Page 148 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

σ ig i0 z i
∗
˜ ˆ − (ε − g i0 ˆε i )z i tanh(
+ θ i θ i ∗ i ) + σ ai (ε − g i0 ˆε i )ˆε i
i
2 ω i
m i e à im

c i1 2 z i 2
+ 1 − 2tanh ( ) ϕ (¯x i ) − V di (9.33)
ij
2 ω i j=1 1 −¯τ i
τ
c i1 2 z i m i e im 2 e i−1
where Q i (Z i ) = f i (¯x i ,0) + tanh ( ) ϕ (¯x i ) + with Z i =
ij
z i ω i j=1 1−¯τ i μ i−1
[¯ x i ,z i ,e i−1 ]∈ R i+2 is an unknown but well-defined function and thus
approximated by a HONN (9.5)overacompactset as Q i (Z i ) =
W i ∗T (Z i ) + ε i.
Since θ i (t) ≥ 0,t ≥ 0, ˆε i (t) ≥ 0,t ≥ 0 for any non-negative initial con-
ˆ
ditions, one can apply the Young’s inequality on the terms g iz i α i , g iz iz i+1,
g iz ie i , z iQ i (Z i ) similar to (9.15)–(9.18) and derive the relations on the terms
θ i θ i , σ ai (ε − g i0 ˆε i )ˆε i, ε |z i | − ε z i tanh( ) similar to (9.20)–(9.22)in
σ i g i0 ˜ ˆ ∗ ∗ ∗ z i
2 i i i ω i
terms of positive constants g i1 and c i2 ,c i3. Then following a similar anal-
ysis as Step 1,ityields
˜ 2
m i 1 1 2 σ ig i0 θ i σ ai 2
∗
˙ V i ≤− g i0k i − − − z − − (ε − g i0 ˆε i )
i
i
2c i1 4c i2 4c i3 4 2g i0
σ aig i0 ˆε 2 i 2 2 2 2 σ ig i0 θ i ∗2 1 σ ai ε ∗2
i
− + c i2g z + c i3g e + + + + 0.2785ω i ε ∗ i
i1 i
i1 i+1
2 4 2g i0 2g i0
m i e à im
c i1 2 z i 2
+ 1 − 2tanh ( ) ϕ (¯x i ) − V di
ij
2 ω i j=1 1 −¯τ i
2
2 2
≤− γ iV i + ϑ i + c i2g z 2 + c i3g e
i1 i+1 i1 i
m i e ϕ ((¯x i )
à im 2
c i1 2 z i ij
+ 1 − 2tanh ( ) (9.34)
2 ω i j=1 1 −¯ i
τ
where γ i and ϑ i are positive constants as

γ i = min 2(g i0k i − m i /2c i1 − 1/4c i2 − 1/4c i3 ),
i σ i /2,
ai σ ai , ,
∗2
∗2
∗
ϑ i = σ ig i0 θ /4 + 1/2g i0 + σ ai ε /2g i0 + 0.2785ω i ε .
i i i
or
Similar to the discussion in Step 1, for both large or small z i (z i /∈ z i
), if z i+1 and e i can be guaranteed to be bounded (this will be
z i ∈ z i
guaranteed in the next step), then according to the extended Lyapunov
Theorem, z i , ˜ , ˜ε i are UUB for small enough ϑ i, c i1,orlarge γ i on a
θ i
compact set C i. This further ensures the boundedness of θ i, ˆε i,and α i.
ˆ