Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers

TL;DR

This paper presents a method to design sparse distributed feedback controllers that minimize system variance amplification by combining sparsity-promoting penalties with the alternating direction method of multipliers, enabling efficient large-scale optimization.

Contribution

It introduces a novel two-step approach using ADMM to identify sparsity patterns and optimize feedback gains, improving scalability and effectiveness over existing methods.

Findings

Effective sparsity pattern identification via ADMM

Optimized feedback gains with reduced communication links

Demonstrated success on multiple example systems

Abstract

We design sparse and block sparse feedback gains that minimize the variance amplification (i.e., the $H_2$ norm) of distributed systems. Our approach consists of two steps. First, we identify sparsity patterns of feedback gains by incorporating sparsity-promoting penalty functions into the optimal control problem, where the added terms penalize the number of communication links in the distributed controller. Second, we optimize feedback gains subject to structural constraints determined by the identified sparsity patterns. In the first step, the sparsity structure of feedback gains is identified using the alternating direction method of multipliers, which is a powerful algorithm well-suited to large optimization problems. This method alternates between promoting the sparsity of the controller and optimizing the closed-loop performance, which allows us to exploit the structure of the corresponding objective functions. In particular, we take advantage of the separability of the sparsity-promoting penalty functions to decompose the minimization problem into sub-problems that can be solved analytically. Several examples are provided to illustrate the effectiveness of the developed approach.