Verification and Design of Resilient Closed-Loop Structured System

TL;DR

This paper investigates the verification and design of resilient feedback structures in large-scale closed-loop systems, proving computational hardness and proposing algorithms for ensuring system resilience against feedback link failures.

Contribution

It establishes NP-completeness and inapproximability results, and introduces algorithms for verifying resilience and designing sparse feedback matrices under certain conditions.

Findings

Verification is NP-complete for irreducible systems.

Design problem is NP-hard and hard to approximate.

Proposed algorithms include a pseudo-polynomial verifier and a logarithmic approximation design.

Abstract

This paper addresses the resilience of large-scale closed-loop structured systems in the sense of arbitrary pole placement when subject to failure of feedback links. Given a structured system with input, output, and feedback matrices, we first aim to verify whether the closed-loop structured system is resilient to simultaneous failure of any subset of feedback links of a specified cardinality. Subsequently, we address the associated design problem in which given a structured system with input and output matrices, we need to design a sparsest feedback matrix that ensures the resilience of the resulting closed-loop structured system to simultaneous failure of any subset of feedback links of a specified cardinality. We first prove that the verification problem is NP-complete even for irreducible systems and the design problem is NP-hard even for so-called structurally cyclic systems. We also show that the design problem is inapproximable to factor (1-o(1))log n, where n denotes the system dimension. Then we propose algorithms to solve both the problems: a pseudo-polynomial algorithm to address the verification problem of irreducible systems and a polynomial-time O(log n)-optimal approximation algorithm to solve the design problem for a special feedback structure, so-called back-edge feedback structure.