Structural Minimum Controllability Problem for Linear Continuous-Time Switching Systems

TL;DR

This paper investigates the minimal number of manipulated state variables needed for structural controllability in switched linear continuous-time systems, providing a polynomial-time solution and analyzing the complexity of switching signals.

Contribution

It introduces a new necessary and sufficient graph-theoretic and algebraic condition for structural controllability, enabling efficient solutions for large-scale systems.

Findings

A polynomial-time procedure for minimum manipulated variables

NP-hardness of determining minimal switching modes

Characterization of switching signal properties for controllability

Abstract

This paper addresses a structural design problem in control systems, and explicitly takes into consideration the possible application to large-scale systems. More precisely, we aim to determine and characterize the minimum number of manipulated state variables ensuring structural controllability of switched linear continuous-time systems. Towards this goal, we provide a new necessary and sufficient condition that leverages both graph-theoretic and algebraic properties required to ensure feasibility of the solutions. With this new condition, we show that a solution can be determined by an efficient procedure, i.e., polynomial in the number of state variables. In addition, we also discuss the switching signal properties that ensure structural controllability and the computational complexity of determining these sequences. In particular, we show that determining the minimum number of modes that a switching signal requires to ensure structural controllability is NP-hard.