Stability-Preserving Rational Approximation Subject to Interpolation Constraints

• Published in: IEEE transactions on automatic control Vol. 53; no. 7; pp. 1724 - 1730
• Main Authors: Karlsson, J., Lindquist, A.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.08.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied mathematics; Applied sciences; Approximation; Computer science; control theory; systems; Constraint theory; Control system synthesis; Control theory. Systems; Councils; Dealing; Entropy; Exact sciences and technology; infinity; Interpolation; Inverse problems; MATEMATIK; Mathematical analysis; MATHEMATICS; model reduction; Modelling and identification; nevanlinna-pick interpolation; Optimeringslära, systemteori; Optimization; Optimization, systems theory; Polynomials; quasi-convex optimization; rational; rational approximation; Reduced order systems; Specifications; Stability; systems; Tillämpad matematik; Tuning; Rational approximation; model reduction; Minimization; Function approximation; Convex programming; Inverse problem; Interpolation; Rational interpolation; quasi-convex optimization; Reduced order systems; Rational function; Analytical function; stability
• ISSN: 0018-9286; 1558-2523; 1558-2523
• DOI: 10.1109/TAC.2008.929384

A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational analytic interpolants with an a priori bound, has been developed in recent years. In this paper, we consider the limit case when this bound is removed, and only stable interpolants with a prescribed maximum degree are sought. This leads to weighted H 2 minimization, where the interpolants are parameterized by the weights. The inverse problem of determining the weight given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution to specifications. The basic idea could also be applied to the case with bounded analytic interpolants.