Mixed-norm optimization and its applications

Dung K Du, Purdue University

Abstract

Classical H2 and H∞ design methods are well developed in the literature. However, these design procedures suffer from certain inadequacies. The H2 optimal solutions may result in unacceptable H∞ errors at certain frequencies. On the other hand, the H∞ solutions may create considerable errors over a large range of frequencies. In this work, we will tackle this dilemma with two different approaches. For the first approach, we directly impose constraints on the H2 and H∞ errors. The set of all functions satisfying each of the constraints forms a convex set. We then formulate the mixed norm optimization problem as a problem of finding a point in the intersection of certain convex sets. We demonstrate our technique with several numerical examples based on a design problem of approximating Infinite Impulse Response (IIR) Filters by Finite Impulse Response (FIR) Filters. For the second approach, we define a trade-off norm called the [special characters omitted]-norm where m is a positive integer. Interestingly, the classical H2 and H ∞ norms can be considered as special cases of the [special characters omitted] norm when m = 1 and m = ∞ respectively. The corresponding optimization problems are then solved with respect to this new norm using the relaxed commutant lifting technique in operator theory. Our solutions are parameterized by a positive scalar and a positive integer which may allow the designer to select from a family of filters the one which is best suited for a specific application. We apply our relaxed commutant lifting technique to a fractional delay filter design problem and multirate filterbank design.

Degree

Ph.D.

Advisors

Frazho, Purdue University.

Subject Area

Applied Mathematics|Electrical engineering|Operations research

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