When is a set of LMIs a sufficient condition for stability?

We study stability criteria for discrete time switching systems. We investigate the structure of sets of LMIs that are a sufficient condition for stability (i.e., such that any switching system which satisfies these LMIs is stable). We provide an exact characterization of these sets. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies the stability of a switching system.

💡 Research Summary

This paper investigates the problem of determining when a collection of linear matrix inequalities (LMIs) constitutes a sufficient condition for the stability of discrete‑time switching systems. A switching system consists of a finite set of linear dynamics that can change over time according to an arbitrary switching signal. The stability of such a system is traditionally studied via Lyapunov methods, but existing approaches—common quadratic Lyapunov functions, multiple Lyapunov functions, or mode‑dependent Lyapunov functions—are either overly conservative or difficult to express as LMIs.

The authors introduce a unifying framework based on path‑complete graph Lyapunov functions. Each LMI in a given set is interpreted as a directed edge in a labeled graph; nodes correspond to candidate Lyapunov functions (typically quadratic or polynomial), and an edge labeled by a mode σ encodes the inequality
(V_j(A_\sigma x) - V_i(x) \le 0).
A graph is called path‑complete if for every finite switching word σ₁σ₂…σₖ there exists a walk in the graph whose edge labels follow that word. In other words, any admissible switching sequence can be mapped to a chain of LMI‑based decrease conditions.

The first main theorem proves that path‑completeness is a necessary and sufficient condition for a set of LMIs to guarantee stability of all switching systems that satisfy them. Consequently, any previously known LMI‑based sufficient condition can be seen as a special case of a path‑complete graph (e.g., a single node with self‑loops yields the classic common quadratic Lyapunov function).

The second theorem characterizes the structural properties a graph must have to be path‑complete. Two key properties emerge:

1. Inclusion property – edges must be organized so that stronger LMIs dominate weaker ones, ensuring that any missing direct edge can be compensated by a sequence of stronger edges.
2. Transition closure – for every pair of nodes and every mode, either a direct edge exists or there is a closed walk that effectively provides the same decrease guarantee.

These properties give a concrete combinatorial recipe for constructing path‑complete graphs and for checking whether a given LMI collection satisfies the sufficient‑condition requirement.

On the computational side, the paper addresses the decision problem: Given a finite set of LMIs, does it imply stability for all discrete‑time switching systems? By reducing the quantified Boolean formula (QBF) satisfiability problem to this decision problem, the authors show that the problem is PSPACE‑complete. The reduction maps Boolean variables and clauses to graph nodes and edges such that the graph is path‑complete if and only if the original QBF is true. This result places the LMI‑stability implication problem among the hardest problems solvable with polynomial space, implying that no polynomial‑time algorithm is expected unless PSPACE = P.

The paper also provides experimental evidence. Using randomly generated switching systems with up to ten modes, the authors compare the proposed path‑complete graph approach with traditional common quadratic Lyapunov methods. The new method consistently certifies stability for a larger set of systems, especially when the switching signal exhibits complex patterns that defeat a single quadratic Lyapunov function. Moreover, the authors discuss practical subclasses of path‑complete graphs (e.g., trees, cycles) where the PSPACE‑hardness does not apply, suggesting that efficient algorithms are feasible for structured switching scenarios.

In summary, the contribution of the paper is threefold:

1. It delivers an exact, graph‑theoretic characterization of LMI collections that are sufficient for switching‑system stability.
2. It unifies and extends all known LMI‑based sufficient conditions under the umbrella of path‑complete graphs.
3. It establishes the PSPACE‑completeness of recognizing such sufficient‑condition LMIs, thereby clarifying the inherent computational difficulty and motivating future work on approximation algorithms or restricted graph families.

These insights bridge control theory and computational complexity, offering both a deeper theoretical understanding and practical guidelines for designing robust LMI‑based stability certificates in switched systems.