Page 42 - Neural Network Modeling and Identification of Dynamical Systems

30 1. THE MODELING PROBLEM FOR CONTROLLED MOTION OF NONLINEAR DYNAMICAL SYSTEMS

y(t i ), t i ∈[t 0 ,t f ], of these observations together
with the corresponding values of the controlled
inputs u i generate a set of N P ordered pairs, i.e.,

{ u i ,y i },u i ∈ U, y i ∈ Y, i = 1,...,N P . (1.29)

FIGURE 1.7 General structure of the simulated dynami- Given the data (1.29), we need to find an ap-
cal system. From [56], used with permission from Moscow proximation (·) for the mapping (·),imple-
Aviation Institute. mented by the system S such that the following
condition is satisfied:
of the problem is formalized in a mathematical (u(t),ξ(t),ζ(t)) − (u(t),ξ(t),ζ(t)) ε,
model (see also remarks on Examples 1 and 2
on page 18). In the following we consider only ∀u(t i ) ∈ U, ∀ξ(t i ) ∈ , ∀ζ(t i ) ∈ Z, t ∈[t 0 ,t f ].
problems 2, 3, and 4. (1.30)
Let there be some dynamical system S,which
Thus, as follows from (1.30), it is necessary
is an object of modeling (Fig. 1.7).
that the desired approximate mapping (·) pos-
The system S perceives the controlled u(t)
sesses the required accuracy not only when re-
and uncontrolled ξ(t) impacts. Under these in-
producing the observations (1.29), but also for
fluences, S changes its state x(t) according to the
all valid values of u i ∈ U. We refer to this prop-
transformation rule (mapping) F(u(t),ξ(t)) that erty of the map (·) as generalization. The pres-
it implements. At the initial time t = t 0 ,the state ence of entries ∀ξ(t i ) ∈ and ∀ζ(t i ) ∈ Z in (1.30)
of the system S takes the value x(t 0 ) = x 0 .
means that (·) will have the required accuracy
The state x(t) is perceived by the observer
only provided that at any time t ∈[t 0 ,t f ] un-
that transforms it into observation results y(t)
controllable effects ξ(t) on the system S and the
via mapping G(x(t),ζ(t)). These observations
measurement noise ζ(t) do not exceed the al-
y(t) represent the output of the system S. Sen-
lowed limits.
sors that perform measurements of the state The map (·) corresponds to the object un-
introduce some error that we describe by ad- der consideration (the dynamical system S), and
ditional uncontrolled influence ζ(t) (“measure- the map (·) will be referred to as the model
ment noise”). The composition of the maps F(·) of this object. We will also assume that we have
and G(·) describes the relationship between the data of the form (1.29) for the system S,andpos-
controlled input u(t) ∈ U of the system S and its sibly some knowledge about the “structure” of
output y(t) ∈ Y, while taking into account the mapping (·), implemented by the system un-
influence of uncontrolled disturbances ξ(t) and der consideration. In this case, the presence of
ζ(t) on the system under consideration, i.e.,
data of the specified type is mandatory (at least,
they are necessary for testing the model (·)),
y = (u(t),ξ(t),ζ(t)) = G(F(u(t),ξ(t)),ζ(t)).
while knowledge of the mapping (·) might be
Suppose that we have performed N p observa- unavailable or unused for the development of
the model (·).
tions for the system S, i.e.,
It is required to clarify what is meant by the
{y i }= (u i ,ξ,ζ), i = 1,...,N P , (1.28) norm · in the expression (1.30), i.e., how to in-
terpret the magnitude of the difference between
and recorded the current value of the controlled the results given by the maps (·) and (·).
input action u i = u(t i ) and the corresponding One possible definition of the residual (1.30)
output y i = y(t i ) for each of them. The results is the maximum deviation of (·) from (·), i.e.,