## 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 |

### Bibliographical note

Funding Information:This research has been supported by NSF Grant ECS-9157306, Draper Laboratory Grant DL-H-441636, and AFOSR Grant F49620-95-0219. 2The authors thank Prof. Sanjoy K. Mitter for the useful suggestions and enlightening discussions. 3Graduate Student, Laboratory for Information and Decision Systems, MIT, Cambridge, Massachusetts. 4Associate Professor, Laboratory for Information and Decision Systems, MIT, Cambridge, Massachusetts.

## Keywords

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