Page 150 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

σ ng n0 c n1 2 z n m n e à nm 2
˜ ˆ
+ θ n θ n + 1 − 2tanh ( ) ϕ (x) − V dn
nj
2 2 ω n j=1 1 −¯τ n
z n
∗
− (¯ε − g n0 ˆε n )z n tanh( ∗ (9.40)
n ) + σ an (¯ε − g n0 ˆε n )ˆε n
n
ω n
c n1 2 z n m n e à nm 2 e n−1
where Q n (Z n ) = f n (x,0) + tanh ( ) ϕ (x) + is an un-
nj
z n ω n j=1 1−¯τ n μ n−1
∈ R n+2 ,
known but well-defined functions of vector Z n =[x,z n ,e n−1 ]∈ Z n
and thus estimated by a HONN as Q n (Z n ) = W ∗T (Z n )+ε n over the com-
n
pact set .
Consider Assumption 9.1 and the fact θ n (t) ≥ 0,t ≥ 0, ˆε n (t) ≥ 0,t ≥ 0
ˆ
for any non-negative initial conditions, it is readily derived
g n0 θ nz 2
ˆ
2 n T z n
g ndz nv ≤−k ng n0 z − n − g n0 ˆε nz n tanh( ), (9.41)
n
n
2 ω n
∗ 2
g n0 θ z 1
n n
T
∗
z nQ n (Z n ) + g n1p|z n | ≤ (Z n ) n (Z n ) + + |z n | ¯ε . (9.42)
n
n
2 2g n0
Moreover, similar to (9.20)–(9.22), itcanalsobeverified that
σ ng n0 σ ng n0 θ ˜ 2 σ ng n0 θ n ∗2
n
˜ ˆ
θ n θ n ≤− + , (9.43)
2 4 4
σ ang n0 ˆε 2 σ an σ an ¯ε ∗2
∗ n ∗ 2 n
σ an (¯ε − g n0 ˆε n )ˆε n =− − (¯ε − g n0 ˆε n ) + , (9.44)
n
n
2 2g n0 2g n0
z n
∗
∗
∗
¯ ε |z n | −¯ε z n tanh( ) ≤ 0.2785ω n ¯ε . (9.45)
n n n
ω n
Therefore, one can rewrite (9.40)as
˜ 2 2
m n 2 σ ng n0 θ n σ an 2 σ ang n0 ˆε n
∗
˙ V n ≤− k ng n0 − z − − (¯ε − g n0 ˆε n ) −
n
n
2c n1 4 2g n0 2
1 σ ng n0 θ n ∗2 σ an ¯ε ∗2
n
∗
+ + + + 0.2875¯ε ω n
n
2g n0 4 2g n0
m n

c n1 2 z n e à nm 2
+ 1 − 2tanh ( ) ϕ (x) − V dn
nj
τ
2 ω n 1 −¯ n
j=1
m n
à nm
c n1 2 z n e 2
≤− γ nV n + ϑ n + 1 − 2tanh ( ) ϕ (x) − V dn
nj
2 ω n 1 −¯ n
τ
j=1
(9.46)
where γ n and ϑ n are positive constants given as

γ n = min 2(g n0k n − m n /2c n1 ),
n σ n /2,
an σ an , ,