Page 176 - Neural Network Modeling and Identification of Dynamical Systems

5.1 SEMIEMPIRICAL ANN-BASED APPROACH TO MODELING OF DYNAMICAL SYSTEMS 167
In the same manner, for any vector-valued func- from X to R , and let F be a set of vector-valued
ˆ
n y
tion g ∈ G and any positive real ε there exists a functions, everywhere dense in F. Also, let G be a
ˆ
vector-valued function ˆ g ∈ G such that subspace of Lipschitz continuous vector-valued func-
tions from R n y to R n y with the continuous inverse,

g(y 1 ,...,y m ) − ˆ g(y 1 ,...,y m ) <ε, and let G be a set of vector-valued functions, ev-

ˆ
∀y i ∈ Y i ,i = 1,...,m. erywhere dense in G. Then the set of vector-valued

ˆ
ˆ
ˆ
functions H = ˆ g(f(x)) | x ∈ X,f ∈ F, ˆ g ∈ G is ev-
ˆ
ˆ
According to the assumption of the theorem, all
functions g ∈ G satisfy the Lipschitz condition erywhere dense in F.
Proof. According to the assumption of this The-
g(y 1 ,...,y i ,...,y m ) − g(y 1 ,..., ˆy i ,...,y m )

orem, for any vector-valued function g ∈ G there

M i y i −ˆy i exists a continuous inverse g −1 . Hence, for any
continuous vector-valued function f ∈ F the
for some nonnegative real constant M i , referred composition g −1 ◦ f is also continuous. This im-
to as a Lipschitz constant. By applying the trian- ˆ
ˆ
gle inequality, we obtain plies that there exists f ∈ F such that

−1
ˆ
ˆ ˆ f(x) − g (f(x)) <ε 1 , ∀x ∈ X.
g(f 1 (x 1 ),...,f m (x m )) − ˆ g(f 1 (x 1 ),...,f m (x m ))

ˆ ˆ Since the vector-valued function g is Lipschitz
g(f 1 (x 1 ),...,f m (x m )) − g(f 1 (x 1 ),...,f m (x m ))
continuous, the following condition holds:

ˆ ˆ ˆ ˆ

+ ˆ g(f 1 (x 1 ),...,f m (x m )) − ˆ g(f 1 (x 1 ),...,f m (x m ))

−1
ˆ
ˆ
g(f(x))) − f(x) = g(f(x))) − g(g (f(x)))

g(f 1 (x 1 ),...,f m (x m ))

M f(x) − g (f(x)) <Mε 1 ,
ˆ −1
ˆ
− g(f 1 (x 1 ),f 2 (x 2 ),...,f m (x m ))
for some nonnegative real Lipschitz constant M.
ˆ

+ g(f 1 (x 1 ),f 2 (x 2 ),...,f m (x m ))
Moreover, there exists a vector-valued function

ˆ
ˆ ˆ ˆ g ∈ G such that
− g(f 1 (x 1 ),f 2 (x 2 ),f 3 (x 3 ),...,f m (x m ))

n y

ˆ g(y) − g(y) <ε 2 , ∀y ∈ R .
+ ···+

ˆ ˆ ˆ
+ g(f 1 (x 1 ),...,f m−1 (x m−1 ),f m (x m )) − g(f 1 (x 1 ), Finally, we apply the triangle inequality and ob-
tain
ˆ
...,f m (x m ))

ˆ

ˆ g(f(x))) − f(x)
ˆ ˆ ˆ ˆ
+ ˆ g(f 1 (x 1 ),...,f m (x m )) − ˆ g(f 1 (x 1 ),...,f m (x m ))
ˆ ˆ ˆ
m m ˆ g(f(x))) − g(f(x))) + g(f(x))) − f(x)

ˆ

< M i f i (x i ) − f i (x i ) + ε< M i ε i + ε.
<ε 2 + Mε 1 .
i=1 i=1
In the case of a dynamical system model-
Nevertheless, under certain conditions the
ing problem, theoretical knowledge of the plant
universal approximation property holds for
semiempirical models. may be used to select the meaningful state
variables (i.e., state variables interpretable in
n x
Theorem 2. Let X be a compact subset of R ,let domain-specific terms) in contrast to black box
F be a space of continuous vector-valued functions models, such as Elman recurrent networks with