Discrete-Time Conewise Linear Systems with Finitely Many Switches

TL;DR

This paper studies discrete-time conewise linear systems with finitely many switches, providing conditions to verify this property and applying it to stabilize an insulin infusion model.

Contribution

It introduces a novel characterization and testing method for systems with finitely many switches, simplifying stability analysis and control design.

Findings

Finite switches property can be tested via a linear system solution.

Conditions for finite switches are expressed through set intersections and Farkas lemma.

Application demonstrated on insulin infusion control for stability.

Abstract

We investigate discrete-time conewise linear systems (CLS) for which all the solutions exhibit a finite number of switches. By switches, we mean transitions of a solution from one cone to another. Our interest in this class of CLS comes from the optimization-based control of an insulin infusion model for which the fact that solutions switch finitely many times appears to be key to establish the global exponential stability of the origin. The stability analysis of this class of CLS greatly simplifies compared to general CLS as all solutions eventually exhibit linear dynamics. The main challenge is to characterize CLS satisfying this finite number of switches property. We first present general conditions in terms of set intersections for this purpose. To ease the testing of these conditions, we translate them as a non-negativity test of linear forms using Farkas lemma. As a result, the problem reduces to verify the non-negativity of a single solution to an auxiliary linear discrete-time system. Interestingly, this property differs from the classical non-negativity problem, where any solution to a system must remain non-negative (component-wise) for any non-negative initial condition, and thus requires novel tools to test it. We finally illustrate the relevance of the presented results on the optimal insulin infusion problem.