State-space measures for stability robustness

Stability robustness measures for a perturbed linear feedback system are derived based on state-space models of the system. The system may be a continuous-time or discrete-time system. The perturbations are modeled as additive perturbation matrices. Necessary and sufficient conditions for the stability of the perturbed closed-loop system for all perturbations of norm bounded by some positive number are obtained. The destabilizing perturbations of minimal norm are characterized. It is shown by an example that there are cases when the destabilizing perturbations of minimal norm are all complex. The results are expressed in terms of induced operator norms. These are later specialized to the Euclidean norm and expressed in terms of singular values. A simple example is also included to illustrate an application of the results of this note.