Stability Robustness of Almost Linear State Equations

Sufficient conditions for stability robustness of finite dimensional autonomous systems are discussed. The system is comprised of a stable linear nominal part, and different types of unstructured norm bounded perturbations. It is known that if the perturbations are arbitrary nonlinear, whose norm is 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. Here, quantitative sufficient conditions for stability robustness are introduced for the case where the perturbations are almost linear, in the sense that both the size and the derivative of the perturbations are bounded. These conditions describe a trade off 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, are shown to be limiting cases of the almost linear type. Finally, as an illustration of this result, the same quantitative sufficient conditions for stability are applied to a system with slowly varying linear perturbations.