Page 251 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

252 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

Figure 16.3 KP model of hysteresis.

is a positive constant. p m (r) is an integrable density function
where p m 0
R
calculated from the experimental data, which satisfies rp m (r)dr < ∞ for
0
any p m (r) ≥ 0. The density function p m (r) generally vanishes for large values
of r.
The identification and control of PI hysteresis models have been ex-
tensively studied. Most of the inverse model based hysteresis controllers are
designed by using the PI model because of its good explicit expression.

16.2.3 Krasnoselskii-Pokrovskii (KP) Model
Krasnoselskii-Pokrovskii (KP) hysteresis model [9]islessusedasaPreisach
type model since the KP model is very similar to the Preisach model. KP
model can be described by using weighted superposition of possibly a con-
tinuum of basic hysteretic elements called hysterons illustrated in as shown
in Fig. 16.3, which makes some improvements on the Preisach operator.
Therefore, its nature, processing methods are similar with Preisach model.
+
Consider a pair of thresholds (α,β), defining ridge function δ : R →
[−1, 1] as

⎧
⎪ −1, if x < 0
⎪
2x
⎨
δ(x) = −1 + , if 0 ≤ x ≤ a (16.5)
a
⎪
⎪
1, if x > a
⎩
where a is the distance shown in Fig. 16.3.
Then the KP operator γ α,β [u(t),ζ] is defined as:
⎧ _ _
⎪ max{γ(t ),δ(u(t) − α)}, if u(t)> u(t )
⎨
_
_
γ α,β [u(t),ζ](t) = min{γ(t ),δ(u(t) − β)}, if u(t)< u(t ) (16.6)
⎪ _ _
γ(t ), if u(t) = u(t )
⎩