Page 87 - Neural Network Modeling and Identification of Dynamical Systems

2.4 TRAINING SET ACQUISITION PROBLEM FOR DYNAMIC NEURAL NETWORKS 75
(s)
{p s },s ∈ I s , allocate the subdomain R ,s ∈ In Eq. (2.111), ϕ(·) is a nonlinear vector func-
XU
I s , in the domain R XU defined by the relation tion of the vector arguments x, u and the scalar
(2.102); in this case argument t. It is assumed to be given and be-
longs to some class of functions that admits the
N p existence of a solution of Eq. (2.111) for given
) (s)
R = R XU . (2.110) x(t 0 ) and u(t) in the considered part of the space
XU
s=1 of states for the plant.
The behavior of the plant, determined by its
Now we can state the task of forming a train- dynamic properties, can be influenced by set-
ing set as a collection of ε-representatives that ting a correction value for the control variable
covers the domain R XU (2.102) of all possible u(x,u ). The operation of forming the required
∗
values of pairs x,u . value u(x,u ) for some time t i+1 from the val-
∗
The relation (2.110)isthe ε-covering condi- ues of the state vector x and the command con-
tion for the training set P of the domain R XU . ∗
trol vector u at the time instant t i
Aset P carrying out an ε-covering of the domain
R XU will be called ε-informative or, for brevity, u(t i+1 ) = (x(t i ),u (t i )) (2.112)
∗
simply informative.
If the training set P has ε-informativity, this we will perform in the device, which we call the
means that for any pair x,u ∈ R XU there is correcting controller (CC). We assume that the
at least one example p i ∈ P which is an ε- character of the transformation (·) in (2.112)is
representative for a given pair. determined by the composition and values of
With respect to the ε-covering (2.110)ofthe the components of a certain parameter vector
domain R XU , the following two problems can be w = (w 1 w 2 ...w N w ).Theset(2.111), (2.112)from
formulated: the plant and CC is referred to as a controlled
system.
1. Given the number of examples N p in the The behavior of the system (2.111), (2.112)
training set P, find their distribution in the with the initial conditions x 0 = x(t 0 ) under the
domain R XU which minimizes the error ε.
control u(t) is a multistep process if we assume
2. A permissible error value ε is given; obtain a
that the values of this process x(t k ) are observed
minimal collection of a number of N p exam-
at time instants t k , i.e.,
ples which ensures that ε is obtained.
{x(t k )},t k = t 0 + k t ,
2.4.2.3 Example of Direct Formation of (2.113)
Training Set k = 0,1,...,N t , t = t f − t 0 .
N t
Suppose that the controlled object under con-
sideration (plant) is a dynamical system de- In the problem (2.111), (2.112), as a teaching
scribed by a vector differential equation of the example, generally speaking, we could use a
form [91,92] pair

˙ x = ϕ(x,u,t). (2.111) (e) (e) (e)
(x 0 ,u (t)), {x (t k ), k = 0,1,...,N t } ,
n
Here, x = (x 1 x 2 ... x n ) ∈ R is the vector of state
m (e) (e)
variables of the op-amp; u = (u 1 u 2 ... u m ) ∈ R where (x ,u (t)) is the initial state of the sys-
0
is a vector of control variables of the op-amp; tem (2.111) and the formed control law, respec-
m
n
R , R are Euclidean spaces of dimension n and tively, and {x (e) (t k ), k = 0,1,...,N t } is the mul-
m, respectively; t ∈[t 0 ,t f ] is the time. tistep process (2.113), which should be carried