Switched Linear Ensemble Systems and Structural Controllability

TL;DR

The paper addresses structural controllability for ensembles of switched, sparse linear systems sharing a common sparsity pattern. It derives a necessary and sufficient condition for controllability under a fixed number of switches $(k)$ and ensemble size $(q)$, and a finite-$k$ criterion that guarantees controllability for all $q$, with a max-flow formulation enabling polynomial-time tests. The authors connect the conditions to maximum-flow and, via graph-theoretic constructions, provide algorithms to verify them and compute the minimal required number of switches $k^*$. These results generalize prior work on LTI ensembles by showing a weaker, switched-system condition and offer scalable tools for analyzing and designing large ensembles. The framework paves the way for efficient structural analysis and synthesis of switched, distributed control systems in engineering applications.

Abstract

This paper introduces and solves a structural controllability problem for ensembles of switched linear systems. All individual subsystems in the ensemble are sparse, governed by the same sparsity pattern, and undergo switching at the same sequence of time instants. The controllability of an ensemble system describes the ability to use a common control input to simultaneously steer every individual system. A sparsity pattern is called structurally controllable for pair \((k,q)\) if it admits a controllable ensemble of \(q\) individual systems with at most \(k\) switches. We derive a necessary and sufficient condition for a sparsity pattern to be structurally controllable for a given \((k,q)\), and characterize when a sparsity pattern admits a finite \(k\) that guarantees structural controllability for \((k,q)\) for arbitrary $q$. Compared with the linear time-invariant ensemble case, this second condition is strictly weaker. We further show that these conditions have natural connections with maximum flow, and hence can be checked by polynomial algorithms. Specifically, the time complexity of deciding structural controllability is \(O(n^3)\) and the complexity of computing the smallest number of switches needed is \(O(n^3 \log n)\), with \(n\) the dimension of each individual subsystem.

Figures (3)

• Figure 1: Left: A sparsity pattern $\mathbb{S}$. Right: The digraph $G$ associated with $\mathbb{S}$.
• Figure 2: Illustration of the graph $G$ (left) and the corresponding flow graph $\mathcal{G}$ (right).
• Figure 3: Illustration of the digraph $\hat{\mathcal{G}}$ for $G$ given in Figure \ref{['fig:small_graph']} and $(k,q) = (1,3)$. Edges are colored according to their corresponding edges in the digraph $\mathcal{G}$ in Fig. \ref{['fig:small_flow']}. For any edge $(u,v)$ in $\mathcal{G}$, all edges in $\phi^{-1}(u,v)$, with $\phi$ the graph homomorphism defined in \ref{['eq:def_phi']}, are plotted in the same color. Specifically, the red edges are those in $\phi^{-1}(s,\lambda_1)$, the blacks are in $\phi^{-1}(s,\nu_1)$, the blues are in $\phi^{-1}(s,\nu_2)$, the greens are in $\phi^{-1}(\lambda_1,\mu_1)$, the grays are in $\phi^{-1}(\lambda_1,\mu_2)$, the magentas are in $\phi^{-1}(\nu_1,\mu_2)$, the browns are in $\phi^{-1}(\mu_1,t)$, and the oranges are in $\phi^{-1}(\mu_2,t)$. For each color, the number of edges in $\hat{\mathcal{G}}$ is determined by the weight of the corresponding edge in $\mathcal{G}$. Every edge of $\hat{\mathcal{G}}$ has weight $1$.

Theorems & Definitions (21)

• Definition 1
• Lemma 1
• Definition 2
• Theorem 1
• Proposition 2
• Remark 1
• proof
• Theorem 3
• Proposition 4
• proof