Two-time-scale averaging of systems involving operators and its application to adaptive control of hysteretic systems

Motivated by the adaptive control problem for systems with hysteresis, a two-time-scale averaging framework is presented in this paper for systems involving operators, by extending the work of Teel and co-workers. The developed averaging theory is applied to the analysis of a model reference adaptive inverse control scheme for a system consisting of linear dynamics preceded by a Prandtl-Ishlinkskii (PI) hysteresis operator. The fast component of the closed-loop system involves the coupling of an ordinary differential equation and a hysteresis operator derived from the PI operator and its inverse, while the slow component is the parameter update rule. The stability of the boundary-layer system and that of the average system are established under suitable conditions, which implies practical regulation of the parameter error and tracking error under the adaptive scheme.