Page 194 - Neural Network Modeling and Identification of Dynamical Systems

5.3 SEMIEMPIRICAL ANN-BASED MODEL DERIVATIVES COMPUTATION 185
2
2
2
∂e ∂e ∂ e ∂ e ∂ e By the triangle inequality we obtain
e, , , , ,
∂w ∂ ˆ x ∂w 2 ∂w∂ ˆ x ∂ ˆ x 2
are contained in some compact set. ˜ y(τ m ) − ˆ g(ˆ x(τ m ,w),w)

The model outputs as well as their derivatives
− ˘y(τ m ) − ˆ g(ˇ x(τ m ,w),w)
depend on the values of model states; hence

their estimates are influenced by the global = ˜ y(τ m ) −˘y(τ m )
truncation error of numerical solutions to cor- + ˆ g(ˇ x(τ m ,w),w) − ˆ g(ˆ x(τ m ,w),w)

responding initial value problems. Since the

˜y(τ m ) −˘y(τ m )
vector-valued function ˆ g has r + 2 continuous

derivatives, it satisfies the Lipschitz condition + ˆ g(ˇ x(τ m ,w),w) − ˆ g(ˆ x(τ m ,w),w)
along with its two derivatives on the compact r 2 min{r 1 ,r 2 }
= O( t ) + O( t ) = O( t ), (5.31)
r 1
set X.Using (5.28), we obtain

ˆ g(ˆ x(τ m ,w),w) − ˆ g(ˇ x(τ m ,w),w)
thus, the error estimate has the order of accuracy
r 1
= O( t ), min{r 1 ,r 2 }.

∂ ˆ g(ˆ x(τ m ,w),w) ∂ ˆ g(ˇ x(τ m ,w),w) Using (5.29)and (5.31) and the fact that the
−
∂w ∂w Frobenius norm is submultiplicative and consid-

r 1
= O( t ), ering that all these quantities are bounded, we

∂ ˆ g(ˆ x(τ m ,w),w) ∂ ˆ g(ˇ x(τ m ,w),w) obtain
−
∂ ˆ x ∂ ˇ x

r 1
= O( t ),
e(˜y(τ m ), ˆ x(τ m ,w),w)
2 2
∂ ˆ g i (ˆ x(τ m ,w),w) ∂ ˆ g i (ˇ x(τ m ,w),w) min{r 1 ,r 2 }
− − e(˘y(τ m ), ˇ x(τ m ,w),w) = O( t ),

∂w ∂w
2 2
∂e(˜y(τ m ), ˆ x(τ m ,w),w)
r 1
= O( t ), i = 1,...,n x ,
∂w
2 2
∂ ˆ g i (ˆ x(τ m ,w),w) ∂ ˆ g i (ˇ x(τ m ,w),w)
− ∂e(˘y(τ m ), ˇ x(τ m ,w),w) min{r 1 ,r 2 }
∂w∂ ˆ x ∂w∂ ˇ x
− = O( t ),
∂w
r 1
= O( t ), i = 1,...,n x ,
∂e(˜y(τ m ), ˆ x(τ m ,w),w)
2 2
∂ ˆ g i (ˆ x(τ m ,w),w) ∂ ˆ g i (ˇ x(τ m ,w),w) ∂ ˆ x
−
∂ ˆ x ∂ ˇ x
2 2
∂e(˘y(τ m ), ˇ x(τ m ,w),w) min{r 1 ,r 2 }
− = O( t ),
= O( t ), i = 1,...,n x . ∂ ˇ x
r 1
(5.29) 2
∂ e(˜y(τ m ), ˆ x(τ m ,w),w)

∂w 2
According to assumptions on the interpola-
2
tion method for target values of observable out- ∂ e(˘y(τ m ), ˇ x(τ m ,w),w) min{r 1 ,r 2 }
− = O( t ),
puts with respect to the time variable, and due to ∂w 2
the fact that these vector-valued functions have 2
∂ e(˜y(τ m ), ˆ x(τ m ,w),w)
r continuous derivatives (r< r 2 ), the interpola- ∂w∂ ˆ x

tion error satisfies 2
∂ e(˘y(τ m ), ˇ x(τ m ,w),w) min{r 1 ,r 2 }
− = O( t ),
r 2 ∂w∂ ˇ x
˜y(t) −˘y(t) = O( t ), t ∈[0, ¯ t]. (5.30)