Sparse feedback synthesis via the alternating direction method of multipliers

We study the design of feedback gains that strike a balance between the H₂ performance of distributed systems and the sparsity of controller. Our approach consists of two steps. First, we identify sparsity patterns of feedback gains by incorporating sparsity-promoting penalty functions into the H₂ 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, we identify sparsity structure of feedback gains using the alternating direction method of multipliers, which is a powerful algorithm well-suited to large optimization problems. This method alternates between optimizing the sparsity and optimizing the closed-loop H₂ norm, which allows us to exploit the structure of the corresponding objective functions. In particular, we take advantage of the separability of sparsity-promoting penalty functions to decompose the minimization problem into sub-problems that can be solved analytically. An example is provided to illustrate the effectiveness of the developed approach.