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Adaptive Neural-Fuzzy Control of Mobile Robots 241

T n r ×(n i ×n r )
ˆ
and S =[ˆs , ˆs , ... , ˆs ] ∈ R with
σ 1σ 2σ n r σ

∂s i
(n i ×n r )×1
ˆ s = ∈ R ,
iσ
i = 1, ... , n r
∂σ c=ˆc,σ=ˆσ
and the residual term d u is bounded by
T T
∗ ˆ ˆ ∗ ˆ ˆ ∗ ˆ
|d u |≤ c · S W + σ · S W + W · S ˆc
c
c
σ

∗ ˆ ∗
+ W · S ˆσ + W 1 (6.18)
σ
Proof The Taylor series expansion of S(x, c , σ ) with respect to (x, ˆc, ˆσ)
∗
∗
can be expressed as

∗
∗
S(x, c , σ ) = S(x, ˆc, ˆσ) − S ˜c − S ˜σ + O(x, ˜c, ˜σ) (6.19)
ˆ
ˆ
σ
c
where O(x, ˜c, ˜σ) denotes the sum of the high order terms in the Taylor series
expansion.
Using (6.19), we obtain
T
T
∗
∗
∗
W S(x, ˆc, ˆσ) − W S(x, c , σ )
ˆ
T

∗ T
∗
ˆ
˜
ˆ
= (W + W ) S(x, ˆc, ˆσ) − W [S(x, ˆc, ˆσ) − S ˜c − S ˜σ + O(x, ˜c, ˜σ)]
c
σ
T

T

˜ ˆ
T ˆ
T ˆ
∗
˜
ˆ
= W S + (W − W) S ˜c + (W − W) S ˜σ − W O(x, ˜c, ˜σ)
ˆ
˜
c σ

T
T
T
T
T
∗
∗
ˆ ˆ
˜ ˆ
˜ ˆ
˜ ˆ
ˆ ˆ
= W S + W S ˜c − W S (ˆc − c ) + W S ˜σ − W S (ˆσ − σ )
c
c
σ
σ
T
∗
− W O(x, ˜c, ˜σ)

˜ T ˆ ˆ ˆ ˆ T ˆ ˆ
= W (S − S ˆc − S ˆσ) + W (S ˜c + S ˜σ) + d u (6.20)
c
c
σ
σ
where the residual term d u is given by
∗ ∗ ∗ T
ˆ
T ˆ
d u = W (S c + S σ ) − W O(x, ˜c, ˜σ)
˜
σ
c
∗
∗
∗
Noting that W = W − W , ˜c =ˆc − c , and ˜σ =ˆσ − σ , Equation (6.20)
ˆ
˜
implies that
T

T
∗
∗
ˆ
ˆ
ˆ ˆ
d u = W S − W S − (W − W ) (S − S ˆc − S ˆσ)
∗ T ˆ
ˆ
c
σ
∗ ∗
ˆ
T ˆ
ˆ
− W [S (ˆc − c ) + S (ˆσ − σ )]
c
σ
T
T
T

T
T
∗
∗
∗
ˆ ˆ
∗ ˆ
∗ ˆ
∗ ˆ
ˆ ˆ
= W S c + W S σ − W S ˆc − W S ˆσ + W (S − S )
c
c
σ
σ

∗
∗
∗
with S = S(x, c , σ ).
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 241 — #13