Discrete-Time Conewise Linear Systems with Finitely Many Switches

TL;DR

This work studies discrete-time conewise linear systems (CLS) whose solutions exhibit a finite number of cone switches, a property that enables stability analysis to reduce to linear dynamics after the last switch. It derives a necessary and sufficient condition for global exponential stability (GES) under the finite-switch assumption by restricting to a no-switch invariant set 𝔽 and proving the origin is GES iff it is GES on 𝔽; it further proves exponential input-to-state stability (ISS) under disturbances. A central contribution is a general sufficient condition for finite-switch behavior based on intersections of finitely many sets, and a tractable reformulation via Farkas lemma that reduces the problem to non-negativity of a single solution to an auxiliary linear discrete-time system, with a detailed treatment for the two-cone case. The authors develop non-negativity analysis (including a theorem and small-n closed-form tests) to enable practical verification, and apply the framework to both a second-order example and a discretized insulin-infusion problem, obtaining stability certificates where classical Lyapunov methods fail. Overall, the paper provides a coherent methodology to certify finite-switch CLS stability and demonstrates its relevance to optimization-based control problems such as insulin infusion, with potential extensions to broader hybrid systems.

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.

Figures (4)

• Figure 1: Top: glucose evolution (open-loop with zero input in black dash-dotted line, closed-loop with $K$ in red solid line, optimal open-loop of Good2019 in blue dashed line). Bottom: Insulin flow (closed-loop with $K$ in (\ref{['eq:matrices-insulin-K']})).
• Figure 2: Different situations in the proof of \ref{['corron3']}.
• Figure 3: Phase portrait in red with $x_0$ on the unit circle.
• Figure 4: Phase portrait in red with $x_0=[0, -1]^\top$ ($x_0$ in ${\@fontswitch\mathcal{C}}_1$).

Theorems & Definitions (32)

• Definition 1: Switching times and number of switches
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Lemma 1
• proof
• Proposition 1
• proof : Proof of \ref{['res:Simp']}