ℓ1-Minimization with Magnitude Constraints in the Frequency Domain

N. Elia and M. A. Dahleh
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N. Elia: MIT
M. A. Dahleh: MIT

Journal of Optimization Theory and Applications, 1997, vol. 93, issue 1, No 2, 27-51

Abstract: Abstract In this paper, we study the $$\ell _1 $$ -optimal control problem with additional constraints on the magnitude of the closed-loop frequency response. In particular, we study the case of magnitude constraints at fixed frequency points (a finite number of such constraints can be used to approximate an $$H_\infty $$ -norm constraint). In previous work, we have shown that the primal-dual formulation for this problem has no duality gap and both primal and dual problems are equivalent to convex, possibly infinite-dimensional, optimization problems with LMI constraints. Here, we study the effect of approximating the convex magnitude constraints with a finite number of linear constraints and provide a bound on the accuracy of the approximation. The resulting problems are linear programs. In the one-block case, both primal and dual programs are semi-infinite dimensional. The optimal cost can be approximated, arbitrarily well from above and within any predefined accuracy from below, by the solutions of finite-dimensional linear programs. In the multiblock case, the approximate LP problem (as well as the exact LMI problem) is infinite-dimensional in both the variables and the constraints. We show that the standard finite-dimensional approximation method, based on approximating the dual linear programming problem by sequences of finite-support problems, may fail to converge to the optimal cost of the infinite-dimensional problem.

Keywords: Robust control; multiobjective control; optimal control; $$\ell _1 $$ –control; computational methods (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1023/A:1022641516007

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