An alternative characterization of the structured singular value

The size of the smallest, structured, destabilizing perturbation for a linear, time-invariant, system can be calculated via the structured singular value (μ). It can be bounded above by the solution of a linear matrix inequality (LMI). This paper gives an alternative characterization which is particularly suited to the case when the system (or matrix) is not of full rank. The approach is based on a Cauchy-Binet expansion of the determinant formula. It is used to study the case when the LMI upper bound is not tight. An alternative perturbation analysis framework, based on the Frobenius norm of the perturbation, is introduced. The solution of this problem can be used to bound the μ in the low rank case, and in the four block example of Doyle, gives a significantly better upper bound for μ than the LMI bound