Page 167 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Strict-Feedback Systems With Non-linear Dead-Zone 163

m 1r M 1 2 2 2 2
2
≤− k 1 − − ˙
z + c 12r g z + r[g 1 (x 1 )N(ξ 1 ) + 1]ξ 1
1 M 11 2
2c 11 4c 12
rˆε 1 |z 1 |σ 11 σ 12r M θ ˜ 2 1 σ 13r M ˜ε 1 2 σ 12r M θ 1 ∗2 σ 13r M ε ∗2
1
+ − − + +
ˆ ε 1 |z 1 |+ σ 11 2 2 2 2
m 1 e à 1m

c 11 2 z 1 2
+ 1 − 2tanh ( ) k (x 1 ) − V d1
1j
2 ω 1 j=1 1 −¯τ 1
2 2 2
˙
≤− γ 1V 1 + ϑ 1 + c 12 γ g z + r[g 1 (x 1 )N(ξ 1 ) + 1]ξ 1
m 11 2
m
c 11 2 z 1 e ˜ ωà 1m 2
+ (1 − 2tanh ( )) k (x 1 ) (10.23)
1j
2 ω 1 1 −Èτ 1
j=1
where g i1 is the upper bounds of g i (¯x i ),and γ 1 and ϑ 1 are positive constants
2

γ 1 = min 2(k 1 − m 1r /2c 11 − 1/4c 12 ), 1r M σ 12 , a1r M σ 13 , , ϑ 1 = σ 11r M +
M
∗2
∗2
σ 12r M θ /2+σ 13r M ε /2. Since k ij (¯x i ) are bounded on arbitrarily large com-
1
1
2
pact set C i and the fact −1 ≤ 1 − 2tanh (z i /ω i ) ≤ 1 holds, the last term of
2
2
2
(10.23) is bounded. The term c 12r g z is also bounded as long as z 2 is
M 11 2
bounded (will be guaranteed in next step). Then for small ϑ 1, c 11, c 12,or
˜
large γ 1,the errors z 1, θ 1, ˜ε 1 can be proved to be bounded according to
Lyapunov’s Theorem and Lemma 10.2 [12].
Step i (2 ≤ i < n). Consider z i = x i − α i−1,then
˙ z i =˙x i −¨α i−1 = f i (¯x i ) + g i (¯x i )(z i+1 + α i ) + h i (t, ¯x i (t − τ i (t))) −¨α i−1
(10.24)
Then the following control laws are developed
2
ˆ
θ isgn(z i ) T ˆ ε z i
i
α i = N(ξ i ) k iz i + (Z i ) i (Z i ) + (10.25)
i
2η 2 ˆ ε i |z i |+ σ i1
i
2 2
2
i
T
˙
ξ i = k iz + θ i ˆ |z i | (Z i ) i (Z i ) + ˆ ε z i (10.26)
i 2 i
2η
i ˆ ε i |z i |+ σ i1

˙ |z i | T
ˆ
ˆ
θ i = i 2 (Z i ) i (Z i ) − σ i2 θ i (10.27)
i
2η
i
˙ (10.28)
ˆ ε i = ai |z i | − σ i3 ˆε i
where i > 0, ai > 0, k i > 0,η i > 0and σ i1 ,σ i2 ,σ i3 > 0 are design parame-
ters.