On the Characterization and Evaluation of Stability of Nonlinear Systems

The problem of stability for nonlinear feedback systems is examined. A hybrid approach that combines Input-Output and state space theories is proposed. Within this framework the standard notions of dynamic gain and dynamic incremental gain of a nonlinear operator are examined and their inability to accurately reflect the dynamic behavior of a nonlinear system is demonstrated. The concepts of dynamic gain and dynamic incremental gain over sets are introduced and their connection to a physical system´s dynamic behavior is established. The theory is developed to enable systematic computation of the incremental gain of an operator over a set, through nonsmooth optimal control theory. Examples are given that elucidate the above issues.