Page 254 - Autonomous Mobile Robots

240 Autonomous Mobile Robots

Remark 6.4 The approximation error (x), is a critical quantity and can be
reduced by increasing the number of the fuzzy rules n r . According to the univer-
sal approximation theorem, it can be made as small as possible if the number
of fuzzy rules n r is sufficiently large.

From the analysis given above, we see that the system uncertainties are
∗
∗
converted to the estimation of unknown parameters W , c , σ , and unknown
∗
bounds .
∗
∗
∗
∗
As the ideal vectors/constants W , c , σ , and are usually unknown,
∗
we use their estimates W, ˆc, ˆσ, and ˆ instead. The following lemma gives the
ˆ
T
ˆ T
∗
∗
∗
properties of the approximation errors W S(x, ˆc, ˆσ) − W S(x, c , σ ). The
0
definition of induced norm of matrices is given here first.
Definition 6.1 Foranm × n matrix A ={a ij }, the induced p-norm, p = 1, 2
of A is defined as
m

A 1 = max |a ij | column sum
j
i=1

T
A 2 = max λ i (A A)
i
Usually, A 2 is abbreviated to A .
The Frobenius norm is defined as the root of the sum of the squares of all
elements
2 2 T
A = a = tr(A A)
F ij
with tr(·) the matrix trace, that is, sum of diagonal elements.
Lemma 6.1 [34, 43] The approximation error can be expressed as
T
T
∗
∗
∗
ˆ
W S(x, ˆc, ˆσ) − W S(x, c , σ )

˜ T ˆ ˆ ˆ ˆ T ˆ ˆ
= W (S − S ˆc − S ˆσ) + W (S ˜c + S ˜σ) + d u (6.17)
c
σ
c
σ
∗
∗
˜
ˆ
where S = S(x, ˆc, ˆσ), W = W − W , ˜c =ˆc − c , and ˜σ =ˆσ − σ are defined
ˆ
∗
T n r ×(n i ×n r )
as approximation error, and S =[ˆs , ˆs , ... , ˆs n r c ] ∈ R with
ˆ
c
1c
2c

∂s i
(n i ×n r )×1
ˆ s =
∈ R , i = 1, ... , n r
ic
∂c c=ˆc,σ=ˆσ
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 240 — #12