Page 239 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Neural Dynamic Surface Control for Pure-Feedback Systems With Input Saturation 239

pressed as follows:

s ˙ 1 = s 2 + y 2 − k 1s 1
s ˙ 2 = s 3 + y 3 − k 2s 2 − s 1 − β 2
˙
.
.
.
(15.48)
˙
˙ s i = s i+1 + y i+1 − k is i − s i−1 − β i , i = 3,...n − 1
.
.
.
s ˙ n =−k ns n − s n−1 − ˜ W φ (¯x n ) + ε − ε N tanh(s n /δ).
T
b
Moreover, we can verify the fact that

s
˙
˙
˙ y 2 = β 2 − ¯z 2 =− y 2 − −k 1 ˙ 1 +¨y d − λ˙e
τ 2 (15.49)
y 2
=− + B 2 s 1 ,s 2 ,y 2 ,y d , ˙y d , ¨y d
τ 2

where B 2 s 1 ,s 2 ,y 2 ,y d , ˙y d , ¨y d =− −k 1 ˙s 1 +¨y d − λ˙e , which is a continuous
function.
Similarly, for i = 2,...n − 1, we have
y i+1 y i+1
s
s
˙ y i+1 =− + k i ˙ i −¨ i−1 =− + B i+1 s 1 ,...,s i+1 ,y 2 ,...y iy d , ˙y d , ¨y d .
τ i+1 τ i+1
(15.50)
Consider the Lyapunov function as
n−1
1
2 2 1 2 1 T −1 1 2
˜
˜
V = s + y i+1 + s + W W + ˜ ε . (15.51)
N
n
i
2 2b 2 2 ε
i=1
Then the derivative of the Lyapunov function can be obtained as
n−1
1 1

T −1 ˙ ˙
s
s
ˆ
˙ V = s i ˙ i + y i+1 ˙y i+1 + s n ˙ n + ˜ W W + ˜ ε N ˆε N
b ε
i=1
n−1
n y 2
=
−k is 2 i + s iy i+1 − i+1 + B i+1y i+1
τ i+1
i=1 i=1
1
T −1 ˙ ˙
ˆ
+ s n − ˜ Wφ (¯x n ) + ε − ε N tanh(s n /δ) + ˜ W W + ˜ ε N ˆε N
ε
n n−1 2
y i+1
2

≤ −k is + s iy i+1 − + B i+1y i+1 + s n ε N − ε N tanh(s n /δ)
i
τ i+1
i=1 i=1
1
T ˙
ˆ
− σ ˜ W W + ˜ ε N ˆε N
ε