Page 166 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

m 1r 2 M 2 rz 1e ˙μ
≤ z + z 1rQ(Z 1 ) + g 1 (x 1 )rz 1z 2 + g 1 (x 1 )z 1rα 1 −
1
2c 11 μ
r ˜ |z 1 | T
θ 1
˜ ˆ
− 2 (Z 1 ) 1 (Z 1 ) + σ 12rθ 1 θ 1 − r˜ε 1 |z 1 | + σ 13r˜ε 1 ˆε 1
1
2η
1

m 1 e à 1m
c 11 2 z 1 2
+ 1 − 2tanh k (x 1 ) − V d1 (10.18)
1j
2 ω 1 j=1 1 −¯τ 1
c 11 2 z 1 m 1 e à 1m 1 2
where Q(Z 1 ) = f 1 (x 1 ) + tanh ( ) k (x 1 ) −¨y d is an unknown
z 1 r ω 1 j=1 1−¯τ 1 1j
4
function with Z 1 =[x 1 ,z 1 , ˙y d ,r]∈ R . According to Lemma 9.1, Q(Z 1 ) is
well defined everywhere including the point z 1 = 0, thus it can be approx-
imated by using HONN without the singularity problem encountered in
[11–13].
Applying Young’s inequality, one can obtain the following inequalities:
∗
rθ |z 1 | T
∗T
1
z 1rQ(Z 1 ) =z 1rW 1 (Z 1 ) + z 1rε 1 ≤ (Z 1 ) 1 (Z 1 )
1 2 1
2η 1
η
2 (10.19)
+ r 1 + ε 1N |z 1 |
2
σ 12r M θ ˜ 2 σ 12r M θ 1 ∗2
1
˜ ˆ
σ 12rθ 1 θ 1 ≤− + (10.20)
2 2
σ 13r M ˜ε 2 1 σ 13r M ε ∗2
1
σ 13r˜ε 1 ˆε 1 ≤− + (10.21)
2 2
z 2
2
2
g 1 (x 1 )rz 1z 2 ≤ 1 + c 12r g z 2 (10.22)
M 11 2
4c 12
where θ = W ∗T W is a positive scalar, ε 1N is the upper bound of NN
∗
∗
1 1 1
approximation error, i.e., |ε 1 | ≤ ε 1N .
Moreover, it can be verified from (10.13)–(10.16)that g 1 (x 1 )rz 1 α 1 =
g 1 (x 1 )rN(ξ 1 )ξ 1 and ˆ (t), ˆε 1 (t) ≥ 0,t ≥ 0 hold for any initial conditions
˙
θ 1
ˆ
θ 1 (0), ˆε 1 (0) ≥ 0and0 ≤ ab ≤ a,∀a,b > 0. Then one can rewrite (10.18)
a+b
as
m 1r 2 M 2 z 2 1 2 2 2
˙
˙ V 1 ≤ z + + c 12r g z + g 1 (x 1 )rN(ξ 1 )ξ 1
M 11 2
1
2c 11 4c 12
ˆ
rz 1e ˙μ rθ 1 |z 1 | T
− + (Z 1 ) 1 (Z 1 )
1
μ 2η 2
1
σ 12r M θ ˜ 2 σ 12r M θ 1 ∗2 σ 13r M ˜ε 1 2 σ 13r M ε 1 ∗2
1
+ rˆε 1 |z 1 | − + − +
2 2 2 2

m 1 e à 1m
c 11 2 z 1 2
+ 1 − 2tanh ( ) k (x 1 ) − V d1
1j
2 ω 1 j=1 1 −¯τ 1