Necessary and sufficient conditions for invariance of convex sets for discrete-time saturated systems

Abstract

A convex analysis-based characterization of invariance and contractivity of compact convex sets for discretetime saturated systems is presented. Necessary and sufficient conditions for the existence of convex set-induced Lyapunov functions is provided. The results generalize the quadratic Lyapunov theory for saturated systems, apply also to asymmetric saturations and can be extended to affine nonlinearity maps. A numerical example illustrates the improvements of our method with respect to other classical ones.

New condition for invariance of ellipsoidal sets for discrete-time saturated systems

“…To present the main result of the paper, the following results about how to find the maximal contractively invariant sets for discrete-time saturated systems given in [9], are introduced in this section, where necessary and sufficient conditions for invariance of convex sets are established.…”

“…The objective of this work is to render these conditions computationally tractable and give the explicit conditions as matrix inequalities, using the ellipsoidal sets. Now, we recall the main result of the work [9]. First, some definitions should be given before the main results.…”

“…Thus, we introduce the necessary and sufficient condition given in [9], which is the basis of our work in this paper. Theorem 1 ([9]): Given the system (1), the C-set Ω is such that αΩ is λ-contractive for every α ∈ [0, 1] if and only if for every J ∈ I and E ∈ K, there exist…”

“…hold for all x ∈ Ω and all η ∈ R n . Proof: The proof of this theorem is given in [9]. The theoretical results given by Theorem 1 might be very complex and hardly manageable for the analysis of stability of saturated systems from the computational point of view.…”

“…They allow to characterize the level set of a local quadratic Lyapunov function, see [1], [8], [15]. The work in this paper is the extension of the results in [9], specifying the convex set as an ellipsoid.…”