ℓ1-Minimization with Magnitude Constraints in the Frequency Domain

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.