STRUCTURED SINGULAR-VALUES AND STABILITY ANALYSIS OF UNCERTAIN POLYNOMIALS, .1. THE GENERALIZED-MU

One of the primary goals in this two-part series is to establish a link between the structured singular value and results in stability analysis of uncertain polynomials. Another primary goal is to develop an improved method for accurately and efficiently computing the structured singular value for an important class of problems in robust stability analysis. To achieve these goals, we first introduce a generalized framework of structured singular values and next show how stability problems for uncertain polynomials may be studied in this framework. Part 1 of this series is devoted entirely to the generalized structured singular values, specifically for the case when a certain matrix representing the 'nominal system' is of rank one. An analytical expression for the generalized notion is derived in this caw which involves solving a convex optimization problem in one real variable and renders the structured singular value readily solvable. In particular, when the general framework is specialized to that of the standard structured singular value, the expression is solved explicitly. Also for several additional important cases, explicit solutions are obtained. The framework as well as results will then be used in Part 2 of this series to study stability problems for a class of polynomials whose coefficients are affine functions of real or complex uncertainties. We demonstrate that the generalized structured singular value is a suitable notion for these problems and its solution unifies a number of results obtained previously via alternative approaches. For several problems of interest, we further demonstrate that stability conditions based upon the structured singular value and those in the spirit of Kharitonov theorem can be derived from one another, and hence establish a link between two drastically different type of results.