Page 264 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Inverse Model Based Control of Uncertain Systems With Backlash 265

Figure 17.2 Block diagram of closed-loop control system.

⎧
⎪ ˙ x i = x i+1 , i = 1,··· ,n − 1
⎨
n n

˙ x n = − a jx n−j+1 + w(t) + ˜ a (17.26)
j=1 j=1 jx n−j+1
w(t) = BI(u(t))
⎪
⎩
T
n
where x =[x 1 ,x 2 ,··· ,x n ] ∈ R is the state vector, ˆa j is the estimated pa-
rameters, ˜a j is the estimation error, w(t) = BI(u(t)) is the backlash dynamics.
The control design objective can be described as: to find an appropri-
ate control u such that the output y of system (17.26) tracks a bounded
reference input y d even in the presence of non-smooth backlash dynamics.
17.4.1 Inverse Model of Backlash
To compensate the effect of backlash [13], in this section, a compensa-
tion approach based on the inverse model of backlash in (17.2)willbe
constructed, and then incorporated into the feedback control design. The
proposed closed-loop control system structure is given in Fig. 17.2,where
u d is the controller output signal, BIV(·)and BI(·) represent the inverse of
the backlash and the backlash, respectively, G(s) is the identified control
plant (17.26), and y = x 1 is the system output, u is the overall control ac-
tion to be applied on the plant (input of the backlash), and u d is the output
of the feedback control to be designed later.
Based on the above identification results, the inverse of the backlash can
be expressed as follows

⎧
1
u d + d 2 , if ˙u d > 0
⎪ ˆ
⎪
⎪
⎨ ˆ l
⎪
u(t) = BIV(u d ) = 0, others (17.27)
⎪ 1
⎪
u d − d 1 , if ˙u d < 0
⎪ ˆ
⎪
⎩
ˆ l
The inverse characteristic given by (17.27) is also piecewise. Hence, we
can design the following dynamic inverse of asymmetric backlash as
1
u(t) = u d (t) + ρW(t) (17.28)
ˆ l