Page 38 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Sliding Mode Control of Non-linear Servo Systems With LuGre Friction Model 29

Please refer to [21] for a similar proof of the above Lemma.
According to Lemma 2.1, the parameter updating laws for each un-
known parameter can be given as

˙
˜ σ
ˆ z
ˆ σ 0 =−k σ 0 0s − γ σ 0 0 (2.37)
|ω|
˙
˜ σ
ˆ σ 1 = k σ 1 ˆ z 1s − γ σ 1 1 (2.38)
g(ω)
˙
ˆ
ζ =−k ζ ωs − γ ζ ζ ˜ (2.39)
˙
˙
ˆ J =−k J δs − γ J J ˜ (2.40)
˙
˜
ˆ
T d =−k T s − γ T T d (2.41)
> 0, γ ζ > 0,
where k σ 0 > 0, k σ 1 > 0, k ζ > 0, k J > 0, k T > 0, γ σ 0 > 0, γ σ 1
γ J > 0, γ T > 0 are the constant tuning parameters involved in .
2.4.3 Stability Analysis
Theorem 2.1. Considering system (2.1) with friction model (2.3)–(2.5), ob-
server (2.28) and control (2.29) with adaptive law (2.36), then the closed-loop
system is asymptotically stable and the tracking error converges to zero.

Proof. Define the following Lyapunov function

1 2 1 2 1 2 1 2 1 2 1 2
V = Js + σ 0 ˜z + σ 1 ˜z + ˜ σ + ˜ σ + ζ ˜
1
1
0
0
2 2k 0 2k 1 2k σ 0 2k σ 1 2k ζ
1 1 2
˜ 2
+ T + ˜ J (2.42)
d
2k T d 2k J
Calculating the derivative of V,wehave
˙
˙
˙
˙
˜ ˜
˙ V = Js˙ + 1 σ 0 ˜z 0 ˜ z 0 + 1 σ 1 ˜z 1 ˜ z 1 + 1 ˜ σ 0 ˜σ 0 + 1 ˜ σ 1 ˜σ 1 + 1 ˙
s
ζζ
k 0 k 1 k σ 0 k σ 1 k ζ
˙ 1
1 ˜ ˜ ˜˜
+ T d T d + JJ
k T k J
d
2 |ω| 1 ˙ 1 ˙ 1 ˙
=−ks + σ 0 ˜z 0s − σ 1 ˜ z 1s + σ 0 ˜z 0 ˜ z 0 + σ 1 ˜z 1 ˜ z 1 +˜σ 0 (ˆz 0s + ˆ σ 0 )
g(ω) k 0 k 1 k σ 0
T d ) + ˜ J(δs + J)
˜
˜
ˆ
+˜σ 1 ( 1 ˙ ˆ σ 1 − |ω| ˆ z 1s) + ζ(ωs + 1 ˙ ζ) + T d (s + 1 ˙ ˆ 1 ˙ ˆ
d
k σ 1 g(ω) k ζ k T k J
(2.43)