Page 169 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Strict-Feedback Systems With Non-linear Dead-Zone 165

m i 1 2 2 2 ˆ ε i |z i |σ i1
˙
˙ V i ≤− k i − − z + c i2g z +[g i (¯x i )N(ξ i ) + 1]ξ i +
i i1 i+1
2c i1 4c i2 ˆ ε i |z i |+ σ i1
σ i2 θ ˜ 2 1 σ i2 θ i ∗2 σ i3 ˜ε 2 i σ i3 ε ∗2
i
− + − + − V di
2 2 2 2
m i e à im
c i1 2 z i 2
+ 1 − 2tanh ( ) k (¯x i )
ij
2 ω i j=1 1 −¯τ i
2
≤− γ iV i + ϑ i + c i2g z 2 +[g i (¯x i )N(ξ i ) + 1]ξ i
˙
i1 i+1

m i e à im
c i1 2 z i 2
+ 1 − 2tanh ( ) k ((¯x i ) (10.35)
ij
2 ω i j=1 1 −¯τ i

where γ i and ϑ i are positive constants given as γ i = min 2(k i − m i /2c i1 −
∗2 ∗2
1/4c i2 ), i σ i2 , ai σ i3 , , ϑ i = σ i1 + σ i2 θ /2 + σ i3 ε /2. The last term of
i i
(10.35) is bounded as k ij (¯x i ) is bounded on compact set C i and −1 ≤
2
2
1 − 2tanh (z i /ω i ) ≤ 1 holds. The term c i2g z 2 is bounded provided z i+1
i1 i+1
is bounded. Then for small ϑ i, c i1, c i2,orlarge γ i,the errors z i , ˜ , ˜ε i are
θ i
bounded. These analyses and claims can be conducted for each subsystem i
(2 ≤ i < n).
Step n. This is the last step to determine the real control v. Consider
z n = x n − α n−1,wehave

˙ z n =˙x n −¨α n−1 = f n (x) + g n (x) d(t)v(t) + ρ(t) + h n (t,x(t − τ n (t))) −¨α n−1
(10.36)
where ˙α n−1 can be represented as a function of x,∂α n−1 /∂x 1 ,··· ,∂α n−1 /
∂α n−1 ˙ ∂α n−1
∂x n−1 ,φ n−1 as stated in [11], where φ n−1 = ξ n−1 + ˙ y d +
∂ξ n−1 ∂y d
n−1 ∂α n−1 ˙ ˆ n−1 ∂α n−1 ˙ ˆ ε k is computable.
k=1 θ k + k=1 ∂ ˆε k
∂ ˆ θ k
Then the real control is proposed as
2
ˆ
θ nsgn(z n ) T ˆ ε z n
n
v = N(ξ n ) k nz n + (Z n ) n (Z n ) + (10.37)
n
2η 2 n ˆ ε n |z n |+ σ n1
2 2
ˆ
θ n |z n | T ˆ ε z
n n
2
˙
ξ n = k nz + (Z n ) n (Z n ) + (10.38)
n
n
2η 2 ˆ ε n |z n |+ σ n1
n

|z n |
˙ T ˆ
θ n (Z n ) n (Z n ) − σ n2 θ n (10.39)
ˆ = n
n
2η 2
n
˙
ˆ ε n = an |z n | − σ n3 ˆε n (10.40)
where n > 0, an > 0, k n > 0,η n > 0and σ n1 ,σ n2 ,σ n3 > 0 are design pa-
rameters.