Composite Systems with Uncertain Couplings of Fixed Structure: Scaled Riccati Equations and the Problem of Quadratic Stability

Abstract

We consider large scale systems consisting of a finite number of separate uncertain subsystems which interact via uncertain couplings of arbitrarily prescribed structure. The couplings are viewed as structured perturbations of the block-diagonal system representing the collection of the separate nominal subsystems (the “nominal system”). We define spectral value sets and stability radii for these time-invariant structured perturbations and derive formulas for their computation. Scaled Riccati equations are introduced to obtain explicit formulas for the stability radii with respect to time-varying and possibly nonlinear perturbations of the given structure. From these we derive necessary and sufficient conditions under which the stability radii with respect to time-invariant and time-varying perturbations are equal. These results are obtained by constructing joint quadratic Liapunov functions of optimal robustness. With their help we prove necessary and sufficient conditions for quadratic stability and sufficient conditions for the validity of a generalized Aizerman conjecture.