Page 145 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ANDSC of Strict-Feedback Systems With Non-linear Dead-Zone 141

Q 1 (Z 1 ) is well defined everywhere including the point z 1 = 0, thus it can
be approximated using HONN (9.5) without encountering the possible
singularity problem over a compact set (i.e., Q 1 (Z 1 ) = W ∗T (Z 1 ) + ε 1).
1
2
T
Since z (Z 1 ) 1 (Z 1 ) ≥ 0and z 1 tanh(z 1 /ω 1 ) ≥ 0 for all z 1 ∈ R, the
1 1
ˆ
fact θ 1 (t) ≥ 0,t ≥ 0, ˆε 1 (t) ≥ 0,t ≥ 0 holds for any bounded initial conditions
θ 1 (0) ≥ 0, ˆε 1 (0) ≥ 0 based on adaptive laws (9.10)–(9.11). Consequently,
ˆ
one can obtain the following inequalities:
ˆ θ 1 T
g 1z 1 α 1 = g 1z 1 −k 1z 1 − z 1 (Z 1 ) 1 (Z 1 ) − ε 1 tanh( z 1 )
2 1 ω 1 ,
ˆ
2 g 10 θ 1 2 T
≤ −g 10k 1z − z (Z 1 ) 1 (Z 1 ) − g 10 ˆε 1z 1 tanh( z 1 )
1 2 1 1 ω 1
(9.15)
z 1Q 1 (Z 1 ) = z 1W 1 ∗T 1 (Z 1 ) + z 1 ε 1
∗ 2
g 10 θ z T 1
1 1
∗
≤ (Z 1 ) 1 (Z 1 ) + + |z 1 |ε , (9.16)
1
1
2 2g 10
z 2 1 2 2
g 1z 1z 2 ≤ + c 12g z , (9.17)
11 2
4c 12
z 2 1 2 2
g 1z 1e 1 ≤ + c 13g e (9.18)
11 1
4c 13
where g 11 is the upper bound of the control function g 1 (·) and c 12 ,
c 13 > 0 are constants. Moreover, the time derivative of V w1 + V a1 along
(9.10)–(9.11) can be derived as
g 10 ˙ 1
˜ ˜
˙ V w1 + ˙ V a1 = θ 1 θ 1 − ∗ ˙
(ε − g 10 ˆε 1 )ˆε 1
1

1
a1
g 10 2 T σ 1g 10
˜ ˆ
˜ z (Z 1 ) 1 (Z 1 ) +
≤− θ 1 1 1 θ 1 θ 1 (9.19)
2 2
z 1
∗
∗
− (ε − g 10 ˆε 1 )z 1 tanh( ) + σ a1 (ε − g 10 ˆε 1 )ˆε 1
1
1
ω 1
It is easy to verify the following relations:
σ 1g 10 θ ˜ 2 σ 1g 10 θ ∗2
σ 1g 10 1 1
˜ ˆ
θ 1 θ 1 ≤− + , (9.20)
2 4 4
σ a1g 10 ˆε 2 σ a1 σ a1 ε ∗2
∗ 1 ∗ 2 1
σ a1 (ε − g 10 ˆε 1 )ˆε 1 =− − (ε − g 10 ˆε 1 ) + , (9.21)
1 1
2 2g 10 2g 10
z 1
∗
∗
∗
ε |z 1 | − ε z 1 tanh( ) ≤ 0.2785ω 1 ε . (9.22)
1 1 1
ω 1