On a behavioral theory for nonlinear systems Ishan Pendharkar, Harish K. Pillai, Department of Electrical Engineering, Department of Electrical Engineering, Indian Institute of Technology Bombay, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India Powai, Mumbai-400076, India Abstract— In this paper, a theory of nonlinear dynamical under the fold of behavioral theory. In the following section, systems is developed in the behavioral-theoretic framework we give a brief introduction to behavioral systems theory, developed by Willems et al. The following problem is considered: stressing on the role and importance of QDFs. given a system obeying a nonlinear law (which we call a nonlinearity), parametrize a class of linear differential behaviors II. BACKGROUND that, when interconnected with the nonlinearity, yield a stable autonomous system. Due to high level of generality in behavioral We assume that the basic terminology and concepts of theory, the results stated here unify a number of classical results behavioral theory of dynamical systems are known. A de- on nonlinear systems analysis. The method suggested can be tailed exposition of these can be found in, amongst others, used for a large class of problems. [5]. A behavior can be expressed as the “kernel” of a certain differential operator. We assume that this kernel is in the I. INTRODUCTION loc sense of locally integrable functions, 1 . We denote the A behavior is a subset of the space of all trajectories family of dynamical systems (behaviors)L that have q “mani- that is characterized by certain laws. These laws are, in fest” variables by Lq. A behavior that is “controllable” in the many cases, defined by algebraic or differential equations. behavioral sense can be given by an “image representation”. A behavior is called linear if the laws are defined by linear We shall denote the family of controllable behaviors with q q algebraic or linear differential equations. When a linear manifest variables by Lcon. behavior is autonomous, (i.e. has no “free” variables), it is In Lyapunov theory, optimal control etc., we often en- well known that trajectories in the behavior form a finite counter quadratic functionals of variables and their deriva- dimensional linear subspace of the space of all trajectories. tives (e.g., Lyapunov function, the cost functional, the La- Stability results for autonomous behaviors defined by linear grangian etc.). In [11] a two variable polynomial matrix was q q differential equations are well known. used to represent such quadratic functionals. Let R × [ζ; η] It is of interest to investigate stability of autonomous denote the set of real polynomial matrices in the indetermi- behaviors defined by nonlinear differential equations. This nates ζ and η. An element Φ in this set is given by paper is an attempt to develop such a theory in the behavioral- Φ(ζ; η) = Φ ζkηl (1) theoretic framework developed by Willems et al. The non- X kl linear differential equation defines a law. We call the set of k;l all trajectories that satisfy this law as the nonlinear behavior. The sum ranges over non-negative integers k; l with Φkl In the present paper, we only consider nonlinear behaviors being constant matrices. This sum is assumed finite (i.e. only that can be “broken up” into a linear and a nonlinear part. a finite number of Φkl are nonzero). Such a Φ induces a By this, we mean that the nonlinear system can be looked quadratic differential form (QDF) at as an interconnection of a linear differential behavior (B) loc q loc QΦ : (R; R ) (R; R) (2) and a behavior that is not necessarily linear (N). We call L1 !L1 N N a nonlinearity since is defined by a nonlinear law. We defined by consider the following general problem: given a nonlinearity N, parametrize a class of linear differential behaviors B dkw(t) dlw(t) (Q (w))(t) = ( )T Φ ( ): (3) that when interconnected with N yield a stable autonomous Φ X dtk kl dtl k;l system. loc q In many cases of practical interest, a nonlinearity can be In the above equation, 1 (R; R ) denotes the space of q loc L defined by a quadratic relation amongst its trajectories and R -valued 1 -functions of the real variable t. We call Φ their derivatives. A notationally convenient and elegant way symmetric ifL Φ(ζ; η) = ΦT (η; ζ) and denote the set of such q q to define such a quadratic relation is by using a Quadratic symmetric two variable polynomial matrices by Rs× [ζ; η]. Differential Form (QDF). The theory of QDFs, developed in It can be easily shown that it is enough to consider just the present form by Willems, Trentelman et al, has been very symmetric two variable polynomial matrices and hence in q q successful in formulating and generalizing several system this paper we will always assume that Φ Rs× [ζ; η]. q q 2 theoretic ideas. By taking the QDF approach, we show that A QDF QΦ induced by Φ Rs× [ζ; η], can be associated many results of nonlinear systems analysis can be brought to some behavior B Lq 2as a measure of “generalized 2 con power”. We say that B Lq is dissipative with respect to special QDFs (see for instance [11]). We quote the following 2 con the QDF QΦ (or in short Φ-dissipative) if important result from [11]. Theorem 2.1: Consider a polynomial matrix Φ(ζ; η) 1 q q 2 Z QΦ(w)dt 0 compactly supported w B (4) R × [ζ; η]. Assume that Φ(ζ; η) can be written as ≥ 8 2 s −∞ T T q Φ(ζ; η) = R (ζ)R(η) S (ζ)S(η) Since B Lcon, we can use an observable image repre- − 2 d sentation of B = imM( dt ) to define a new two variable with R(ζ) a nonsingular matrix and R(ξ);S(ξ) right co- polynomial matrix prime matrices. Then, the following are equivalent: T 1) Φ( i!; i!) 0 for all real ! and R(ξ) is strictly Φ0 = M (ζ)ΦM(η) (5) Hurwitz,− i.e. ≥det R(ξ) has all its roots in the open left Then the condition for Φ-dissipativity given by equation (4) half complex plane. can be rewritten as 2) There exists a positive definite storage function on the d R( dt ) 1 states of the behavior B obtained as im d Z QΦ (`)dt 0 compactly supported ` (6) S( ) 0 ≥ 8 dt −∞ with respect to the supply rate defined by matrix It has been shown in [11] that equation (6) holds if and only diag(Ir ; Ir ). Here rR and rS denote the number of R − S if rows of R(ξ) and S(ξ) respectively. Moreover, every Φ0( i!; i!) 0 ! R (7) storage function on the states of the behavior B is − ≥ 8 2 positive definite. q q Given a Φ R × [ζ; η], one expects a subset of behaviors q 2 Notwithstanding the fact that Theorem 2.1 is an impor- in Lcon to be Φ-dissipative. We denote the set of all Φ- q tant result, storage functions need not be state functions dissipative controllable behaviors by LΦ. Thus LΦ L . ⊂ con for general dissipative behaviors. We therefore formulate a Closely associated with the behavior and the supply func- generalization of Theorem 2.1 that avoids state construction tion are generalizations of the concepts of “stored energy” q q completely by considering only manifest variables. Such a and “dissipated power”. A QDF Q∆ with ∆ R × [ζ; η] is 2 s generalization is crucial to the results discussed in this paper. called a “dissipation function” if We shall, however, tackle the relatively simpler case where B B Q∆(w) 0 and (8) has only two manifest variables, i.e. is a single-input- ≥ single-output (SISO) system. 1 Z QΦ(w)dt = III. DISSIPATIVE BEHAVIORS WITH POSITIVE −∞ 1 DEFINITE STORAGE FUNCTIONS Z Q∆(w)dt compactly supported w B (9) 8 2 Consider a single input single output (SISO) system whose −∞ transfer function is given by Here Q∆(w) 0 means that the QDF Q∆ is point wise ≥ non-negative on trajectories w(t). A QDF QΨ is said to p(s) d G(s) = (10) be a “storage function” if QΨ(w) QΦ(w) w B. q(s) dt ≤ 8 2 Moreover, given a supply function QΦ, there is a one-to-one This transfer function defines a relationship between the input relation between the storage and dissipation function given u(t) and output y(t) of the system. All such (u; y) define by [11]: a behavior B. In the following sections, we identify this d behavior B with the transfer function G(s). If p(ξ) and QΨ(w) = QΦ(w) Q∆(w) w B dt − 8 2 q(ξ) are coprime, B is controllable and can be given by The above equation is also known as the dissipation equality. the following observable image representation: It is well known that dissipation functions (and hence also the u q( d ) = dt ` (11) storage functions) associated with a Φ-dissipative behavior d y p( dt ) are not unique. More details about dissipative behaviors can with the scalar ` being a free latent variable. Let the behavior be found in [11], [8], [9]. In [11] it is shown that the set of B be Φ-dissipative for some Φ R2 2[ζ; η]. Define a all possible storage functions forms a convex set. Hence we × symmetric polynomial Φ (ζ; η) in ζ;2 η by can talk of a “maximum” and a “minimum” storage function. 0 Associated with a behavior B are certain special latent q(η) Φ0(ζ; η) = q(ζ) p(ζ) Φ(ζ; η) variables that are called the states.