ISS-like properties in Lie-bracket approximations and application to extremum seeking

Abstract

In this paper, we consider a class of nonlinear time-varying systems that are subject to uncontrolled inputs. Those inputs may represent, for instance, disturbances, time-varying parameters, or model uncertainties. Exploiting the stability properties of an extended system, a so-called Lie-bracket system, we derive stability properties of the original system. Those stability results, in the spirit of input-to-state stability, are then applied to the case where the dynamical system is an extremum seeking system. Two scenarios are analyzed. We first consider that the uncontrolled input is a deterministic noise corrupting the cost measurement. We then examine the case where the uncontrolled input induces time-variations of the cost optimizer.