Input-restricted stability of continuous and discrete time nonlinear feedback systems

In this paper we develop the concept of inputrestricted 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 specialised into simpler-to-compute corollaries and expanded to additional theorems which provide useful additional insights. The paper ends with two salient specialisations of key results developed herein: one is a type of input-restricted smallgain 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 specialisations, 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.