Page 118 - Neural Network Modeling and Identification of Dynamical Systems

3.4 ANN-BASED CONTROL OF DYNAMICAL SYSTEMS 107
In contrast to the reference model, the control number of “degrees of freedom” in the model
action on the plant consists of the command sig- used may be required. This number grows as
∗
nal u (t i ) and an additional signal the number of configurable parameters of the
neurocontroller increases. Computational ex-
u(t i ) = (x(t i−1 ),u (t i )), periments show that even for relatively simple
∗
problems the necessary number of variables can
in which the character of the function (·) de-
be of the order of several tens.
pends, as above, on the composition and values
To cope with this situation, we need a mathe-
of the parameters w in it.
matical model for the adjusting controller, which
So, the error function E(·) depends on the
parameter vector w, and by varying its com- has less computational complexity in solving the
ponents, we can choose the direction of their problem (3.26) than the traditional NLP problem
change so that E(w) decreases. mentioned above. One of the possible variants
of such mathematical models is the ANN. The
As we can see from Fig. 3.4, the error func-
adjusting controller, implemented as the ANN,
tion E(w) is defined at the outputs of the plant.
will hereafter be named neurocontroller.
It has already been noted above that the goal
More details on the main features of the struc-
of solving the problem of adjusting the dynamic
properties of a plant is to minimize the function ture and use of the ANN will be discussed be-
E(w) with respect to the parameters w, i.e., low. For now we only note that using this ap-
proach to represent the mathematical model of
∗
E(w ) = minE(w). (3.26) the adjusting controller allows us to reduce the
w
computational complexity of the problem (3.26)
Generally speaking, we could treat the prob- to about O(N w ) [14,28,29], i.e., it grows in pro-
lem (3.26) as a traditional optimization problem, portion to the first power of the number of vari-
namely, as a nonlinear programming problem ables N w . There is also the opportunity to reduce
(NLP), which has been well studied theoreti- this complexity [27].
cally, for the solution of which there are a signif- In the adopted scheme, as already noted, the
icant number of algorithms and software pack- minimized error function E(w) is defined not at
ages. With this approach, however, there is a cir- the outputs of the adjusting controller (realized,
cumstance that substantially limits its practical for example, in the ANN form), but at the out-
applicability. puts of the plant. But, as will be shown below,
Namely, the computational complexity of to organize the process of selecting the parame-
such algorithms (based, for example, on gradi- ters of the ANN, it is necessary to know the error
2
ent search) is of the order of O(N ) [67,68], i.e., E(w) directly at the output of the adjusting con-

w
it grows in proportion to the square of the num- troller.
ber of variables in the problem being solved. Hence we need to solve the following prob-
Because of this, the solution of NLP problems lem. Let there be an output of the plant model,
with a large number of variables occurs, as a which differs from the desired (“reference”) one.
rule, with severe difficulties. Such a situation for We must be able to answer the following ques-
traditional NLP problems can arise even when tion: how should the inputs of the plant model
N w is of the order of ten, especially in cases when be changed so that its outputs change in the di-
even a single calculation of the objective func- rection of reducing the error E(w)?
tion E(w) is associated with significant compu- The inputs of the model which are adjusted in
tational costs. such a way become target outputs for the neuro-
At the same time, to track the complex non- controller. The parameters w in the ANN vary
linear dynamics of the plant, a considerable to minimize the deviation of the current ANN