A Convex Approach to Sparse H infinity Analysis & Synthesis
For control theorists and engineers dealing with large-scale distributed systems, this work provides a less conservative robustness measure and synthesis method, though it is an incremental extension of existing SDP techniques.
The paper addresses the conservatism of the H-infinity norm for large-scale systems by introducing a cardinality-constrained variant that limits the number of disturbance channels. It proposes an SDP relaxation and a rounding heuristic, demonstrating effectiveness through numerical examples.
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.