Input-Restricted Stability of Continuous and Discrete Time Nonlinear Feedback Systems

In this article, we develop the concept of input-restricted stability, which determines whether a feedback interconnection remains stable only for inputs in a given subset of all possible inputs in a specified signal space. Graph separation concepts and continuity are employed to derive an input-restricted feedback stability theorem, which guarantees input-restricted stability of a feedback interconnection if both systems in the interconnection fulfil some given criteria related to their input–output relationships. Significantly, this result is applicable to both continuous and discrete time systems, unlike many existing local stability results. This theorem is then specialized into simpler-to-compute corollaries and expanded to additional theorems which provide useful additional insights. This article ends with two salient specializations of key results developed herein: one is a type of input-restricted small-gain stability theorem with one system bounded by a linear gain and the other by a quadratic gain; and the other is a type of input-restricted passivity theorem. For both of these specializations, which are not stable for all energy bounded inputs, an example is provided where the feedback interconnection is shown to be stable when the energy of exogenous inputs is below a given threshold.