[Paper Review] On the mean stability of a class of switched linear systems

Masaki Ogura, Clyde F. Martin|arXiv (Cornell University)|Sep 21, 2014

Stability and Control of Uncertain Systems17 references3 citations

TL;DR

This paper establishes the equivalence between pth mean stability and the existence of a homogeneous Lyapunov function of degree p for discrete-time stochastic switched linear systems, using the L^p-norm joint spectral radius. It further shows that as p→∞, pth mean stability converges to absolute asymptotic stability of an associated deterministic switched system, and provides a computable characterization for Markovian switched systems via matrix spectral radius conditions.

ABSTRACT

This paper investigates the mean stability of a class of discrete-time stochastic switched linear systems using the $L^p$-norm joint spectral radius of the probability distributions governing the switched systems. First we prove a converse Lyapunov theorem that shows the equivalence between the mean stability and the existence of a homogeneous Lyapunov function. Then we show that, when $p$ goes to $\infty$, the stability of the $p$th mean becomes equivalent to the absolute asymptotic stability of an associated deterministic switched system. Finally we study the mean stability of Markovian switched systems. Numerical examples are presented to illustrate the results.

Motivation & Objective

• To establish a converse Lyapunov theorem linking pth mean stability to the existence of a homogeneous Lyapunov function of degree p.
• To investigate the limiting behavior of pth mean stability as p→∞ and relate it to absolute asymptotic stability of a deterministic switched system.
• To extend stability characterization to Markovian switched systems under invariance assumptions.
• To provide a computationally tractable criterion for pth mean stability using matrix spectral radius via linear matrix inequalities or eigenvalue problems.

Proposed method

• Introduces the L^p-norm joint spectral radius as a key tool to analyze pth mean stability of stochastic switched linear systems.
• Proves a converse Lyapunov theorem showing that pth mean stability is equivalent to the existence of a homogeneous Lyapunov function of degree p, under mild invariance conditions.
• Derives a limiting equivalence: as p→∞, pth mean stability becomes equivalent to absolute asymptotic stability of an associated deterministic switched system.
• Constructs a matrix T_p of size Nd^p × Nd^p using Kronecker products and transition probabilities to compute the spectral radius ρ(T_p), which characterizes pth mean stability.
• Applies the framework to Markovian switched systems by defining a linear operator L on state trajectories and relating its spectral properties to the joint spectral radius.
• Uses Corollary V.3 to compute ρ_p,A = ρ(T_p)^{1/p} for practical stability checks, enabling numerical verification via standard eigenvalue solvers.

Experimental results

Research questions

• RQ1 Is pth mean stability of a stochastic switched linear system equivalent to the existence of a homogeneous Lyapunov function of degree p?
• RQ2 What happens to pth mean stability in the limit as p→∞, and how does it relate to deterministic switched system stability?
• RQ3 Can the pth mean stability of Markovian switched systems be characterized using the L^p-norm joint spectral radius under invariance assumptions?
• RQ4 How can the stability condition be reduced to a computable spectral radius condition on a structured matrix T_p?
• RQ5 Can stabilizing feedback gains be systematically designed for Markovian switched systems using the proposed framework?

Key findings

• For general even p, pth mean stability is equivalent to the existence of a homogeneous Lyapunov function of degree p, and this equivalence holds for general p under a specific invariance condition.
• As p→∞, pth mean stability becomes equivalent to absolute asymptotic stability of the associated deterministic switched system, extending known results on joint spectral radius limits.
• For Markovian switched systems, pth mean stability is characterized by ρ_p,A < 1, which is equivalent to ρ(T_p) < 1, where T_p is a Kronecker-structured matrix derived from transition probabilities and system matrices.
• The spectral radius ρ_p,A can be computed as ρ(T_p)^{1/p}, enabling efficient numerical computation via standard eigenvalue solvers.
• Numerical results show that with feedback gain f = [0.36, 0.50], the first moment stability of a 3-mode Markovian system is achieved, with ρ_1,A+bf = 0.9554 < 1.
• Sample paths illustrate stabilization: the original system (dashed) diverges, while the controlled system (solid) converges to zero in expectation, confirming theoretical results.