Exponentially Stable Linear Time-Varying Discrete Behaviors

Abstract

We study implicit systems of linear time-varying (LTV) difference equations with rational coefficients of arbitrary order and their solution spaces, called discrete LTV-behaviors. The signals are sequences, i.e. functions from the discrete time set of natural numbers into the complex numbers. The difference field of rational functions with complex coefficients gives rise to a noncommutative skewpolynomial algebra of difference operators that act on sequences via left shift. For this paper it is decisive that the ring of operators is a principal ideal domain and that nonzero rational functions have only finitely many poles and zeros and grow at most polynomially. Due to the poles a new definition of behaviors is required. For the latter we derive the important categorical duality between finitely generated left modules over the ring of operators and behaviors. The duality theorem implies the usual consequences for Willems' elimination, the fundamental principle, input/output decompositions and controllability. The generalization to autonomous discrete LTV-behaviors of the standard definition of uniformly exponentially stable (u.e.s.) state space systems is unsuitable since u.e.s. is not preserved by behavior isomorphisms. We define exponentially stable (e.s.) discrete LTV-behaviors by a new analytic condition on its trajectories. These e.s. behaviors are autonomous and asymptotically stable. Our principal result states that e.s. behaviors form a Serre category, i.e., are closed under isomorphisms, subbehaviors, factor behaviors and extensions or, equivalently, that the series connection of two e.s. input/output behaviors is e.s. if and only the two blocks are. As corollaries we conclude various stability and instability results for autonomous behaviors. There is presently no algebraic characterization and test for e.s. of behaviors, but otherwise the results are constructive.

A constructive test for exponential stability of linear time-varying discrete-time systems

“…We complete the stability results of [4] and use the notions, in particular the linear timevarying (LTV) discrete-time behaviors and the exponential stability (e.s. ), of this paper, but extend the theory from the coefficient field C(z) of rational functions to the larger field C << z >> of locally convergent Laurent series with at most a pole at 0.…”

“…2. Section 3 presents the theory of discrete-time LTV behaviors for the coefficient field C << z >>, but we expose the necessary modifications of [4] only. In particular, behaviors and their morphisms, autonomous behaviors and their e.s.…”

“…In particular, behaviors and their morphisms, autonomous behaviors and their e.s. are recalled from [4] and adapted. Essential properties of these are stated in Corollary 3.10 on modulebehavior duality, in Corollary 3.12 on closure properties of the class of e.s.…”

“…193-368] for comprehensive surveys of exponential stability of state space systems. Part II of the book [3] contains a detailed theory of general LTV behaviors and their stability that was modified in the papers [4,5]. We also refer to the recent papers [1,2,7,10,13].…”

“…The sequences a(t −1 ), t ∈ N, t > σ(a), are the time-varying coefficients of the difference equations of the present paper that in [4] were used for a ∈ C(z) = C(z −1 ). The coefficient functions a(t −1 ), a ∈ K, are of at most polynomial growth on each closed interval [σ 1 , ∞), σ 1 > σ(a), i.e., there are c > 0 and m ∈ N such that |a(t −1 )| ≤ ct m for t ≥ σ 1 [5, (29)].…”