Page 249 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Figure 16.1 State transitions of the Preisach hysteresis.

used hysteresis models. Several typical systems with hysteresis dynamics will
also be briefly introduced.

16.2 HYSTERESIS MODELS
Modeling, identification, and control of hysteretic system have recently
attracted significant attentions. During the past decades, several hysteresis
modes have been reported. Several well-known models will be briefly in-
troduced.

16.2.1 Preisach Model
The dynamics of Preisach hysteresis are shown in Fig. 16.1. Consider a
pair of thresholds (α,β) with α ≥ β, Preisach operator [7] γ α,β [·,·] can be
defined as

⎧
⎪ −1, if u(t)<β
⎨
γ α,β [u,ζ]= 1, if u(t)>α (16.1)
γ α,β [u,ζ](t ) if β ≤ u(t) ≤ α
⎪ _
⎩
_
where u ∈ C([0, T]), ζ ∈{1,−1}, t ∈{0,T}, γ α,β [u,ζ]= ζ and t =
lim ε→0 t − ε. Then, Preisach model can be defined as

y(t) = f [u(t)]= μ(α,β)γ α,β [u,ζ](t)dαdβ (16.2)
P 0

2
where P 0 {(α,β) ∈ R : α ≥ β} is the Preisach plane, μ(α,β) is the
Preisach density function.

16.2.2 Prandtl-Ishlinskii (PI) Model
PI model [8] is a kind of phenomenological model derived from Preisach
model. Essentially, the Preisach model and the PI model can be converted