Page 260 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Inverse Model Based Control of Uncertain Systems With Backlash 261

and then define {y 0 (k)} as the output sequence of the system (17.8) under
the following input signals [11]
⎧
⎪ m 1 + δ, k = hT + 2n
⎨
u 0 (k) = m 1 + jδ, k = hT + (2 + j)n; j = 1,··· , s (17.10)
0, others
⎪
⎩
where δ satisfies

(s − 1)(m s+1 − m s ) m s+1 − m s
<δ < (17.11)
s 2 s
and m 1 ,··· ,m s+1 will be given by (17.9). Moreover, from Fig. 17.1,we
know that when u 0 = m 1 (= u m ), w 1 = BI(u m ) = w m.
Thus for arbitrary k > 0, the input and output relationship given in
(17.8) can be represented as

−1
−1
A(z )y 0 (k) = B(z )w l (k) (17.12)
The unmeasurable w l (t) will be further addressed by using the DPPR of
Backlash.

17.3.2 Discontinuous Piecewise Parametric Representation of
Backlash
In this section, the parametric piecewise linear expression [10]isusedto
approximate the unknown w l (k) = BI(u) existed in (17.11). Parameterized
piecewise linear expressions are constructed as follows: consider the fact that
w l = BI(u) is a piecewise continuous function defined on [u m , u M ],then

within the domain u m , u M , the sampling points {(u i ,w i )}, i = 1,2,···
are divided into s > 2 non-overlapping subintervals u m = m 1 < m 2 < ··· <
m s+1 = u M ,with l = m i ,and l i = m i+1 (i = 1,2,··· ,s).
i
Then the piecewise linear function w l (t) of backlash defined on each
partition can be expressed as [10]
s

w l (u) = p 0 + p j σ j (0, u − l , l j − l ) (17.13)
j j
j=1
where p j (j = 0,...,s) are the unknown coefficients, l j and l are the up-
j
per and lower bound of the j-th interval with l = m i ,and l i = m i+1 (i =
i
1,2,··· ,s),and σ j (0,u − l , l j − l ) is the activation function, which fulfills
j
j