Discrete-Time Conewise Linear Systems With Finitely Many Switches

• 成果类型: Article
• 署名作者: Daafouz, Jamal; Loheac, Jerome; Morarescu, Irinel-Constantin; Postoyan, Romain
• 署名单位: Universite de Lorraine; Centre National de la Recherche Scientifique (CNRS); Institut Universitaire de France
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2024.3512205
• 发表日期: 2025
• 页码: 3665-3680
• 关键词: Stability analysis; insulin; Symmetric matrices; Linear systems; Quantum cascade lasers; Numerical stability; computational modeling; Closed loop systems; Lyapunov methods; control theory; Conewise linear systems (CLS); Farkas lemma; insulin infusion; Lyapunov stability; optimization-based control

摘要

In this article, 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 nonnegativity test of linear forms using Farkas lemma. As a result, the problem reduces to verify the nonnegativity of a single solution to an auxiliary linear discrete-time system. Interestingly, this property differs from the classical nonnegativity problem, where any solution to a system must remain nonnegative (componentwise) for any nonnegative 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.