Stability analysis of piecewise affine discrete-time systems

Abstract

This paper presents an implicit model for piecewise affine functions as the interconnection of a linear function and ramp nonlinearities. We show how to verify positive-definiteness of piecewise quadratic functions by exploiting sector properties of the ramp function. These properties are characterized by a set of identities instead of the usual sector inequalities adopted in the study of Lurie systems. Based on this setup, we then formulate conditions for stability of discrete-time piecewise systems using piecewise quadratic Lyapunov functions in terms of linear matrix inequalities. The results are illustrated in numerical examples.

Stabilization of Discrete-Time Piecewise Affine Systems in Implicit Representation

“…Assumption 1 ensures the existence of inverse for (I − F 4 Φ i ) in ( 5) and can be verified by applying the following result, whose proof is found in [4]. Proposition 1.…”

“…Equation (2b) defines the activation of affine terms through function ϕ depending on the state value. Hence, there is a direct relation of this equation and the partition regions Γ i of an explicit representation (see [4] for details). In particular, for each region of the explicit representation, some entries of ϕ will be zero and others will be equal to y i .…”

“…Proof. Since f 5 is assumed to be nonpositive, then V (x) has a finite quadratic upper bound given by ϵ 2 ∥x∥ 2 [4].…”

“…To overcome the need for the preliminary reachability analysis, an implicit representation, based on the use of ramp functions, has been recently proposed in [4]. From this representation, it is possible to express piecewise quadratic Lyapunov candidate functions in a compact way and to cast Lyapunov conditions as generalized quadratic inequalities involving ramp functions and their arguments.…”

“…Thanks to the advantages of the representation presented in [4], this paper investigates the global stabilization problem using this implicit representation. With this aim, starting from the Lyapunov inequalities formulated for the stability analysis, we obtain quasi-LMI conditions allowing the synthesis of stabilizing gains through the solution of semidefinite programming problems.…”