Interconnection Between Schur Stability and Structured Singular Values

Abstract

In this paper, we present new results on the interconnections between Schur stability and structured singular values of real-valued matrices, denoted as R^n×n . Most new findings are obtained for n = 2 and n = 3. These novel insights into the relationship between Schur stability and structured singular values are developed by applying various tools from linear algebra, system theory, and matrix analysis. Schur stability ensures that all eigenvalues lie within the unit circle in the complex plane, which is fundamental for the boundedness and stability of system responses. Structured singular values, on the other hand, provide a measure of robustness, stability, and performance against structured perturbations in system parameters, offering valuable insights into the stability margins and performance limits under such uncertainties.