Algebraic Approach to Checking Strict Positive Real Property of Uncertain Real Rational Functions

Abstract

Parameter space robustness analysis has been gaining momentum in recent years, ever since the emergence of Kharitonov’s theorem in the control audience. Though the central issue is understandably the stability problem, considerable attention has also been paid to positive real property of rational functions in association with adaptive identification, adaptive control, absolute stability problems and so forth. A typical result among studies on strict positive real(SPR) property in the parameter space reads that we have only to check the SPR property of a few extreme plants in order to assure the same property for an interval plant family. Here, an interval plant is understood to be the ratio of two interval polynomials. In this paper, we attempt to generalize this problem in such a way that a plant is expressed as the ratio of two polytopes of polynomials. This enable us to cope with transfer functions whose denominator coeflïcients(and/or numerator ones) are mutually dependent. In the first half of the present paper, a necessary and sufficient condition is derived to ensure the SPR property of such a plant family. It is shown that we need to test Hurwitz propety of a certain edge polynomials as well as the SPR property of some “extreme” plants. The latter half is devoted to considering a method to check the obtained condition concretely. We propose an algebraic approach which consists of polynomial positivity tests and the usual Routh-Hurwitz stability test. The approach features that we can dispense with the matrix inversion and the eigenvalue computation otherwise required.KeywordsConvex CombinationAlgebraic ApproachStable PolynomialExtreme PolynomialHurwitz MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.