Stability analysis methods for delta operator polytopic polynomials

Abstract

SUMMARYStability analysis of a linear uncertain polynomial system in parameter space is one of the central themes of recent control theory. It often requires an enormous number of stability checks with conventional stability test methods. A delta operator polytopic polynomial, a convex combination of some delta operator vertex polynomials, is one of such cases. In this paper, robust stability is addressed for polytopic delta operator polynomials and several necessary and sufficient stability conditions are derived. The delta operator, an operator used to express discrete time systems, is known to have significant features: numerical advantage in implementation and ability to smoothly connect the z-operator with the Laplace operator. Based on this last feature and on the celebrated Edge theorem, we first derive three kinds of exact stability conditions for the uncertain delta operator polynomials. We then extend the directional stability radius method, which was developed for diamond polynomials, so that it can also be applied to polytopic polynomials. This extension gives rise to the fourth stability test. Furthermore, it is shown from the result of the numerical experiments with these stability analysis methods that one of these four methods, which uses eigenvalues of matrices, turns out to be most efficient for stability analysis of the polytope.

Derivation of Robust Stability Ranges for Disconnected Region with Multiple Parameters

“…Schur stability of a segment polynomial is discussed in [7]. A method for stability analysis of delta-operator segment polynomial is presented in [8].…”

Robust D-Stability Analysis of Linear Systems with Real Structured Uncertainties by the Stability Feeler

“…Schur and Hurwitz stability analysis methods for polytopes of polynomials are given in [26] and [27], respectively. Robust stability analysis methods for polytopes of delta operator polynomials are given in [30]. Largest hypersphere containing only stable polynomials in the coefficient space is derived by the method in [28].…”