Page 280 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

or decreasing) and if the first piecewise section indicates increasing trend
(Fig. 18.2C (a)), then all odd number Preisach planes denote the increas-
ing sections (Fig. 18.2C (a, c)) and even number Preisach planes denote
the decreasing sections (Fig. 18.2C (b, d)). Consequently, different triangle
Preisach planes represent different piecewise monotonic sections, so that all
the details are memorized and the wiping-out phenomenon is avoided.
Let [u min ,u max ] be the practical input bounds of the hysteresis operator,
which is usually a strict subset of [α 0 ,β 0 ]. If we discretize [u min ,u max ] into
m levels, a Preisach density function in a compact set can be discretized to
m
a m dimensional function at time instant k. With a finite input set {u i } ,
i=1
the output of discrete Preisach operator at time instant k can be expressed
as in [13]
m

x(k) = μ i (k)γ α,β [u i ,ζ(α i ,β i )](k) (18.17)
i=1
where μ i (k) is the Preisach density function for k ∈[0,T].
m
Assume the input is {u i } i=1 , then each monotonous section can be
n L m
L
n
expressed as a subset {u j } of {u i } ,sothat i=1 i = m.For each
j=1 i=1
monotonous section, the following theorem is established:
Theorem 18.1. [22] For each piecewise monotonic section, the Preisach plane is
defined as P i, then the input sequence U and output sequence X are expressed as
follows:

T
U = u 1 u 2 ... u n L ,
T

X = x 1 x 2 ... x n L .
We augment the input U and output X into matrices as

⎡ ⎤ ⎡ ⎤
ˆ u 11 0 0 ... 0 ˆ x 11 0 0 ... 0
⎢ 0 ... ⎥ ⎢ 0 ... ⎥
⎢ ˆu 21 ˆ u 22 0 ⎥ ⎢ ˆx 21 ˆ x 22 0 ⎥
⎢ ⎥ ⎢ ⎥
ˆ U = ⎢ ˆu 31 ˆ u 32 ˆ u 33 ... 0 ⎥, ˆ X = ⎢ ˆx 31 ˆ x 32 ˆ x 33 ... 0 ⎥
⎣ ... ... ... ... ... ⎦ ⎢ ... ... ... ... ⎦
⎥
⎣ ...
⎥
⎢
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ u n L 1 u n L 2 u n L 3 ... u n L n L ˆ x n L 1 x n L 2 x n L 3 ... x n L n L
(18.18)
u i x i
where ˆu ij = , ˆx ij = ,j = 1,2,...,i. Then, the Preisach density function μ can
i i
be obtained as
−1
μ = ˆ X ˆ U −1 ˆ ω , (18.19)