On Minimum Cost Sparsest Input-Connectivity for Controllability of Linear Systems

TL;DR

This paper develops polynomial-time algorithms and approximation methods for designing minimal cost, sparsest input connections that ensure controllability in linear systems with known input influence sets.

Contribution

It introduces efficient algorithms and a 2-approximation method for optimal input-connection design under cost constraints, expanding controllability design techniques.

Findings

Polynomial-time solutions for certain system classes.

A 2-approximation algorithm for the general problem.

Demonstrated effectiveness through illustrative examples.

Abstract

We deal with algorithmic techniques for minimal cost input-connectivity while maintaining controllability of linear systems. The input matrix is assumed to be constrained in the sense that the set of states that each input (if present) can influence is known a priori, and that each interconnection between an input and a state is associated with a certain cost. In this setting we determine a set of input-connections that lead to the minimum cost and ensures that the resulting system is structurally controllable. We also identify a sparsest set of input-connections with minimum cost while maintaining structural controllability of the system. A large class of systems are identified for which these problems are solvable in polynomial time using efficient algorithms. A 2-approximation solution is presented for the general case. Graph-theoretic tools are employed to tackle the above class of constrained design problems. Illustrative examples are included to demonstrate the efficacy of the techniques developed here.