A Convex Approach to Sparse H infinity Analysis & Synthesis

TL;DR

This paper introduces a convex, semidefinite programming-based method to analyze and synthesize robust control for large-scale systems with limited disturbance channels, improving over traditional H infinity analysis.

Contribution

It proposes a novel convex relaxation for a cardinality-constrained H infinity norm, enabling tractable analysis and control synthesis for large-scale, distributed systems.

Findings

SDP relaxation provides tight upper bounds on robustness.

Heuristic rounding yields effective lower bounds.

Method demonstrates improved robustness analysis in numerical examples.

Abstract

In this paper, we propose a new robust analysis tool motivated by large-scale systems. The H infinity norm of a system measures its robustness by quantifying the worst-case behavior of a system perturbed by a unit-energy disturbance. However, the disturbance that induces such worst-case behavior requires perfect coordination among all disturbance channels. Given that many systems of interest, such as the power grid, the internet and automated vehicle platoons, are large-scale and spatially distributed, such coordination may not be possible, and hence the H infinity norm, used as a measure of robustness, may be too conservative. We therefore propose a cardinality constrained variant of the H infinity norm in which an adversarial disturbance can use only a limited number of channels. As this problem is inherently combinatorial, we present a semidefinite programming (SDP) relaxation based on the l1 norm that yields an upper bound on the cardinality constrained robustness problem. We further propose a simple rounding heuristic based on the optimal solution of SDP relaxation which provides a lower bound. Motivated by privacy in large-scale systems, we also extend these relaxations to computing the minimum gain of a system subject to a limited number of inputs. Finally, we also present a SDP based optimal controller synthesis method for minimizing the SDP relaxation of our novel robustness measure. The effectiveness of our semidefinite relaxation is demonstrated through numerical examples.