Page 147 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ANDSC of Strict-Feedback Systems With Non-linear Dead-Zone 143

Step i (2 ≤ i < n). Consider the definition z i = x i − s i−1,the filter(9.6),
and the filter error (9.7), it follows
s i = e i + α i , i = 1··· ,n − 1, (9.26)
e i
s ˙ i =− , i = 1··· ,n − 1. (9.27)
μ i
Therefore, from (9.4)and (9.27), it follows

s s
˙ z i =˙x i −¨ i−1 = f i (¯x i ,0) + h i (¯x i , ¯x i (t − τ ij (t))) + g i (¯x i , ¯x i (t − τ ij (t)))x i+1 −¨ i−1
s
= f i (¯x i ,0) + h i (¯x i , ¯x i (t − τ ij (t))) + g i (z i+1 + s i ) −¨ i−1
e i−1
= f i (¯x i ,0) + h i (¯x i , ¯x i (t − τ ij (t))) + g i (z i+1 + e i + α i ) +
μ i−1
(9.28)
The virtual control α i for the i-th subsystem can be specified as

ˆ
θ i T z i
α i =−k iz i − z i (Z i ) i (Z i ) − ε i tanh , (9.29)
i
2 ω i
˙
i 2 T
ˆ
ˆ
θ i = [z (Z i ) i (Z i ) − σ i θ i ], (9.30)
i
i
2
z i
ˆ ε i =
ai [z i tanh( ) − σ ai ˆε i ], (9.31)
˙
ω i
where k i > 0, ω i > 0,
i > 0,
ai > 0and σ i > 0, σ ai > 0 are design parame-
ters.
Select a Lyapunov-Krasovskii function as
m i
1 2 c i1 e à im t − (t−ς) 2 g i0 2
V i = z + e ϕ (¯x i (ς))dς + θ ˜ i
ij
i
2 2 1 −¯ i t−τ ij (t) 2
i
τ
j=1
1 2
∗
+ (ε − g i0 ˆε i ) (9.32)
i
2g i0
ai
where c i1 > 0 is a constant and g i0 is the lower bounds of g i (·), respectively.
Taking the time derivative of V i along (9.28)–(9.31)yields

m i 2 e i−1
˙ V i ≤ z + z i f i (¯x i ,0) + g i [z i+1 + e i + α i ]+
i
2c i1 μ i−1
m i à im
c i1 e 2 g i0 ˙ 1
∗
˙
˜ ˜
+ ϕ (¯x i ) − V di + θ i θ i − (ε − g i0 ˆε i )ˆε i
ij
i
2 1 −¯ i
i
ai
τ
j=1
m i 2 g i0 2 T
≤ z + z iQ i (Z i ) + g iz iz i+1 + g iz i α i + g iz ie i − ˜ z (Z i ) i (Z i )
i θ i i i
2c i1 2