . © . © © © © © « © © © 140 LIST OF REFERENCES © © © © © © © © © © © © © © © © o © © 1^*9 LIST OF ILLUSTRATIONS Figure Page 2.1 Various Representations of the System Solution 2.2 Classification of Singular Points for Second Order Systems . . . . . . . . . . . . . . 14 Classification of Singular Points for Third Order Systems . . . . . . . . . . . . . 15 Stability Behavior of Limit Cycles in Second Order Systems . . . . . . . . . . . . . 19 Geometric Interpretation of the Polncare-Bendixson Theorem . . . . . . . . . . . 22 Geometric Interpretation of the Liapunov Stability Theorem . . . . . . . 31 36 2.3 2.4 2.5 2.6 9 2.7 Geometric Interpretation of the Zubov Equation . . 2.8 Approximate Region of Asymptotic Stability Determined by Variable Gradient Method , . . . . 45 Inner Bound for van der Pol Equation via Variable Gradient Method . . . . . . . . . . 54 . 58 Bound for Second Order Limit Cycle Using Contact Curves . . . . . . . . . . . . . . 60 Bound for van der Pol Example of Section 3«3 . 66 Liapunov Function Positive Definite with Respect to the Closed Curve W(jc) = 1 . . . . 69 Annular Region Determined by Proposed Liapunov Function . . . . . . . . . . . 78 V = Constant Curves for the Liapunov Function given in Equation (3.102) . . . 81 Block Diagram of System Considered in Szego and Geiss (1963) . . . . . . . . . . . . 90 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 Contact Curve when a(x) is a Linear Form vi . . . vii LIST OP ILLUSTRATIONS— Continued Page Figure 4,2 Typical Trajectories for System (4,1) 9 0 0 , , , , 4,3 Limit Cycle for System (4,88) , , . , 0 , 0 , , o 4,4 Typical Behavior for System (4,146) , 0 , 0 0 0 0 , 1 31 A,1 Block Diagram of Nonlinear Feedback System 0 , O 0 14 2 A,2 Inner Bound for Limit Cycle of System (A,8 ) S O S O \ o 93 118 144 ABSTRACT In this work the Second Method of Liapunov is used in conjunction with the Poincare-Bendlxson Theorem to ex amine methods of determining or specifying limit cycles in J second and third order systems. For second order systems the variable gradient method is used to find contact curves in the construction of the annular region required by the Poincare-Bendlxson Theorem. In this manner it is possible to bound the limit cycle to a region of the state plane. With regard to finding the exact limit cycle a method is formulated by defining a Liapunov function referred di rectly to the limit cycle. Since the limit cycle is unknown a priori. the Liapunov function is defined as a known function of an unspecified closed curve. The form of the Liapunov function is chosen according to the geometrical requirements of the Poincare-Bendixson Theorem. Thus, for the two dimen sional case, the Liapunov function represents two concentric closed curves and, in three dimensions, a toroid. Making use of the hypothesis of the Poincare-Bendlxson Theorem that all the trajectories cross the region bounding the limit cycle in an inward direction, the time derivative of the Liapunov function is forced to be negative definite with respect to viii ix the limit cycle. closed This condition forces the unspecified curve to be the limit cycle. It is shown that the above formulation is very general for both second and third order systems. Unfortunately, it is difficult to establish a routine method of solution in the third order case. However# a method of solution is developed for the special case of systems where the trajectories tend to a stable surface containing the limit cycle and exhibit a certain symmetry. In summary, this work. there are three major contributions of First, a scheme is developed for applyingthe variable gradient method to the problem of finding contact curves for limit cycles in second order systems. Second, the special class of third order systems Which have a limit cycle contained in a stable surface is Investigated and a method developed for its solution. Third, the general problem of enclosing the limit cycle in a toroid is formu lated and its properties investigated. I CHAPTER 1 INTRODUCTION AND ORGANIZATION 1.1 Introduction As modern control systems become more sophisticated and include nonlinear elements, various phenomena occur which cannot be explained by linear theory. One such phenomenon is the existence of sustained periodic oscilla tions which are independent of the initial conditions ap plied to the system. This type of oscillation exists in engineering practice (in relay control systems as well as in nuclear reactors and guidance systems), and in most cases it is difficult to eliminate from the system because of the lack of understanding of its basic mechanism. In other cases the sustained oscillation is desired, as in electronic oscillator circuits. The investigation of the fundamental properties of sustained oscillations (limit cycles) is the subject of this work. To understand the properties of limit cycles, it is necessary to formulate the problem in terms of nonlinear differential equations, and then study the equations by general mathematical techniques. The mathematical techniques can be divided into two classes % (1 ) the perturbation or analytic methods which are basically time domain approximation schemes and (2 ) the topological methods consisting of the geometrical or state space approach to the limit cycle problem* concerned only with the latter approach* This study is The Poincare- Bendixson theory is basic to the topological approach and is very general in scope, but unfortunately, very difficult to apply* This difficulty stems from the problem of specify ing the solution curves of the differential equation without actually finding the complete solution* In this work the Second Method of Liapunov Is used to specify the solution curves of the systems* The Second Method is a state space technique for determining the sta bility of a solution by examining the solution curves with respect to that given solution* The principal feature of the Second Method is that it abstracts only the stability information from the system equations and does not require the entire solution* Unfortunately, the method represents only a sufficient condition and requires the existence of a testing function (Liapunov function) for which only some what restricted techniques of construction are available* The purpose of this work is to investigate methods of constructing Liapunov functions which can be used in the study of systems having limit cycles, A general formulation is proposed in which a Liapunov function is found displaying the properties of the Poincare-Bendlxson Theorem, It is shown that the formulation provides a consistent approach to both second order and third order systems* 3 1,2 Organization of the Work This work is organized around the investigation of limit cycles by Liapunov methods, and contributions are made both for second order and third order systems, with empha sis on the latter. Approximately half of the work deals with background material and second order systems. Chapter 2 presents the basic properties of singular points and limit cycles in second and third order systems, A summary of the basic concepts of the Second Method of Liapunov useful for studying limit cycle problems is also included in Chapter 2, In Chapter 3» Liapunov techniques are used to investigate limit cycles in second order systems. After the known methods are discussed, a new Liapunov function is proposed which is defined in terms of the limit cycle. The Liapunov function is shown to be a formulation of the PoincareBendixson Theorem, The second half of this work, consisting of Chapter 4, is concerned with the third order problem. First, it is shown that the existing methods considered in Chapter 3 are applicable to the limit cycle problem only because of the simple structure of the two dimensional plane and cannot be extended to three dimensions. -third order problem are given. Then two formulations for the The first approach, based on a paper of Szego and Geiss (1963)» is valid for the class of systems which have a limit cycle contained in a stable sur face, A method is developed for finding the required surface 4 for a special class of systems. The second approach is an extension to three dimensions of the Liapunov function formulation proposed in Chapter 3• In this case, the Liapunov function represents a toroid enclosing the limit cycle. This formulation is also shown to be valid for systems whose trajectories spiral about the limit cycle as they approach It. Such a behavior cannot be handled by the first method. Chapter 5 summarizes the results obtained, and de fines areas for further research. Examples are presented throughout the work whenever they can serve to better Illus trate a point. 1.3 Notation Wherever possible, vectors are represented by under lined lower case Roman letters, as x = (x^,x2 ,....x^) or as 5 = c o K x ^ . x ^ , ... .x^). W. An exception is the gradient symbol The transpose of a vector is designated by a prime, as 2** rule. Upper case Roman letters designate scalars as a The n dimensional state space Is designated by En , and its Euclidean norm by ||x || . of class C A function f (x) is said to be if it is continuous and has continuous first par tial derivatives. In the notation G : g(x) = 0, G represents the state space orbit or curve of the equation g(x) = 0. In general, theorems and definitions are titled and Indented so as to set them off from the text. Where many definitions are given, only the most significant are set off„ while the rest are simply underlined. The phrase "second order limit cycle" means "limit cycle of a second order system." A similar condition is used for third order systems. In Section 2.5 a. lower case v is used in defining the scalar function associated with the Zubov equation. This is done for convenience only in Section 2.5® Elsewhere, scalar functions associated with the Zubov equation are designated by upper ease letters. CHAPTER 2 GENERAL BACKGROUND 2.1 Introduction In this chapter the background necessary for the understanding of this work is presented. Section 2.2 is concerned with the general representation of a control system in terms of a set of first order nonlinear differen tial equations. The behavior of the system solutions are examined in Section 2.3 with emphasis bn the limit cycle concept for second and third order systems. In Section 2.4 the notion of stability in the sense of Liapunov is defined and then discussed with reference to the stability of limit cycles. The fundamental concepts of the Second Method of Liapunov are also included in this section. Section 2.5 concludes the chapter by presenting two general methods for generating Liapunov functions which are of importance in this work. The topics discussed in Chapter 2 present background material for this study. No attempt is made at completeness, and only those topics which are needed in later chapters are included. 6 7 2,2 State Space Representation The control systems considered In this work are assumed to satisfy the following conditions t 1) The state of the system, at any time t , Is completely defined by the variables x1 (t ), x2(t) 2) xn (t). (2.1) The variables given above are related by a set of n first order differential equations of the form x = f1 (x1 .x2 X >. 1 = 1,2,...,n, (2.2) where the dot Indicates differentiation with respect to the Independent variable t , and the f^ are sufficiently smooth to guarantee a unique continuous solution through every Initial condition. The variables ,...,x^ are called state variables, and the n-tuple (x^,x2 ,...,x^) represents a point In the n dimensional Euclidean space called the state space. En . f^ — x ^ , 1 « l,2,...,n - 1 , the state space Is called a phase space. If (2.3) The system of differential equations of the form (2.2), written In vector notation as Z = Liz). (2.4) 8 Is called an autonomous system. Physically, this means that the control system Is neither time-varying nor forced. Thus, only the characteristic behavior of the unforced system Is Investigated in this work. The solutions of Equation (2.4) are given various names depending on the space and form in which they are represented. If the solution is expressed in the form g(x, t) = C, (2.5) and represented in the (x, t)-space, it is called an integral curve. If solution (2.5) is parametrized by t in the (jc)-space (i.e., in the state space, En ), it is called a trajectory or a motion. The direction of increasing time is called the direction of the trajectory. For the case where solution (2.5) is represented in En as a locus of points without reference to a parametrization, it is called an orbit or oath. Figure 2.1 illustrates the various repre sentations of the solution of the system equations. In addition to the conditions given above, the systems considered in this work are also assumed to display an oscillatory equilibrium state, e.g., a limit cycle. The properties of limit cycles are taken up in the next section. 2.3 Singularities and Limit Cycles in Nonlinear Systems A physical system is said to be in a state of equi librium if its response assumes either a stationary value or 9 X(0) INTEGRAL CURVE TRAJECTORY (ORBIT) Figure 2,1 Various Representations of the System Solution 10 a fixed, periodic oscillation. Furthermore „ nonlinear systems often have several equilibrium states with related stability properties. Thus„ in order to examine systems with limit cycles„ it is necessary to understand the prop erties of equilibrium states in general, Such an under standing is obtained by considering certain aspects of the mathematical theory of nonlinear differential equations. Only those topics are considered here which contribute directly to the understanding of the class of systems of interest in this work. Firste properties of stationary equilibrium states are examined in terms of the mathematical notion of a sin gular point, The local behavior of the system trajectories is discussed for the various types of singular points in second and third order systems. Then limit cycles are dis cussed In terms of the mathematical concept of a limit set, culminating in the Poineare-Bendixson Theorem, A stationary equilibrium state of a system is charac terized by the fact that the solution remains constant and does not vary with time. Such a condition is represented as a fixed point in En and is'called a singular point. This concept is expressed more formally in the following def initions Definition 2.1 Singular Point A point x is called a singular (critical9 stationary) point of Equation (2.4) if 11 f(xo ) = 0. (2.6 ) A point which Is not a singular point is called a regular point. It should be recalled from Chapter 1 that only the principal definitions are set off from the text and numbered. Other definitions are simply underlined. For the purposes of this discussion it is assumed that the singular points are isolated and that a singular point occurs at the origin of the state space. Any singular point can be moved to the origin by a simple translation of coordinates. Thus, the nonlinear system of interest in this work is given by x = f(x), f(0) = 0. (2.7) The general behavior of trajectories near singular points for second and third order systems is considered next. Coddington and Levinson (1955) and Hartman (1964) consider second order systems of the form x = Ax + h(x), (2.8) where A is a 2 x 2 matrix, and h(x) represents a power series satisfying (2.9) and show point the at the behavior of the trajectoriesnear the singular origin is characterized bytheeigenvalues. 12 except for the special cases given below, of the linearized system, x = A^. (2.10) Hayashi (1964) considers the above conditions for third order systems. Singular points having eigenvalues which are either zero, imaginary, or nondistinct are called singular points of higher order, and their investigation requires the con sideration of the nonlinear terms of the system equations (Coddington and Levinson 1955# Hayashi 1964). For the pur poses of this study this class of problems can be handled by The Second Method of Liapunov and will not be considered any farther. A singular point x0 is stable if in some neighborhood of xQ all the trajectories tend to x0 as t -»- *>. Also the sign of the real part of the eigenvalues determines the stability of the singular point. The various types of singular points for Equation (2.8), as characterized.by the eigenvalues of the linearized system (2.10), are given below for second and third order systems. For second order systems the classification of singular points is given as follows (Coddington and Levinson 1955# Hartman 1964): 1) If the eigenvalues are real and of the same sign, the singular point is a node. 2) If the eigenvalues are real and of opposite signs, 13 the singular point is a saddle point. 3) If the eigenvalues are complexconjugates„ the singularity is a focus. 4) If the eigenvalues are imaginary, the singu larity is either a center or a focus. The various classifications defined above are shown in Figure 2,2. For third order systems the classification becomes (Hayashi 1964)s 1) If all the eigenvalues are real and of the same sign, the singularity is a node. 2) If all the roots are real and not of the same sign, the singularity is a saddle point, 3) If one root is real and two are complex conju gates such that the real part of the conjugate roots is of the same sign as thereal root, the singularity is a focus. 4) If one root is real and two are complex conju gates such that the real part of the conjugate roots is of the opposite sign of the real root, the singularity is a saddle-focus. 5) If the two conjugate roots are imaginary, the singularity is either a center, focus or a saddle-focus. This classification is shown in Figure 2.3° 14 (a) NODE (STABLE) (b) FOCUS (STABLE) (c) SADDLE POINT Figure 2.2 Classification of Singular Points for Second Order Systems 15 (b) (a) NODE SADDLE (c) FOCUS k±> (d) SADDLE FOCUS Figure 2,3 (e) CENTER Classification of Singular Points for Third Order Systems 16 In addition to the stationary states described above a nonlinear system may also have oscillatory equilibrium states. Oscillatory states which are dependent upon the Initial conditions of a system, as In conservative systems, are always associated with centers; hence, they represent a property of a particular type of singularity, and not of a phenomenon directly attributed to the nonlinearity of the system. Such oscillatory states do not comprise the main effort in this study. Isolated periodic oscillations which are independent of the initial conditions of the system repre sent the main topic of interest in this work. Oscillatory states of this type are called limit cycles (defined more precisely below) and are unique to nonlinear systems. Limit cycles, as contrasted to singular points, do not appear explicitly in the form of the system equations. Consequently, their properties can only be determined by examining the limiting behavior of the system trajectories as t The limiting behavior of a trajectory is given precise meaning through the mathematical notion of a limit set (Cesari 1963# Coddington and Levinson 1955* Hartman 1964). Definition 2.2 Positive Limit Set A positive limit (or limiting) set L(C+ ) of a solution C+ !2 = x(t), t ^ 0, of Equation (2.4) is a set of points xQ for which there exists a sequence t o < t, < ... , where t — ► » as n 1 n 17 such that £( tn ) - ^ x 0 as n - * ™ . (2.11) For example, the solution C+ :x(t) = e atslntit, has thelimit set L(C+ ) = 0. a > 0, Hence, for (2.12) the casewhere the limit set Is a closed curve, Definition 2.2 leads directly to the general definition of a limit cycle. Definition 2.3 (+) Limit Cycle A periodic orbit C0 ijc ■ x^(t) , -<» < t < 00 which is contained in some limit set L(C+ ), C* c£ C0 , Is called a (+) limit cycle. Several statements concerning Definition 2.3 are in order. It should be noted that this definition Is not restricted to second order systems, therefore, the limit cycle concept is valid for third order systems. Also, thecondition C+ C0 implies that the oscillation is Isolated; hence, not every closed orbit is a limit cycle (e.g., the closed orbits associated with a center). Since the limit cycle is a property of the system trajectories and does not appear explicitly in the form of the system equations, its existence and stability properties are difficult to determine. For second order systems the motion is sufficiently restricted so that very general re sults about the limit cycle behavior are possible. This is 18 not true In the third order case. In the remainder of this section the classical existence and stability properties of limit cycles In second and third order systems are given. The body of knowledge concerning the topological behavior of trajectories of nonlinear second order systems of the form x1 = P (x1 ,x2 ) x2 = Q(x1 ,x2 ), (2.13) with suitable restrictions (Coddlngton and Levinson 1955)• Is called the Polncare-Bendlxson Theory. Only second order systems which possess limit cycles are considered. A funda mental property of this class of systems Is the division by the limit cycle of the state plane Into two disjoint re gions so that trajectories which originate In one region cannot cross the limit cycle, and consequently, must remain In that region for all t ^ 0 (Hartman 1964). If the tra jectories In some neighborhood of the limit cycle tend to the limit cycle as t -*• 00, the limit cycle is said to be stable (more precisely, orbitally stable, see Definition 2.5 in the next section). In addition to the stable limit cycle other types of two dimensional behavior are unstable and semistable limit cycles (Pontryagln 1962). The three types of behavior are illustrated In Figure 2.4. Pontryagln (1962) also states that a limit cycle which is stable for t > 0 Is unstable for t < 0; hence, the stability of a (a) STABLE (b) UNSTABLE (c) & (d ) SEMISTABLE Figure 2.4 Stability Behavior of Limit Cycles in Second Order Systems 20 limit cycle and of the other equilibrium states is reversed by running time backwards. This reversal of stability with a change in the direction of time is also a property of higher order systems. In many systems only one singularity is enclosed by the limit cycle. In this case the singu larity must be either a node or focus whose stability is opposite to the stability of the interior of the limit cycle (Coddington and Levinson 1955)• The Polncare-Bendixson Theorem represents a very general hypothesis for the existence of a stable limit cycle in second order systems (Cesari 1963, Coddington and Levinson 1955, form is due Hartman 1964, Struble 1962). The following to Struble (1962)% Theorem 2.1 Polncare-Bendixson Theorem If C+ is bounded and L(C+ ) consists only of regular points, then L(C+ ) is a periodic orbit and either 1) C+ = L(C+ ), or 2) C+ "spirals" to L(C+ ) on one side of L(C+ ). Although the theorem is general, it is not in a form which can be easily applied. The following geometric interpre - tation, due to Poincare, reveals a more convenient form (Hayashl 1964, Stoker 1950)i let R be a ring-shaped region such that R does not contain any singular points. If all the system trajectories cross the boundary of R in a direction toward its interior, then the above hypothesis is satisfied, and R contains a stable limit cycle 21 (see Figure 2.5) • It should "be mentioned that even with the above geometrical Interpretation It Is difficult to construct a suitable region R. This theorem Is considered again in Chapter 3, where Liapunov functions are used to construct R e Very few general results are available for the exis tence of limit cycles in third order systems. Fuller (1952) showed by means of a counter example that the PoincareBendixson Theorem cannot be extended directly to third order systems by making R a toroid. The existence of limit cycles in the several third order systems considered in the litera ture were determined by showing, in addition to crossing the toroid in an inward direction, that the system trajectories also circulate inside the toroid and converge to a single curve (Friedrichs 19^6, Rauch 1950» Perello 1965), In this work limit cycles are examined via the Second Method of Liapunov, Although this approach restricts the class of problems considered, it is suitable for a large class of practical problems, The next section presents the basic concepts of the Second Method of Liapunov, 2,4 Stability and the Second Method of Liapunov The Second Method of Liapunov is a method for determining the stability (in the sense of Liapunov) of a trajectory (unperturbed trajectory) with respect to its neighboring trajectories. The method proposed in this work 22 LIMIT CYCLE REGION R Figure 2o5 Geometric Interpretation of the Poincare-Bendixson Theorem Is to let the unperturbed trajectory be the limit cycle so that the Second Method of Liapunov can be used to examine Its stability properties. This formulation presents two conditions not encountered In the application of the Second Method to stationary equilibrium states. The first Is the restrictiveness of stability in the sense of Liapunov (Liapunov stability) compared to orbital stability, and the second condition is the incomplete knowledge of the equi librium state. I.e., of the limit cycle. These conditions are discussed here only to the extent of showing how they arise in the theory. A complete investigation is deferred to the later chapters where the necessary groundwork is developed. The section concludes with a presentation of the fundamental concepts of the Second Method of Liapunov. Consider first the meaning of stability in the sense of Liapunov and its relation to other types of stability, namely, orbital stability. Since stability is meaningful physically only if the unperturbed trajectory is an equi librium state, let x^(t) represent an equilibrium state of the system x = f (jc). (2.14) Then the stability of xfi(t) in the sense of Liapunov is de fined in the following manner: 24 Definition 2.4 Stability In the Sense of Liapunov The equilibrium state (t ) Is stable in the sense of Liapunov if for every e > 0, there exists a 6(e) > 0 such that any solution x(t) of the system (2.14) for which x(0) - xe (0)|| < 6, (2.15) < e (2 .16 ) satisfies ||a(t) - xe (t)|| for all t ^ 0. Note that Liapunov stability implies a time dependent com parison between the perturbed solution x(t) and the equi librium solution x (t ). This comparison is not evident when e defining the stability of a singular point, since the equi librium solution is a constant. For oscillatory equilibrium states, such a time dependent comparison classifies, for example, nonlinear conservative systems as unstable. This is true because for nonlinear conservative systems adjacent orbits have different periods so that the distance between two orbits may be quite large when measured at specific instants of time, whereas, in reality the orbits are geo metrically close together. To include such cases in the definition of stability, orbital stability is defined as follows: 25 Definition 2. 5 Orbital Stability (as t — » oq) Let Ce : x = xe (t) be an equilibrium state of the system (2.14). Then Is orbltally stable as t » If for every € > 0, there exists a 6(e) > 0 such that If xQ Is within a distance 6 of Ce , then all solutions C+ : x = x (t) of (2.14) remain within a o o distance e of C for t ^ 0. e Thus, orbital stability removes the time dependent comparison In Definition 2.4 and compares the nearness of the two tra jectories geometrically. In summary, the most general form of stability for oscillatory equilibrium states is orbital stability. But for systems which have a single limit cycle enclosing one singularity at the origin, a condition commonly satisfied in physical systems, the limit cycles are Liapunov stable. This statement is verified for second order systems in Chapter 3 and for third order systems In Chapter 4. The Second Method, where applicable, offers an important ad vantage over the classical procedures of estimating the limit cycle and determining its stability properties, be cause it is more systematic, and therefore, easier to apply. After discussing the second condition associated with limit cycle problems mentioned in the introduction, the remainder of the section is devoted to the presentation of the funda mental concepts of the Second Method. 26 As stated previously, the method of investigating limit cycle behavior in this work is to let the limit cycle be the unperturbed trajectory and apply the Second Method of Liapunov. Consider first a method of reducing a general unperturbed trajectory to a normalized form (Krasovskil 1963). Let Z = &(z) (2.17) represent the system of interest, and &(t) the unperturbed trajectory. Then the transformation x = 2 -X (2.18) takes system (2.17) Into x = f(x), (2.19) where f(x) = &(x +X)- £(X) , (2.20) x =0 (2.21) so that is the new form for the unperturbed trajectory. Hence, in principle, all problems can be transformed to a form which requires only the investigation of a singular point at the origin. However, if the unperturbed trajectory is a limit cycle, such a transformation is not meaningful practically 27 because It requires that the limit cycle be known before hand. In most systems not only Is the limit cycle unknown, but It cannot even be expressed In a closed form. Thus, the Liapunov Theory, I.e., the Second Method, formulated for the general unperturbed trajectory Is valid in a practical sense only If the unperturbed trajectory Is known a priori. as for example, a singular point of some given system. For the case where the unperturbed trajectory Is a limit cycle, the theory is only formal and does not indicate any methods of Investigating the limit cycle be havior. The fundamental concepts of the Second Method are presented next where the system Is assumed to have the form (2.19) with Equation (2021) as the unperturbed trajectory. General references for this material are Hahn 1963, Krasovskll 1963 , LaSalle and Lefschetz 1961, and Malkin 1952. The Second Method of Liapunov is based on finding a function, V(x), which is a measure of the distance from any point in the state space to the equilibrium state. By evaluating V(x) along the system trajectories, x(t), stability can be established if V(t) = V(x(t)) decreases monotonically with time. (2.22) A principal advantage of the Second Method is that V(x) is not unique. Functions which are definite with respect to sign are suitable as distance functions. 28 Definition 2.6 Positive (Negative) Definite A scalar function, V(x), Is positive (negative) definite In a region N If for all x In N, x ^ 0, V(x) > 0 (< 0) (2.23) and V(0) = 0. V(x) is called positive (negative) semldefInlte if the strict Inequality (2 .23 ) Is replaced by the weaker condition V(x) ^ 0 U 0). Furthermore, definite functions from which information about the system stability can be determined are called Liapunov functions. Definition 2.7 Liapunov Function A scalar function, V(x), defined on a neighborhood, N , of the equilibrium state x = 0 Is called a Liapunov function for the system x = f(x), f (0) = 0, (2.24) If the following conditions are satisfied; 1) V(%) is of class C | 2) V(jc) is positive definite; 3) V(jc) is negative semldef inlte along each trajectory in N. The name Liapunov function originally referred to the con ditions imposed on a definite scalar function, V(x), in the Stability Theorem of Liapunov (Theorem 2.2) to show stability in the sense of Liapunov. But a Liapunov function as now 29 used In the literature means a definite function associated with the hypotheses of various theorems which present re finements of the Stability Theorem of Liapunov. Hence, conditions (2) and (3) In the above definition are often altered to conform with the hypothesis of the particular theorem under consideration. Normally, these changes appear as qualifying conditions In the statement of the Liapunov function. I.e., ”a Liapunov function satisfying . . .". Geometrically, the Liapunov function V = V(x) Is represented as a well behaved n + 1 dimensional surface, L , with an Isolated minimum at the origin of the (x, V) space. V(%), when evaluated along a trajectory. Is represented as a curve on this surface. Lefschetz (1963) states that In a sufficiently small neighborhood of the origin, £ can be pro jected in a continuous manner onto the state space as a family of nested level surfaces, V(i) = C, enclosing the origin. (2.25) Thus the stability problem has a com plete geometrical representation In the state space where evaluating V(%) along a solution takes on the significance of the trajectory crossing successively smaller surfaces of constant V. The next theorem represents a sufficient condition for stability. Theorem 2.2 Stability Theorem of Liapunov If there exists a Liapunov function, then the singular point x = 0 of system (2.24) Is stable (In the sense of Liapunov). It Is Instructive to consider a geometrical proof of this theorem; a two dimensional Interpretation Is shown In Figure 2.6. As preliminary notation, let S denote a h sphere consisting of points jc satisfying J|x|| < h. Assume that there exists a Liapunov function V(x) defined on a neighborhood N. Then there exists a sufficiently small constant, C > 0, such that V(%) = C represents (2.26) a closed surface, S, which lies wholly In N. It also follows from the above assumption that V = VV'f < 0. For any (2.2?) on S, V V represents the outward normal of S, and f the direction of the trajectory on S. Thus, geometrically, condition (2.2?) signifies that the direction of every tra jectory onS is either toward the interior to S. face S. Now of S or tangent choose £so that the sphere Se contains the sur Then clearly any 6, where will satisfy the stability definition. Is contained in S, 31 S: V(x) = C Figure 2.6 Geometric Interpretation of the Liapunov Stability Theorem 32 Liapunov stability is physically meaningful only for the case where the trajectories eventually approach the equilibrium state. This type of behavior of the system tra jectories is called asymptotic stability. Definition 2.8 Asymptotic Stability The origin of system (2.24) is called asymptotically stable if in addition to being Liapunov stable„ the trajectories also tend to the origin as t —> «>. The next theorem gives a sufficient condition for asymptotic stability of a singular point. Theorem 2.3 If there exists a Liapunov function V(x) such that V is negative definite along the system trajectories„ then the singular point x = 0 of system (2.24) is asymptotically stable. This hypothesis can be weakened to having V a negative semidefinite function which is never Identically zero along the system trajectories. The discussion thus far has been concerned with local stability. Theorem 2.3 insures only that the origin is asymptotically stable for sufficiently small perturbations In initial conditions but gives no indication of their allowable magnitude. Clearly, it is of considerable importance to be able to estimate the domain of asymptotic stability for a given system. This concept is used in Chapter 2 to find an inner bound for limit cycles in second order systems. 33 Definition 2.9 Region of Asymptotic Stability The set of all Initial points, xq , In the state space for which the singular point, x = 0, Is asymp totically stable Is called the region of asymptotic stability of x = 0. Other names used for this definition are domain of asymp totic stability and domain of attraction. LaSalle and Lefschetz (1961) give the following theorem for estimating the region of asymptotic stability. Theorem 2.4 Let V(x) be a scalar function of class C*. designate the region where V(x) < I . Let A fi Assume that Aj% is bounded and that within A ^ : 1) V(z) > 0 ,for x ^ 0, V(0) = 0; 2) V(x) < 0, for all x ^ 0. Then the origin is asymptotically stable, and every solution In A^ tends to the origin as t In conclusion, the theorems given inthis <®. section are concerned only with sufficient conditions for stability. Thus, the failure to find a Liapunov function to prove sta bility in no way implies that the system is unstable. Theo rems for proving instability are also sufficiency theorems (see for example, Hahn 1963 or Krasovskli 1963 ). Also, different Liapunov functions for a given system imply dif ferent regions of stability; hence, one function may give a 34 better answer than another« The next section gives prac tical methods of finding Liapunov functions„ 2o5 Methods of Constructing Liapunov Functions In the previous section it was shown that the Second Method is based on the concept of examining stability in terms of a Liapunov function. Thus the application of the Second Method is dependent upon the construction of an appropriate Liapunov function. Unfortunately, the theory presented in Section 2,4 specifies only the existence of such a function but not a method for its construction. It is the purpose of this section to present two methods of constructing Liapunov functions so that a region of asymp totic stability of a singular point can be found. Consider first the Zubov method (Kerr 1964, Margolis and Vogt 1963» Zubov 1961), This method is the result of an existence theorem (Theorem 2,5) which gives both necessary and sufficient conditions for a region A to be a region of asymptotic stability. In this ease the required Liapunov function is a solution of a first order partial differential equation (the Zubov equation), The discussion given here follows Margolis and Vogt (1963), Theorem 2,5 Let A be an open domain containing the origin, and let 6A be the boundary of A, Then A is the region of asymptotic stability of the system If, and only If, there exists two functions v(x) and p(x) satisfying: 1) v(%) Is positive definite and continuous with continuous first partial derivatives. 2) p(x) Is positive definite and continuous. 3) For 2 In A, x ^ 0, 0 < 4) v(jc) — ►! as x-*»6A. 5) v(^) satisfies v(x) v(x) < 1. = -p(x)(l - v(x))• (2.29) The condition that p(x) be positive definite can be relaxed to p(x) just being positive semidefinlte (Kerr 1964). Insight into the established in light meaning of Equation (2.29)iseasily of the previous stabilitytheorems. In a neighborhood of the origin, Equation (2.29) is approximated by v(x) « -9(x). (2.30) Thus, v(x) is negative definite and v(x) is positive definite. That is, v(x) is a Liapunov function, and by Theorem 2.3 the origin is asymptotically stable. In the region where v(j[) > 1, v(x) is positive definite; hence, the system is unstable. For v(][) = 1, v(x) = 0implies that the system trajectory remains on the surface v(x) t ^ 0. =1 for all These conditions are shown in Figure 2.7. 36 REGION OF ASYMPTOTIC STABILITY, A Figure 2.7 Geometric Interpretation of the Zubov Equation 37 The transformation V = -ln(l - v ) , 0 ^ v < 1, (2.31) reduces Equation (2.29) to V(x) = -3(x), (2.32) which requires that V(x) be negative definite in the entire space. The boundary of the region ofasymptoticstability is given by V( x) — ► 00. (2.33) Form (2.32) is sometimes more convenient to use than (2.29). The following example from Hahn (1963 ) illustrates the Zubov procedure, Let the system be *1 = -X1 + 2i12i2 x2 = -x2 , (2.34) and choose p(a) = 2(x12 + *22 ). (2 .35 ) The Zubov equation then becomes + 2*i 2x 2) + % (-I2 ) = -2(X12 + x22)(1 • V)> (2*36) or in terms of Equation (2.32), The solution for the latter equation is x 2 V(x) = x 22 + -- -3=---- . X1X2 (2.38) Hence, the stability boundary, obtained by letting V approach infinity. Is x1x2 = 1 . (2.39) This example is Illustrated in Figure 2.8, after the same problem is solved using the variable gradient method. For systems having stability boundaries not expres sible in closed form, and where f(x) in Equation (2.28) can be expanded in a convergent power series about the origin, Zubov (1961 ) proposed the following iteration method of solution. The procedure is given in terms of second order systems only for convenience» Theoretically, it is valid for systems of any order. Let the system equations be written as ^ = f = ^ + f 2 + f 3 + ... x 2 = g = e1 + 62 + g 3 + where f^, g^represent the terms, etc.. (2.40) linear terms, f , g^ the quadratic The solution v(x^,x^) of Equation (2.29) is approximated by the series 39 v(x1 ,x2 ) = v 2 ^x! »x2 ^ + v 3*xi ,x2^ + •e e, (2.41) where v (x ,x ) Is a homogeneous form of the mth degree, m 1 2 Also, let the sum of the first m - 1 terms of the series (2.41) be denoted as m (x1 ,xp ) = ^ 1=2 v (x ,x0), 1 ■L (2.42) ^ and assume that 3(x ^ ,x^) In Equation (2.29) Is a quadratic form. The quadratic approximation Is found by the following procedure. Let the quadratic form V2*X1 ,X2^ = aXl 2 ^ bXlX2 + CX22, where a, b, and c are unknown constants. (2.43) Then substitute Equation (2.43) into Equation (2.29) and keep only the quadratic terms, i.e., = -9(x). o ^ Equation (2.44) (2.44) 2 is solved for the unknownconstants, and c,to give the desired form of v^. For a, b, the next approx imation, substitute v (3) = v 2 + v3 into Equation (2.29) to get (2.45) 40 + f x ~ ^ gl + g2^ = -9(2) (1 - v^3 h . (2.46) The quadratic terms will cancel, since they satisfy Equation (2.44). Now, discard all terms in Equation (2.46) which are not homogeneous of the third degree to obtain which is solved for the unknown constants in v . The mth order approximation is + | ^ si - W 3^ * ( 2 -48) where R (x.,x_) is the homogeneous form of the mth degree m 1 2 given by m-1 m-1 V h 1* , ’ - » v 2 l2 -49> i=2 i=2 The approximate region of asymptotic stability for any iter ation is obtained by finding the largest closed curve, v^m ^(x^,x^) = constant about the origin which is contained by the curve, (2 .50 ) Margolls and Vogt (1963 ) present the required theorems which establish this procedure. In summary, the Zubov method Is an exact formulation for finding region of asymptotic stability. Its major dis advantage is that no criteria exist for choosing 0 in Equation (2.29). This problem is still not solved as evi denced by the recent literature (Fall side et, al. 1965 )• The approximation scheme is systematic and is easily imple mented on a digital computer, with the major disadvantage being the slow convergence of the series solution (Margolls and Vogt 1963 , Rodden 1964)0 In contrast to the Zubov method the variable gra dient method is an approximate scheme for finding Liapunov functions which are tailored to the nonllnearlty of the sys tem so that the largest region of stability can be deter mined with a minimum of effort. It is based on choosing n undetermined elements in the gradient function,VV, such that v = v v ’f (2.52) is forced to be at least semidefinlte in some region about the origin. A general outline of the method is given next (Schultz and Gibson 1962 ). 42 Let V V have the following form: aH (z)xi + ai 2 (z)l2 + + ain(z)Zn a21(z)ll + a 22(z)X2 + + a2n(z)ln an i ^ ) xi + " + an n (z ) x VV = where the functions a^j(x) are unknown. (2.53) Then V(x) is formed as n V(x) = V V » f = V " (VV) f , i=l where the (VV)^ are the rows of VV. (2.54) The next and most cru cial step of the procedure is to choose the unknown a so that V(x) is at least negative semidefinite. The Liapunov function, V(x), is found by integrating V V , 2 i.e0 , i^ •(x2~ • ••=xn=:0) V(x) = ^ V V « d x = ^ o (x) (VV)1dx1 + o x1 ,x2 ,(x^=...=xn= 0 ) < W ) 2d x 2 + . . . n + \ ( W ) n dx n . (2.55) 43 subject to the condition j(v v ), d(vv) n, 1 / j. (2.56) This result insures a unique V and also insures that the path of integration is arbitrary. Once V and V have been found, the theorems of the previous section can be applied to de termine the stability properties of a region about the singular point. Consider the solution of the system given by Equations (2.34) using the variable gradient method. In this case the unspecified V(jc) takes the form x22(-a22 + 2a12xl2*" If and (2.57) are made zero. Equation (2.57) becomes V(jc) = - X j ^ a ^ d - 2x 1 x 2 ) - x22 (a22). (2.58) Thus, V(2 ) is negative definite for (2.59) For convenience, let a 11 * a 22 2 . (2.60) 44 Then the integration In Equation (2.55) gives the Liapunov function V(x) = (2.61 ) + x/. The approximate region of asymptotic stability for this Liapunov function is the largest circle x^2 + x^2 = constant (2.62) which is inscribed in the curve xlx2 ’ 2* This result, bility (2.63) along with the exact region ofasymptotic sta asdetermined by the integration of the Zubov equa tion, is shown in Figure 2.8. Rodden (1964) solved the above example using the Zubov iteration scheme. His results, carried up to the tenth degree in v, clearly indicate the slow convergence of the iteration scheme. In fact, the v ^ ^ approximation is (2 ) not significantly better than the v approximation. Also, the v (2 ) approximation is only slightly better than the re gion obtained using the variable gradient method. The two methods of finding the region of asymptotic stability of a singular point presented in this section are applied to the limit cycle problem in Chapter 3* The re sults carry over directly since a two dimensional unstable limit cycle is actually the boundary of the region of i+5 EXACT BOUNDRY APPROXIMATE BOUNDRY Figure 2.8 Approximate Region of Asymptotic Stability Determined by Variable Gradient Method asymptotic stability of the origin. A Liapunov function is developed in the next chapter which is referred directly to the limit cycle instead of the singular point at the origin. CHAPTER 3 CONSTRUCTION OP LIAPUNOV FUNCTIONS FOR SECOND ORDER LIMIT CYCLE PROBLEMS 3.1 Introduction In the previous chapter two general procedures are indicated which, when combined with the Second Method of Liapunov, provide a convenient means for solving second order limit cycle problems. These procedures are s (1) the estimation of the region of asymptotic stability of a sin gular point and (2) the construction of the annular region, R, used in the Poincare-Bendlxson Theorem. Section 3.2 is concerned with estimating the region of asymptotic stability via the Zubov and variable gradient methods, while in Section 3.3 Liapunov methods are used to find contact curves for the construction of the annular region, R, Unfortunately, the Liapunov methods considered in Sections 3.2 and 3.3 are dependent on the structure of the two dimensional space and are inadequate for the study of limit cycles in third order systems. In Section 3*4 a new formulation is proposed where the Liapunov function is de fined directly in terms of the limit cycle. This formulation represents a restricted form of the Poincare-Bendlxson Theorem and is valid for third order systems. It is shown. 48 for the second order limit cycle problem, that this formu lation reverts to the solution of the Zubov equation. The application to third order problems is considered in Chapter 4 e 3®2 Estimation of the Region of Asymptotic Stability In this section the methods discussed in Section 2,5 are applied to the second order limit cycle problem. More specifically, the variable gradient and the Zubov methods are used to find an interior bound to the limit cycle. The Zubov method, being an iterative scheme, specifies a bound which tends to the limit cycle from the interior as more iterations are taken, A fundamental property of second order systems, stated in the previous chapter, is as follows: the limit cycle divides the state plane into two disjoint regions, an inner region R and an outer region R , For the systems of 1 o interest, i,e., systems which have only one limit cycle and one singular point Interior to the limit cycle, the sta bility properties Of R^ are the same as those of the singu larity, In this case the stability properties of R^ can be reversed simply by replacing t with -t in the system equa tions, Hence, the methods of Section 2,5 are equally appli cable when the limit cycle is either stable or unstable. 49 Consider first the Zubov method. The Zubov equation, V(x) = -ts(x) (1 - V(x) ), (3.1) is valid for finding the region of asymptotic stability of a stable singular point. If the system under consideration has a stable limit cycle, then the enclosed singular point is unstable. In this case either the system equations can be reversed in time or else t can be replaced with -t in Equation (3.1) to obtain v(jc) = P(x)(l - V(x)) (3.2) In any case the limit cycle represents the boundary of the region of asymptotic stability and is given by V(x) = 1 (3.3) This discussion is summed up in the following theorem Theorem 3.1 Let 2. = I (*) (3.4) represent a second order system having one stable limit cycle which encloses one singularity. Then the limit cycle is given by V(x) = 1 which Is a solution of (3.5) 50 v(x) = e (x )(l - v (x )) (3.6) For the systems which satisfy Theorem 3.1, the limit cycle can always be found either by the direct integration of the Zubov equation, if this is possible, or by applying the Zubov Iteration procedure discussed in Section 2.5. Both Margolis and Vogt (1963 ) and Rodden (1964) have implemented the Zubov iteration procedure on a digital computer and have given examples which show the solution of the Zubov equation approaching the limit cycle as more iterations are taken. Unfortunately, the convergence of the iterative solution is slow with the rate of convergence depending on the form chosen for p(x). The variable gradient method, on the other hand, is not a systematic Iterative procedure» The principal advan tage of the variable gradient method is that a reasonably good bound, in the sense that it approximates the form of the limit cycle orbit, can usually be found with a minimum of effort. To Illustrate the method, consider the van der Pol equation. (3.7) where e is taken as unity for convenience. Since the damp ing term in Equation (3.7) is negative for small x, and positive for large x, then it appears that a stable limit cycle exists and the origin is unstable. It should be noted 51 that the previous statement is Just a guess and not a proof for the existence of the limit cycle. To apply the variable gradient scheme, Equation (3.7) must first be expressed in state variable form. Again for convenience choose the system coordinates to be the phase variables, i.e., X1 = X2 • 2 x2 = -x^ + *2 ~ X1 X2" (3.8) The first step of the variable gradientprocedure is to ob tain thegeneral form for V which in this case is V = ^ (-a21 ) + " a 22 + a 21 “ *21*1 ^ + X22(ai2 + a 22 " a22Xl2)- (3*9) It is easily shown that the origin is unstable by examining the sign of the eigenvalues of the linearized part of system (3.8). Thus V is forced to be positive semidefinite. This is accomplished by letting the coefficients of the x^x^ and 2 x2 terms be zero, and then determining the region where the 2 coefficient of the x^ term is positive, i.e., by letting *12 = -*22 + *22X1 2 (3*10) a22 = constant (3.11) The relation between and is determined from Equation (2 .56 ), which in this case reduces to al2 = a ^ (a21Il )* Hence, (3-13) becomes a— g a21 = -*22 + “ f2*! * Eliminating in Equation (3.12) gives (3-l4) , in terms of *22' 88 -fh- *11 = 2*22 - 3a 22Xl2 + Thus, constant , CL2\ t ai2 are (3*15) specified in terms of the Substituting Equations (3 .10), (3.14), and (3.15) into Equation (3.9) gives 2 V = CL22X1 d x 2 3 (3.16) Clearly, V is positive semldefinite for xJ < v 5. For convenience, let “ I* (3.17) Then the gradient of V is Integrating y v as shown in Equation (2.55) results in the Liapunov function + I 1?2, (3.19) The appropriate inner bound to the limit cycle is obtained by finding the largest value of V * constant for which the closed curve (3.19) is contained in the region specified by the inequality (3.17). This bound is shown in Figure 3.1. Another common form for expressing the van der Pol equation is in terms of Lienard coordinates. In this case the van der Pol equation becomes (3 .20 ) The van der Pol equation (3.20) is actually a special case of the more general Lienard equation, for which the coor dinates were originally developed. and McRuer (1961). See, for example, Graham The advantage of using Lienard coor dinates for the van der Pol equation is that the limit cycle becomes a smoother curve and approximates a circle fairly closely. This property is due to the fact that x^ is 54 LIMIT C YC LE INNER BOUND Figure 3.1 Inner Bound for van der Pol Equation via Variable Gradient Method 55 determined by an integration process in Equation (3e20)„ whereas in the phase coordinates of Equation (3«8) it is the result of a differentiation. Carrying this idea a step far ther » it would appear that the variable gradient method would give a simpler curve for the inner bound if applied to the Lienard coordinates. This is exactly what happens , with the inner bound taking the form of a circle. The variable gradient method is considered again in the next section where both an inner bound and an outer bound are obtained for the limit cycle, 3,3 Application of the Poincare-Bendixson Theorem This section is concerned with a method of bounding the limit cycle between two .concentric closed curves. The procedure employs the concept of contact curves, originally proposed by Poincare„ for finding a suitable annular region, R, required in the application of the Polncare-Bendixson Theorem (Hayashi 1964, Stoker 1950), By combining the vari able gradient method with the concept of contact curves, a method is developed for bounding a limit cycle between two closed curves representing Liapunov functions. The idea of applying Liapunov functions in this manner is due to Szego (1962)9 but the development of the scheme around the vari able gradient method is original. First, the general prop erties of contact curves are given, and then they are applied to the limit cycle problem. 56 For presenting the general properties of contact curves, let xi = fi (-) = f2(Z> (3.21) represent the second order system under consideration, and let a(a) be a scalar function. The slope of the system trajectories is given by dx9 f9 -T-2 = 7^. dxl fl (3.22) and the slope of <x(x) = C, C a constant. Is <ix is. ax ^.23) ax2 With this background, a contact curve can be defined in the following manner: Definition 1*1 Contact Curve The locus of points at which the direction field of system (3.21) is tangent to a scalar function, a(i), i.e., where (3.24) is called a contact curve. 57 Note that If (3.25) then a(x) = C represents a solution of (3.21). Given next is the geometrical interpretation for various forms of (%(%). If a(x) is the linear function. a(x) = ax + big (3 .26 ) then the contact curve is an isocline, since at every point of the contact curve, the direction field is tangent to Equation (3.26) which has a constant slope (see Figure 3.2). Each set of constants a,b in Equation (3.26) defines an isocline. For the case where 0 (5 ) is a quadratic form, let <1(5) = x ^ + x22 (3.2?) and assume that the contact curve is a closed curve contained in a bounded region of the state plane (note that a(x) = C is also a closed curve). In this case the contact curve represents a curve along which the direction field varies as If the curve (3.29) 58 TRAJECTORIES CONTACT CURVE <*= CONSTANT CURVES Figure 3.2 Contact Curve when a(x) Is a Linear Form 59 inscribes the contact curve, and the curve a(x) = CBai (3.30) circumscribes it, then every periodic orbit of system (3.21) must necessarily lie in the annular region Cmln < *(3) < (3-31) Otherwise there would exist points of the contact curve outside a(x) = C (or inside a(x) = C , ), contradicting max min the definition of these curves (see Figure 3.3). This prop erty of contact curves also applies to closed curves, a(%) * C,of closed higher degree.If the contact andbounded,then only an inner can be found. curve is not bound, <x(x) =C . , mm Examples illustrating this classical method are found in Hayashi (1964) , Stoker (1950), and MacFarlane (1964). If a(]c) is a Liapunov function, then the concepts of the Second Method of Liapunov can be applied to construct contact curves. Although the approach is straight forward, it was not until 1962 that Szego (1962 ) used Liapunov func tions to find a contact curve for the van der Pol equation. Szego1s ideas are extended here to include the variable gradient method as a means of generating contact curves. To obtain a contact curve using the Second Method, a Liapunov function has to be found such that its time deriva tive, evaluated along the trajectories of the system, can be 60 CONTACT z CURVE m ax LIMIT ' CYCLE Figure 3.3 Bound for Second Order Limit Cycle Using Contact Curves 61 expressed in the form, v(z) = ;(z)(i - U(2 )). (3 .32) U(z) = l (3.33) where is a closed curve and 3(z) is a semldefinite function. Equation (3.33) represents the desired contact curve since along this curve, V - % fi + (3-34) is zero* which implies that the direction field is tangent to the closed curve given by the scalar function (3.35) V(x) = C. Note that if the constant, C, in Equation (3.35) is made sufficiently large so that this curve completely encloses the contactcurve, trajectories then V is negativesemldefinite and all crossV(^) = C In an inwarddirection. this curve must also enclose the limit cycle. Hence, The smallest outer bound for the limit cycle, denoted by the curve V U > = Cmax. (3.36) is found by choosing C = C so that the curve (3.35) clrmax cumscribes the contact curve. A similar argument is used to 62 find the Inscribed curve, v(— > = cm l n . (3.37) i.e., the largest inner bound for the limit cycle. It should be noted that the above inner and outer bounds depend on the contact curve. Since contact curves are not unique, then the bounds obtained using one curve may give a better answer than for another. It should also be noted that for some systems the contact curves may be un bounded so that only an inner bound, as determined in the previous section, can be found (Hayashi, 1964). The variable gradient scheme is considered next as a means for finding a suitable form for V. Since the method to be presented is basically a trial and error scheme, it is best demonstrated by an example. Consider the van der Pol equation examined in Section 3.2, i.e.. (3.37) Following the usual variable gradient procedure, V in terms of the unknown a's is (3.38) 63 Since the simplest form a closed curve can assume Is a quadratic form without the cross term, it seems logical to look first for a contact curve of this form. V = x 12 (-Ax 1 2 - B x 22 + 1). Thus assume (3.39) where U = Ax 2 + Bx 2 . (3.40) Equation (3*39) requires that the a's in Equation (3.38) satisfy “ll + *21 " *22 " *21Xl2 " 0 (3.41) *12 + *22 ' *22=1^ = -Bll2 -&2i e Since a,, appears 11 (3‘42) (3.^3) only in Equation (3.41),let *11 = *22 " *21(1 " Xl2)> (3>4) = a 22 + 1 • x^2 (l + A) + (3.45) which becomes after Equation (3.43) is substituted into Equation (3.44). Note that <x^ is only a function of x^. If is assumed to be a constant. Equation (2.56) of Section 2.5 can be used to find 64 *12 = 5 ^ (a2ixi) = 3Axi2 ' 1* Then Equation (3.42), In terms of (3.^6) • is 3Ax1 2 - 1 + a 22 - a 22x1 2 =-Bx1 2. To satisfy Equation (3.47) choose (3.47) to be unity. B = 1 - 3A. Then (3.48) Hence, Equation (3.40) becomes U = Ax 2 + (1 - 3A)x2 = 1 (3.49) which represents a contact curve for all 0 < A < 1. (3.50) For the sake of illustration, let A = J. Then the contactcurve (3. 51) is the circle l xi 2 + i x22 = ^ (3-52) and the Liapunov function, determined by integrating V, as indicated in Equation (2.55) Is V = i?Xl6 - A X/ + 4X!3X2 + X2 " X1X2 + K 2* ( 3 ' 5 3 ) 65 The inscribing and circumscribing Liapunov functions» as well as the contact curve in Equation (3<>52), are shown in Figure 3°^° Note that in the extreme case, 1,6*, for A = i (3.54) 3 the contact curve (3.49) degenerates into two parallel lines and the problem reverts to the one already solved in Section 3.2, The Liapunov functions considered in both this sec tion and Section 3.2 treat the limit cycle problem in terms of the singularity at the origin. In the next section a Liapunov function Is proposed which is defined in terms of the limit cycle, 3.4 Liapunov Functions Definite with Respect to t M Limit .Cycle The methods presented in the previous sections of this chapter are basically methods for examining the sta bility of the singular point at the origin. The Liapunov functions for this case are positive definite with respect to the origin and give meaningful results for the second order limit cycle problem only because of the simple struc ture of the two dimensional plane, Furthermore, these methods cannot be carried over to third order limit cycle problems. This aspect is discussed in Chapter 4, 66 V(x) = CONTACT CURVE m ax m in LIMIT CYCLE Figure 3.4 Bound for van der Pol Example of Section 3*3 6? In this section a Liapunov function definite with respect to the limit cycle is defined and its properties developed. It is proposed that this Liapunov function represents a general approach to the limit cycle problem, but in this section only second order limit cycles are studied. Discussion of the proposed Liapunov function for third order systems is deferred until Chapter 4. The geometrical properties of a Liapunov function for studying second order limit cycles are established by examin ing the Poincare-Bendixson Theorem. From this theorem it is seen that the desired Liapunov function should represent two closed curves bounding the limit cycle, with all trajec tories crossing these curves in a direction toward the limit cycle. Furthermore, the Liapunov function must have an iso lated minimum along the limit cycle since this is the stable state to which the system trajectories tend as t Unfortunately, this last condition cannot be satis fied since the limit cycle is not known & priori. To cir cumvent this difficulty the Liapunov function is expressed as a known function V = V(w - 1) (3.55) of an unspecified closed curve W(z) = 1. (3.56) such that for each value of V, Equation (3.55) represents 68 two closed curves which bound the curve (3*56)• Furthermore, It is assumed that V(0) = 0 (3.57) The curve (3*56) is forced to be the limit cycle by imposing the condition on the Liapunov function that all trajectories tend to this curve as t —►«», i.e., V is negative definite with respect to this curve. In terms of equations the Liapunov function, when differentiated with respect to time, takes the form (3.58) If W(a) is chosen to have the form W(z) = h(j), (3.59) such that (3.60) is negative definite with respect to the curve (3.56), then Equation (3.59) specifies the limit cycle. A Liapunov func tion of the form described above is shown in Figure 3.5. The Justification of the procedure outlined above is considered next. After formally defining a Liapunov function definite with respect to the limit cycle, it is shown that such a function can indeed be used to examine the stability 69 LIAPUNOV SURFACE W(x) = Figure 3.5 Liapunov Function Positive Definite with Respect to the Closed Curve W (x) 70 of limit cycles. The proposed method is then verified. With this background it is shown that the class of second order problems outlined in Chapter 2 has limit cycles which are Liapunov stable so that the Second Method approach is valid. Finally, second order limit cycles are examined to illustrate the proposed method. Assume that the nth order system, Z = 1 (z) (3.61) has one limit cycle which encloses only one singular point located atthe origin. a Liapunov In a manner similar to Definition 2.7# function definite with respect to the limit cycle is defined as follows: Definition 1.2 Liapunov Function Definite with Respect to the Limit Cycle Let L represent the limit cycle of system (3.6l). The scalar function V(,$) is a Liapunov function definite with respect to the limit cycle in a region N, if it satisfies: 1) V(^) is positive definite with respect to the limit cycle, i.e., V(z) =0, for x eL V(z) >0, for x f L; (3 .62 ) 2) V(%) is of class C1 ; 3) V(%) is negative definite with respect to the limit cycle. 71 The following theorem. If properly Interpreted, shows that a Liapunov function definite with respect to the limit cycle Is valid for examining stability of limit cycles, theorem JU2 If there exists a Liapunov function definite with respect to the limit cycle, then the limit cycle is stable. To prove this theorem, assume that there exists a Liapunov function V(%) definite with respect to the limit cycle L :^ = (t ), and choose a number €. > 0 such that V(x) is defined in the region IU ‘ ijl < E. (3.63) From Definition 3.2, V(a) > 0 on the surface II* - * e ll {3-6k) =e* Hence, let V(js) ^ X > 0 for Since V(%) ||x - xe || = € • (3.65) is continuous, there exists a6 > 0, with 6 < e such that V(x) <: | for ||z - x^|| ^ 6. (3.66) Choose ^(t) and a suitable %^(0) such that ||z(0) - z e(0)|| $ 6. (3.6?) 72 Since V(x) Is a nondecreasing function along x(t), then V(x(t)) ^ V(x(0)) < ^ (3 .68 ) as long as x(t) - xe (t) || < e. (3.69) But £(t ) cannot penetrate the surface (3.64) since on this surface v(%) a x. (3.70) Thus, Z(t) - JEe (t) ^ 6 for all t ^ 0 and Definition 2.4 Is satisfied. (3.7D To verify that the tra jectories tend to the limit cycle as t — ► *>, note from Definition 3.2 that V < 0. (3,72) v(t) « V(z(t)) (3.73) Then decreases monotonlcally along z(t) when condition (3*67 ) Is satisfied. Thus V(t) tends to a constant a ^ 0 as t — ►«>. Clearly, V(t) ^ a for t ^ 0. Suppose that a > 0. (3-74) Then x(t) Is bounded away from ^ ( t ) . 73 and hence, V is bounded away from zero. Denote a lower bound on V for all t ^ 0 as V < - 3 < 0. (3.75) Then t V(t) = \ V(s)ds < -3t. But this result contradicts statement (3.7*0. (3.76) Hence, a = 0 and (assuming a proper choice of 2 ^(0 )) Z(t) — ► x e (t) as t —♦ ®°. (3.77) To show that the proposed procedure defines the limit cycle, the following theorem is given. Theorem 3.1 Let V = V(W - 1) (3.78) be a Liapunov function of system (3.61) definite with respect to the closed curve W(a) = 1, (3.79) and form (3*80) If W(z) = h(x) (3.81) 74 Is chosen such that 1■ » - 8z> Is negative definite In the neighborhood of the curve (3.79), then Equation (3.81) defines the limit cycle of the system. The proof of this theorem follows directly from the reali zation that If In the neighborhood of the curve (3.79) the Liapunov function satisfies the conditions V > 0, for W ^ 1 V = 0, for W = 0, (3.83) then the following must be true: aV > 0, for W > 1 M w - i) Av M W - 1) = 0, for W = 1 a(wV- i) < ° ’f°r w < 1- (3-84) Consequently, any function h(x) of Equation (3.81) which has the form h(jt) < 0,for W > 1 h(jc) = 0,for W = 1 h(z) > 0,for W < 1 (3.85) guarantees that V Is negative definite with respect to 75 curve (3*79). In particular choose W = e(x)(W - 1), (3.86) where 9 (x) Is positive definite In the neighborhood of curve (3.79). But Equation (3.86) Is the Zubov equation. Thus, by Theorem 3.1* Equation (3.79) Is the limit cycle of the system. Note that Equation (3.86) Is not unique, but It does specify a unique limit cycle. The properties considered above are general and are not restricted to any particular system order. However, the remainder of this section is devoted to only second order systems. The next theorem Justifies that the limit cycles of the class of systems of interest are indeed Liapunov stable. Theorem 2*4 If system (3.61) is a second order system, then its limit cycle is Liapunov stable. This theorem is verified by using the properties developed above. Assume that the limit cycle given by W(j) = 1 is stable the (orbitally or otherwise). (3.8?) Then by Theorem 3.1 Zubov equation, W = -9(W - 1), must be satisfied. (3.88) Recall that the Zubov equation implies 76 only that the singularity at the origin exhibitsLiapunov stability and is not in any way related to thetype bility of the limit cycle, U = Now of sta let (3.89) w be a function definite with respect to the limit cycle. u - (.w +..i ) (w - (3 .90 ) i)w. w2 Then By substituting Equation (3 .88 ) into Equation (3-90), U becomes u = - i i « —iJ L l.(w - I) 2 (3 .9 1 ) tr which is negative definite with respect to the limit cycle. Hence, U is a Liapunov function. Summarizing the above proof, the limit cycle of system (3*61) is completely de fined by the Zubov equation whether the limit cycle is orbltally stable or Liapunov stable. The Zubov equation implies the existence of a Liapunov function definite with respect to the limit cycle, and the Liapunov function in turn implies the limit cycle is Liapunov stable. The solution of the general second order problem via the proposed Liapunov function is outlined next. Because of the simple structure of the two dimensional space, it is possible to specify a form for the Liapunov function which is suitable for examining local and global stability for 77 any second order limit cycle enclosing only one singularity. Such a form is 2 v = (" - 1) . (3.92) Note that another possible choice for the Liapunov function is V = (w - l)2 , (3.93) but this equation is restricted to V < 1, (3.94) and hence, suitable for examining only local stability. In Chapter 4 where it is difficult to obtain a third order global form analogous to Equation (3*92), Liapunov func tions valid only in the local sense are used. The form given in Equation (3.92) is used In the following discussion. For every constant V > 0, the level curves of Equation (3 *92 ) given by W(z) - 2_±_z ± J(V-±-Z)Z . 1 (3.95) represent two closed curves which constrain the limit cycle to the annular region ,-------j j— - U-W) 14-2 - - 1 « 1 (see Figure 3.6)* To show the existence of a limit cycle. 78 LIMIT CYCLE V(x) = CONSTANT Figure 3.6 Annular Region Determined by Proposed Liapunov Function 79 It must be shown that there exists a function W(x) which makes V negative definite with respect to the limit cycle. Differentiating Equation (3.92) with respect to time gives v = - (1 + W)w . (3. 97) w2 This equation assumes the negative definite form V = - P U t W )(1 w W)2 (3.98) if W is chosen to be W * 3(1 - W). (3.99) Thus, Equation (3.99) represents the auxiliary equation which in this case is simply the Zubov equation, and the limit cycle is W(x) = 1. (3.100) Since the formulation is reduced to the solution of the Zubov equation which was already considered previously, no specific examples are considered in this section. It is seen from the above discussion that a dis tinctive property of the proposed formulation is that the given Liapunov function is reduced to a set of auxiliary equations which define the limit cycle. For the second order case the auxiliary equation is the Zubov equation which represents the stability of a singular point at the 80 origin. Two auxiliary equations are obtained In the third order case which can be Interpreted as representing the stability of two surfaces whose Intersection defines the limit cycle. Third order systems are considered In Chapter 4. To conclude this section, an Interesting property of the proposed Liapunov function Is discussed which Is not apparent In Liapunov functions defined with respect to the origin. In the general Liapunov formulation Liapunov func tions defined with respect to the origin need not be posi tive definite but only satisfy the condition that V and V have opposite signs, I.e., that the Inequality VV < 0 Is satisfied (Krasovskll, 1963 )♦ (3.101) Since for this case the Liapunov function Is always definite, Inequality (3.101) Implies only that one sign Is associated with the Liapunov function when It Is evaluated along a system trajectory. In the case of Liapunov functions definite with respect to the limit cycle. Inequality (3.101) takes on a broader meaning. For example, a Liapunov function which is not positive definite usually takes one sign for all closed curves interior to the limit cycle and the opposite sign for curves exterior to the limit cycle (see Figure 3.7). However, the previously developed geometrical meaning is lost in this process. 81 LIMIT CYCLE : < V = - 1/2 ' (W = 2) V =1 (W =1/ 2) Figure 3.7 V = Constant Curves for the Liapunov Function given In Equation (3.102) 82 To illustrate this property, the method developed in this section is applied to the general Liapunov function v - I ■~ w. (3-102) Differentiating Equation (3.102) with respect to time gives V = (3.103) wz which satisfies the inequality (3.101) if W is chosen as W = 9(1 - W). (3.104) It is interesting to note that the auxiliary Equation (3.104) is again the Zubov equation; hence, the limit cycle is W(a) = 1. (3.105) If 9 (5 ) in Equation (3.104) is made equal to W(x), then Equation (3.103) becomes v = - U "- w), (3.106) and (3.107) V * -V is satisfied. The latter equation is considered in greater detail in Chapter 4. that if V =V(%) is Furthermore, it can easily be shown a Liapunov function, which nite in sign but satisfies Equation (3.107). is not defi then 83 u = v2 (3.108) Note that Equation (3.109) is similar to Equation (3=107). In summary, a Liapunov function is proposed which is referred directly to the limit cycle instead of the singu larity at the origin. This formulation is basically a re stricted form of the Po1neare-Bendlxson Theorem where the bounding curves of the annular region, R, take on the exact shape of the limit cycle. The Liapunov function, V(x)„ is represented by the bounding curves, and for successively smaller values of V = constant the geometrical represen tation of these curves is a sequence of nested annular re gions tending to the limit cycle. For second order limit cycles the proposed formulation does not lead to any new methods of solution but does indicate how the limit cycle problem Is reduced to an auxiliary one, i.e., the Zubov equation. This property is very important in the third order ease where no established methods similar to the Zubov equation exist. The remainder of this work is con cerned with the third order limit cycle problem. CHAPTER 4 LIAPUNOV FUNCTIONS FOR THIRD ORDER LIMIT CYCLE PROBLEMS 4.1 Introduction In this chapter Liapunov methods for estimating limit cycles in third order systems are considered. In \ Section 4.2 a survey of the literature related to this sub ject is presented. This survey is followed by a discussion showing that Liapunov functions positive definite with re spect to the singular point at the origin are very restric tive when applied to three dimensional limit cycle problems. For the purposes of this work the third order limit cycle problem is divided into two classes. In Section 4.3 third order systems are considered which can be reduced to second order systems where the existing Liapunov techniques apply. The reduction is effected by locating a stable sur face (If one exists) which contains the limit cycle (Szego and Geiss 1963)0 Unfortunately, no criterion exists for judging whether a stable surface exists for a particular third order system. In Section 4.4 the general limit cycle problem is discussed in terms of Liapunov functions definite with re spect to the limit cycle. It is shown for the third order problem that the Liapunov function represents a toroid en closing the limit cycle and exhibits the properties of the 84 three dimensional version of the Poincare-Bendixson Theorem discussed in Chapter 2, where the trajectories enter the toroid and circulate around it as they tend to a fixed closed curve. This Liapunov function can also be inter preted as the sum of two Liapunov functions, each definite with respect to a stable surface such that the intersection of these surfaces defines the limit 6ycle„ Using the latter interpretation, it is shown that the proposed formulation is a generalization of the problem considered by Szego and Gelss (1963) in Section 4.3. Examples are given to illus trate the various aspects of the theory. 4.2 Previous Work The literature on the investigation of limit cycles in third order systems via Liapunov functions consists of two works. Both of the works are considered in this section and their limitations examined. The section is concluded with a discussion of why the Liapunov schemes considered in Sections 3.2 and 3.3 cannot be extended to the study of limit cycles in third order systems. In his book Zubov (1963) determined the region of asymptotic stability for the limit cycle of the third order system, 86 X1 = X1 + X2 ' X1 (X12 + X22) X2 = -X1 + X2 - X2 (X1 2 + X22) (4.1) X3 = •X3 > by solving Equation (2.29). by Zubov is as follows: The method of solution given Using the transformation (2.31), the Zubov equation is reduced to (4.2) V = -e(x). Then 0 (x) is chosen as _ 2 li - (X1 0(x) = X2 }] [ l + xl 2 x22] + 21* (4.3) Xl2 + X 22 to obtain the partial differential equation % X1 + X2 - X1 (X1 -2 [l - d x 3x3 + X2 > (XjZ + -X1 + X2 - X2 (X12 + X22) x 2 2 )] [l + 2 " X1 (4.4) + x 2 2] 2x 2 . 2 + X2 The solution to this equation is [l - ( : / + x ^ ) V + x. xl2 + x22 (4.5) 8? which represents a toroid in x in the x ^ , x^-plane. 2 1 about the limit cycle + x 2 2 1 (4.6) The region of asymptotic stability, determined by letting V ( x ) « (4.7) is seen from Equation (4.5) to be the entire space exclusive Although this result is significant, no indication is given on how to formulate the equation (4.8) V = -e(x) for a particular problem. Recall that this situation is similar to the difficulty apparent in the previous chapter where no criterion exists for choosing P(%) in the two dimensional Zubov equation. For second order problems other Liapunov methods such as the variable gradient scheme can be used to develop insight in choosing p(x); but un fortunately, these methods cannot be used in conjunction with the third order limit cycle problem. This difficulty is partially overcome in the Liapunov formulation given in Section 4.4. The approach taken by Szego and Geiss (19^3) is to find an asymptotically stable surface which contains the limit cycle. Then by constraining the system to this 88 surface, the problem Is reduced to one of two dimensions. Szego considered the third order system x 2 x 3 x3 = '*1 " x2 • x3 + e(1 • xl2 2x1x2 )x2 " + (4.9) where £ > 0 . He then showed that all the system trajec tories approach the surface x, = -Xj + €(1 - x 1 2 )x 2 (4.10) (the appropriate theorem to verify that the trajectories Indeed approach this surface Is given In Section 4.3). The system when constrained to the surface (4.10) becomes (4.11) Xg = -x1 + €(1 - x1 )x2 which Is the familiar van der Pol equation having a stable limit cycle. A few remarks are In order concerning the properties and limitations of this formulation. vided into two distinct parts: ically stable surface, and behavior on this surface. The solution Is di (1 ) finding the asymptot (2 ) examining the limit cycle Since the surface is 89 asymptotically stable, the system trajectories cannot Inter sect It but tend to It as t — ►<». Hence, the method excludes all systems In which the trajectories spiral around the limit cycle as they tend to the limit cycle. If a system has a stable surface, then any motion constrained to this surface must have the effect of reducing the degree of free dom of the system by one. In some cases this property ap pears as a pole-zero cancellation In the system equations. This Is exactly what happens In the example given by Szego and Gelss (1963) since system (4.9), In terms of a single third order equation. Is x* + (1 - e )x + (1 - e )x + x = -e(x2x + 2xx2 + x 2x ) , (4.12) which becomes (s2 - €s + 1)(s + l)x = -€(s + 1 )x2x (4.13) when expressed In terms of the differential operator s. Note that after the cancellation of the common terms. Equation (4.13) results In the van der Pol equation (see Figure 4.1). Thus, system (4.9) represents a very artifi cial third order limit cycle problem. This method Is con sidered further In the next section. It Is clear from the above discussion that the appli cation of Liapunov functions to the study of third order limit cycle problems Is Just beginning to be explored. To conclude this section, the difficulty of applying existing R= 0 x 2y=x^x s+1 (s2 -es+lXs+1) Figure 4.1 Block Diagram of System considered by Szego and Geiss (1963) 91 Liapunov methods to third order systems with limit cycles is discussed. The properties of second order systems which do not carry over to higher dimensions and the dependence of Liapunov functions on these properties are considered first. Then a counter example is given to show that Liapunov functions definite with respect to the origin are too re strictive in third order systems. Assume that the second order system of interest has only one limit cycle enclosing the origin and one singu larity at the origin. A limit cycle in a two dimensional space divides the plane into two disjoint regions, such that no trajectory can pass from one region into the other. Thus it must enclose a singularity (a node or a focus), and all trajectories which originate Inside the limit cycle are either stable with respect to the origin or approach the limit cycle as t — ► *>. Similarly, all trajectories origi nating outside the limit cycle are unstable with respect to the origin or approach the limit cycle. Since there are only two possible choices in each region, the stability of the limit cycle is completely determined by examining the stability of the trajectories with respect to the origin, i.e., the singular point. Consequently, Liapunov functions which are definite with respect to the singular point are appropriate in this case. Now consider a Liapunov function definite with re spect to the origin in a three dimensional space. By 0 92 analogy, the Liapunov function represents a closed surface about the origin. Thus„ the Liapunov problem becomes one of bounding closed curves by closed surfaces. Although such a procedure can be used for certain systems, the fact remains that only a limited amount of information is obtained on the position of the limit cycle. It should also be noted that the singular point at the origin of a second order system with a limit cycle exhibits some type of stability behavior. However, in many third order systems with limit cycles, the singularity at the origin is a form of saddle point for which stability in the usual sense has no meaning. For this case, a Liapunov function definite with respect to the ori gin cannot be used for finding an inner bound to the limit cycle. An example illustrating this point is system (4,1), i.e., the system considered by Zubov (1963)0 The eigen values of the linearized part of system (4,1) indicate that the origin is a saddle-focus with the stable eigenvector near the origin being the x^-axis. By combining this result with the information that the x_-axis is excluded from the 3 region of asymptotic stability of the limit cycle (see Equa tion 4*7)» it Is deduced that the origin is the stable state for any trajectory originating on the x-axls. Thus, 3 there does not exist a closed surface about the origin on which the direction field of the system points In an outward direction. Typical trajectories for this system are shown in Figure 4,2* Consequently, Liapunov functions representing 93 LIMIT C Y C L E Figure 4.2 Typical Trajectories for System (4.1) 94 closed surfaces about the origin are In general inadequate for examining the behavior of third order limit cycles.Following the reasoning of Section 2.5 that the Liapunov function be referred to the limit cycle» it be comes evident that the geometrical form required for the third order problem is a toroid enclosing the limit cycle. Liapunov functions representing toroids are considered in Section 4.4. In the next section the method proposed by Szego and Geiss (1963) is examined in detail. 4.3 Reduction of Third Order Problem The present section is concerned with third order systems which have a limit cycle contained in a stable sur face. By finding the stable surface„ the system is reduced to a second order system for which the Liapunov techniques discussed in Chapter 3 apply. This procedure is a special ease of a theorem (see Theorem 4.1) given by Szego and Geiss (1963 ) which implies that if a stable n - 1 dimensional sur face M in E*1 can be found„ then the problem is reduced, for stability purposes» to an n - 1 dimensional problem on the surface M. First the theorem of Szego and Geiss (1963) 'is given as the foundation for the procedure developed later in the section. Then a Liapunov function definite with respect to a surface is defined. After formulating the third order problem, the computational aspects of finding a stable sur face are considered. The’section is concluded with a third 95 order example illustrating the techniques discussed. Except for Theorem 4,1 the techniques and properties developed in this section are original. The following theorem presents a Liapunov approach to the problem of finding a stable surface of a given systems Theorem 4,1 (Szego and Geiss 19^3) Consider the nth order dynamical system x = f(x), (4.14) f (0 ) = 0 Let 1) V(x) be a continuous scalar function with continuous first partial derivatives in the n whole space E 2) 6 (v ) be a continuous scalar function, 3) M be the manifold on which V(x) = 0, Assume that: 4) 0 (V(jc)) = 0 in all points of M, 6 (V(x)) ^ 0 for x ^ M. 5) The equation VV'f(jc) = -e(v) (4.15) is satisfied in the whole space E*1. 6) V(jc)9(V(x)) ^ 0 n in the whole space E . 7) The trivial solution V * 0 of equation (4.16) 96 v = -e(v) (4.1?) is globally asymptotically stable. 8) a) a( ^(x.M)) < ||V(x)|| b) a(f(x,M)) < ||V(x)|| < b(^>(x,M)) (4.18) where ^ (%,M) is the Euclidean distance of the point x from the set M, a(r) and b(r) are positive definite scalar functions, and a(r) is such that 11m a(r) = ». r —*oo Then if 8a) is satisfied, 11m ^(x(t), M) = 0 t -*■00 (4.19) for all initial conditions, and if 8b) is satisfied, M is globally asymptotically stable. Since Liapunov methods are used to find M, it is con venient to define a Liapunov function related to this surface. Definition 4.1 Liapunov Function Definite with Respect to a Surface The scalar function V(jc) is a Liapunov function definite with respect to the surface M if 1) V(^) ^ 0 , for x £ M V(%) = 0, for x € M; (4.20) 2) V is of class C^; 3) V is identically zero only for points of M, and satisfies VV ^ o. (4.21) 97 Using this concept, the general formulation for find ing a stable surface in a third order system can be given as follows: Suppose that a third order system given by *1= V x2 = W w V W has a limit cycle contained in the surface M : = 0, (4.23) Suppose further that Equation (4.23) represents a stable surface, i.e., it satisfies V = -0(V ) , (4.24) where V is a suitable Liapunov function definite with respect to the surface. Once Equation (4.23) is found, the follow ing method is used to constrain the system equations to this surface. say Solve Equation (4.23) for one of its variables, , to obtain ]y = g (x 1 ,x 2 ). (4.25) It should be noted that G may represent more than one sur face. Since Equation (4.25) represents the constraining surface, then must be Identical to = f^(x1 ,x2 ,G). (4.2?) Therefore, the constrained second order system is X1 =: fl^xl ,x2 #G^ x2 = f2 (x ^ •x2 »G). (4.28) System (4.28) can now be examined for limit cycles using the methods of Chapter 3* Certain computational aspects of the above formu lation are discussed next. After presenting the difficulties involved in obtaining a solution to the general problem, a special class of problems is examined in detail and a method of solution outlined. example. The method is illustrated with an Szego and Geiss (1963) do not consider any methods of solution. Their contribution consists only of Theorem 4.1 and two examples illustrating its meaning. Consider first the difficulties in formulating a general solution to the problem of finding a stable surface in a third order system. Initially, the existence of a sta ble surface must be established. Unfortunately, Just as in the case of limit cycles, a stable surface represents a property of the system trajectories which does not appear explicitly in the system equation, and hence, its existence 99 Is very difficult to establish. The existence of the stable surface can be incorporated into the existence of a Liapunov function so that if a Liapunov function definite with re spect to the limit cycle can be found, then the system has a stable surface. difficulty. Note that this does not eliminate the The next step in the general solution is to find the required Liapunov function by solving V = -0(V), (4.29) where the function 9(V) satisfies V0(V) % 0. (4.30) Unfortunately, 6(V) is unspecified and no criterion exists for choosing it. This condition is reminiscent of the dif ficulty associated with the Zubov equation. The approach taken in this work is to treat a special case of Equation (4.29) and to examine its properties so that a solution may be obtained for a limited class of problems. In the remainder of this section 9(V) is taken as 9(V) = XV, where X is a considered realnumber. This form restricts the problems tothose systems whosetrajectories degree of symmetry. (4.31) exhibit some Both of the third order systems con sidered in the literature fall into this class of problems 100 (see Section 4.2) ; thus, Equation (4.31) Is worthy of further study. The following discussion Is centered on developing a method for finding a Liapunov function V(jc) satisfying V = XV (4.32) for third order systems containing a limit cycle. It is shown that if V(2 ) contains linear as well as nonlinear terms, then X is an eigenvalue of the linear part of the system equations. Furthermore, in the case of systems representable in terms of the block diagram configuration shown in Figure 4.1, this form of V(x) Implies that polezero cancellation occurs in the system. The example of Szego and Gelss (1963) considered previously is a special case of this result. Then an iterative scheme is developed for solving Equation (4.32). It should be noted from Definition 4.1 that V(x) is not required to be positive definite with respect to a surface since the product W is negative definite. To show that pole-zero cancellation occurs in the system if the stable surface contains both linear and higher order terms, consider the system x + (a + 0)x + apx = f(x,x), (4.33) where f is a nonlinear function expressible in a power series starting with terms of the second degree, i.e., all 101 the linear terms of f have already been incorporated into the left hand part of the equation. The discussion is given in terms of a second order system only for convenience. It is valid for nth order systems since none of the arguments given is restricted only to second order systems. Equation (4.33), expressed in phase variables, is %2 = -(& + P)%2 ” a ^xi + f(x1 ,x2 ). (4.34) Again, phase variables are chosen for convenience; the dis cussion is independent of the coordinates used. Assume fur ther that the Liapunov function is given as V(jc) * ax1 + bx2 + h(x1 ,x2 ), (4.35) where a, b are constants and h(x^,x2 ) an unknown function. In the following discussion the relation which exists be tween f and h is determined subject to the condition that Equation (4.35) represents the desired Liapunov function. Substituting Equation (4.35) into Equation (4.32) gives ax2 - b(a + 0)*2 “ bagx^ + bf + h = Xax^ + Xbx^ + Xh. By equating like powers of x in (5*36), the equations (4.36) 102 -baft = Xa a - b(a + y) = Xb (4.37) are obtained, with the remaining unknown terms satisfying bf + h = Xh. (4.38) Note that for Equations (4.37), a nontrivial solution exists for a and b , only If X Is an eigenvalue of the linear part of System (4.33). -P. In this case the eigenvalues are -a and For X = -a. Equation (4.34) becomes V(x) = x1 - 3x 2 + h(x^,x2 ) (4.39) and Equation (4.38) is -Pf + h = -ah. (4.40) Solving Equation (4.40) for f yields r = h + flh, (4.4i) P where h Is a general function. If Equation (4.41) is expressed in terms of the differential operator, s, and substituted into Equation (4.33) also expressed in terms of s , It Is seen that the result (s + a ) (s + P)x = 1 exhibits pole-zero cancellation. (s + a)h (4.42) The problem considered by Szego and Gelss (1963 ) in Section 4.2 is a special case of this result. Summarizing, it is shown that if V(x) contains both linear and nonlinear terms, then X is an eigenvalue of the linear part of the system and f must be of such a form that pole-zero cancellation occurs. The method proposed for solving Equation (4.32 ) is essentially the Iteration scheme discussed in Section 2.5* To outline the method here, recall that a homogeneous form of degree m is defined as m Vm (4.43) k=0 where the constants a^ are unspecified. For convenience, the discussion is given in terms of two variables and can be easilyextended to ing for now that the general caseof n X in Equation (4.32) iteration consists of finding . variables. is known, Assum the first This is done by: letting m = 1 in Equation (4.43), substituting it into Equation (4.32), and determining the unspecified coefficients by equating like terms. For the next iteration, y(2) = V1 + V2 is formed, where ratic form. is known and (4.44) is an unspecified quad Following the same procedure as in the previous step, the unknown coefficients in are found by equating like terms of the second degree and discarding all higher order terms. This iterative procedure is continued until 104 the desired Liapunov function, (4.45) v = Z vi- is found such that V satisfies Equation (4.32) with no excess terms discarded. Before this scheme can be applied, X must be determined. It is shown next that X is specified by the homogeneous form of lowest degree appearing in V. Although this statement is verified using a second order system, it is valid for systems of any order. X = X Li + X Thus let N (4.46) represent the system, where the subscripts L and N designate the linear and nonlinear terms, respectively. Furthermore, assume that the stable surface is specified completely by a known function homogeneous of degree m and n with m < n (such a simplified formulation is used only for convenience and does not limit the generality of the discussion). Thus, a suitable Liapunov function is (4.4?) where V m and V n are defined as (4.48) 105 and the are unspecified constants. This Liapunov func tion Is definite with respect to the surface If It satisfies Equation (4.32), I.e., W M v m + Vn ). ikM) where \ -& XL+Ss +TTyL +FTyr (U-50) Equating terms of the same degree of homogeneity In Equation (4.49) gives S XL + = XVm as the homogeneous form of smallest degree. (4-51) Equation (4.51) can be expressed as a (m - j )xm~^"’1y^x T3T J L a x^'^jy^'^y = J=0" J L x> a xm"^y^. J=0 J (4.52) By equating like powers of x and y In Equation (4.51), a set of m + 1 equations Is obtained In terms of X and the m + 1 unknown coefficients a^. This set of equations has the form Aa * Xa, (4053) where A Is an (m + l)x(m + 1) constant matrix and a Is the 106 vector a = col(aQ (4.54) Equation (4.53) represents an eigenvalue problem associated with the homogeneous form (4.48) and specifies X. Consequently, the above result replaces the problem of specifying X by one of determining the smallest power of x In V. The determination of the smallest power of x Is strictly a trial and error process. Fortunately, the number of trials can be minimized by evaluating beforehand all the possible eigenvalues associated with a particular homo geneous form. By doing this. It Is then possible to reverse the process and consider all the possible homogeneous forms which are associated with a particular eigenvalue. In this manner, various eigenvalues can be ruled out simply by examining the associated homogeneous forms to see If they are admissible as a solution to Equation (4.32). The veri fication that the eigenvalues can be computed separately for each homogeneous form Is given next. Consider again the second order system (4.46) and assume that the Liapunov function of the form V = V m + V, n m < n Is adequate to specify the stable surface. (4.55) Substituting Equation (4.55) Into Equation (4.32) gives (4.56) 10? or J x 5*1! + + dym(yL + yN^ + d x (xL + XN^ + S f irL + V “ XVm + XVn* (4-57) By equating all terms of the mth degree of homogeneity, the equation + It X * xvm ( 4 -58) Is obtained which defines X , and hence, specifies V • m Equating homogeneous terms of degree n gives Un + where n + = XVn ’ n (4'59) represents those terms of 51 % + r r yN which are homogeneous of degree n. 5 1 % + l f yN - °n + ^ (4-6o) Note that XN + 5 T yN ’ °- (4-6l) Equation (4,59) Is a nonhomogeneous linear algebraic equa tion whose solution can be given In terms of a homogeneous solution and a particular solution (Hohn 1958). homogeneous solution Implies that But the is satisfied. Equation (4.62) is interpreted as the eigen value problem considered previously and defines a X which must agree with one X specified in Equation (4.58). Summarizing, if V = V + V m n (4.63) satisfies Vm + Vn = Xl (Vm + V - (4-64> then it must also be true that Vm “ XlVm V n « in . (4.65) where only the linear terms of the system equations are used in Equation (4.65). A method of obtaining the desired X is considered in the following discussion. The discussion is given in terms of the second order system (4.46) but can easily be general ized to higher order systems. As the first step in the pro cedure, the eigenvalues associated with the homogeneous form in Equation (4.43) are evaluated for m = 1,2,...,p, where p is sufficiently large. Then starting with an eigenvalue of V^, say X^, all the homogeneous forms found in the previous step are considered which have the same eigenvalue. Note 109 that n Is taken sufficiently large to Insure that all the homogeneous forms associated with a particular X have been found. By using the reasoning developed in the example of system (4.72), it can be determined qualitatively whether or not this combination of homogeneous forms is admissible as a solution of (4.66) V = x1v This procedure is continued first for all eigenvalues of then for the eigenvalues of V^, V^,..., etc., until an admissible combination is found. For the purposes of lllus- tration assume that the eigenvalue, X^, associated with the homogeneous form V , is a suitable candidate, and form (4.6?) V = V2 + h where h contains all the other admissible homogeneous forms of the admissible combination. Using the complete system equations, substitute Equation (4.6?) into Equation (4.32) to obtain v2 + h - x2(v2 + h) • Recall that V and X are known. (4.68) Since (4.69) then Equation (4.69) becomes 110 (4.70) or h = X (4.71) The remainder of the procedure reverts to the application of the iteration scheme of Section 2.5# i.e., each of the homo geneous forms in Equation (4.67) is substituted, in increas ing order of homogeneity, into Equation (4.71), and like powers of x and y are equated to determine the unspecified constants. The above procedure is illustrated by finding the sta ble curve for the system *1 - 4 = 1 + =2 (4.72) System (4.72) can be represented by the block diagram con figuration of Figure 4.1 and, since no pole-zero cancel lation is evident, then V(jt) contains no linear terms. The eigenvalues associated with the homogeneous form (4.73) are *■ = -1. - -2. (4.74) Ill Since the poles of the linearized part of system (4.72) (which incldently are the same as the eigenvalues associated with the homogeneous form V ) are at s = -1, -1/2, then the only admissible choices of the X associated with are X = ”3/2 and -2. For Illustrative purposes consider X = -3/2 first. The eigenvector associated with this value Is V2 = 2V 2 + X22 <4 -75) and Equation (4.71) becomes h = - ^h - 2 x ^ - (4.76) By letting h = h^ and substituting It Into Equation (4.76), the problem can be examined In general terms. h 3 = d ^ (" 2X1 + x2 ) + ^ Thus ("X2 + xl > = - -|h^ - 2x14 - 2x 1^x 2. (4.77) 3 ^(x/) X2 1 (4.78) Since the term )h is of thefifth degree and does not cancel Equation(4.77), then dh^/dXg Is forced condition Is true for any h . m any other term In to bezero. This But the remaining term is of the third degree and will not cancel the fourth degree terms on the right hand side of Equation (4.77)• Therefore, Equation (4.76) is never satisfied, and, consequently, X = -3/2 is an incorrect choice. Next try X = -2. In this case V 2 = cx22 . (4.80) where c is a constant, and h is h = -2h - 2x 2x ^ Since Equation (4.81) c . (4.81) contains afourth degree term, h is taken as h * h^. (4.82) Then Equation (4.81) becomes + x2 ) + ^ ( - % 2 + : / ) „ -2h - ZXgX^c. (4.83) For the same reason as before, the condition 3h, r— T" “ 0 (4.84) 2 is imposed. Since all the terms in Equation (4.83) are of the fourth degree, a solution is possible. shown that It can be easily satisfies Equation (4.83)# Thus a solution of (4.86) V = -2V is (4.8?) To conclude this section a stable surface containing a limit cycle for the third order system. (4.88) is found using the procedure outlined above. It is a straight forward process to show that the origin is stable (X = -1, -1, -3), thus, a stable surface containing the origin may exist. Following the procedure given earlier, consider the homogeneous form + d x ^ 1 3 + ex^Xj + fx 2 3 * (4.89) Using only the linear terms of system (4.88), the admissible X •s are determined by solving the equations 114 -2a = Xa - 2b = Xb - 2c l/3b = Xc - 4d 2 /3c = Xd - 4e = Xe /3e - 6f = Xf (4.90) simultaneously which give X = -2, -2, -2, -4, V(x) cannot have both V and -4,-6. (4.91) terms since none of their associated eigenvalues are the same. Next, the eigenvalues associated with are explored in a general sense to determine if any of them lead to an admissible form for V. Consider X = -6. Assuming that represents the homogeneous form of lowest degree in V, the eigenvector associated with this X is V2 = ox32 , (4.92) V = cx 2 + h. (4.93) which results in Substituting Equation (4.93) into V » -6V gives (4.94) 115 h = h - 2cx x 3 2 (4.95) Since the additional term In Equation (4.95) Is of the sixth degree, let h = h^ (4.96) and substitute It Into Equation (4.95) to get -^(-^ + + + 2Xlx2 ) + g ^ ( - x 2 + x3 )+ x2^) = -6h^ - 2cx^x.^* (4.97) Note that Equation (4.97) Is satisfied if (4.98) By inspection It is seen that h6 = -x26 satisfiesEquation (4.99) (4.97)• and that the unknown constant In Equation (4.93) Is c = 3. Thus asolution (4.100) of Equation (4.94) is V - 3x 2 - x2 . (4.101) This equation implies that a stable surface is *3 = ± -4 x 23. (4.102) Consider next the system motion on the surface (4.102)e duces It is easily verified that Equation (4.102) re thelast equation in (4.88) to anidentity,so that the secondorder system constrained tothis surface is x^ = -x^ + x ^ + 2x1x22 (4.103) x2 « -x2 + x23 . (4.104) To determine if the surface (4.102) contains a limit cycle, the stability properties of the constrained system above are examined using the variable gradient method. The general form for V is V = vv 2, 2 . _ 2 ’f “ *1 (-*11 + *11=1 + 2*11=2 > + =1=2(-*12 + *12=/ - *21 * *21=/ + 2*12=22) + =2 ("*22 * *22=2 ^ (4.105) which becomes V = -a11x12(l - I^2 - Zg2) - a22I12(1 - x12 * x22) when (4.106) 117 By examining Equation (4.106), it is seen that if = a 22 = constant, (4.108) and if the plus part of the surface (4.102) is used, then V becomes v= -al l (x;L2 + x22 )(1 - x 12 - x22 ). (4.109) The corresponding Liapunov function is r x2 V = Wvidjc =t x2 + * ^ 2 ' (4.110) ^0 For convenience is taken as a = 2. (4.111) Thus, Equations (4.109) and (4.110) take the forms V = -2(x1 2 + x 2 2 )(1 - x12 + x22 ) and V = X 2 + x22 , (4.113) respectively, from which it is seen that the system has an unstable limit cycle given by x^ + Xg^ = 1. (4.114) Consequently, a very unusual limit cycle is obtained in this case (see Figure 4.3)• The surface 118 x. SURFACE CONTAINING LIMIT CYCLE V z Vs. Figure 4.3 Limit Cycle for System (4.88) 119 3 (4.115) contains no singularities except the origin and appears in the formulation only because it has the same slope at the origin as does the positive part of the surface. Note that the results of this section are valid only for systems having a surface which divides the space into two disjoint regions so that trajectories originating in one region cannot cross into the other. The reason for this is that equations of the form (4.116) imply that V = V(t) tends to the surface V ;« 0 exponentially as t (4.117) Also, since in the constrained system the Liapunov equation used is of the Zubov form (Equation (4.112) for example), then the trajectories tend to the limit cycle in an exponential manner. Therefore, systems, whose trajectories spiral around the limit cycle as they tend to it, are excluded by this method. In the next section a formulation is developed where such a system can be treated. 4.4 LlapuQpy Formulation o£ the General Problem For the purposes of this work the three dimensional limit cycle problem is divided according to its formulation 120 into two classes„ The first class consists of systems which can he solved in terms of the stable surface containing the limit cycleo Once the stable surface Is found» the problem reduces to a two dimensional one and the Liapunov techniques discussed in Chapter 3 are applicable. The disadvantage of this formulation is that it applies only to the restrictive class of problems where the trajectories approach but do not cross the stable surface. This problem was treated in detail in Section 4,3 where a method of solution was devel oped for systems whose trajectories exhibit a certain symme try as they tend to the stable surface. Both of the third order systems cited in the literature (see Section 4,2) are examples of the class considered here and are solvable by the method developed in Section 4,3, In this section a method of enclosing the limit cycle in a toroid is developed. The formulation is basi cally an extension of the concepts of Section 2,5 to three dimensions. It is shown that this formulation is a gener alization of the one discussed in Section 4,3, and, hence,! applies to systems whose trajectories spiral about the limit cycle as they tend to it. The existing Liapunov methods, i,e,, Liapunov func tions represented by closed surfaces about the origin, were examined in Section 4,2 and shown to be restrictive with re gard to the third order limit cycle problem. It was also proposed that the desired representation for the Liapunov 121 function is a toroid enclosing the limit cycle* This con clusion is not new since it is the "basic approach taken in the mathematical literature in the study of limit cycles. Unfortunately, the toroids considered in this literature are not defined explicitly in terms of an equation as is needed here but are studied ,in terms of topological properties. The Liapunov functions considered in this section, just as those developed in Section 2,5» are formulated with reference to the Poincare-Bendixson Theorem, As indicated in the discussion of the Poincare-Bendixson Theorem, it is 3 no longer sufficient in E to simply show that the trajec tories cross the toroid in an Inward direction. The ad ditional hypothesis needed is that the trajectories also spiral around the toroid and tend to a fixed closed curve inside it as t — o<x>e The goal of this section is to find a Liapunov function which represents a toroid and satisfies, the Poincare-Bendixson Theorem, Following the ideas given in Section 2*5, it is de sired to develop the properties required of a Liapunov func tion definite with respect to the limit cycle in three dimen sions, Since the limit cycle is not known a priori. a Liapunov function of known form must be constructed about an unspecified closed curve in E 3 which is forced to be the limit cycle when additional restrictions are imposed on the Liapunov function. It has already been stated that the desired form for the Liapunov function is one which 122 represents a toroid. Thus, let V = V(W - 1, S) represent a Liapunov function which is a toroid in E (4.118) 3 en closing the closed curve, C, given by W(x) = 1 C t S(z) = 0 . (4.119) The closed curve C Is defined by the Intersection of a cylindrical type surface W(i) = 1 (4.120) and an unbounded open surface S(x) = 0 passing through the origin. (4.121) For the Liapunov formulation it is desired that V(0,0) = 0, (4.122) i.e., the Liapunov function is zero on the limit cycle. The Poincare-Bendixson hypotheses that the tra jectories cross the toroid in an inward direction and that the trajectories spiral about the toroid and tend to a fixed closed curve are examined next. In terms of the Liapunov formulation the latter condition is automatically satisfied once it is shown that the trajectories cross the toroid in 123 an inward direction. This is true for the following reason. Because of the continuity and definiteness properties of the Liapunov function, it can be represented as a sequence of nested toroids closing down onto the closed curve. If the trajectories cross each surface in an Inward direction, then the limiting toroidal surface must approach a limit cycle. In terms of equations this means that V must be made nega tive definite with respect to the closed curve. Thus, dif ferentiating Equation (4.118) with respect to time results in (4.123) Then W = g(a) S = h(2) (4.124) are chosen such that * * a(wv- i y g(3) + -ih(z) (4.125) is negative definite with respect to the closed curve. Theorem 3*3 Equations (4.124) define the limit cycle. By It should be noted that since there are two terms in Equation (4.125)• then Equations (4.124) may contain terms which can cel when the substitution is made into Equation (4.125). It is precisely this condition which reveals itself in terms of the trajectories spiraling about the limit cycle. If no such 124 coupling occurs, then the problem reverts to the type con sidered in the previous chapter. The problem considered by Zubov (1963 ), which is discussed in Section 4.2, is of the noncoupled type. Unfortunately, no proof is given here to show the extent to which the Liapunov function approach is applicable in third order systems. However, based on the study of com puter solutions of several representative systems and on the intuition developed in the second order case, it appears that a significant number of systems are Liapunov stable. The next topics of interest are the specification of a form for the Liapunov function representing a toroid and the examination of its properties. The method used in this work for generating the toroid is a generalization of the concepts presented in Section 2 .5 . It is shown in Section 2.5 that a suitable Liapunov function for studying second order systems is V = (W - I)2 . (4.126) Equation (4.126), for any number 0 < V < 1, represents two concentric closed curves bounding the limit cycle. The Liapunov function used in three dimensions is V = (W - l)2 + S . (4.12?) It is easily verified that this Liapunov function represents a toroid when Equation (4.12?) is expressed in the form 125 w = 1 ± l/v - s . (4.128) It Is seen that as the surface S = K, (4.129) where K is a constant, is translated above and below the surface 8 = 0 (4.130) the equation W = 1 ± (4.131) for (4.132) represents two concentric closed curves on the surface (4.129)• Thus, for any given 0 ^ V < 1, the locus of all curves, obtained by letting K take on all values (positive and negative) satisfying Equation (4.132), is a toroid. It should be noted that the Liapunov function is valid for examining only local properties of the limit cycle. The time derivative for the Liapunov function takes the form V - 2(W - 1)W + 2SS, (4.133) and the auxiliary equations required for making V negative definite with respect to the limit cycle are W - -/$(U - 1) + p(2) (4.134) 126 S = -f(S) + q(x) , (4.135) where (W - l)p(x) P(S)S ^ are satisfied. + Sq(x) = 0 (4.136) 0 (4.137) Then V becomes V = -20(W - I)2 - 2Sf(S) (4.138) For the case when no coupling exists, I.e., when p(x) = q(x) = 0, (4.139) Equations (4.134) and (4.135) become W = -P(W - 1) (4.140) S = -P(3) , (4.141) and the problem Is of the type considered In Section 4.3 where f(S) = 9(8). (4.142) Unfortunately, when coupling Is present, the problem becomes very complex. In the first place both W(jc) and p(x) can be functions of all three variables. The degree of added com plexity Is realized by recalling that the problem of choos ing p(%) has not yet been resolved for the two dimensional case. The other principal difficulty Is associated with 12? the equation S = -fts) + q(x) (4.143) In the uncoupled case It Is shown In Section 4.3 that for 0(8) = -XS, (4.144) the constant X Is an eigenvalue associated with the homo geneous forms present In S. Once X Is known an Iteration procedure can be used to find S. In the coupled case S = XS + q(x) (4.145) Is an appropriate form for the solution of third order sys tems having a saddle-focus as the singularity at the origin; but unfortunately, X no longer represents the eigenvalue as sociated with the homogeneous forms in S (this point is il lustrated In the examples). The reason for this Is that both the XS and q(x) terms of Equation (4.145) have homo geneous forms of the same degree, and It is not possible to determine a priori how the homogeneous forms are distributed tributed between these terms. Because these difficulties have not been resolved, no method of solution Is presented here. The basic contri bution In this section Is the formulation of the general third order limit cycle problem. It has been shown that the examples treated in the literature are special cases of this formulation. Hence, by means of the theory developed here, 128 It Is possible to justify the arbitrary choices made by Zubov (1963 ) In the solution of his problem discussed In Section 4.2. Furthermore, the formulation also revealed the existence of a type of behavior which has not been con sidered previously In the literature on Liapunov methods. Finally, it is felt that the formulation given is a natural one, since It appeals directly to the Polncare-Bendixson Theory. Several examples are given next which Illustrate the oscillatory type behavior of the trajectories about the limit cycle and which consider W(jc) as a function of three variables. Consider first the system represented by ^ ^ + *2 + x3 - + x22 ) *2 " "X1 + X2 + X3 ' X2 (X1 2 + X22) = x1 + x2 - x^ - + x2 )(x1 2 + x22 ), (4.146) and let the Liapunov function be V = (W - l)2 + S2. (4.147) Differentiating Equation (4.147) with respect to time gives V = 2(W - 1)W By examiningthe eigenvalues of determined that + 2SS. (4.148) the linearizedsystem it is the origin is a saddle-focus;hence, it 129 appears that an appropriate set of auxiliary equations are W = - P U ) ( W - 1) + p(x) (4.149) S = + XS + q(z) (4.150) where (w - l)p(z) + Sq(i) = 0. (4.151) By combining the condition (4.151) Into the auxiliary equations, they become W « iz)(W - 1) - r(x)S S = XS + r(%)(W - 1). (4.152) It can be shown that a solution to this set of equations Is W(x) = x ^ + x22 S(a) = (4.153) (4.154) where r(s) = X = -1 B U ) - x12 + x22. Thus, the limit cycle Is given by (4.155) (4.156) (4.15?) 130 X. 1 2 + x 2 = 1 2 (4.158) (4.159) Several features Illustrated by this example should be noted. Since coupling exists, the trajectories spiral around the limit cycle as they approach it (see Figure 4.4). Also, -1 in Equation (4.156) is not an eigenvalue of the linear ized system equations, i.e., of the linear homogeneous form S(a) • The next example exhibits the properties of W(x) being a function of all three coordinates. Consider the system (4.160) Using the Liapunov function V = (W - i)2 + s2 (4.161) V is # V - +2(W - 1)W + 2SS. (4.162) In this case the appropriate auxiliary equations are W = 0(W - 1) (4.163) S = XS. (4.164) 131 TRAJECTORY LIMIT CYCLE Figure 4.4 Typical Behavior for System (4.146) 132 A solution to this set of equations Is 2 i 2 (4 .165 ) W(z) = s(a) = (4.166) - x2 . where (4 .167 ) 6(a) = X « -1. (4.168) Thus, the Liapunov function Is V = + U 3 - x2 ) 1 " Xl 2 - 2 (X2 + X 3 ) . (4 .169 ) To show that Equation (4.169) represents a toroid write this equation In the form, 2 1 + l (x2 + X3 ) ' “ 1 * r ' (x3 " V ' and assume that V Is some given positive number V (4.170) ^ 1. Also, let x3 ' X2 “ K * (4.171) where K Is any number satisfying (4.172) Then for every constant K Equation (4.170) represents two concentric closed curves, 133 xl 2 + 2 (x2 + x3 )2 “ 1 in the surface (4.171). ±\lVo ' k2, (4.173) The two curves in Equation (4.173) coincide on the surfaces x3 " X2 = * ^ which indicate the top and bottom of the toroid. (4-174) CHAPTER 5 SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH 5*1 Summary In this work methods are Investigated for construct ing Liapunov functions which can be used to determine limit cycles and study their stability behavior. In Chapter 3 two approaches are taken in the study of the second order limit cycle problem. First, the es tablished methods of constructing Liapunov ',functions are applied to -limit', cycle problems. The notable contribution here is the application of the variable gradient method to the generation of contact curves needed in the construction of a suitable annular region, R, in the Poincare-Bendixson Theorem. This scheme provides a method for bounding the limit cycle to a certain region of the state plane. Second, a Liapunov function is constructed which represents the hypothesis of the Poincare-Bendixson Theorem directly. In this formulation a Liapunov function is de fined as a known function of a generic closed curve. The form of the Liapunov function is specified by the requlre, ° ments of the Poincare-Bendlxson Theorem. By forcing V to be negative definite with respect to the limit cycle, i.e., satisfying the condition that all trajectories cross into the interior of the.annular region, R, an auxiliary equation . ' '13^. \ 135 Is obtained which specifies the limit cycle, It is shown that for second order systems the auxiliary equation is the Zubov equation. Chapter 4 is concerned with limit cycles in third order systems. The first topic considered is the possi bility of reducing the third order problem to one of two dimensions by finding a stable surface which contains the limit cycle. The idea was expressed in a short paper by Szego and Geiss (19&3) where they presented a theorem (see Theorem 4.1) regarding such a reduction. This problem is discussed more thoroughly in this work where some im portant characteristics of the formulation are presented. First, a Liapunov function definite with respect to the stable surface is defined in which the aggregate of tra jectories making up the surface is considered as the unper turbed trajectory in the Liapunov stability definition. Then by Szego*s theorem, the surface is stable if a Liapunov function V can be found satisfying V = -9(V) (5.1) V0(V) £ 0. (5.2) It is interesting to note that V need not be a positive definite function when Equation (5.2) is satisfied. Also, it is shown that the theorem is valid only if the surface 3 separates E into two disjoint regions such that trajectories '136 cannot cross from one region Into the other. T h u s , the method Is limited to the case where the trajectories ap proach the limit cycle in the direction of the surface. Unfortunately, this geometrical structure cannot be recog nized from the system equations. On the computational side of the problem, a method of solution is given when the system satisfies (5.3) V = XV, where X is some real number. Equation (5*>3) imposes a sym metry restriction on the type of systems considered. It is shown that X is completely specified by the terms of lowest degree in V. Thus, X can be thought of as an eigenvalue associated with a homogeneous form with respect to the linearized system. For the special case where the lowest degree terms in V are the linear terms, then X is an eigen value of the linear form, and pole-zero cancellation occurs in the given system if it can be represented in terms of the usual feedback diagram. By using the above method for find ing X, an iterative scheme is developed and used for solv ing Equation (5.3) Also developed in Chapter 4 is a general Liapunov formulation for the third order limit cycle problem. This development is an extension of the results of Section 5.4 to the three dimensional case where the limit cycle is enclosed by a toroid. Because of the extra degree of freedom, the ;i37 three dimensional formulation is more complex than the one given previously. In the first place„ the Liapunov func tion is a function of both the surfaces W(x) = 0 and S(x) = 0;;whose- intersection specifies the required closed o curve. Furthermore„ in forcing V to be negative definite with respect to the limit cycle, two auxiliary equations o o in W (x) and S(x) are obtained instead of one as in Section 3.4. Since coupling can exist between the auxiliary equa tions, the system behavior specified by this formulation is more general than the, formulation proposed by Szego and Gelss (1963). 5.2 Suggestions for Future Research Because of the generality of the proposed formulation, there are many possibilities for extending this work. For second order systems the principal area of research is in the development of criteria for choosing an appropriate p(x) in the Zubov equation. Although the method proposed by Szego and Gelss (1963) can treat only a special class of problems, it still Is important because it appears in the general formulation. Consequently, it is desirable to find methods of choosing 9(x) in Equation (4.15). The development of a method of solution for the general formulation represents a fruitful area of investi gation. For the ease where 138 S = xs + p(x), (5.4) It Is desirable to find a method of determining X. Once X Is known, then It Is possible to develop an Iterative scheme for solving the auxiliary equations, This, of course, as sumes that a suitable choice is made for p (x ) . It should be noted an exact one in the sense that the proposed formulation is that the auxiliary equations specify the limit cycle exactly. Using the ideas of the two dimensional formulation, it seems that the limit cycle can be approximated by taking only a few terms of a series solution of the auxiliary equations. Although this is possible, no criteria exists for estimating how close the approximation is to the true solution. Another approach to this problem is to formulate an approximate scheme for estimating the limit cycle instead of the exact scheme proposed here. A means of accomplishing this is to formulate the problem so that an equation is ob tained which does not specify the limit cycle exactly but gives information regarding whether or not a particular toroid encloses the limit cycle. It should be noted that this type of information is not used in the exact formu lation, because once the auxiliary equations are solved for the limit cycle, the fact that the Liapunov function repre sents a toroid is of secondary importance. APPENDIX A SECOND ORDER SYSTEMS WHICH EXHIBIT LIMIT CYCLES EXPRESSIBLE IN CLOSED FORM Api Introduction At present» there exists in the literature few exam ples of second order systems which have limit cycle solu tions expressible in closed form. The examples which are given exhibit a high degree of symmetry and can be deter mined by inspection from the system equations. The Zubov equation (2.29) represents an ideal pro cedure for synthesizing systems with known limit cycles. The procedure consists of selecting an equation to rep resent the desired limit cycle and then forcing it to be a solution of the Zubov equation. The latter step is a trial and error procedure where the system equations are chosen so that the Zubov equation is satisfied. Unfortunately8 in many cases a trial and error method applied directly to the Zubov equation for some desired limit cycle equation cannot be realized because of the number of simultaneous conditions which must be satisfied. Also the synthesized system which exhibits the desired limit cycle solution may be very com plex even though the limit cycle is given by a simple expression. . \X39* 140 In this appendix a method is presented where the limit cycle of a system is approximated or bounded by a curve expressible in closed form; then a new system is generated which has this closed curve as its limit cycle. Using this procedure, the disadvantages given previously are minimized. An outline of the method is given, followed by an illustrative example. In third order systems there is no way of obtaining a closed form approximation to the limit cycle for a given system. Hence, any procedure for finding a third order system which has as its limit cycle a desired closed curve is entirely a trial and error process with no guarantee that such a system can be found. For this reason third order limit cycles are not considered here. A.2 A Synthesis Method Let (A.l) represent a system which exhibits a stable limit cycle, L, not expressible in closed form. By applying the variable gradient method to system (A.l) it is possible to find an inner bound b(x) = 1 to the limit cycle which approximates its shape (A.2) 141 (see Chapter 3)• It is desired next to generate a system, *1 = gi (xr x2 ) X2 * 62^Xl e X2^• ■ (*.3) which exhibits (A.2) as a limit cycle and resembles the original system (A.l). Thus, let V = b(x). (A.4) and substitute Equation (A.4) into the Zubov equation to obtain o 1 + 2 = -P(t> - 1). (A.5) The positive semidefinite function p(x) is determined by the variable gradient approximation. Equation (A,5) are g^ and The only unknowns in and are determined by trial and error using system (A.l) as a guide. An example illustrating the procedure described above is developed in detail. Starting with the block diagram in Figure A.l, a nonlinear element, f(x), is chosen along with a transfer function, G(s), such that when f(x) is linearized using the describing function method, the root locus of the linearized system exhibits a ^-crossing (Graham and McRuer 1961, Truxal 1955)• By choosing f(x) and the poles of G(s) so that the system is unstable for low gain and the root locus crosses into the left half plane for high gain, the 142 R=0 y = f(x) Figure A.l Block Diagram of Nonlinear Feedback System -X 143 system will then have a stable limit cycle. The choices 3 f (x ) = -2^— (A .6) G (s ) = — jj S 8^ - S + 1 (A.?) represent such a system, which when expressed In phase variablesv is X1 = X2 ;2 = "xi + x2 - xi % - (A-8) Using the variable gradient method the inner bound, V = lxi6 - 2x^4 + 2il 2 + 2 x ^ * 2 - 2 x 1 x 2 + x22 = 3. is determined for the limit cycle (see Figure A.2). (A.9) In this case V * - ^x1 2 (x12 - 3) (Ado) so that e = i12 . (a .ii) The required condition for (A.9) to represent a limit cycle is the Zubov equation ! ^ Sl where - -*12(V - 3). (A.12) 144 LIMIT C YC LE INNER BOUND Figure A.2 Inner Bound for Limit Cycle of System (A.8) 1^5 B; " 3X1 5 " § X1 3 + 4xi + 2xi2x2 • 2X2 U -13) = 3X1 3 - 2X1 + 2x2 (A-14) and -x1 2 (V - 3 ) = - Ix ^8 + - x ^ - 2x^^ - •|x1 5x2 + 2x13x2 - x12x22 + 3x1 2 . (A.15) Q To obtain the x term on the left hand side of Equation (A.12) let gl = aXl3 6 2 = bX (A.16) where a and b are unknown constants• These constants are evaluated by equating all termsof the eighth power. I.e., ■^a + ^b « -J (A.17) a = - ^ - b. (A.18) to get Thus, Equations (A.16) become 146 g1 = <- £ - b)x1 3 82 » (A.19) It must be remembered that Equations (A.19) have added other terms Into Equation (A.12), but these terms are of lower order and are considered in other iterations. The next highest order term on the righthand side ofEquation (A.12) most general is "! x ^ . To eliminate thisterm, the form that the system equations can assume are gi = cxi g 2 « dx 3. 1 (A.20) As before, c and d are evaluated by equating the terms of the sixth degree, i.e., ■^c + -^d — — (— ^ — b ) - 2b » — 3 3 3 6 3 (A.21) to give c « ^ - b - d. (A.22) Thus, after two iterations the system equations are g^ = (- ^ - b)x^3 + - b - d)x^ 8 2 * bx^5 + dx^3. Each successive term, (A.23) iteration eliminatesthenext highest order including those introduced byg^ andg^. At times it 14? is better to let g be zero for a particular iteration be cause it may introduce other terms which are difficult or impossible to remove• Most of the unknown constants ac cumulated in the system equations during the iteration process are evaluated when it is necessary to eliminate the remaining terms in Equation (A.12) after no more iterations are possible. For this example the system equations become *1 ” ""(& + + (^ - b - d)x1 - (^ + i)x2 (A.24) x2 = bx]L5 + dxl 3 + (- ^ + 2b - d)xi + lx1 2x2 ’ ^2 + 1^X2 t subject to the conditions 6b - 21 = 3 24b + 6d = -1. (A.25) If the one remaining free parameter, b, is set to zero, system (A.24) becomes, *1 “ X2 = W -W - 2X12x2 "K " + 2x i + x 2 + X2 (A'26) which has the limit cycle lx & _ 2% 4 + 2% 2 + 3% 91 31 1 3 1 2 as a solution. - 2x x 12 x 2 * 3 2 (A.27) It is interesting to note system (A.26) and its prototype» system (A,8), contain terms of only the first and third degree. LIST OF REFERENCES Cesar!, L a Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. New Yorks Academic Press, Inco, 19&3* Coddlngton, E,A. and N 0 Levinson, Theory of Ordinary Differential Equations. New York: McGraw-Hill Book Company, Inc., 1955® Fall side, F e<) M. R. Patel, and M. Ether ton, "Control Engineering Applications of V. I, Zubov*s Construction Procedure for Liapunov Functions," IEEE Trans. on Automatic Control, Vol. 10, No. 2 (1965)» 220-21. Friedrichs, K. 0,, "On Nonlinear Vibrations of Third Order," Studies in Nonlinear Vibration Theory. New York Univ. 1946, 65-103. Fuller, F. B , , "Note on Trajectories in a Solid Torus," Ann. of Math. (2) 56, 438-439 (1952). Graham, D. and D. McRuer. Analysis of Nonlinear Control Systems. New York: John Wiley and Sons, Inc., 1961. Hahn, W. Theory and Application of Liapunov*s Direct Method. Englewood Cliffs, N. J.s Prentlce-Hall, inc., 1963. Hartman, Philip. Ordinary Differential Equations. John Wiley and Sons, Inc., 1964. New York: Hayashi, Chihiro. Nonlinear Oscillations in Physical Systems. New York: McGraw-Hill Book Company, Inc., 19o4, Hohn, F. E. Elementary Matrix Algebra. MacMillan Company, 1958. New York: The Kerr, C. N, , "Control Engineering Applications of V, I. Zubov* s Construction Procedure for Liapunov Functions," IEEE Trans. on Automatic Controls, Vol. 9, No. 2 (1964), 196. Krassovskil, N.N. Stability of Motion. Stanford University Press, I963T ” 149 Stanford* California: 150 LaSalle» J. P« and Solomon Lefschetz. Stability by Liapunov8s Direct Method With Applications. New Yorks Academic Press, 1961. Lefschetz» Solomon. Differential Equations : Geometric Theory. New York? Interscience PublishersB 19&3. MacParlane, A. G. J. Engineering Systems Analysis. Reading., Mass.? Addison-Wesley Publishing Company, Inc., 1964. Malkin, I. G. Theory of Stability of Motion. AEG Translation No. 3352, 1958. Washington,D,C „s Margolis, S. G, and W. G. Vogt. ”Control Engineering Applications of V. I. Zubov’s Construction Procedure for Liapunov Functions,” IEEE Trans. on Automatic Control, Vol. 8, No. 2 (1963). 104-113. Perello , Carlos. Periodic Solutions of Ordinary Differential Equations With and Without Time Lag. Providence, Rhode Islands Brown University, Ph.D. Thesis, 1965. Pontryagln, L. S. Ordinary Differential Equations. Reading, Mass.: Addlson-Wesley Publishing Company, Inc., 1962 . Rauch, L. L . , "Oscillation of a Third Order Nonlinear Autonomous System, " in Contributions to the Theory of Nonlinear Oscillations. Ann. of Math Studies, No. 20 (1950), 39- 88. Hodden, J. J. Applications of Liapunov Stability Theory. Stanford , California? Stanford University Ph.D. Thesis, 1964. Schultz, D. Go and J. E. Gibson, "The Variable Gradient Method for Generating Liapunov Functions," Trans. AIEE, Application and Industry, Vol. 81 (1962), 203-210. Stoker, J. J. Nonlinear Vibrations in Mechanical and Electrical Systems„ New Yorks Interscience Publ ., I950. Struble, R. A. Nonlinear Differential Equations. McGraw-Hill Book Company, Inc., 19o2. New York? 151 Szego, Go Poo "A Contribution of Liapunov8s Second Methods Nonlinear Autonomous Systems/' ASME J. Basic Engineering, December 1962, 571~578o Szego, Go P. and G„ R. Geiss, "A Remark on 8A New Partial Differential' Equation for the Stability Analysis of Time Invariant Control Systems8 »" J 0 SIAM Control, Series A, Vol. 1, No. 3 (1963). Truxal, J. G. Control System Synthesis. New Yorks McGraw-Hill Book Company, Inc., 1955« Zubov, V. I. Mathematical Methods of Investlgating Automatic Control Regulation Systems. New Yorks Pergamon Press, 1963.

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