Page 86 - Neural Network Modeling and Identification of Dynamical Systems

74 2. DYNAMIC NEURAL NETWORKS: STRUCTURES AND TRAINING METHODS

combination of x,u . By a reaction of this kind, In a similar way, we introduce one more exam-
we will understand the state x(t k+1 ),towhich ple of p j ∈ P:
the dynamical system (2.99) passes from the
state x(t k ) with the value u(t k ) of the control ac- p j ={ x (j) (t k ),u (j) (t k ) ,x (j) (t k+1 )}. (2.107)
tion, written
The source data of the examples p i and p j will
F(x,u,t) be considered as not coincident, i.e.,
x(t k ),u(t k ) −−−−−→ x(t k+1 ). (2.103)
(i)
(i)
x (t k ) = x (j) (t k ), u (t k ) = u (j) (t k ).
Accordingly, some example p from the training
set P will include two parts, namely, the input In the general case, the dynamical system re-
(this is the pair x(t k ),u(t k ) ) and output (this is sponses to the original data from these examples
the reaction x(t k+1 )) of the dynamical system. do not coincide, i.e.,
2.4.2.2 Informativity of the Training Set x (t k+1 ) = x (j) (t k+1 ).
(i)
The training set should (ideally) show the dy-
namical system responses to any combinations We introduce the concept of ε-proximity for
of x,u satisfying the condition (2.102). Then, a pair of examples p i and p j . Namely, we will
according to the Basic Identification Rule (see consider examples of p i and p j ε-close if the fol-
page 73), the training set will be informative, lowing condition is satisfied:
that is, allow to reproduce in the model all the (i) (j)
specific behavior of the simulated DS. 5 x (t k+1 ) − x (t k+1 ) ε, (2.108)
Let us clarify this situation. We introduce the
notation where ε> 0 is the predefined real number.
N p
We select from the set of examples P ={p i }
i=1
(i) (i) (i) a subset consisting of such examples p s for
p i ={ x (t k ),u (t k ) ,x (t k+1 )}, (2.104)
which the ε-proximity relation to the example
where p i ∈ P is the ith example from the train- p s is satisfied, i.e.,
ing set P. In this example (i) (j)
x (t k+1 ) − x (t k+1 ) ε, ∀s ∈ I s ⊂ I. (2.109)
(i)
(i)
(i)
x (t k ) = (x (t k ),...,x (t k )),
1 n (2.105) Here I s is the set of indices (numbers) of those
(i)
(i)
(i)
u (t k ) = (u (t k ),...,u (t k )). examples for which ε-proximity is satisfied with
1 m
respect to the example p s , while I s ⊂ I ={1,...,
(i)
The response x (t k+1 ) of the considered dynam- N p }. 6
ical system to the example p i is We call an example p i ε-representative if for
the whole collection of examples p s , ∀s ∈ I s ,
(i)
(i)
(i)
x (t k+1 ) = (x (t k+1 ),...,x (t k+1 )). (2.106) that is, for any example p s ,s ∈ I s , the condition
1 n of ε-proximity is satisfied. Accordingly, we can
now replace the collection of examples {p s },s ∈
5 It should be noted that the availability of an informative I s ,byasingle ε-representative p i ,andtheer-
training set provides a potential opportunity to obtain a ror introduced by such a replacement will not
model that will be adequate to a simulated dynamical sys- exceed ε. Input parts of collections of examples
tem. However, this potential opportunity must still be taken
advantage of, which is a separate nontrivial problem, the
successful solution of which depends on the chosen class of 6 This means that the example p i is included in the set of
models and learning algorithms. examples {p s },s ∈ I s .