Abstract :
The concept of structured singular value was introduced by Doyle [1] as a tool for the analysis and synthesis of linear control systems with structured uncertainties. It is a key to the design of control systems under joint robustness and performance specifications and it nicely complements the H?? approach to control system design [2,3]. It has already been used successfully in various application areas [4-7]. The question of how to numerically evaluate, in a reliable manner, the structured singular value of a matrix has been explored by several authors [1,8-11] but is not entirely resolved yet. As the largest singular value of a square matrix is in fact its structured singular value with respect to the ´trivial´ structure (i.e., no structure), the question arises as to wheather some kind of ´structured singular value decomposition´ can be meaningfully defined. Such a decomposition, besides its conceptual interest, may shed new light into the question of the numerical evaluation of the structured singular value. In this paper, we present a few ideas and results that we hope will lead to a decomposition of interest. Throughout the paper, given any square complex matrix A, we denote, by ??-(A) its largest singular value and by AH its complex conjugate transpose. Given any complex vector x, xH indicates its complex conjugate transpose and ||x|| its Euclidean norm. The unit sphere in Cn is denoted by ??B. A block-structure of size m is any m-tuple K = (k1,...,km) of positive integer.[1] Given a block-structure K of size m, we make use of the projection matrices Pi = block diag(Ok 1,..,Ok i-1, Ik i, Ok i+1,...,Ok m), where, for any positive integer k, Ik is the k ?? k identity matrix and Ok the k ?? k zero matrix.
