1-minimization with magnitude constraints in the frequency domain

N. Elia, M. A. Dahleh

Research output: Contribution to journalArticle

5 Scopus citations


In this paper, we study the ℓ1-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 script H sign-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.

Original languageEnglish (US)
Pages (from-to)27-51
Number of pages25
JournalJournal of Optimization Theory and Applications
Issue number1
StatePublished - Apr 1997
Externally publishedYes


  • Computational methods
  • Multiobjective control
  • Optimal control
  • Robust control
  • ℓ-control

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