Page 96 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Figure 5.3 Control actions u for different controllers.

Moreover, the tedious backstepping procedure in ANDSC is also avoided.
Moreover, the control action of the proposed method is smoother than
other three controllers (Fig. 5.3). The performances of ANDSC and AN-
RISE also illustrate that the ESN used in this chapter can achieve better
estimation performance, which further contributes to the improved con-
trol response. In the experiments, the system output signals are measured by
using encoders, which could subject to measurement noise. Thus, there are
small steady-state errors in the experiment results as other RISE methods
(e.g., [9]).
Moreover, a step reference with an amplitude x d = 0.6is used to further
validate the transient performance (e.g., overshoot). To fulfill condition
−δμ(0)< e(0)< −δμ(0), the PPF is selected as μ(t) = (1−0.01)e −3.5t +0.01
with δ = 1. Comparative results are shown in Fig. 5.4, from which one can
see that the transient performance of the proposed control scheme is better
than the other three control methods.
All the aforementioned experimental results clearly show that the pro-
posed ANRISE can guarantee both the transient and steady-state error per-
formances. Moreover, compared with other discontinuous friction model,
the use of the continuous differentiable friction model together with the
ESN could lead to smoother control responses. Moreover, the approxi-