Page 131 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Finite-Time Neural Control of Servo Systems With Non-linear Dead-Zone 127

ii) The terminal sliding mode manifold s can converge to zero within a finite
1
time if the design parameters δ 1 and δ 2 are chosen to satisfy δ 1 ≥ ε N and
ϕ 0
1 ∗T
δ 2 ≥ W φ(X) F .
ϕ 0
iii) The tracking error e will converge to zero within a finite time.
Proof. i) Choose the following Lyapunov function:
1 2 1 2
V(t) = s + θ ˜ (8.33)
2k 0 ϕ 0 2γ
Differentiating (8.33) with respect to time t and using (8.27)and (8.25)
with Young’s inequality on the term sW φ(X) yields
∗
1 1 ˙
s
˜ ˜
˙ V(t) = s˙ + θθ
k 0 ϕ 0 γ
2 ˜ T 1 ˙ 1 2 r+1 1 ε 2
˜ ˆ
≤ s θφ (X)φ(X) − θθ − (k 1 − )s − k 2 |s| − δ|s|+ +
γ 2 4ϕ 0 2ϕ 2
0
(8.34)
Substituting (8.25)into(8.34)yields
1 2 r+1 σ(2λ−1) 2 1 ε 2 σð ∗2
˙ V(t) ≤−(k 1 − )s − k 2 |s| − δ|s|− θ ˜ + + 2 + θ
2 2λ 4ϕ 0 2ϕ 0 2
1 2 σ(2λ−1) 2 1 ε 2 σð ∗2
≤−(k 1 − )s − θ ˜ + + 2 + θ
2 2λ 4ϕ 0 2ϕ 0 2
≤−ηV + ξ
(8.35)
where η = min{(2k 1 − 1)k 0 ϕ 0 ,σ(2λ − 1)} is positive for k 1 > 1/2,λ > 1/2,
2
2
and ξ = 1/(4ϕ 0 ) + ε /(2ϕ ) + σðθ ∗2 /2 is a bounded constant.
0
Because the NN approximation is feasible in a compact set, the resulting
stability is true in semi-global sense. From (8.33)–(8.35), we can conclude
that both s and θ are semi-globally uniformly ultimately bounded. Consid-
˜
ering (8.11) and the boundedness of W , it can be obtained that e, ˙e,and
∗
ˆ W are also uniformly ultimately bounded, and thus the control input v(t) is
bounded according to (8.24). Since y d , ˙y d ,and ¨y d are bounded, the bound-
edness of x r and ˙x r is guaranteed by (8.12)and (8.16). From (8.27), it can
be concluded that ˙s is uniformly ultimately bounded due to the bounded-
ness of d(t). Therefore, all signals of the closed-loop system are uniformly
ultimately bounded in a compact set.
ii) From (8.26), we can conclude that the sigmoid function φ i (X) is
a a
bounded by 0 <φ i (X)< L 0, i = 1,··· ,L 1,with L 0 = max{| +d|,| +d|}.
b b+1
Then, φ(X) is bounded by

φ(X) ø L 0 L 1 (8.36)