Page 195 - Neural Network Modeling and Identification of Dynamical Systems

186 5. SEMIEMPIRICAL NEURAL NETWORK MODELS OF CONTROLLED DYNAMICAL SYSTEMS

2
∂ e(˜y(τ m ), ˆ x(τ m ,w),w) − e(˘y(τ m ), ˇ x(τ m ,w),w)

∂ ˆ x I
2
(M + 1) max ω e(˜y(τ m ), ˆ x(τ m ,w),w)

m
2 m=0,...,M
∂ e(˘y(τ m ), ˇ x(τ m ,w),w) min{r 1 ,r 2 }
− = O( t ).
∂ ˇ x 2 − e(˘y(τ m ), ˇ x(τ m ,w),w)
(5.32) = O( t min{r 1 ,r 2 }−1 ). (5.34)
Next, we analyze the errors of estimates for By the triangle inequality, from (5.33)and(5.34)
integrands in equations (5.8), (5.12), and (5.13). we obtain
Considering that all the terms are bounded and ¯ t

that the Frobenius norm is submultiplicative

and taking into account (5.28)and (5.32), we get e(˜y(t), ˆ x(t,w),w)dt

the same order of accuracy min{r 1 ,r 2 }. 0

Denote the maximum integration step as M
I

τ = max (τ m − τ m−1 ). Since the true inte- − ω e(˘y(τ m ), ˇ x(τ m ,w),w)
m
m=1,...,M m=0
grands have r continuous derivatives with re-
¯ t
spect to the variable of integration, and since

r 3 r, the estimates of definite integrals given e(˜y(t), ˆ x(t,w),w)dt

by the r 3 th-order numerical integration method 0
r 3
would have the global error of the form O( τ ) M
I
in case the true integrand values were used for − ω e(˜y(τ m ), ˆ x(τ m ,w),w)

m
the computations. According to the assump- m=0
tion of this theorem, the numerical integration
M

I
method linearly depends on the values of the in- + ω e(˜y(τ m ), ˆ x(τ m ,w),w)

m
tegrand. Denote the weights of this linear com- m=0
I
bination by ω . Thus, we have
m M

I
− ω e(˘y(τ m ), ˇ x(τ m ,w),w)
¯ t m

m=0

e(˜y(t), ˆ x(t,w),w)dt r 3 min{r 1 ,r 2 }−1
= O( τ ) + O( t )

0
min{r 1 −1,r 2 −1,r 3 }

M = O( t ). (5.35)
I r 3

− ω e(˜y(τ m ), ˆ x(τ m ,w),w) = O( τ ),
m
Using the same argument for definite inte-
m=0
(5.33) grals in (5.12)and (5.13), we obtain the final re-
sult for individual trajectories, i.e.,
and also min{r 1 −1,r 2 −1,r 3 }

ˇ
E(w) − E(w) = O( t ),

M

I
ω e(˜y(τ m ), ˆ x(τ m ,w),w) ∂E(w) ˇ

m − ∂E(w) min{r 1 −1,r 2 −1,r 3 } ),
= O( t

m=0 ∂w ∂w

M
2 2 ˇ
I ∂ E(w)
− ω e(˘y(τ m ), ˇ x(τ m ,w),w) − ∂ E(w) min{r 1 −1,r 2 −1,r 3 } ).
= O( t
m
2 2
m=0 ∂w ∂w
(5.36)
M

I
ω e(˜y(τ m ), ˆ x(τ m ,w),w)
m
m=0