Page 165 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Strict-Feedback Systems With Non-linear Dead-Zone 161

2 2
ˆ
k 1z 2 1 θ 1 |z 1 | T ˆ ε z z 1e ˙μ
1 1
˙
ξ 1 = + (Z 1 ) 1 (Z 1 ) + − (10.14)
1
r 2η 2 ˆ ε 1 |z 1 |+ σ 11 μ
1

˙ ˆ |z 1 | T ˆ
θ 1 = r 1 2 (Z 1 ) 1 (Z 1 ) − σ 12 θ 1 (10.15)
1
2η
1
˙
ˆ ε 1 = r a1 |z 1 | − σ 13 ˆε 1 (10.16)
where z 1 is the transformed error defined in (10.9), r can be calculated
based on e(t), μ(t),and 1 > 0, a1 > 0, k 1 > 0,η 1 > 0and σ 11 ,σ 12 ,σ 13 > 0
are design parameters. It should be noted that the fact ˆε 1 (t) ≥ 0,t ≥ 0 holds
for any initial conditions ˆε 1 (0) ≥ 0 based on (10.16). Thus the term ˆε 1 |z 1 |+
σ 11 in (10.13) is always positive (i.e., ˆε 1 |z 1 |+ σ 11 > 0), and there is no
singularity problem in the proposed control design.
Consider the following Lyapunov-Krasovskii function
m 1
1 2 c 11 e à 1m 1 t − (t−ς) 2 1 2 1 2
˜
V 1 = z + e k (x 1 (ς))dς + θ + ˜ ε 1
1
1j
1
τ
2 2 1 −¯ 1 t−τ 1j (t) 2 1 2 a1
j=1
(10.17)
, ¯ i are positive scalars
τ
where c 11 > 0, > 0 are positive constants, and τ im i
˜
ˆ
defined in Assumption 10.1,and ˜ε i = ε − ε i ,i = 1,···n−1and θ i = θ −θ i ,
∗
∗
i i
∗
i = 1,···n are parameter errors between the bounded constants ε = ε iN +
i
2
η /2, θ = W ∗T W , and their estimations ˆε i, θ i.
∗
∗
ˆ
i i i
i

Consider h ij (t, ¯x i ) ≤ k ij (¯x i ) with k ij (¯x i ) ≥ 0,thetimederivativeof V 1

along (10.13)–(10.16) can be given as
⎛ ⎞
m 1

˙ V 1 ≤z 1r f 1 (x 1 ) + g 1 (x 1 )(z 2 + α 1 ) + ⎠
h 1j (x 1 (t − τ 1j (t))) − e ˙μ/μ −¨y d
⎝
j=1
m 1 à 1m
c 11 e 2 2
+ k (x 1 (t)) − k (x 1 (t − τ 1j (t))) − V d1
1j
1j
τ
2 1 −¯ 1
j=1
1 ˙ 1
˙
˜ ˜
+ θ 1 θ 1 + ˜ ε 1 ˜ε 1
1 a1
m 1r 2 M 2 c 11 m 1 2
≤ z + h (x 1 (t − τ 1j (t)))
1j
1
2c 11 2 j=1

+ z 1r f 1 (x 1 ) + g 1 (x 1 )(z 2 + α 1 ) − e ˙μ/μ −¨y d

c 11 m 1 e à 1m 2 2
+ k (x 1 (t)) − k (x 1 (t − τ 1j (t))) − V d1
1j
1j
2 j=1 1 −¯τ 1

|z 1 | T
− θ 1r 2 (Z 1 ) 1 (Z 1 ) − σ 12 θ 1 −Èε 1r |z 1 | − σ 13 ˆε 1
ˆ
˜
1
2η
1