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Filter design: A finite dimensional convex optimization approach. (English) Zbl 1043.93019

Summary: We consider the design of transfer functions (filters) satisfying upper and lower bounds on the frequency response magnitude or on phase response, in the continuous and discrete time domains. The paper’s contribution is to prove that such problems are equivalent to finite-dimensional convex optimization problems involving linear matrix inequality constraints. Nowadays such optimization problems can be efficiently solved. Note that this filter design problem is usually reduced to a semi-infinite-dimensional linear programming optimization problem under the additional assumption that the filter poles are fixed (for instance, when considering FIR design). Furthermore, the semi-infinite-dimensional optimization is practically solved using a gridding approach on the frequency. In addition to be finite-dimensional, our formulation allows us to set or to set not the filter poles. These problems were mainly considered in signal processing. Our interest is to propose an approach dedicated to automatic control problems. In this paper, we focus on the following problems: design of weighting transfers for \(H_{\infty}\) control and design of lead-lag networks for control. Numerical applications emphasize the usefulness of the proposed results.

MSC:

93B40 Computational methods in systems theory (MSC2010)
93B51 Design techniques (robust design, computer-aided design, etc.)
90C25 Convex programming
15A39 Linear inequalities of matrices

Software:

LMI toolbox; LMITOOL
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References:

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