Optimal controller for dithered systems with backslash or hysteresis (Q1367788)

A dither i.e. an introduction of a high-frequency periodic signal into a nonlinear control system with memory is discussed. The aim is generally to simplify the system due to averaging and to improve its stability properties. Backslash and hysteresis nonlinearities are considered. The analysis is given for piecewise decreasing and increasing input signals. Stability is understood as a boundedness of the operator of the system in some Sobolev spaces. The main results state that (1) under certain conditions, the operator of the smoothed system is bounded in certain Sobolev spaces, and (2) if the dither's frequency is high enough and if a smoothed system is bounded in the same Sobolev sense, then the dithered system is \(L_2\)-bounded. The smoothed system is understood as a one with a nonlinearity averaged in time along the period supposing the dither's amplitude is large enough so that it determines uniquely the output.