Page 212 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ESO Based Adaptive Sliding Mode Control of Servo Systems With Input Saturation 211

will converge to zero. Then based on the sliding mode control theory, the
dynamics of s will have invariant properties when the state variables x 1 and
x 2 reach the sliding surface s = 0. From (13.17), we can conclude that e c1
and e c2 converge to zero, which means that the state variables x 1, x 2 will
track the desired signals x 1d , x 2d . This completes the proof.

13.4 SIMULATIONS
In the simulation, the initial conditions and parameters of the servo system
(13.3)are setas θ m (0),ω m (0) = (0,0), J = 0.5, D 1 = 0.3, K t = 1, T f = 10
and the sampling time is t = 0.01 s. In the ESO design, we set a 0 =−20,
b 0 = 5, and the ESO gains l i are calculated by using the pole placement
and given by l 1 = 60, l 2 = 1200, and l 3 = 8000. The control saturation
constraint is chosen as v max = 12, and the controller parameters are chosen
as λ 1 = 10, α 1 = 1, α 2 = 0.5, α 3 = 0.25, and τ = 1as [14]. In addition, in
order to verify the superiority and effectiveness of the proposed method,
the following three different control methods are tested and compared in
the simulations:
1) First method: the traditional sliding mode control without saturation
compensation, in which the controller is given by (13.19), with con-
∗
troller gain k = 50.
2) Second method: the adaptive sliding mode control without saturation
compensation, in which the controller is expressed by (13.20), and the
adaptive parameter and the boundary parameter are set as k m = 13 and
μ = 0.1.
3) Third method: the adaptive sliding mode control with saturation com-
pensation proposed in this paper, in which the controller is also given
in (13.20), and the parameters of adaptive laws are chosen as the same
as the second method. The main difference is that the disturbance term
in the observer design includes the saturation compensation is not ac-
tivated or not.
In the simulations, the desired signal to be tracked is y = sin(2t).
Comparative simulation results of the three methods are shown in
Figs. 13.1–13.4.From Fig. 13.1 and Fig. 13.2, it can be seen that all the
three methods can track the desired signal after 1 s transient response.
Specifically, the third method can achieve the best tracking performance
and transient response. Moreover, as shown in Fig. 13.3 and Fig. 13.4,the
chattering phenomenon in the first method is much more serious than the
other two methods since the compensation of situation by using ESO is