On the Analysis of the Eigenvalues of Uncertain Matrices by u and v: Applications to Bifurcation Avoidance and Convergence Rates

Based on the structured singular value μ and the skewed structured singular value v, this technical note presents several useful relationships between an uncertain matrix expressed in a linear fractional form and its eigenvalues. The results are used to derive 1) sufficient conditions to avoid bifurcations for systems with parametric uncertainties and 2) bounds on the convergence rate for uncertain stable matrices and uncertain Markov matrices. Illustrative examples are also provided.