Page 241 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Neural Dynamic Surface Control for Pure-Feedback Systems With Input Saturation 241

any B 0 > 0and p > 0, are compact. Hence, × i is also compact. There-

fore, |B i+1 | has a maximum M i+1 on × . Then, we know that for any
i
positive number η, such that
y 2 B 2 η
i+1 i+1
+ ≥ B i+1y i+1 (15.57)

2η 2
Consequently, we can obtain
n−1 2 2 2 2
B
y
n
2 1 2 1 M i+1 2 M i+1 i+1 i+1 η
˙ V ≤ −α 0s + y − + + α 0 y + +
i 4 i+1 4 2η i+1 2 2
i=1 i=1 2ηM i+1
σ
+ 0.2785ε δ + W N 2
∗
N
2
n−1 B 2 M 2 y 2
n
2 2 σ 2
≤ −α 0s + −α 0y − 1− i+1 i+1 i+1 + 0.2785ε N δ + W
i i+1 M 2 2η 2 N
i=1 i=1 i+1
n
2 σ 2
≤ −α 0s + 0.2785ε N δ + W . (15.58)
i 2 N
i=1
Hence, we can conclude ˙ V ≤ 0if

2
0.2785ε N δ + σW /2
N
|s i | ≥ (15.59)
α 0
Then, one can claim that the ultimate boundedness of s i will converge
to a small invariant set

2
0.2785ε N δ + σW /2
N
= {|s i | ≤ γ s } for γ s = . (15.60)
α 0
From (15.23), the error dynamics are given by ˙e+λe =˙s 1, which further
implies the boundedness and convergence of the tracking error e as [17].
This finishes the proof.
15.4 SIMULATIONS
In order to show the efficacy of the proposed control, and its superior track-
ing performance, we consider three different control methods: S1) neural
dynamic sliding mode control with saturation compensation (the proposed
method); S2) neural dynamic sliding mode control without saturation com-
pensation (the NN used in S1) is turned off ); S3) neural dynamic surface
control without saturation compensation [18].
Then, the tracking performances of these three control schemes are pro-
vided for a spring mass and damper system (as shown in Fig. 15.1 and [19]),