Filter design: a finite dimensional convex optimization approach

Abstract

Abstract 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 contribution is to prove that such problems are equivalent to finite dimensional convex optimization problems involving linear matrix inequality constraints. At now, 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 to set or 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 ∞ control and design of lead‐lag networks for control. Numerical applications emphasize the interest of the proposed results. Copyright © 2003 John Wiley & Sons, Ltd.