Calculation of the structured singular value with a reduced number of optimization variables

For the 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. It is shown that for elementwise, magnitude-bounded 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 n ²-1 independent optimization parameters. A simple extension of this result shows that a reduction in the number of independent optimization variables is also possible for more general block-structured uncertainties. A more efficient implementation of the vector optimization method developed by M.K.H. Fan and A.L. Tits (1986) is also proposed. Several examples are included to illustrate the results