Page 225 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

1) PID control: the classical PID control is given by u = k pe+k i edt +k d ˙e,
where the gains are set as k p = 20, k i = 0.05, and k d = 4.
2) Neural-network sliding mode control (SMC) [20]: which is given by

u 0 1 T
u =− =− −¨ y d + ˆ W φ(X) + μsgn(s 2 ) + α˙s 1 + k 1s 2 (14.35)
b 0 b 0
where the variables and parameters are set as s 1 = e, b 0 = 6, α = 2,
k 1 = 10, and μ = 0.1.
3) Neural-network non-singular terminal sliding mode control (NTSMC)
[21], which is given by

u 0 1 T 1 p/q
u =− =− −¨ y d + ˆ W φ(X) + μsgn(s 2 ) + α˙s 1 + |s 2 | sgn(s 2 )
b 0 b 0 β
(14.36)
where the variables and parameters are set as s 1 = e, α = 2, β = 0.2,
k 1 = 10, p = 5, q = 7, b 0 = 6, and μ = 0.1.
The proposed NTSMFC can be implemented by using NN parameters
are = 0.1, r 1 = 2, r 2 = 10, r 3 = 1, r 4 =−10. The parameters of funnel
boundary (14.12) are chosen as δ 0 = 100, δ ∞ = 0.3, and a 0 = 3. And the
control (14.21)isusedwith α = 2, β = 0.2, k 1 = 10, p = 5, q = 7, b 0 = 6,
and μ = 0.1.
For fair comparison, all control parameters are fixed for various refer-
ence signals. The initial states of the system are x 1 (0) = 0, x 2 (0) = 0. The
unknown system dynamics are select as f (x,t) = 0.2x 2sin(x 2 ), and the satu-
ration bound is set as v max = 1as [22]. In the following, a sinusoidal wave
y d = 0.5sin(t) is used as the reference signal. Comparative simulation re-
sults are shown in Fig. 14.2, where both the tracking performance and
the corresponding tracking errors are all provided. As shown in Fig. 14.2,
when tracking the sinusoidal wave, PID control has the largest overshoot
and more sluggish transient response. Moreover, we can also see that the
NTSMFC has the smallest tracking error and fastest convergence speed
among those four controllers; NTSMC has the largest overshoot at the be-
ginning of simulations, and PID scheme has a significant steady tracking
error. To study the control responses quantitatively, the following perfor-
mance indices are adopted:
t f
1) IAE = |e(t)|dt, which is the integrated absolute value of the error to
0
measure the intermediate tracking error result.
t f
2) ITAE = t|e(t)|dt, which is the integral of the time multiplied by the
0
absolute value of the error, and used to measure the tracking perfor-