Page 144 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

z 1
˙ ˆ ε 1 =
a1 [z 1 tanh( ) − σ a1 ˆε 1 ], (9.11)
ω 1
where k 1 > 0, ω 1 > 0,
1 > 0,
a1 > 0and σ 1 > 0, σ a1 > 0 are design pa-
rameters.
Consider the following Lyapunov-Krasovskii function

V 1 = V c1 + V d1 + V w1 + V a1
1 2 m 1 e à 1m t − (t−ς) 2
= z + c 11 e ϕ (x 1 (ς))dς (9.12)
2 1 2 j=1 1−¯τ 1 t−τ 1j (t) 1j
g 10 ˜ 2 1 ∗ 2
θ +
+ 2
1 1 2g 10
a1 (ε − g 10 ˆε 1 )
1
where c 11 > 0, > 0 are positive parameters, g 10 is the lower bound of
g 1 (·). The scalars ˜ε i = ε − ε i and θ i = θ − θ i are parameter errors with
˜
∗
∗
ˆ
i
i
∗
∗
θ = W ∗T W and ε being bounded positive scalars of (9.5).
∗
i i i i
In view of the fact τ ij (t) ≤ τ im, ˙τ ij (t) ≤¯τ i < 1, then it follows

c 11 m 1 e à 1m 2 − à 1j (t) 2
˙ V d1 = ϕ (x 1 (t)) − (1 −¨τ 1j (t))e ϕ (x 1 (t − τ 1j (t)))
2 j=1 1−¯τ 1 1j 1j
c 11 m 1 e à 1m t − (t−ς) 2
− e ϕ (x 1 (ς))dς
2 j=1 1−¯τ 1 t−τ 1j (t) 1j
(9.13)
Consider Assumption 9.2 and (9.8) and use Young’s inequality, then the
time derivative of V c1 + V d1 can be given as

˙ V c1 + ˙ V d1 ≤ z 1 f 1 (x 1 ,0) + g 1 [z 2 + α 1 + e 1 ]+ h 1 (x 1 ,x 1 (t − τ 1j (t))) −¨y d
m 1 à 1m
c 11 e 2 2
+ ϕ (x 1 ) − ϕ (x 1 (t − τ 1j )) − V d1
1j
1j
2 1 −¯ 1
τ
j=1

m 1 2
≤ z + z 1 f 1 (x 1 ,0) −¨y d
1
2c 11
m 1 e à 1m
c 11 2 z 1 2
+ tanh ( ) ϕ (x 1 ) + g 1z 1z 2
1j
z 1 ω 1 j=1 1 −¯τ 1
m 1 e à 1m

c 11 2 z 1 2
+ g 1z 1 α 1 + g 1z 1e 1 + 1 − 2tanh ( ) ϕ (x 1 )
1j
2 ω 1 j=1 1 −¯τ 1
− V d1
m 1 2
=
z + z 1Q 1 (Z 1 ) + g 1z 1z 2 + g 1z 1 α 1 + g 1z 1e 1
1
2c 11

m 1 e à 1m
c 11 2 z 1 2
+ 1 − 2tanh ( ) ϕ (x 1 ) − V d1 (9.14)
1j
2 ω 1 j=1 1 −¯τ 1
c 11 2 z 1 m 1 e à 1m 2
where Q 1 (Z 1 ) = f 1 (x 1 ,0) −¨y d + tanh ( ) ϕ (x 1 ) is an un-
1j
z 1 ω 1 j=1 1−¯τ 1
3
known function of Z 1 =[x 1 ,z 1 , ˙y d ]∈ R . According to Lemma 9.1,