Numerical Computation of Structured Singular Values for Companion Matrices

Abstract

In this article, the computation of µ-values known as Structured SingularValues SSV for the companion matrices is presented. The comparison of lower bounds with the well-known MATLAB routine mussv is investigated. The Structured Singular Values provides important tools to analyze the stability and instability analysis of closed loop time invariant systems in the linear control theory as well as in structured eigenvalue perturbation theory.

Computing mu-values for Representations of Symmetric Groups in Engineering Systems

“…In the following we give definition of local extremizer of a structured spectral value set. Definition 3.2 [14]. A matrix ∆ ∈ ∆ * B so that ∆ possesses a unit 2-norm while ( 0 M ∆) has a maximum eigenvalue which maximizes (locally) Λ ∆ * B 0 (M ) is known a local extremizer of structured spectral value set.…”

“…In following theorem we replace full blocks in local extremizer with rank-1 matrices, in turn, this allow us to work to Forbenius norm instead with matrix 2-norm. Theorem 3.4 [14]. Let's suppose that…”

“…The value of upper bound is obtained as, that is, µ upper P D = 1.0000 while the same lower bound is obtained, that is, µ lower P D = 1.0000. Now, by using algorithm [14], we obtain the perturbation structure * ∆ * with ∆ * = 0 0 0 −0.5000 − 0.8660i .…”

“…In Figure 1, we show the comparison of lower bounds computed by algorithm [14] with the bounds (Lower and Upper) computed by mussv function for matrix valued function B(w) for w = 1 : 4, where w ∈ Ω and Ω denotes the frequency range of interest which is usually R + . The frequency response w is the quantitative measure of output of (M −∆) system.…”

“…The value of lower bound remains same, that is, µ lower P D = 1.0000. Now, by using algorithm [14], we obtain the perturbation structure * ∆ * with…”