Page 259 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

where,

n−i n−1 n−j j−i
j−i
α n−i = (−1) n−i C + a n−j T (−1) C
i j=i 0 j
n−1 n−j j−i j−i (17.5)
β n−i = b n−jT (−1) C
j=i 0 j
are the unknown coefficients, and T 0 is the sampling interval, y(z) and w(z)
are the z-transform of sampling sequence {y(k)} and {w(k)}, respectively.
Hence, for system (17.3), theequivalentdifferenceequationisgivenby
⎧
−1
−1
⎪ A(z )y(k) = B(z )w(k)
⎨
−1
A(z ) = 1 + α 1z −1 + ··· + α nz −n (17.6)
⎪ −1 −1 −2 −n
⎩ B(z ) = β 1z + β 2z + ··· + β nz
To address the unmeasurable w(t), we need to further represent the
backlash dynamics (17.2) by using the idea of discontinuous piecewise para-
metric representation (DPPR) [10]. Considering the non-linearity of the
backlash as shown in Fig. 17.1, we know that its input u(t) and output w(t)
fulfills u m ≤ u(t) ≤ u M and w m ≤ w(t) ≤ w M , respectively. Moreover, there
are three parameters l,d 1 ,d 2 to be identified together with α i ,β i.
Hence, we first consider the left half plane of the curve w l (green line
[light gray in print version] in Fig. 17.1), which can be described as

l(u(t) + d 1 ), if ˙u < 0and w l (t) = l(u(t) + d 1 )
w l (t) = (17.7)
w M , if ˙u ≤ 0and u(t)< u M

In the process of parameter identification, we first make the input u(t)
of the system satisfy the first condition as described in (17.7), i.e., w(t)
moves on the curve w l (t) but does not switch to the other side (right side
of curve w r (t), blue line [dark gray in print version] in Fig. 17.1)exceptfor
the point (u M , w M ).Inthiscase,Eq.(17.6) can be written as

−1
−1
A(z )y(k) = B(z )w l (k)
(17.8)
w l (k) = BI(u(k))
To use the method of DPPR [10] that has been explained in Chapter 6

in more details, we can divide the input partition u m , u M into s > 2
non-overlapping subintervals,

u m = m 1 < m 2 < ··· < m s+1 = u M (17.9)