Page 132 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

where ·( denotes the Euclidean norm of a vector φ(X) =[φ 1 (X),··· ,
T
φ L (X)] .
According to the property of Frobenius norm, we can obtain

T
˜ W φ(X) F ≤ ˜ W F φ(X) . (8.37)
Select a Lyapunov function as

1 2
V 1 = s . (8.38)
2k 0 ϕ 0
Differentiating (8.38) and using (8.27), we have

1 T r
ˆ
˙ V 1 = s k 0d(t)[−θsφ (X)φ(X) − k 1s − k 2 |s| sgn(s) − δsgn(s)]
k 0 ϕ 0
+k 0 (W ∗T φ(X) + ε)
2 r+1
∗T
1
≤−k 1s − k 2 |s| + 1
W φ(X)
|s| + |s|ε − δ |s|
ϕ 0 F ϕ 0
(8.39)
1 1 ∗T
Since δ 1 ≥ ε N and δ 2 ≥ W φ(X) F ,Eq.(8.39) is reduced to
ϕ 0 ϕ 0
2 r+1
˙ V 1 ≤−k 1s − k 2 |s|
r+1
r+1
2
2
≤−2k 1k 0 ϕ 0 ( 1 1 s ) − k 2 (2k 0 ϕ 0 ) 2 ( 1 1 s ) 2 , (8.40)
2 k 0 ϕ 0 2 k 0 ϕ 0
=−k 1V 1 − k 2V 1 ¯ k 3
¯
¯
r+1
¯ ¯ ¯ r+1
where k 1 = 2k 1k 0 ϕ 0, k 2 = k 2 (2k 0 ϕ 0 ) 2 ,and k 3 = . Then, we can obtain
2
¯
¯
˙ V 1 + k 1V 1 + k 2V ¯ k 3 ≤ 0 (8.41)
1
According to Lemma 8.1, it can be concluded that the fast terminal
sliding manifold s can converge to zero within a finite time t 1 given by
¯
¯
1 k 1V 1 1− ¯ k 3 (t 0 ) + k 2
t 1 = ln . (8.42)
¯
¯
¯
k 1 (1 − k 3 ) k 2
From (8.42), we can see that the reaching time t 1 depends on the con-
stants k 0, k 1,and ϕ 0.
iii) Once the sliding mode surface s = 0 is achieved, it will remain on
it and the system (8.11) has the invariant properties. On the sliding surface
s = 0, we can conclude
γ
˙ e =−λ 1e − λ 2 |e| sgn(e). (8.43)