Calculation of the Structured Singular Value with a Reduced Number of Optimization Variables

Optimal scaling techniques have become widely accepted tools in the analysis and design of systems in the presence of structured uncertainties. Among these are the block similarity scaling techniques, the so-called "structured singular value" introduced by Doyle, which has been shown to apply to general block structured uncertainties. For the special case of an n × n uncertainty matrix with n² nonzero 1 × 1 blocks, the structured singular value technique with similarity scaling suffers from the disadvantage of having to expand an n × n matrix problem to an n² × n² matrix optimization problem with n² - 1 free variables. For this same class of uncertainties with scalar blocks, an alternative approach proposed by Kouvaritakis and Latchman employs a "nonsimilarity" scaling technique which preserves the original matrix dimension (n × n) and requires only 2(n - l) optimization parameters. The aim of this paper is to show that for scalar block structured uncertainties, the structure of the problem may be exploited to yield a similarity scaling method which uses no more than 2(n - 1) rather than n² - 1 optimization parameters. A simple extension of this result shows that a reduction in the number of free variables is also possible for general block structured uncertainties. A more efficient implementation of the vector optimization method developed by Fan and Tits is also proposed. Several examples are included to illustrate the results.