On the solutions of the rational covariance extension problem corresponding to pseudopolynomials having boundary zeros

• Published in: IEEE transactions on automatic control Vol. 51; no. 2; pp. 350 - 355
• Main Authors: Nurdin, H.I., Bagchi, A.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.02.2006; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Australia Council; Automatic control; Boundaries; Boundary zeros; bounded solutions; Computer science; control theory; systems; Constraint optimization; Constraint theory; Control theory. Systems; Covariance; Exact sciences and technology; Filters; Information technology; Interpolation; Mathematical models; Nevanlinna-Pick interpolation; Nonlinear equations; Optimization; Poles; poles and zeros; rational covariance extension; Studies; Sufficient conditions; System theory; Utilities; Boundary value problem; Stability; Covariance; rational covariance extension; Nevanlinna-Pick interpolation; Unit circle; bounded solutions; Convex programming; poles and zeros; Pole zero method; Bounded solution; Boundary zeros
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2005.863503

In this note, we study the rational covariance extension problem with degree bound when the chosen pseudopolynomial of degree at most n has zeros on the boundary of the unit circle and derive some new theoretical results for this special case. In particular, a necessary and sufficient condition for a solution to be bounded (i.e., has no poles on the unit circle) is established. Our approach is based on convex optimization, similar in spirit to the recent development of a theory of generalized interpolation with a complexity constraint. However, the two treatments do not proceed in the same way and there are important differences between them which we discuss herein. An implication of our results is that bounded solutions can be computed via methods that have been developed for pseudopolynomials which are free of zeros on the boundary, extending the utility of those methods. Numerical examples are provided for illustration.