Page 155 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ANDSC of Strict-Feedback Systems With Non-linear Dead-Zone 151

The unknown time-varying delays are given as τ 1 = 0.5(1 + sin(t)) and
τ 2 = 0.5(1 + cos(t)), and the desired tracking trajectory is taken as y d (t) =
0.5(sin(t) + sin(0.5t)). It is noted that the functions f i (x,x(t − τ)),i = 1,2
in (9.52) can not be bounded by some non-negative functions of the de-
layed states x i (t − τ i ). However, after transforming it into the form (9.4),
Assumption 9.2 can be fulfilled by selecting the bounding function as
2
2
ϕ 11 (x 1 ) = 2x and ϕ 21 (¯x 2 ) = 2|x 1 |x . In addition, the control functions
1 2
g i (·),i = 1,2in system (9.52) also contain x i (t) and x i (t − τ i ) simultaneously.
For such cases, it is not possible to use recent results to derive a suitable
controller. However, the proposed ANDSC can be employed. By using the
proposed methods, appropriately large feedback control gains k 1 = k 2 = 20
and adaptive learning parameters
1 =
2 =
a1 =
a2 = 10 are selected to
sufficiently satisfy the condition (9.49). Other control parameters can be
specified as σ 1 = σ 2 = σ a1 = σ a2 = 0.01, ω 1 = ω 2 = 0.1, and μ 1 = μ 2 = 0.01.
−x
The HONN sigmoidal functions are chosen as σ(x) = 2/(1 + e ),and a
systematic online tuning approach leads to the final choice of NN with
8 neurons, i.e. L 1 = L 2 = 8. (A further increase in the number of nodes
cannot significantly help improve the performance and overlearning effects
might be observed.) Simulation results are depicted in Fig. 9.2 with the ini-
ˆ
tial condition θ 1 (0) = θ 2 (0) = 0, ˆε 1 (0) =ˆε 2 (0) = 0, and x 1 (0) = 1,x 2 (0) = 0.
ˆ
It is shown that a good tracking performance can be achieved as depicted
in Fig. 9.2A. The control signal evaluation is provided in Fig. 9.2B, the
adaptive parameters and NN weights profiles are illustrated in Fig. 9.2C
and Fig. 9.2D, respectively. As it can be seen, all signals in the closed-loop
are bounded. Moreover, the adaptive parameters θ 1 (t),θ 2 (t), ˆε 1 (t),and ˆε 2 (t)
ˆ
ˆ
are all positive as guaranteed by the proposed adaptive laws. It is noticed
that the dead-zone characteristic parameters and the information on the
upper bounds of h i (·), g i (·) are not used in the control implementation.

9.5 CONCLUSION

An adaptive neural dynamic surface control is presented for a class of gen-
eral non-linear time-delay systems with an unknown non-linear dead-zone
input. The difficulty from the non-linear dead-zone is handled by repre-
senting it as a time-varying system with a bounded disturbance and then
constructing the adaptive control without using dead-zone characteristic
parameters and inverse model compensation. The proposed backstepping
design incorporates the dynamic surface control, eliminates the problem
of “explosion of complexity” in the conventional backstepping synthe-