Page 39 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Substitute (2.37)–(2.41)into(2.43), and we have

2
˙
˙ V =−ks + σ 0 ˜z 0s − σ 1 |ω| ˜ z 1s + 1 σ 0 ˜z 0 ˆ z 0 − ω − |ω| z 0 + 1 σ 1 ˜z 1
g(ω) k 0 g(ω) k 1

˙ |ω| γ σ 0 2 γ σ 1 2 γ ζ ˜ 2 γ J 2 γ T ˜ 2
˜
ˆ z 1 − ω − z 1 − ˜ σ − ˜ σ − ζ − J − T
g(ω) k σ 0 0 k σ 1 1 k ζ k J k T d
2 σ 0 |ω| 2 σ 1 |ω| 2 γ σ 0 2 γ σ 1 2 γ ζ ˜ 2 γ J 2 γ T ˜ 2
˜
=−ks − ˜ z − ˜ z − ˜ σ − ˜ σ − ζ − J − T
k 0 g(ω) 0 k 1 g(ω) 1 0 k σ 1 1 k ζ k J k T d
k σ 0
1 ˙ |ω|
+ σ 0 ˜z 0 ˆ z 0 − ω − ˆ z 0 − k 0s
k 0 g(ω)

1 ˙ |ω| |ω|
+ σ 1 ˜z 1 ˆ z 1 − ω − ˆ z 1 + k 1 s (2.44)
k 1 g(ω) g(ω)
Then, substituting (2.28)into(2.44)yields
σ 0 |ω| σ 1 |ω| γ σ 0 γ σ 1 γ ζ γ J 2
2
2
2
2
2
2
˙ V =−ks − ˜ z − ˜ z − ˜ σ − ˜ σ − ζ ˜ − ˜ J
k 0 g(ω) 0 k 1 g(ω) 1 k σ 0 0 k σ 1 1 k ζ k J
γ T
˜ 2
− T ≤ 0 (2.45)
d
k T
From (2.45), we can obtain
˙ V ≤−ks 2 (2.46)
Integrating both sides of (2.46)from0to t,wehave
t
2
V(t) − V(0) ≤−k s dτ (2.47)
0
such that
t
2
V(t) + k s dτ ≤ V(0) (2.48)
0
t 2
Consider the facts k s dτ ≥ 0, and V(t) ≥ 0from (2.42), then one can
0
verify that
t
2
k s dτ ≤ V(0) (2.49)
0
and
V(t) ≤ V(0) (2.50)
t 2
From (2.49)and (2.50), we can conclude that k s dτ and V(t) are
0
bounded, which further implies that s, ˜z 0, ˜z 1, ˜σ 0, ˜σ 1, ζ are bounded.
˜