Page 149 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ANDSC of Strict-Feedback Systems With Non-linear Dead-Zone 145

Step n. This is the last step to obtain v(t). Consider z n = x n − s n−1 and
s ˙ n−1 =− e n−1 , one may obtain
μ n−1
e n−1
s
˙ z n =˙x n −¨ n−1 = f n (x,0) + h n (x,x(t − τ nj (t))) + g n d(t)v(t) + ρ(t) +
μ n−1
(9.35)
The final control v is given as

ˆ
θ n T z n
v =−k nz n − z n (Z n ) n (Z n ) − ε n tanh( ), (9.36)
n
2 ω n
˙
n 2 T
ˆ
ˆ =
θ n [z (Z n ) n (Z n ) − σ n θ n ], (9.37)
n
n
2
z n
˙ ˆ ε n =
an [z n tanh( ) − σ an ˆε n ], (9.38)
ω n
where k n > 0, ω n > 0,
n > 0,
an > 0and σ n > 0, σ an > 0 are design param-
eters.
The following Lyapunov function is used:
m n t
1 2 c n1 e à nm − (t−ς) 2 g n0 2
V n = z + e ϕ (x(ς))dς + θ ˜ n
n
nj
τ
2 2 1 −¯ n 2
n
j=1 t−τ nj
1
∗
+ (¯ε − g n0 ˆε n ) 2 (9.39)
n
2g n0
an
where c n1 > 0 is a positive constant, g n0 is the lower bound of control func-
tion g n (·), = min(d l0 ,d r0 ) ≤ d(t) and p = (d l1 + d r1 )max{b r ,−b l }≥ |ρ(t)| are
the information on the bounds for the dead-zone dynamics, the scalar ˆε n is
∗
the estimation of the bounded constant ¯ε = ε + g n1p.
∗
n n
Thetimederivativeof V n along (9.35)–(9.38) can be given as

e n−1
˙ V n ≤z n f n (x,0) + g n [dv + ρ]+ h n (x,x(t − τ n (t))) +
μ n−1
m n
c n1 e à nm 2 2
+ ϕ (x(t)) − ϕ (x(t − τ nj ))
nj
nj
2 1 −¯ n
τ
j=1
c n1 m n e à nm t − (t−ς) 2 g n0 ˙
˜ ˜
− e ϕ (x(ς))dς + θ n θ n
nj
2 j=1 1 −¯τ n t−τ nj (t)
n
1
∗ ˙
− (¯ε − g n0 ˆε n )ˆε n
n

an
m n 2 g n0 2 T
≤ z + z nQ n (Z n ) + g ndz nv + g n1p|z n | − ˜ z (Z n ) n (Z n )
n θ n n n
2c n1 2