Page 240 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

n n−1 y 2
≤
−k is 2 + s iy i+1 − i+1 + B i+1y i+1 + ε N (|s n |−s n tanh(s n /δ))
i
τ i+1
i=1 i=1
1
∗ T ˆ ˙
+ ε s n tanh(s n /δ) − s n ˆε N tanh(s n /δ) − σ ˜ W W +
N ˜ ε N ˆε N
ε
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
T
ˆ
+ ε N (|s n |−s n tanh(s n /δ)) − σ ˜ W W. (15.52)
By using the following property with respect to function tanh(·),we
have
x

0 ≤ |x| − xtanh ≤ 0.2785δ. (15.53)
δ
Using the fact
2

∗
˜
˜
−σ ˜ W W ≤−σ ˜ W T ˜ W + W ∗ ≤−σ W + σ W W
T ˆ

2 2

˜ σ ˜ σ 2
≤−σ W + W + W
2 2 N
2
σ
≤− W + W 2 (15.54)
˜

σ
2 2 N
and substituting (15.53)and (15.54)into(15.52), we can obtain
n n−1 y 2 σ
2
˙ V ≤
−k is 2 + s iy i+1 − i+1 + B i+1y i+1 + 0.2785ε N δ + W .
i N
τ i+1 2
i=1 i=1
(15.55)
1 2
2
Using the fact s + y ≥ s iy i+1,wehave
i
4 i+1
n n−1 1 y 2
2
˙ V ≤
−k is 2 + s + y 2 − i+1 + B i+1y i+1
i i 4 i+1
i=1 i=1 τ i+1
σ 2
+ 0.2785ε N δ + W . (15.56)
N
2
We can choose the parameters as k i = 1 + α 0 ,i = 1,...,n − 1; k n = α 0,
2
1
and 1 = + M i+1 + α 0,where α 0 and η are positive constants and |B i+1 | ≤
τ i+1 4 2η

2 2 2

M i+1. As pointed out in [1], the sets := y d , ˙y d , ¨y d : y +˙y +¨y ≤ B 0
d d d

2
1 2
T

−1 ˜
and := i−1
s + y 2 + s + ˜ W W + 1 ˜ ε 2 ≤ 2p, i = 2,...,n,for
i i i+1 b n ε N
j=1