Page 253 - Autonomous Mobile Robots

Adaptive Neural-Fuzzy Control of Mobile Robots 239

For clarity, let us define the weight vector and fuzzy basis function vector
respectively as
T
]
W =[w 1 , w 2 , ... , w n r
T
]
S(x, c, σ) =[s 1 , s 2 , ... , s n r
n i n r n i T T T T

where s l = µ l(x i )/[ µ k(x i )], c =[c , c , ... , c ] , and
i=1 A i k=1 i=1 A i 1 2 n r
T T
T
T
σ =[σ , σ , ... , σ ] . Then, Equation (6.13) can be represented as
1 2 n r
T
y = W S(x, c, σ) (6.15)
Remark 6.2 For Equation (6.15), W and S(x, c, σ) are the weights and the
(normalized) basis functions in NN terminology, while they are the outputs of
the rules and the weighted firing strength in fuzzy logic terminology. Because
of the difficulty in deriving the rules in fuzzy systems for systems with little
physical insights, we would hereby like to present the adaptive laws to design
the outputs of the “rules” numerically using adaptive (NN) control techniques.
It has been proven that, if the number of the fuzzy rules n r is sufficiently
large, a fuzzy logic system (6.15) is capable of uniformly approximating any
given real continuous function, h(x), over a compact set x ⊂ R n i to any
arbitrary degree of accuracy in the form
T
∗
∗
∗
h(x) = W S(x, c , σ ) + (x), ∀x ∈ x ⊂ R n i (6.16)
∗
∗
where W , c , and σ ∗ are the ideal constant vectors, and (x) is the
∗
∗
∗
approximation error. The following assumption is made for W , c , σ ,
and (x).
∗
∗
∗
Assumption 6.2 The ideal NF vectors W ,c , σ , and the NF approximation
error are bounded over the compact set, that is,
∗
∗
∗
W ≤ w m , c ≤ c m , σ ≤ σ m , | (x)|≤ ∗
∀x ∈ x with w m ,c m , σ m , and being unknown positive constants.
∗
∗
∗
Remark 6.3 The optimal weight vector W ,c , and σ ∗ in (6.16) is an
∗
∗
“artificial” quantity required only for analytical purposes. Typically, W ,c ,
∗
n i
and σ are chosen as the value of W that minimizes (x) for all x ∈ x ⊂ R ,
that is,

T
∗
∗
∗
(W , c , σ ) := arg min sup |h(x) − W S(x, c, σ)|
W,c,σ
x∈ x
© 2006 by Taylor & Francis Group, LLC
FRANKL: “dk6033_c006” — 2006/3/31 — 16:42 — page 239 — #11