Page 134 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

dynamics, we further simulate the following dead-zone input
⎧
⎪ (1 − 0.3sin(v))((v − 0.5) v(t) ≥ 0.5
⎨
u(t) = DZ(v(t)) = 0 −0.25 < v(t)< 0.5
⎪
(0.8 − 0.2cos(v))(v + 0.25)
⎩ v(t) ≤−0.25
(8.48)
Moreover, in order to compare the tracking performance of the pro-
posed scheme, four different control schemes, including adaptive ro-
bust finite-time neural control (ARFTNC), NN-based terminal sliding
mode control (NNTSMC) [25], NN-based linear sliding mode control
(NNLSMC) [29] and PID control are performed in the experiments. It
should be noted that in NNTSMC and NNLSMC, neural network is
employed to approximate the unknown non-linearities, while the compen-
sation for the dead-zone dynamics is not considered. For fair comparison,
the initial states of the system and NN parameters are set the same, i.e.,
(x(0), ˙x(0)) = (0,0), = 0.05, a = 2, b = 10, c = 1, and d =−10. The de-
tails of the four controllers are given as:

1) Adaptive Robust Finite-Time Neural Control (ARFTNC)
In the proposed control scheme, the fast terminal sliding manifold is
selected as (8.20), where the parameters are set as γ = 9/11, λ 1 = 5, and
λ 2 = 1. The designed controller is given by (8.24), and the parameters are
set as k 1 = 0.5, k 2 = 0.1, r = 9/11, δ 1 = δ 2 = 0.01, and ζ = 0.001.
2) NN-Based Terminal Sliding Mode Control (NNTSMC)
In this scheme, the terminal sliding manifold is defined as (8.19), with
γ = 9/11, λ 0 = 6. The controller is addressed as
T
r
v(t) = ˆ W φ(X) + k 0 |s| sgn(s) + (δ 1 + δ 2 )sgn(s) (8.49)
where k 0 = 0.6, r = 9/11, δ 1 = δ 2 = 0.01, and ζ = 0.001.
3) NN-Based Linear Sliding Mode Control (NNLSMC)
In this scheme, the linear sliding manifold is chosen as (8.9), with λ 0 = 6.
The controller is expressed as
T
v(t) = ˆ W φ(X) + k 0s + (δ 1 + δ 2 )sgn(s) (8.50)
where k 0 = 0.6, δ 1 = δ 2 = 0.01, and ζ = 0.001.
4) PID Control
The tested PID controller was the same as those given in Chapter 4,
where the parameters are determined by using a heuristic tuning approach
for a given reference signal.