Page 275 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Control of Hammerstein Systems With Hysteresis Non-linearity 277

fined as
⎧
⎪−1 if u(t)<β,
⎪
⎨
γ α,β [u,ζ] 1 if u(t)>α, (18.3)
⎪
⎪
γ α,β [u,ζ](t ) if β u(t) α,
⎩ −
−
where u ∈ C([0,T]), ζ ∈{1,0} or {1,−1}, t ∈[0,T],γ α,β [u,ζ](0 ) = ζ,and
t = lim t − ε.
−
ε→0
Define
2
P 0 {(β,α) ∈ R : β α},
where P 0 is the Preisach plane, and thus each (α,β) ∈ P 0 is limited by
γ α,β .For u ∈ C[0,T] and a Borel measurable initial configuration ζ 0 : P 0 →
{1,0}, the output of the Preisach operator can be obtained as [13]

γ
x(t) = f (u(t)) = μ(α,β) ˆ α,β [u,ζ 0 (α,β)](t)dαdβ (18.4)
P 0
where μ is the Preisach density function.
The objectives of this chapter are: 1) to estimate the order n of lin-
ear system (18.1); 2) to identify the unknown coefficients {a 1 ,a 2 ,...,a n },
{b 1 ,b 2 ,...,b n } of (18.1); 3) to identify the dynamics of x(t) = f (u(t)) in
(18.4); 4) to design a composite control for the Hammerstein system shown
in Fig. 18.1 to achieve output tracking control.

18.3 IDENTIFICATION OF HAMMERSTEIN SYSTEM WITH
HYSTERESIS

In this section, we address the identification of Hammerstein system shown
in Fig. 18.1. The order n of linear transfer function (18.1)willbefirstly
determined by using Hankel matrix approach. Then the coefficients of
G(z) can be identified in terms of blind identification by using the output
measurement y(t) only. The unknown input variable x(t) is determined
based on the identified transfer function G(z) and the measurable output
y(t). Finally, the hysteresis Preisach model will be identified by introducing
a novel method with the input u(t) and the calculated variable x(t).

18.3.1 Estimation of System Order
We can set the input of the system as a pulse sequence, then the pulse
output response can be denoted as y(1),y(2),...y(L). The Hankel matrix