Page 172 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

on
, and then consider the Lyapunov function as
"
1 2 − (t−ς) 2
n n m i e à im t
V = V i = z + c i1 e k (¯x i (ς))dς
i
ij
i=1 2 i=1 j=1 1 −¯τ i t−τ ij (t)
#
1 2 1 2
+ θ ˜ + ˜ ε (10.48)
i i
i ai
Recalling the previous design procedure from Step 1 to Step n,itcanbe
obtained
m 1r M 1 2 n−1 m i 1 2 2
2
˙ V ≤− k 1 − − z − k i − − − c i−1,2g z
1 i−1,1 i
2c 11 4c 12 i=2 2c i1 4c i2

m n 2 2
− k n − − c n−1,2g ˙
n
n−1,1 z + r[g 1 (x 1 )N(ξ 1 ) + 1]ξ 1
2c n1
n−1 n σ i2 θ ˜ 2
˙
˙
+ [g i (x)N(ξ i ) + 1]ξ i +[g n (x)dN(ξ n ) + 1]ξ n − i
i=2 i=1 2
σ i3 ˜ε i σ i2 θ i σ i3 ε i
n 2 n ∗2 ∗2
− + α σ i1 + +
i=1 2 i=1 2 2g i0
$ m i e %
n à im n
c i1 2 z i 2
+ 1 − 2tanh ( ) k (¯x i ) − V di
ij
i=1 2 ω i j=1 1 −¯τ i i=1
n
˙
≤− γV + ϑ + α[G i (x)N(ξ i ) + 1]ξ i
i=1
$ m i e %
n à im
c i1 2 z i 2
+ 1 − 2tanh ( ) k (¯x i ) (10.49)
ij
i=1 2 ω i j=1 1 −¯τ i
$
2
with γ = min 2 k 1 − m 1r /2c 11 − 1/4c 12 ,2(k i − m i /2c i1 − 1/4c i2 −
M

c i−1,2g 2 ),i = 1,··· ,n − 1,2 k n − m n /2c n1 − c n−1,2g 2 , i σ i2 , ai σ i3 ,i =
i−1,1 n−1,1
%
i
n σ i2 θ i ∗2 σ i3 ε ∗2
1,··· ,n, , ϑ = α i=1 σ i1 + 2 + 2g i0 . If the control parameters
fulfill (10.47), the variables γ and ϑ are all positive. Moreover, consider the
2
fact 1 − 2tanh (z i /ω i ) ≤ 1 and the functions k ij (¯x i ) are bounded on any

compact set
, the last term of (10.49) is bounded by a positive constant
M i ≥ 0, which can be described as

m i e à im
2 z i 2
1 − 2tanh ( ) (10.50)
k (¯x i ) ≤ M i
ij

ω i
j=1 1 −¯τ i
γt
Multiplying both sides of (10.50)by e yields
γt
d(Ve ) n c i1M i γt n γt
˙
≤ ϑ + e + α[G i (x)N(ξ i ) + 1]ξ ie (10.51)
dt i=1 2 i=1