Page 179 - Neural Network Modeling and Identification of Dynamical Systems

170 5. SEMIEMPIRICAL NEURAL NETWORK MODELS OF CONTROLLED DYNAMICAL SYSTEMS

Note that since ˜ t< ¯ t, the following inequalities by various numerical methods; consequently,
hold: theoretical domain-specific knowledge as well
ε −M ¯ t ε −M ˜ t as the knowledge of the properties of these nu-
f
f
e < e , merical methods provides useful insights as to
3M g 3M g
ε −M ¯ t ε −M ˜ t which method should be applied to each spe-
f
f
e < e .
g
g
3M ¯ t 3M ˜ t cific problem. Meanwhile, this knowledge is
completely ignored by purely empirical discrete
Hence, the well-posedness theorem for initial
time state space models of the form (2.13), be-
value problems (see Theorem 55 in [16]) implies cause in these models the state at the next time
that on the whole segment [0, ˜ t] there exists a
instant is given directly by the black box para-
unique solution ˆ x of the initial value problem
for the system of ODEs with the right hand side metric family of functions F which performs the
functions of both the ODE right hand side eval-
ˆu s
f and initial condition ˆ x(0) = ˆ x . It also implies
that this solution is uniformly close to the solu- uation and the numerical ODE solver. Thus, the
tion of the original initial value problem, i.e., family of functions F is implicitly forced to learn
the numerical method for each problem, which
2ε leads to unnecessary complication of the model.
x(t) − ˆ x(t) <ε ∀t ∈[0, ˜ t].
3M g The semiempirical approach not only overcomes
this problem, but also provides additional flexi-
According to the assumption of this theorem,
bility, because it allows to use the experimental
the solution x exists on the whole segment [0, ˜ t]
data set with variable time step for the model
and is contained in X along with the closure of
training and testing; it also allows to use dif-
its ε-neighborhood; therefore the solution ˆ x is
contained in X. ferent numerical ODE solvers for different tra-
Since the set of vector-valued functions G is jectories from the data set; finally, the trained
ˆ
everywhere dense in the space G,for anyvector- model may be used in combination with another
valued function g ∈ G and any positive real ε g numerical ODE solver and with any time step.
ˆ
there exists a vector-valued function ˆ g ∈ G such One example of successful application of this
that approach is the Runge–Kutta Neural Network
(RKNN) model, based on the explicit fourth-
g
∗
∗
∗
g(x ) − ˆ g(x ) <ε ∀x ∈ X.
order Runge–Kutta method [18].
ε
g
If we assume that ε = , then we obtain the fol-
3
lowing inequality:
5.2 SEMIEMPIRICAL ANN-BASED
y(t) −ˆy(t)
MODEL DESIGN PROCESS
= g(x(t)) − ˆ g(ˆ x(t))
g(x(t)) − ˆ g(x(t)) + ˆ g(x(t)) − ˆ g(ˆ x(t)) The semiempirical ANN model design proce-
g g
<ε + M x(t) − ˆ x(t) dure consists of the following stages:
ε g 2ε
< + M = ε, t ∈[0, ˜ t]. 1. Design of the original theoretical state space
3 3M g continuous time model for the system.
Similar results for the case of purely empirical 2. Acquisition of experimental data of system
recurrent neural networks were given in [17]. behavior (5.1).
The initial value problem for the system of 3. Evaluation of theoretical model accuracy us-
ODEs defined by the model (5.2) can be solved ing the available experimental data.