Differential stability and robust control of nonlinear systems

This paper introduces a notion of distance between nonlinear dynamical systems which is suitable for a quantitative description of the robustness of stability in a feedback interconnection. This notion is one of several possible generalizations of the gap metric, and applies to dynamical systems which possess a differential graph. It is shown that any system which is stabilizable by output feedback, in the sense that the closed-loop system is input-output incrementally stable and possesses a linearization about any operating trajectory (i.e., about any admissible input-output pair), has a differential graph. A system which possesses a differentiable graph is globally differentiably stabilizable if the linearized model about any admissible input-output trajectory is stabilizable. It follows that if a nonlinear dynamical system is globally incrementally stabilizable, then it is (globally incrementally) stabilizable by a linear (possibly time-varying) controller. A suitable notion of a minimal opening between nonlinear differential manifolds is introduced and sufficient conditions guaranteeing robustness of stability are provided.