Page 170 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Choose the Lyapunov function as
m n
1 c n1 à nmn t 1 1
e
2
2
k (¯x(ς))dς +
V n = z + e − (t−ς) 2 nj θ ˜ + ˜ ε n 2
n
n
2 2 1 −¯ n 2 n 2 an
τ
j=1 t−τ nj
(10.41)
2
where c n1 > 0and ˜ε n = ε − ε n with ε = ε nN + g n1p + η /2 being the upper
∗
∗
n
n
n
bounds of NN and dead-zone error, and g n1 > 0and p ≥ |ρ(t)| are the upper
bounds of g n (·) and the dead-zone slope, respectively. The time derivative
of V n along (10.36)–(10.41) can be derived as
1 1
˙
˙
˜ ˜
˙ V n ≤z n f n (x) + g n (x)(dv + ρ) + h n (t,x(t − τ n )) −¨α n−1 + θ n θ n + ˜ ε n ˜ε n
n an

c n1 m n e à nm 2 2
+ k (x(t)) − k (x(t − τ nj )) − V dn
nj
nj
2 j=1 1 −¯τ n
m n 2
≤ z + g n (x)z ndv + z nQ(Z n ) + g n1p|z n |
n
2c n1
m n e à nm

c n1 2 z n 2
+ 1 − 2tanh ( ) k (x) − V dn
nj
2 ω n j=1 1 −¯τ n
˜
θ n |z n | T
˜ ˆ
− (Z n ) n (Z n ) + σ n2 θ n θ n −Èε n |z n | + σ n3 ˜ε n ˆε n (10.42)
n
2η n 2
m n e à nmn 2
c n1 2 z n
where Q(Z n ) = f n (x) + tanh ( ) k (x) −¨α n−1 is an unknown
nj
z n ω n j=1 1−¯τ n
function approximated by a HONN with Z n =[x,z n ,∂α n−1 /∂x 1 ,··· ,
∂α n−1 /∂x n−1 ,φ n−1 ]∈ R 2n+1 .
The following inequalities can be verified
∗ 2
θ |z n | T η n
n
z nQ(Z n ) + g n1p|z n | ≤ (Z n ) n (Z n ) + ( + ε nN + g n1p)|z n |,
n
2η 2 2
n
(10.43)
σ n2 θ ˜ 2 σ n2 θ n ∗2
n
˜ ˆ
σ n2 θ n θ n ≤− + , (10.44)
2 2
σ n3 ˜ε n 2 σ n3 ε ∗2
n
σ n3 ˜ε n ˆε n ≤− + . (10.45)
2 2
Moreover, from (10.37)–(10.38), we have g n (x)dz nv = g n (x)dN(ξ n )ξ n,then
˙
it follows
˜ 2
m n 2 ˆ ε n |z n |σ n1 σ n2 θ n
˙
˙ V n ≤− k n − z +[g n (x)dN(ξ n ) + 1]ξ n + −
n
2c n1 ˆ ε n |z n |+ σ n1 2
σ n2 θ n ∗2 σ n3 ˜ε 2 n σ n3 ε ∗2
n
+ − + − V dn
2 2 2