A general framework for the calculation of stability radius with D stability region for a parametrically uncertain system via non-linear optimisation

A general approach to the calculation of stability radius defined in the p norm for an uncertain system with uncertain parameters q entering nonlinearly into the characteristic polynomial is presented in this paper. The stability margin problem is transformed into a standard, computationally tractable non-linear constrained optimisation problem, for which a commercial software package has been developed in the Matlab optimisation toolbox. We will explain the principle and the steps taken to cast the problem into the standard optimisation problem. The advantages of this approach include: 1) the desired D stability region can be of any form and be connected or disjoint. This leads to a unified treatment of the continuous and discrete cases and the Hurwitz and Schur stability regions are special cases; 2) the uncertain parameters appearing in the uncertain plant may be interval, linear affine, multi-linear affine, or even non-linear; 3) the quantitative measure of stability margin may be any continuous function of the uncertain parameter q such as the p norm; 4) the number of uncertain parameters may exceed three if the coefficients of a characteristic polynomial is non-linear. Examples are provided to show the merits of this analysis approach.