Rational functions whose zeros and poles are constrained by conditions arising in control

Author

Rebecca L. Sparks, University of Rhode Island

Date of Award

2001

Degree Type

Dissertation

First Advisor

Orlando Merino

Abstract

We shall study three different, yet related mathematical problems. The first is given as follows. Beta problem. Given β > 0, determine if there exists a rational function H which has no poles in [special characters omitted] so that the rational function [special characters omitted]has no zeros and no poles in [special characters omitted]. This problem in an equivalent form has been studied by Blondel, et al., and its statement can be found in, Open Problems in Mathematical Systems and Control Theory. Previous work on this problem has yielded that there exists β* such that if β < β*, then no solution to the Beta Problem exists. Furthermore, it has been shown that 10−5 < β* < [special characters omitted]. In this dissertation we improve upon the upper bound on β*. We prove that β* ≤ 7 − 4[special characters omitted] ≈ 0.0718 by constructing a solution to a limiting case of the Beta Problem. The Beta Problem is a special ease of the following problem. Avoidance problem. Given a rational function [special characters omitted], and given nonnegative integers n and m, determine if there exists a rational function [special characters omitted] such that[special characters omitted] An algorithm is proposed to construct numerical solutions to the Avoidance Problem by solving an optimization problem. Numerical Experience is reported. By solving the Avoidance Problem, one also solves the simultaneous stabilization problem of control theory. It is well known that "[the] simultaneous stabilization problem is recognized as one of the hard open problems in linear systems theory." With the proposed algorithm, it is possible to construct solutions to the simultaneous stabilization problem when they exist. The third problem to be studied is an interpolation problem. Interpolation problem. Given z1, … , zk ∈ [special characters omitted] with zℓ ≠zj for ℓ ≠j, w1, … , wk ∈ [special characters omitted], wℓ ≠0 for all ℓ = 1, … , k and nonnegative integers n, m, determine if there exist polynomials P and Q such that [special characters omitted] An algorithm is proposed to obtain numerical solutions to the Interpolation Problem by solving an optimization problem. Numerical experience is reported, and properties of solutions to the optimization problem are proved.

Recommended Citation

Sparks, Rebecca L., "Rational functions whose zeros and poles are constrained by conditions arising in control" (2001). Open Access Dissertations. Paper 1781.
https://digitalcommons.uri.edu/oa_diss/1781