Page 85 - Neural Network Modeling and Identification of Dynamical Systems

2.4 TRAINING SET ACQUISITION PROBLEM FOR DYNAMIC NEURAL NETWORKS 73
2.4 TRAINING SET ACQUISITION 2.4.2 Direct Approach to the Process of
PROBLEM FOR DYNAMIC Forming Data Sets Required for
NEURAL NETWORKS Training Dynamic Neural
Networks
2.4.1 Specifics of the Process of Forming 2.4.2.1 General Characteristics of the
Data Sets Required for Training Direct Approach to the Forming of
Dynamic Neural Networks Training Data Sets

Getting a training set that has the required We will clarify the concept of informative
level of informativeness is a critical step in solv- content of the training set, and we will also es-
ing the problem of forming the ANN model. timate its required volume to provide the neces-
If some features of dynamics (behavior) do not sary level of informativeness. First, we will per-
find reflection in the training set, they, accord- form these actions in the framework of a direct
ingly, will not be reproduced by the model. In approach to solving the problem of the forma-
one of the fundamental guidelines for the identi- tion of a training set; in the next section, the con-
cept will be extended to an indirect approach.
fication of systems, this provision is formulated
as the Basic Identification Rule: “If it is not in the Consider a controllable dynamical system of
the form
data, it cannot be identified” (see [89], page 85).
The training data set required for the for- ˙ x = F(x,u,t), (2.99)
mation of the dynamical system ANN model where x = (x 1 ,x 2 ,...,x n ) are the state variables,
should be informative (representative). For the
u = (u 1 ,u 2 ,...,u m ) are control variables, and t ∈
time being we will assume that the training set T =[t 0 ,t f ] is time.
is informative if the data contained in it are suf- The variables x 1 ,x 2 ,...,x n and u 1 ,u 2 ,...,u m ,
ficient to produce an ANN model that, with the taken at a particular moment in time t k ∈ T ,char-
required level of accuracy, reproduces the be- acterize, respectively, the state of the dynamical
havior of the dynamical system over the entire system and the control actions on it at a given
range of possible values for the quantities and time. Each of these values takes values from the
their derivatives that characterize this behavior. corresponding area, i.e.,
To ensure the fulfillment of this condition, when
forming a training set, it is required to obtain x 1 (t k ) ∈ X 1 ⊂ R,...,x n (t k ) ∈ X n ⊂ R;
(2.100)
data not only about changes in quantities, but u 1 (t k ) ∈ U 1 ⊂ R,...,u n (t k ) ∈ U m ⊂ R.
also about the rate of their change, i.e., we can
assume that the training set has the required in- In addition, there are, as a rule, restrictions on
formativeness if the ANN model obtained with the values of the combinations of these vari-
its use reproduces the behavior of the system not ables, i.e.,
only over the whole range of changes in the val-
x = x 1 ,...,x n ∈ R X ⊂ X 1 × ···X n ,
ues of the quantities characterizing the behavior (2.101)
of the dynamical system but also their deriva- u = u 1 ,...,u m ∈ R U ⊂ U 1 × ···U n ,
tives (and also all admissible combinations of
as well as on blends of these combinations,
both quantities and the values of their deriva-
tives).
x,u ∈ R XU ⊂ R X × R U . (2.102)
Such an intuitive understanding of the infor-
mativeness of the training set will be further re- The example included in the training set
fined. should show the response of the DS to some