### Abstract

In this paper, we study the ℓ_{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 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 language | English (US) |
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Pages (from-to) | 27-51 |

Number of pages | 25 |

Journal | Journal of Optimization Theory and Applications |

Volume | 93 |

Issue number | 1 |

DOIs | |

State | Published - Apr 1997 |

Externally published | Yes |

### Keywords

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

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## Cite this

_{1}-minimization with magnitude constraints in the frequency domain.

*Journal of Optimization Theory and Applications*,

*93*(1), 27-51. https://doi.org/10.1023/A:1022641516007