Page 193 - Neural Network Modeling and Identification of Dynamical Systems

184 5. SEMIEMPIRICAL NEURAL NETWORK MODELS OF CONTROLLED DYNAMICAL SYSTEMS

need the corresponding initial value problem so- solutions to the initial value problems given
lutions at time instants τ m ; therefore we require by the explicit one-step r 1 th-order method have
the numerical solver to visit these nodes. Only the global truncation error of the form O( t ).
r 1
the errors of initial value problem solution esti- Thus, we obtain the following inequalities for
mates at these nodes influence the total error of the discrete time approximations of these solu-
the definite integral estimates; hence the other tions:
time instants are not treated explicitly in this

analysis. ˆ x(τ m ,w) − ˇ x(τ m ,w) = O( t ),
r 1

Since the vector-valued functions u, f,and f ˆ
∂ ˆ x(τ m ,w) ∂ ˇ x(τ m ,w)
r 1
satisfy the conditions of the theorem on exis- − = O( t ),
∂w ∂w
tence and uniqueness of solutions to initial value
2 2
problems (see Theorem 54 in [16]), there ex- ∂ ˆ x i (τ m ,w) ∂ ˇ x i (τ m ,w)
r 1
− = O( t ),
ist unique solutions to the corresponding initial ∂w 2 ∂w 2
value problems: x and ˆ x. Moreover, according
i = 1,...,n x .
to the assumptions of this theorem, these solu-
(5.28)
tions exist on the whole segment [0, ¯ t] and are
contained in X. Since both the control u and the
Note that we assume all the matrix norms to be
evolution function f have r continuous deriva-
Frobenius norms.
tives, the state space trajectory x of unknown dy-
namical system has r + 1 continuous derivatives Hence, for all positive real ε there exists such
with respect to time. The observation function a small time step size t = t(ε) t that

g has r continuous derivatives, which implies ˆ x(τ m ,w) − ˇ x(τ m ,w) ε. According to the as-

that the output y also has r continuous deriva- sumption of this theorem, solutions ˆ x are con-
tives. Similarly, since the control has r continu- tained in the compact set X along with the clo-
ous derivatives and the evolution function f has sure of their ε-neighborhood; therefore the ap-
ˆ
r + 2 continuous derivatives, the state space tra- proximate solutions ˇ x are contained in X.Initial
jectory ˆ x of the semiempirical model has r + 1 conditions of the initial value problems (5.16)
continuous derivatives with respect to time. Two and (5.17) are identical to zero. Meanwhile, the
2
∂ ˆ x
∂ ˆ x
additional continuous derivatives of the vector- corresponding solutions ∂w and ∂w 2 are rep-
ˆ
valued function f also provide the r continuous resented by (at least) continuous vector-valued
derivatives for right hand sides of (5.16)and functions of time, defined on compact time seg-
(5.17); hence the corresponding sensitivities ∂ ˆ x ments. Therefore, the abovementioned solutions
∂w
2
and ∂ ˆ x have r + 1 continuous derivatives with are contained in some compact subsets of Eu-
∂w 2
respect to time. Also, the observation function ˆ g clidean space. The vector-valued function g is
has r + 2 continuous derivatives, which implies the continuous function of states; therefore val-
that e has r + 2 continuous derivatives. Thus, all ues of y are contained in some compact set. Ac-
the integrands in (5.8), (5.12), (5.13) also have r cording to the assumption of the absence of mea-
continuous derivatives. Since r is strictly posi- surement noise, we have ˜y(t) = y(t), i.e., the val-
tive, the integrands are at least continuous, and ues of ˜y belong to some compact set. The vector-
therefore the corresponding definite integrals do valued function ˆ g and its two derivatives with
exist. respect to parameters w are the continuous func-
Since the vector-valued functions ˆ x, ∂ ˆ x ,and
∂w tions of the states and the parameter values (note
2
∂ ˆ x that the parameters also belong to the compact
∂w 2 have r + 1 continuous derivatives with re-
spect to time, and since r 1 r, the numerical set W). Hence, the values of