Page 192 - Neural Network Modeling and Identification of Dynamical Systems

5.3 SEMIEMPIRICAL ANN-BASED MODEL DERIVATIVES COMPUTATION 183
Theorem 4. Let U be a compact subset of R ,let with the closure of its ε-neighborhood for some posi-
n u
n x
X be a compact subset of R , and let Y be a subset tive real ε.
of R . Assume that an unknown dynamical system Suppose that the estimates of error function val-
n y
has the form ues E(w) (5.7) as well as the values of its derivatives
¯
2
∂ ¯ E(w) (5.10), ∂ ¯ E(w) (5.11) with respect to parame-
∂w ∂w 2
dx(t) ters w are computed numerically, using the following
= f(x(t),u(t)),
dt procedure. The initial value problems for systems of
y(t) = g(x(t)), ODEs (5.2), (5.16), (5.17) are solved by an explicit
one-step method which has the order of accuracy r 1 if
where the vector-valued functions f: X × U → R n x the true solution has r 1 + 1 continuous time deriva-
and g: X → Y have r continuous derivatives with tives; time steps of the solver are selected in such a
respect to all of their arguments for some positive in- way that all the points of the control discontinuity
teger r. match the nodes. The target values of observable out-
Assume that we are given a finite set of con- puts ˜y (p) (t (p) ) are interpolated with respect to the
(p) P (p) (p) k
trol signals u , such that u :[0, ¯ t ]→ time variable using the method which has the or-
p=1
U are piecewise-defined vector-valued functions of der of accuracy r 2 if the true functions have r 2 + 1
time that have r continuous derivatives on each continuous time derivatives. Values of definite inte-
subdomain. Suppose that x (p) :[0, ¯ t (p) ]→ X and grals (5.8), (5.12), (5.13) are estimated by a method
y (p) :[0, ¯ t (p) ]→ Y represent states and outputs of which linearly depends on the integrand values at
an unknown dynamical system for the corresponding some nodes and has the order of accuracy r 3 if the
controls u (p) . integrand has r 3 continuous derivatives with respect
Let the finite data set of the form (5.1)begiven. to the variable of integration; the numerical integra-
Assume that this data set contains the exact val- tion step does not exceed t. Finally, suppose that the
ues for controls u (p) (t (p) ), initial states ˜ x (p) (0) = following inequalities hold: r 1 r, r 2 <r, r 3 r.
k
x (p) (0), and observable outputs ˜ y (p) (t (p) ) = Then there exists a maximal time step size t =
k t(ε) t, such that if time step sizes used by
y (p) (t (p) ) over the whole time segment t (p) = ¯ t (p) .
k K (p) numerical solvers for initial value problems (5.2),
Denote the maximum time step t = (5.16), (5.17) do not exceed t, then the estimates
(p) (p)
max max t k − t k−1 . of the error function and its derivatives with re-
p=1,...,P k=1,...,K (p) spect to the parameters have the asymptotic error
Let W be a compact subset of R . Suppose that min{r 1 −1,r 2 −1,r 3 }
n w
O( t ).
the parametric families of vector-valued functions
ˆ f: X × U × W → R n x and ˆ g: X × W → Y which Proof. Since the total error function (5.7)isa
define the semiempirical state space continuous time summation of errors for each individual trajec-
model of the form (5.2) have r + 2 continuous deriva- tory (5.8) over a finite set of trajectories, its esti-
tives with respect to all of their arguments, including mate has the same order of accuracy as the esti-
the parameters w ∈ W. mates for individual trajectory errors. The same
Let ˆ x (p) :[0, ¯ t (p) ]→ X and ˆy (p) :[0, ¯ t (p) ]→ Y argument applies to the total gradient (5.10)and
represent states and outputs of a semiempirical model the Hessian (5.11). Thus, we analyze the accu-
given initial conditions ˆ x (p) (0) = ˜ x (p) (0). Suppose racy of individual trajectory estimates and omit
that for the parameter values w, for each initial condi- the corresponding index for convenience.
tion ˜ x (p) (0) and control u (p) , the corresponding ini- Nodes used for the computation of definite
tial value problem solution ˆ x (p) exists on the whole integrals are denoted as τ m , m = 0,...,M (τ m ∈
time segment [0, ¯ t (p) ] and is contained in X along [0, ¯ t]). In order to evaluate the integrands, we