[Paper Review] Deterministic and probabilistic algorithms for stabilizing discrete-time switched linear systems

Atreyee Kundu, Niranjan Balachandran+1 more|arXiv (Cornell University)|May 8, 2014

Formal Methods in Verification30 references3 citations

TL;DR

This paper presents deterministic and probabilistic algorithms for synthesizing stabilizing switching signals in discrete-time switched linear systems using weighted digraphs. By modeling switching signals as walks on a graph and applying graph-theoretic and martingale-based analysis (including Azuma's inequality), the authors establish sufficient conditions for stability and provide a probabilistic bound on the likelihood of stabilizing switching sequences, extending prior work on asymptotic switching conditions without requiring average dwell time constraints.

ABSTRACT

In this article we study algorithmic synthesis of the class of stabilizing switching signals for discrete-time switched linear systems proposed in [12]. A weighted digraph is associated in a natural way to a switched system, and the switching signal is expressed as an infinite walk on this weighted digraph. We employ graph-theoretic tools and discuss different algorithms for designing walks whose corresponding switching signals satisfy the stabilizing switching conditions proposed in [12]. We also address the issue of how likely/generic it is for a family of systems to admit stabilizing switching signals, and under mild assumptions give sufficient conditions for the same. Our solutions have both deterministic and probabilistic flavours.

Motivation & Objective

• To address the algorithmic synthesis of stabilizing switching signals for discrete-time switched linear systems with possibly unstable subsystems.
• To determine under what conditions such stabilizing signals exist, particularly when subsystems are not all Schur stable.
• To develop deterministic and probabilistic algorithms for detecting or designing stabilizing switching sequences based on asymptotic properties of the switching signal.
• To quantify the likelihood of existence of stabilizing switching signals under mild assumptions using probabilistic tools like Azuma’s inequality.
• To extend the framework of [12] by providing computationally tractable, graph-based methods for verifying and constructing stabilizing switching signals.

Proposed method

• Model the switched system as a weighted directed graph (digraph), where nodes represent subsystems and edges represent transitions between them.
• Define a class of stabilizing switching signals via infinite walks on the digraph that satisfy asymptotic conditions derived from [12], avoiding pointwise dwell-time constraints.
• Use graph-theoretic tools to identify cycles in the subgraph of stable subsystems ($\mathcal{P}_S$) with length at least $\lfloor \Phi(|\mathcal{P}_S|) \rfloor$.
• Apply Doob’s decomposition to the walk’s weight process to separate it into a martingale and a strictly decreasing compensator, enabling probabilistic analysis.
• Leverage Azuma’s inequality on the martingale component to derive an upper bound on the probability that the total weight exceeds zero, ensuring stability with high probability.
• Design Algorithm 1 to detect cycles in $\mathcal{P}_S$ with sufficient length, which correspond to stabilizing switching sequences.

Experimental results

Research questions

• RQ1 Under what conditions does a family of discrete-time switched linear systems admit a stabilizing switching signal that satisfies the asymptotic conditions of [12]?
• RQ2 How can such a stabilizing switching signal be algorithmically detected or synthesized in a computationally efficient manner?
• RQ3 What is the probability that a randomly selected switching signal (represented as a walk on the weighted digraph) leads to system stability?
• RQ4 How does the structure of the weighted digraph—particularly the in-degrees of stable subsystems—affect the existence and likelihood of stabilizing switching sequences?
• RQ5 Can probabilistic tools like Azuma’s inequality be effectively applied to bound the failure probability of stabilizing switching signals in the absence of average dwell-time constraints?

Key findings

• A stabilizing switching signal exists if the subgraph of stable subsystems ($\mathcal{P}_S$) contains a cycle of length at least $\lfloor \Phi(|\mathcal{P}_S|) \rfloor$, where $\Phi$ is a function of the system's asymptotic properties.
• Algorithm 1 can detect such cycles in the subgraph of stable subsystems, providing a deterministic method for constructing stabilizing switching signals.
• The probability that a random walk on the weighted digraph results in a stabilizing switching signal is bounded above by $\exp\left(-\frac{1}{2}\left(\frac{(α-β)\sqrt{n}}{A+B}\right)^2\right)$, where $\alpha < \beta$ are weight parameters and $n$ is the cycle length.
• The bound on the failure probability decays exponentially with the square root of the cycle length, indicating that longer cycles significantly increase the likelihood of stability.
• The proposed method avoids the need for semidefinite programming or average dwell-time constraints, offering a numerically simpler alternative to existing approaches.
• The framework applies even when some subsystems are unstable, provided the stable ones form a sufficiently rich and connected subgraph in the digraph representation.