Robust Stability of Polynomials with Affine Coefficient Perturbations and Degree Dropping

A fundamental problem in control theory is concerned with stability of a given linear system. The system designer often wants to know if all roots of the systems characteristic polynomial are located in a pre-specified region in the complex plane (the left-half plane, or the open unit disc are important examples of such regions). In many applications the coefficients of the characteristic polynomial are functions of independent physical parameters (as, for example, coefficients of friction, spring constants, etc.). The design of a control system is generally based on a simplified model, and the true values of the physical parameters may differ from the assumed values. Consequently, it is of interest to derive the maximum allowable perturbations of the parameters while maintainig the stability of the system. In many recent papers this problem has been addressed for a special case of families of monic polynomials. In this paper we consider families of polynomials whose coefficients are affine functions of the parameters, and we do not make any special assumptions concerning the leading coefficient. This particular setting enables one to investigate systems which have a singular behavior for some values of the parameters. In particular we cover the case where the leading coefficient of some polynomials vanish. Conditions under which the "degree dropping" does not affect the evaluation of stability radii are provided.