Page 257 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Figure 17.1 Backlash non-linearity.

bust inverse model controller is designed to achieve the output tracking
and compensation of backlash dynamics. Simulation results show that the
proposed control system can retain good tracking performance and strong
robustness.

17.2 SYSTEM DESCRIPTION AND PROBLEM FORMULATION

17.2.1 Uncertain System With Input Backlash
In this chapter, we consider a typical modular non-linear systems, consisting
of an input backlash and a linear plant, which can be described as
⎧
⎪ ˙ x i = x i+1 ,i = 1,...,n − 1
n
⎪
⎪
˙ x n = (−a j )x n−j+1 + w(t)
⎨
j=1 (17.1)
⎪ w(t) = BI(u(t))
⎪
⎪
y(t) = x 1 (t)
⎩
T
where x =[x 1 ,··· ,x n ] is the system state vector, y(t) is the system output,
n is the system order, and BI(·) is the backlash non-linear function, w(t)
is the output of backlash module (e.g., realistic output driving force of
actuator), which is unmeasurable, and u(t) is the controller output (e.g.,
input of backlash module).
The dynamics of backlash can be shown in Fig. 17.1, which can be
described by
⎧
⎪ l(u(t) − d 2 ), if ˙u > 0and w(t) = l(u(t) − d 2 )
⎨
w(t) = BI(u(t)) = l(u(t) + d 1 ), if ˙u < 0and w(t) = l(u(t) + d 1 )
⎪ _
w(t ),
⎩ others
(17.2)