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

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 con...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2013-09, Vol.58 (9), p.2426-2431
Main Authors: Fu Lin, Fardad, Makan, Jovanovic, Mihailo R.
Format: Article
Language:English
Subjects:
Algorithms
Alternating direction method of multipliers (ADMM)
communication architectures
continuation methods
Convergence
ell _{1} minimization
Equations
Linear programming
Minimization
Optimal control
Optimization
separable penalty functions
Sparsity
sparsity-promoting optimal control
structured distributed design
Studies
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Summary: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.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2013.2257618