The Gap between Complex Structured Singular Value Μ and Its Upper Bound Is Infinite

Here and below, ‖A‖ always denotes the induced (by the Euclidean norm in Cn) operator norm of A, i. e. its maximal singular value. Entries of the matrices A and ∆ are complex, so we deal with complex structured singular value. The first equality above is the definition of μ, the second is a simple exercise in linear algebra. Note, that if in the definition of μ we take infimum over all matrices, not only diagonal, we get exactly the norm ‖A‖. On the other hand, if we take infimum over a smaller set of scalar matrices λI, λ ∈ C, we get exactly the spectral radius r(A) of the matrix A. So μ(A) can be estimated as r(A) μ(A) ‖A‖. The structured singular value μ was introduced in connection with robust control with structured uncertainties, see [1, 2] (we should also mention papers by M. Safonov [9, 10] where the multivariable stability margin Km(G), which is essentially the reciprocal of μ, was introduced). Without going into a lot of details (a reader interested in a detailed introduction into the subject, with all references and complete history, should look somewhere else, [13] will be a good reference), let us just remind the reader main ideas. Consider the system on Fig. 1 with uncertainty ∆ in the feedback loop. Here G is causal stable LTI (Linear Time Invariant) plant. An important notion in robust stability is the so called stability margin. Suppose, that our uncertainty ∆ belongs to some class U of stable causal