Page 177 - Neural Network Modeling and Identification of Dynamical Systems

168 5. SEMIEMPIRICAL NEURAL NETWORK MODELS OF CONTROLLED DYNAMICAL SYSTEMS

abstract context units. Therefore, the initial val- tion f, as well as the neural network–based ap-
ues of these state variables might be estimated proximations for unknown dependencies.
during the same experiment carried out to col- We consider the following family of continu-
lect the training data set via additional measure- ous time state space semiempirical models:
ments or calibration procedures. As mentioned
d ˆ x(t,w)
in Chapter 2, the initial values of state variables = f(ˆ x(t,w),u(t),w),
ˆ
may also be subject to optimization along with dt (5.2)
the model parameters; however, even in this ˆ y(t,w) = ˆ g(ˆ x(t,w),w),
case such estimates might serve as a good initial
where ˆ x:[0, ¯ t]× R n w → R n x is an estimate of
guess. Thus, the experimental data set required
the state space trajectory, ˆy:[0, ¯ t]× R n w → R n y
for the semiempirical model training includes is an estimate of the observable outputs trajec-
the estimates of state variables at the initial time
ˆ
tory, and f: R ×R ×R n w → R n x and ˆ g: R ×
n x
n x
n u
instant. Moreover, this data set also contains the
R n w → R n y are parametric families of functions,
values of sampling time instants t. The experi-
such as layered feedforward neural networks or
mental data set has the following form:
semiempirical neural network–based function
approximators.

˜ x (p) (0),u (p) (0), ˜y (p) (0) , The following theorem describes the capabil-
ities of this family of models.

P

K (p)
(p) (p) (p) (p) (p) Theorem 3. Let U be a compact subset of R ,let
n u
t ,u (t ), ˜y (t ) . (5.1)
k k k
k=1 n x
p=1 X be a compact subset of R , and let Y be a subset
of R . Also, let F be a subspace of the space of con-
n y
(p) n x
Here, we assume that t ≡ 0. tinuous vector-valued functions from X × U to R ,
0
Since the state variables of semiempirical locally Lipschitz continuous with respect to all of its
ˆ
models are interpretable in domain-specific arguments, and let F be a set of vector-valued func-
terms, it is usually possible to utilize additional tions, everywhere dense in F. Similarly, let G be a
theoretical knowledge of the internal structure subspace of continuous vector-valued functions from
n y
of the simulated object. This knowledge might X to R , locally Lipschitz continuous with respect
ˆ
allow us to determine the specific form of some to all of its arguments, and let G be a set of vector-
relationships between state variables, controls, valued functions, everywhere dense in G. Then, for
and observable outputs with sufficient accu- all vector-valued functions f ∈ F and g ∈ G and all
racy. These accurately known relationships are positive real ¯ t and ε there exist a positive real δ and
ˆ
ˆ
ˆ
ˆ
to be embedded in mappings f and ˆ g,justas vector-valued functions f ∈ F, ˆ g ∈ G such that for
s
s
in the case of function approximation. For ex- any x ∈ X, any ˜ x contained in the δ-neighborhood
s
ample, in the problem of aircraft motion mod- of x , any ˜ t ∈ (0, ¯ t], and any measurable, locally inte-
eling, discussed in Chapter 6, the relationships grable function u:[0, ˜ t]→ U such that the solution
between state variables and observable outputs x:[0, ˜ t]→ X of the initial value problem for the sys-
u
are known theoretically with sufficient accuracy; tem of ODEs with the right hand side f (t,x(t)) ≡
s
hence the function ˆ g from the observation equa- f(x(t),u(t)) and initial condition x(0) = x exists on
the whole segment [0, ˜ t] and is contained in X along
tion exactly matches the theoretical model coun-
with the closure of its ε-neighborhood, the following
terpart g and lacks any parameters to be tuned.
conditions hold:
The function f from the state equation in these
ˆ
problems also includes some known dependen- x(t) − ˆ x(t) <ε
cies borrowed from the theoretical model func- y(t) −ˆy(t) <ε ∀t ∈[0, ˜ t],