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Interval polynomials, matrices and roots
Abstract
Stability robustness of linear time invariant systems with parametric uncertainty has received considerable attention recently. We focus our attention on two types of systems descriptions: Input-output and state space descriptions. Such models are obtained by identifying uncertain parameters in the mathematical model of the plant. This gives rise to a system description, in which coefficients of the polynomials in an input-output description (or elements of the matrices in a state space description) are assumed to lie in some closed and bounded set. It is of interest to establish the stability/instability properties of such an uncertain system. In this dissertation, we restrict our attention to systems with interval uncertainty structure; i.e., a family of polynomials with where every coefficient lies in an interval (input-output description) or a family of matrices where every element lies in an interval (state space description). The results in this dissertation can be divided into three parts. In the first part, we consider a family of polynomials and show that there can be no simple boundary checks to establish its robust instability. This work is motivated by the problem of stabilizing a fixed plant from a given compensator class. The second part deals with uncertain interval matrices. We look at the robust stability and quadratic stability of interval matrices. We show that there is no computationally feasible boundary checks to establish the robust stability of a family of interval matrices. We then provide a simple sufficient condition for the quadratic stability of uncertain matrices and go onto explore possible links between Kharitonov's result and Lyapunov theory. In the third part, we identify root regions with interval structure in the complex plane such that these regions generate a polytope of polynomials. This result has applications in robust pole placement. We illustrate this technique by synthesizing a controller for the automatic bus steering problem.
Subject Area
Electrical engineering
Recommended Citation
Padmanabhan, Prasad, "Interval polynomials, matrices and roots" (1994). Doctoral Dissertations Available from Proquest. AAI9420671.
https://scholarworks.umass.edu/dissertations/AAI9420671