Page 168 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Select a Lyapunov-Krasovskii function as

m i
1 2 c i1 e à im i t − (t−ς) 2 1 2 1 2
˜
V i = z + e k (¯x i (ς))dς + θ + ˜ ε i
i
ij
i
2 2 1 −¯ i t−τ ij (t) 2 i 2 ai
τ
j=1
(10.29)
where c i1 > 0and > 0 are design parameters.
Similar to Step 1, taking the time derivative of V i along (10.24)–(10.28)
yields
˜
m i 2 θ i |z i | T
˙ V i ≤ z + z iQ(Z i ) + g i (¯x i )z iz i+1 + g i (¯x i )z i α i − (Z i ) i (Z i )
i 2 i
2c i1 2η i
˜ ˆ
+ σ i2 θ i θ i −Èε i |z i | + σ i3 ˜ε i ˆε i − V di
m i
à im
c i1 2 z i e 2
+ 1 − 2tanh ( ) k (¯x i ) (10.30)
ij
2 ω i 1 −¯ i
τ
j=1
c i1 2 z i m i e à im 2
where Q(Z i ) = f i (¯x i ) + tanh ( ) k (¯x i ) −¨α i−1 with Z i =[¯x i ,z i ,
ij
z i ω i j=1 1−¯τ i
∂α i−1 /∂x 1 ,··· ,∂α i−1 /∂x i−1 ,φ i−1 ]∈ R 2i+1 is an unknown function approx-
imated by HONN. As stated in [11], [19], it should be emphasized that
i−1 ∂α i−1 ∂α i−1 ˙ ∂α i−1 ˆ
i−1 ∂α i−1 ˙
˙ α i−1 = ˙ x k + φ i−1 with φ i−1 = ξ i−1 + ˙ y d + θ k +
k=1 ∂x k ∂ξ i−1 ∂y d k=1 ∂ ˆ θ k

i−1 ∂α i−1 ˙ ˆ
ˆ ε k is a computable function of ¯x i−1 ,y d ,ξ i−1 , ˆε 1 ,··· , ˆε i−1 ,θ 1 ,
k=1 ∂ ˆε k
··· ,θ i−1.
ˆ
The following inequalities also hold
∗ 2
θ |z i | T η i
1
∗T
z iQ(Z i ) = z iW i (Z i ) + z i ε i ≤ (Z i ) i (Z i ) + + ε |z i |
i 2 i iN
2η i 2
(10.31)
σ i2 θ ˜ 2 σ i2 θ i ∗2
i
˜ ˆ
σ i2 θ i θ i ≤− + (10.32)
2 2
σ i3 ˜ε i 2 σ i3 ε ∗2
i
σ i3 ˜ε i ˆε i ≤− + (10.33)
2 2
z 2 i 2 2
g i (¯x i )z iz i+1 ≤ + c i2g z (10.34)
i1 i+1
4c i2
Substituting (10.31)–(10.34)into(10.30) and following similar analysis as
in Step 1, we have