Page 268 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Inverse Model Based Control of Uncertain Systems With Backlash 269

size of the residual error can be reduced by using a large feedback gain k p.
Hence, the closed-loop system is uniformly ultimately bounded.
2) Considering the definition of r givenin(17.38), we know that the
output tracking error e is also bounded as long as the filtered error r is
bounded. Moreover, the error bound of e also depends on the design pa-
rameters λ i and feedback gain k p. This completes the proof.

17.5 SIMULATIONS

In this section, a simulation example is provided to validate the proposed
identification and control method. Here, the model of a linear motor will
be used and the influence of non-linear dynamics will not be considered
in this chapter since we will mainly focus on validating the efficacy of
the proposed identification and control for linear systems perturbing by a
non-smooth backlash input. The studied linear motor model is described
by [9]

⎧
⎨
m¨y = w(t) − F f
(17.47)
w(t) = BI(u(t))
⎩
where y is the load position, m is the total mass of the inertia load and the
core, F f = a f ˙y is linear viscous friction (a f is the viscous friction coefficient),
and w(t) is the output of backlash non-linearity, which is given by

⎧
⎪ 1.2(u(t) − 0.5), if ˙u > 0and w(t) = 1.2(u(t) − 0.5)
⎨
w(t) = 1.2(u(t) + 0.8), if ˙u < 0and w(t) = 1.2(u(t) + 0.8)
w(t_),
⎪
⎩ others
(17.48)
which means that l = 1.2, d 1 = 0.8, d 2 = 0.5in the backlash.
From Eq. (17.47), we can represent it in the following state-space form

˙ x 1 = x 2
(17.49)
θ 1 ˙x 2 = w(t) − θ 2x 2

where x 1 = y is the motor position, and x 2 =˙y is the speed. The unknown
parameter vector is θ =[θ 1 ,θ 2 ]=[m, a f ].
The parameters m and a f used in this simulation are set as m =
2
0.1 (V/m/s ), a f = 0.27 (V/m/s), respectively. To implement the proposed