Page 279 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Control of Hammerstein Systems With Hysteresis Non-linearity 281

Figure 18.2 The hysteresis and the Preisach planes. (A) A hysteresis curve; (B) A Preisach
plane with stair-stepping curve; (C) The separated Preisach planes; (D) Illustration of the
discretization scheme.

identify the Preisach model now. In this section, a new deterministic iden-
tification approach is introduced by using the input u(k) and the calculated
variable x(k).
The classical identification of Preisach model is to divide the Preisach
plane into two subdivisions with stair-stepping curve. A typical hysteresis
curve is illustrated in Fig. 18.2A, and the classical Preisach plane can be
illustrated as in Fig. 18.2B, where the increasing sections (a, c) is horizontal
line and the decreasing sections (b, d) is vertical line in the stair-stepping
Preisach plane.
In this chapter, the increasing sections (a, c) and decreasing sections
(b, d) are illustrated in different triangle Preisach planes P i (1 i L, L rep-
resents the number of the piecewise monotonic sections, see Fig. 18.2C
with L = 4) and thus it does not require the stair-stepping surface [22]. In
this case, each Preisach plane represents one monotonic section (increasing