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This paper investigates the frequency response of an uncertain system whose set of possible transfer functions is represented by a polytope of numerator-denominator polynomial pairs. It will be shown that the frequency response of the edge transfer functions provides most of the information required to determine all possible frequency responses of the system Once the edge response is computed, a simple procedure can be used to obtain the complete response. These results can be used to compute the frequency response at any point in the complex plane, so they are applicable to continuous-time and discrete-time systems. For an interval family of tansfer functions, the edge result in this paper can be combined with known properties of interval polynomials to give an even simpler method of computing the frequency response. This simplification is only valid for frequencies on the jw-axis, so its application is limited to continuous-time systems.

The stability version of the Edge Theorem states that a polytope of polynomials is D-stable if and only if its edge polynomials are all D-stable. Unfortunately, this statement is only true for restricted classes of polytopes and restricted classes of stability regions. If either of these restrictions is removed, the theorem will not be valid. In order to remove these constraints, this paper will present a simple precondition for the Edge Theorem. If this condition is not satisfied then the polytope is not D-stable. If this condition is satisfied then the stability version of the Edge Theorem is valid for all stability regions D and all polytopes of polynomials. The Generalized Edge Theorem of Soh and Berger is stated in a similar form. They consider stability regions D whose complements are simply connected in the complex plane. A special case of this theorem states that a polytope of polynomials is D-stable if and only if the edge polynomials are D-stable. It will be shown by counter example that this statement is not true. The precondition presented in this paper suggests ways of correcting this theorem.

In this paper we will examine how the spectrum (of eigenvalues) of an interval matrix family M depends on the spectrum of its extreme sets. We present three results: 1) We show that the roots of pairwise convex combinations of the characteristic polynomials of the vertices of M provide a bound for the spectrum of M. 2) We state conditions on M which guarantee that the spectrum of M can be determined from the spectrum of its relative boundary. 3) For the special case of M having a real spectrum, we show that this spectrum is completely determined by the eigenvalues of its vertices.

This paper presents a design methodology for the synthesis of a robust controller that accounts for both unmodelled dynamics and structured real parameter uncertainty for multiple-input, multiple-output systems. In a design aimed at constraining both the H ? norm of a certain disturbance transfer matrix and a quadratic gaussian performance index under their respective bounds, a surrogate system may be formed by modelling the structured real parameter uncertainty as additional noise inputs and additional weights at the existing noise inputs and measurement outputs of the system. Application of a Riccati equation approach to this surrogate system then yields a robust controller which, when used in the actual system, will result in a closed-loop system having the same H ? bound and quadratic gaussian performance index bound as the surrogate system, even in the presence of given real parameter variations.