Page 250 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Hysteresis Dynamics and Modeling 251

Figure 16.2 PI model of hysteresis.

to each other. However, the Preisach model and the PI model are differ-
ent in the explicit expression: the Preisach model is mainly composed of
Preisach operator and Preisach density function, while the PI model relies
on the superposition of play operator as shown in Fig. 16.2. Therefore, the
identification method for Preisach model cannot be directly applied.
PI model is the weighted superposition of play operators and a linear
input function to describe the hysteresis non-linearity. The play operator is
the basic hysteresis operator with symmetric and rate-independent proper-
ties. The 1-D play operator can be considered as a piston with a plunger
of threshold r,asshown inFig. 16.2. The output ω is the position of the
center of the piston, and the input is the plunger position u. For any piece-
wise uniformly monotonic input function u(t) ∈ C[0,t], it is monotonic in
every subspace [t i ,t i + 1],where i = 0,1,··· ,N − 1. When threshold r ≥ 0,
the play operator is defined as

ω m (0) = G mr [u](0) = g mr (u(0),0)
γ α,β [u,ζ]= (16.3)
ω m (t) = G mr [u](0) = g mr (u(0),ω m (t i ))

where g mr (u,ω m ) = max{m 0 (u − r),min[m 0 (u + r),ω m ]}, t i ≤ t ≤ t i+1,0 ≤
i < N, ω is initially given and m 0 ∈ R is a ramp variable introduced to
+
adjust the hysteresis shape.
According to the definition of play operator (16.3) and explanations
in [8], the extended PI model can be defined as:

R
y(t) = v m [u](t) = p m0u + p m (r)G mr [u]dr (16.4)
0