# Approximating Constrained Minimum Cost Input-Output Selection for Generic Arbitrary Pole Placement in Structured Systems

## TL;DR

This paper introduces a polynomial-time approximation algorithm for selecting minimum cost input-output sets in structured systems to achieve arbitrary pole placement without fixed modes, addressing an NP-hard problem.

## Contribution

It proposes a novel approximation algorithm by decomposing the problem into three sub-problems and linking them to set cover and perfect matching problems, providing an order-optimal solution.

## Key findings

- The algorithm effectively solves the minimum cost input-output selection problem.
- It reduces complex sub-problems to well-known polynomial-time solvable problems.
- The approach guarantees an order-optimal solution within polynomial time.

## Abstract

This paper is about minimum cost constrained selection of inputs and outputs for generic arbitrary pole placement. The input-output set is constrained in the sense that the set of states that each input can influence and the set of states that each output can sense is pre-specified. Our goal is to optimally select an input-output set that the system has no structurally fixed modes. Polynomial algorithms do not exist for solving this problem unless P=NP. To this end, we propose an approximation algorithm by splitting the problem in to three sub-problems: a) minimum cost accessibility problem, b) minimum cost sensability problem and c) minimum cost disjoint cycle problem. We prove that problems a) and b) are equivalent to a suitably defined weighted set cover problems. We also show that problem c) is equivalent to a minimum cost perfect matching problem. Using these we give an approximation algorithm which solves the minimum cost generic arbitrary pole placement problem. The proposed algorithm incorporates an approximation algorithm to solve the weighted set cover problem for solving a) and b) and a minimum cost perfect matching algorithm to solve c). Further, we show that the algorithm is polynomial time an gives an order optimal solution to the minimum cost input-output selection for generic arbitrary pole placement problem.