Page 178 - Neural Network Modeling and Identification of Dynamical Systems

5.1 SEMIEMPIRICAL ANN-BASED APPROACH TO MODELING OF DYNAMICAL SYSTEMS 169
where ˆ x:[0, ˜ t]→ R n x is a solution of the initial value that
problem for the system of ODEs with right hand
ˆu
ˆ
∗
∗
∗
ˆ
∗
∗
∗
side f (t, ˆ x(t)) ≡ f(ˆ x(t),u(t)) and initial condition f(x ,u ) − f(x ,u ) <ε f ∀x ∈ X ∀u ∈ U.
s
ˆ x(0) = ˜ x ; y(t) = g(x(t)); ˆy(t) = ˆ g(ˆ x(t)).
Next, we estimate the upper bound on the ap-
Proof. For any vector-valued functions f ∈ F and
proximation error for the right hand side of the
f ∈ F, the vector-valued functions f u and f ˆu
ˆ
ˆ
system of ODEs, evaluated at each point of the
satisfy all conditions of the theorem on exis- true state space trajectory x(t) on the whole time
tence and uniqueness of solutions to initial value
segment. We have
problems (see Theorem 54 in [16]), due to the fol-
lowing considerations:
t

u
ˆ u
• the vector-valued function u(t) is measur- sup f (s,x(s)) − f (s,x(s))ds

able, while vector-valued functions f(x,u) t∈[0,˜ t]
0
and f(ˆ x,u) are continuous with respect to t
ˆ
u
∗
u; therefore the compositions f (t,x ) and
ˆ u
u

sup f (s,x(s)) − f (s,x(s)) ds
ˆu ∗
f (t, ˆ x ) are measurable with respect to t for
t∈[0,˜ t]
all ˆ x ; 0
∗
• the vector-valued function u(t) is locally inte- ˜ t ˜ t
u ˆ u f
grable, while vector-valued functions f(x,u) = f (s,x(s)) − f (s,x(s)) ds < ε ds
ˆ
and f(ˆ x,u) are locally Lipschitz continuous 0 0
with respect to u; therefore the compositions f
= ˜ tε .
u
ˆu
∗
f (t,x ) and f (t, ˆ x ) are locally integrable
∗
with respect to t for all ˆ x ;
∗
According to the assumption of this theo-
ˆ
• vector-valued functions f(x,u) and f(ˆ x,u) are rem, the initial condition ˜ x is contained in the
s
continuous and locally Lipschitz continuous δ-neighborhood of x , i.e.,
s
with respect to x; therefore the vector-valued
ˆu
u
∗
functions f (t ,x) and f (t , ˆ x) are also con- s s
∗
x − ˜ x <δ.
tinuous and locally Lipschitz continuous with
respect to x for all t .
∗
Also, we have assumed that all vector-valued
ˆ
ˆ
ˆ
Thus, there exists the unique solution x of the functions f ∈ F and ˆ g ∈ G satisfy the Lipschitz
initial value problem for the system of ODEs condition, i.e.,
with right hand side f u and initial condition

s
x(0) = x ,aswellasthe unique solution ˆ x of ˆ ∗ ˆ ∗ f ∗

x − x , ∀u ∈ U,
f(x ,u ) − f(x ,u ) M
ˆu
the similar problem with right hand side f and
g

s

initial condition ˆ x(0) = ˜ x . According to the as- ˆ g(x ) − ˆ g(x ) M x − x , ∀x ,x ∈ X,
sumption of this theorem, the solution x exists
on the whole segment [0, ˜ t] and is contained in for some nonnegative real Lipschitz constants
f
g
X along with the closure of its ε-neighborhood. M and M . Let us also assume that
Since the set of vector-valued functions F is
ˆ
ε f
everywhere dense in the space F,for anyvector- δ = g e −M ¯ t ,
valued function f ∈ F and any positive real ε f 3M
f
f ε −M ¯ t
there exists a vector-valued function f ∈ F such ε = g e .
ˆ
ˆ
3M ¯ t