Switched Linear Ensemble Systems and Structural Controllability

TL;DR

This paper studies the structural controllability of ensembles of switched linear systems with shared sparsity patterns, providing necessary and sufficient conditions, and polynomial algorithms for verification and minimal subsystem computation.

Contribution

It introduces a new controllability criterion for sparse switched linear ensembles and connects it with maximum flow, enabling efficient polynomial-time algorithms.

Findings

Derived necessary and sufficient conditions for structural controllability.

Connected controllability conditions with maximum flow problems.

Provided polynomial algorithms for verification and minimal subsystem computation.

Abstract

This paper introduces and solves a structural controllability problem for ensembles of switched linear systems. All individual systems in the ensemble are sparse and governed by the same sparsity pattern, and undergo switching among subsystems by following the same switching sequence. 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 \(k\) subsystems. 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 subsystems needed is \(O(n^3 \log n)\), with \(n\) the dimension of each individual system.