Switched Linear Ensemble Systems and Structural Controllability
This work addresses controllability in multi-agent systems, providing theoretical foundations and efficient algorithms for applications like robotics or networks, but it is incremental as it extends prior linear time-invariant ensemble results.
The paper tackles the structural controllability problem for ensembles of switched linear systems, deriving necessary and sufficient conditions for controllability and showing that these conditions can be checked with polynomial-time algorithms, specifically O(n^3) and O(n^3 log n) complexities.
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.