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 ∈ 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 ∈as a measure of “generalized ∈ con power”. We say that B Lq is dissipative with respect to special QDFs (see for instance [11]). We quote the following ∈ con the QDF QΦ (or in short Φ-dissipative) if important result from [11]. Theorem 2.1: Consider a polynomial matrix Φ(ζ, η) ∞ q q ∈ Z QΦ(w)dt 0 compactly supported w B (4) R × [ζ, η]. Assume that Φ(ζ, η) can be written as ≥ ∀ ∈ 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- − ∈ 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( iω, iω) 0 for all real ω 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 ) ∞ states of the behavior B obtained as im  d  Z QΦ (`)dt 0 compactly supported ` (6) S( ) 0 ≥ ∀ 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( iω, iω) 0 ω R (7) storage function on the states of the behavior B is − ≥ ∀ ∈ positive definite. q q Given a Φ R × [ζ, η], one expects a subset of behaviors q ∈ 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 ∈ 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. ∞ Z QΦ(w)dt = III. DISSIPATIVE BEHAVIORS WITH POSITIVE −∞ ∞ DEFINITESTORAGEFUNCTIONS Z Q∆(w)dt compactly supported w B (9) ∀ ∈ 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 ≤ ∀ ∈ 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 − ∀ ∈ 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,∈ 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. These variables obey   p(η) the axiom of state (see [6]) In several problems in systems We further assume that Φ (ζ, η) can be written in the theory, it is important to have storage functions that are 0 following manner positive definite functions on the states of the behavior. This problem has been addressed in behavioral theory for some Φ0(ζ, η) = r(ζ)r(η) s(ζ)s(η) (12) − where r(ζ), s(ζ) are (scalar) polynomials in ζ. This factoriza- 2 2 Hermitian matrix for every ω0 R. Therefore it × ∈ tion of Φ0(ζ, η) is motivated by several interesting examples has two real eigenvalues. It is shown in [4] that the most of supply rates. interesting cases are when Φ( iω, iω) has zero signature for − In several problems in systems theory, like stabilization almost all ω R. What this means is Φ( iω0, iω0) has one of interconnected systems, it is important to have storage positive and one∈ negative eigenvalue for almost− all values of functions that are positive definite functions on the manifest ω0. Note that J is a matrix of this type. It is further shown in variables of the behavior. For Φ-dissipative behaviors with [4] that for such Φs there exists a simple factorization given two manifest variables, we have the following result that by gives necessary and sufficient conditions such that every stor- π(ω)Φ( iω, iω) = KT ( iω)JK(iω) (16) age function of a Φ-dissipative behavior is positive definite − − on the manifest variables. where π(ω) is a polynomial in ω such that π(ω) 0 ω R ≥ ∀ ∈ Theorem 3.1: Consider a behavior B defined as: . K(iω) is a 2 2 polynomial matrix in iω with real coefficients. If K(ξ×) is the matrix one obtains by substituting u q( d ) = dt ` ξ = iω then K(ξ) is a map from the set of all Φ-dissipative  y   p( d )  dt behaviors to the set of all J-dissipative behaviors. That is Let the image representation for B given above be observ- K(ξ): Φ J (17) able. Let B be Φ-dissipative with Φ(ζ, η) a 2 2 symmetric L → L × matrix in ζ and η. Define a polynomial Φ0(ζ, η) as: Let L(ξ) be the adjoint of K(ξ) (i.e. K(ξ)L(ξ) = 2 2 q(η) det(K(ξ))I × ). Then, L(ξ) is an inverse map of K(ξ) Φ (ζ, η) = q(ζ) p(ζ) Φ(ζ, η) 0  p(η)  (see [4]). With this background, let us look at matrices   2 2 Φ(ζ, η) Rs× [ζ, η] such that Assume that Φ0(ζ, η) can be written in the following manner: ∈ Φ(ζ, η) = KT (ζ)JK(η) (18) Φ0(ζ, η) = r(ζ)r(η) s(ζ)s(η) (13) − The motivation for such a factorization will become obvious where r(ξ), s(ξ) are (scalar) polynomials with r(ξ) = 0. 6 in the sequel. We can ask given such a Φ, can we parametrize Then, following are equivalent: the set of Φ-dissipative behaviors such that every behavior 1) r(ξ) and s(ξ) are coprime polynomials in ξ and r(ξ) has positive definite storage functions? As the following is a non-constant strictly Hurwitz polynomial, i.e. r(ξ) theorem shows, there does exist a simple parametrization is not constant for all ξ C and has all its roots in the in terms of certain rational functions that are positive real ∈ open left half complex plane. or strictly positive real. Positive real and strictly positive 2) There exists a positive definite storage function on the real functions are of great interest in network analysis and manifest variables of the behavior B with respect to synthesis. More details about these functions can be found the supply rate induced by Φ(ζ, η). And every storage in, amongst others, [2]. function on the manifest variables of B is positive Theorem 3.2: Consider a Φ-dissipative controllable be- definite. havior B given by We use Theorem 3.1 to arrive at some interesting connections d u q( dt ) with classical results in systems and networks theory.   =  d  ` 2 y p( dt ) Let a behavior B in Lcon be defined by an observable image representation given by equation Consider the J-dissipative behavior defined by d u q( dt ) q˜(ξ) q(ξ)   =  d  ` (14)   = K(ξ)   y p( dt ) p˜(ξ) p(ξ) Let B be dissipative with respect to a supply rate defined Then, every storage function for B is positive definite if and by u(t)y(t) when (u(t), y(t)) B. If we define a matrix only if q˜(ξ) and p˜(ξ) are coprime and the rational function 2 2 ∈ J R × as p˜(ξ)/q˜(ξ) is non-constant and positive real. ∈ 0 1 J = (1/2) (15) Theorem 3.2 can be looked at as a generalization of the  1 0  classical Kalman-Yakubovich lemma (see [1], [7] for details). then, in the notation developed above, B is J-dissipative, or Notice that dissipative behaviors with positive definite stor- in short B J . It has been shown in [4] that there exists age function may, in general, have trajectories along which ∈ L a simple characterization of the set J , namely, for every there is no dissipation. In other words, there may exist trajec- L behavior in J , the Nyquist plot of the corresponding rational tories along which the rate of change of the storage exactly function liesL in the closed right half of the complex plane. equals the supply. Such a situation is undesirable in certain 2 2 Consider a matrix Φ(ζ, η) R × [ζ, η]. The matrix obtained situations. We have the following result that guarantees that ∈ s by substituting ζ = iω0, η = iω0, i.e. Φ( iω0, iω0) is a the rate of change of storage is strictly less than the supply: − − Theorem 3.3: Consider a Φ-dissipative controllable be- Here is any norm on the space Rn. Given a nonlinear havior B given by system,|| • we || can check if it is stable by constructing a Lya- punov function. We now show how to use QDFs as Lyapunov u q( d ) = dt ` functions on the states of B and thereby establish stability  y   p( d )  N dt and asymptotic stability of the equilibrium (u, y) = (0, 0) in Consider the J-dissipative behavior defined by BN. q˜(ξ) q(ξ) Suppose a two variable polynomial matrix Θ(ζ, η) can = K(ξ)  p˜(ξ)   p(ξ)  be found such that all trajectories of N are point-wise non- negative on the QDF QΘ. That is, Then, every storage function for B is positive definite and the rate of change of the storage is strictly less than the QΘ(w, v) 0 (w, v) N (20) supply if the following holds: q˜(ξ) and p˜(ξ) are coprime and ≥ ∀ ∈ the rational function p˜(ξ)/q˜(ξ) is non-constant and strictly We define a two variable matrix Φ(ζ, η) using the QDF positive real. Θ(ζ, η) and the interconnection matrix X in the following Theorem 3.2 and Theorem 3.3 are the most sophisticated manner: T results developed in the paper so far. These results are crucial Φ(ζ, η) = X Θ(ζ, η)X (21) − for developing a theory of nonlinear behaviors. Clearly, Φ(ζ, η) can be associated with a QDF QΦ. We now IV. NONLINEAR BEHAVIORS find linear differential behaviors B that are Φ-dissipative, i.e.

Consider a system that is described by (possibly) nonlinear ∞ laws. We call such a system a nonlinearity. Let us assume Z QΦ(u, y)dt 0 compactly supported (u, y) B ≥ ∀ ∈ that the nonlinearity defines a certain subset (w(t), v(t)) of −∞ (22) loc R R2 1 ( , ) which we denote by N. Consider a controllable We now consider the interconnection of B with N. Then, L B linear differential behavior given by the observable image such an interconnection forces the allowed trajectories in BN representation: to be point-wise lossless with respect to QΦ, i.e. d u q( dt )   =  d  ` QΦ(u, y) = 0 (u, y) BN (23) y p( dt ) ∀ ∈ We interconnect B and N in the following manner: Since B is Φ-dissipative, there exists a storage function QΨ satisfying u(t) w(t) X   =   (19) d y(t) v(t) Q (u, y) Q (u, y) (u, y) B (24) dt Ψ Φ 2 2 ≤ ∀ ∈ Here X R × is called the interconnection matrix. Given ∈ d a N, we want conditions on B such that the interconnected From equation (23) we can see that dt QΨ(u, y) is is non- system is autonomous and stable (or asymptotically stable, positive for all (u, y) BN. Suppose that B also has a storage function that is∈ positive definite on all (u, y) as the case may be). ∈ With this broad viewpoint, we state the following defini- B. We can use such a storage function as a Lyapunov tions: function on the manifest variables of BN. This represents Definition 4.1: Nonlinear behavior (BN): The set of all a generalization of classical Lyapunov theory for behaviors trajectories in the system obtained by interconnecting B and which we state in the following theorem: N will be called the nonlinear behavior. Theorem 4.4: Given a nonlinearity N, suppose a QDF QΘ It is obvious that every trajectory in BN satisfies the laws of can be found such that N is point-wise non-negative on QΘ. both B and N. Since BN is autonomous, it admits a state We obtain another QDF QΦ from the nonlinearity and the space representation of the form x˙ = f(x) with x Rn. interconnection constraints as in equation (21). We find the The integer n is the dimension of the state space system.∈ set of behaviors that are Φ-dissipative. Let B be such a It can be determined from the laws defining B and N. behavior given by: The zero state trajectory x = 0 (which corresponds to the u(t) q( d ) (u, y) = (0, 0) B = dt ` trajectory N) is an equilibrium point.  y(t)   p( d )  The definitions of stability of∈ the equilibrium x = 0 are well dt known: B and N are interconnected to obtain the autonomous non- Definition 4.2: The equilibrium x = 0 is stable if for every linear system BN. Let QΨ a storage function of a B Φ. ∈ L  > 0 there exists a δ > 0 such that whenever x(0) < δ, Then, equilibrium (0, 0) in BN is stable if QΨ is positive || || x(t) <  t > 0. definite on the manifest variables of B. BN is asymptotically || || ∀ d Definition 4.3: The equilibrium x = 0 is uniformly stable if the following holds: QΨ(u, y) < 0 (u, y) BN. dt ∀ ∈ asymptotically stable if it is stable and limt x(t) = 0. →∞ || || v The above constraints can be expressed in a convenient manner as h 0 1 u w =  1 0   y   v  −∆ ∆ − 0 1 0 We call X = as the interconnection matrix. As  1 0  w − in equation (21) we compute another QDF QΦ with Φ(ζ, η) −h given by 0 ζ Φ(ζ, η) = XT Θ(ζ, η)X = (1/2)   − η 0

Fig. 1. Ideal relay with hysteresis Note that Φ(ζ, η) can be written as 0 ζ ζ 0 η 0 (1/2)   =   J   With reference to Theorem 4.4, note that given a N, we η 0 0 1 0 1 only need to determine QΘ and QΦ. Theorem 3.2 and Theo- ξ 0 We denote by K(ξ) the 2 2 polynomial matrix  . rem 3.3 can then be used to parametrize a set of Φ-dissipative × 0 1 linear differential behaviors. Theorem 4.4 guarantees that an From Theorem 3.2, every storage function of B is positive interconnection of the nonlinearity and any behavior in the definite if and only if the rational function corresponding to d parametrized set will result in a stable system. Also, since the behavior im K( dt )(B) is non-constant and positive real. the procedure is independent of exact structure of QΘ or QΦ, Written explicitly, this means that our results are quite general. The same general procedure can p(ξ) be used to analyze a large class of nonlinearities– both with G(ξ) := is non constant Positive Real (25) ξq(ξ) − memory and without. We illustrate the above procedure by considering an example of a nonlinearity with memory and p(ξ), ξq(ξ) are coprime polynomials. Due to the constructive procedure followed in computing V. IDEAL RELAY WITH HYSTERESIS QΦ, all trajectories in BN are point-wise lossless on QΦ. In order to demonstrate the generality and versatility of Thus, if G(ξ) is positive real, the minimum storage function our results, we consider the nonlinearity N to be an ideal QΨ for B satisfies the following inequalities relay with hysteresis. A typical example is shown in figure QΨ(u, y) > 0 nonzero (u, y) BN (1) ∀ ∈ It is easy to see that, for the ideal relay with hysteresis, d QΨ(u, y) 0 (u, y) BN (26) w(t) and v(t) satisfy the condition: dt ≤ ∀ ∈ dv(t) Using Theorem 4.4 we conclude that the equilibrium (0, 0) w(t) 0 dt ≥ of BN is stable. To show that our conclusion is in agree- ment with other methods of analysis, consider the following where the derivative is taken in the sense of loc-functions. 1 example: Written as a matrix, N is point-wise non-negativeL on the Example 5.1: Let H(ξ) = (ξ+2)/(ξ+3). Let p(ξ) = ξ+2 QDF Q (w(t), v(t)) with Θ and q(ξ) = ξ + 3. Then, G(ξ) as given by (25) is: 0 η Θ(ζ, η) = (1/2) ξ + 2  ζ 0  G(ξ) = ξ(ξ + 3) We want to parametrize a class of linear differential behaviors It can be seen that G(ξ) is a positive-real transfer function. that when interconnected with N using a negative feedback Since G(ξ) is positive real, an ideal relay when intercon- yield a stable interconnected system. We assume that every nected with H(ξ) using “negative feedback” yields a stable behavior B in this class is given by an observable image system. This can also be verified using classical methods of representation: analysis (for instance, describing functions, [3]). d  u(t) q( dt )   =  d  ` VI.CONCLUSION y(t) p( dt ) In this paper we have developed a theory for nonlinear Let us denote by B , the nonlinear behavior obtained by N systems from a behavioral viewpoint. Given a nonlinearity interconnection of B and N. with some quadratic constraints on its trajectories and their A negative feedback can be looked at as having following derivatives, we have obtained a parametrization for a class of constraints on B and N linear differential behaviors that can be interconnected with u(t) = v(t) and y(t) = w(t) the nonlinearity to yield a stable autonomous system. We − have constructed some special quadratic differential forms [5] J.W. Polderman, J.C. Willems, “Introduction to and used them as Lyapunov functions for the resulting mathematical systems theory: A behavioral approach” interconnected autonomous system. 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