Stability robustness of almost linear state equations

Sufficient conditions for stability robustness of finite-dimensional autonomous systems are discussed. The system is made up of a stable, linear, nominal part, and different types of unstructured norm-bounded perturbations. It is known that if the perturbations are arbitrary-nonlinear, with norms bounded by the complex stability radius, the system is stable. It is also known that the real stability radius serves as a bound ensuring the stability of the system if the perturbations are linear. The case of equality of these two stability radii is characterized. Quantitative sufficient conditions for stability robustness are introduced for the case where perturbations are almost linear, in the sense that both the size and the derivative of the perturbations are bounded. These conditions describe a tradeoff between the size of the perturbations and their distance from linearity. Each of the first two types of perturbations, the arbitrary-nonlinear and the special case of linear, is shown to be a limiting case of the almost linear type