Mathematics – Optimization and Control
Scientific paper
2011-11-26
Mathematics
Optimization and Control
Submitted to IEEE Trans. Automat. Control; 32 pages, 13 figures
Scientific paper
We design sparse and block sparse feedback gains that minimize the $H_2$ norm of distributed systems. Our approach consists of two steps. First, we identify sparsity patterns of the feedback gains by incorporating sparsity-promoting penalty functions into the $H_2$ problem, where the added terms penalize the number of communication links in the distributed controller. Second, we optimize the state feedback gains subject to the structural constraints determined by the identified sparsity patterns. This polishing step improves the $H_2$ performance of the distributed controllers. In the first step, we identify sparsity structure of the 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 $H_2$ norm, 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. In the second step, we develop Newton's method in conjunction with the conjugate gradient scheme to efficiently compute the optimal sparse feedback matrix. Several examples are provided to illustrate the effectiveness of the developed approach.
Fardad Makan
Jovanović Mihailo R.
Lin Fu
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