Minimum Input Selection for Structural Controllability

TL;DR

This paper addresses the problem of selecting the minimal set of state variables to ensure structural controllability in linear systems, considering forbidden variables, and provides an efficient deterministic solution.

Contribution

It introduces a polynomial-time algorithm for minimum input selection with forbidden variables in structural controllability problems.

Findings

The problem can be solved in $O(n+m \sqrt{n})$ time.

The algorithm guarantees minimal input set selection.

Applicable to large-scale systems with constraints.

Abstract

Given a linear system $\dot{x} = Ax$, where $A$ is an $n \times n$ matrix with $m$ nonzero entries, we consider the problem of finding the smallest set of state variables to affect with an input so that the resulting system is structurally controllable. We further assume we are given a set of "forbidden state variables" $F$ which cannot be affected with an input and which we have to avoid in our selection. Our main result is that this problem can be solved deterministically in $O(n+m \sqrt{n})$ operations.