Minimum Cost Feedback Selection in Structured Systems: Hardness and Approximation Algorithm

TL;DR

This paper investigates the complexity of selecting feedback connections in structured control systems, proving NP-hardness, inapproximability, and proposing algorithms with approximation guarantees, including optimal solutions for special network topologies.

Contribution

It establishes the NP-hardness and inapproximability of the feedback selection problem and provides approximation algorithms and polynomial-time solutions for specific network structures.

Findings

The problem is NP-hard and inapproximable within a constant factor.

A greedy-based algorithm achieves a logarithmic approximation ratio.

Optimal solutions are found for hierarchical network topologies.

Abstract

In this paper, we study output feedback selection in linear time-invariant structured systems. We assume that the inputs and the outputs are dedicated, i.e., each input directly actuates a single state and each output directly senses a single state. Given a structured system with dedicated inputs and outputs and a cost matrix that denotes the cost of each feedback connection, our aim is to select an optimal set of feedback connections such that the closed-loop system satisfies arbitrary pole-placement. This problem is referred to as the optimal feedback selection problem for dedicated i/o. We first prove the NP-hardness of the problem using a reduction from a well known NP-hard problem, the weighted set cover problem. In addition, we also prove that the optimal feedback selection problem for dedicated i/o is inapproximable to a constant factor of log n, where n denotes the system dimension. To this end, we propose an algorithm to find an approximate solution to the optimal feedback selection problem for dedicated i/o. The proposed algorithm consists of a potential function incorporated with a greedy scheme and attains a solution with a guaranteed approximation ratio. Then we consider two special network topologies of practical importance, referred to as back-edge feedback structure and hierarchical networks. For the first case, which is NP-hard and inapproximable to a multiplicative factor of log n, we provide a (log n)-approximate solution, where n denotes the system dimension. For hierarchical networks, we give a dynamic programming based algorithm to obtain an optimal solution in polynomial time.