Prandtl-Ishlinskii operator

The best-known examples of these so-called complex hysteretic non-linearities are the Preisach- or Krasnosel skii-Pokrovskii operator R, the Prandtl-Ishlinskii operator H and the modified Prandtl-Ishhnskii operator M = S H) which is constructed as a concatenation of a Prandtl-Ishlinskii operator H and an asymmetrical scalar function S of Prandtl-Ishlinskii type which models the deviation of the real hysteretic nonhnearity from the class of Prandtl-Ishlinskii operators [332,353,355], All these operators belong to the class of operators with a Preisach memory P [341]. 

If the mappings T in the sensor equation (6.75) and the actuator equation (6.76) are purely hysteretic they can be modeled by a Prandtl-Ishhnskii operator H, a modified Prandtl-Ishlinskii operator M or a Preisach hysteresis operator R depending on the degree of symmetry of the branching behaviour. The calculation of these hysteresis operators and the corresponding compensators from the measured output-input characteristic requires special computer-aided synthesis procedures which is based on system identification methods. Due to a lack of space, this article cannot further comment on these synthesis methods. However, a detailed description of both the synthesis method and the mathematical basics can be found in the literature [332,341,350-352,356]. 

A scalar operator which considers simultaneously complex hysteresis effects, log(t)-type creep effects as well as saturation effects can be constructed by the parallel connection of a Prandtl-Ishlinskii hysteresis operator H and a Prandtl-Ishlinskii log(t)-type creep operator K followed by a concatenation with a memory-free scalar nonlinearity S. In this case the mapping T in (6.75) and (6.76) is given by a so-called modified Prandtl-Ishlinskii creep extension Mk. The corresponding reconstruction model is then given by (6.77) and (6.78) with the compensator Tg... 

---