Structured singular values and stability analysis of uncertain polynomials, part 2: a missing link

Abstract In Part 1 of this paper, a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one. In Part 2 of this series, we demonstrate that the framework developed in Part 1 is particularly suitable for a class of stability problems and its unifies several frequency-based stability conditions obtained via alternative approaches. Generally, these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion. Several stability conditions are derived readily from solution of the structured singular value presented in Part 1, which either extend or coincide with previously known results. In particular, for family of interval and diamond polynomials, we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of Kharitonov theorem lead to one another, thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series. A related robust stabilization problem is also investigated in this framework.

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