Extending hysteresis operators to spaces of piecewise continuous functions

We consider continuous-time hysteresis operators, defined to be causal and rate independent operators mapping input signals to output signals . We show how a hysteresis operator defined on the set of continuous piecewise monotone functions can be naturally extended to piecewise continuous piecewise monotone functions. We prove that the extension is also a hysteresis operator and that a number of important properties of the original operator are inherited by the extension. Moreover, we define the concept of a discrete-time hysteresis operator and we show that discretizing continuous-time hysteresis operators using standard sampling and hold operations leads to discrete-time hysteresis operators. We apply the concepts and results described above in the context of sampled-data feedback control of linear systems with input hysteresis.