Technical science/12. Automation control
systems in manufacturing
Ph.D., Salykova O.S.
Undergraduate of specialty
6M070400- Computing
and software, Bizhanova O.I.
Akhmet Baitursynov
Kostanay State University, Êàçàõñòàí
PROBLEM
DEFINITION OF OPTIMIZATION AND METHODS OF ITS DECISION FOR DESIGN OF THE
AUTOMATED SYSTEM
Investigating
this or that task of optimization, first of all it is necessary to choose a
mathematical method for its decision. The chosen method has to lead to such
results that the volume of information was the greatest, and costs of
calculations – the smallest. The main criterion of a choice of a method is not
only statement of an optimum task, but also also mathematical model which use
in the course of object optimization.
The
methods solving optimum problems a set, but the following is the most popular:
- methods of research
of functions of the classical analysis;
- the methods based
on use of uncertain multipliers of Lagrange;
- calculus of
variations;
- dynamic
programming;
- principle of a
maximum;
- linear programming;
-
nonlinear programming.
Since
recent time the method of geometrical programming was developed and introduced.
This method use for the solution of a certain class of tasks.
As a
rule, it is difficult to recommend only one method which can be used for the
solution of all tasks. One methods are more the general, others – less the
general. The whole group of methods (methods of research of functions of the
classical analysis, a method of multipliers of Lagrange, methods of nonlinear
programming) at certain stages of the solution of an optimum task can be
applied in combination with other methods, for example dynamic programming or
the principle of a maximum.
Dynamic
programming is well adapted for the solution of problems of optimization of multistage
processes, especially in what the condition of each stage is characterized by
rather small number of variables of a state. However in the presence of
considerable number of these variables, i.e. at high dimension of each stage,
application of a method of dynamic programming is difficult owing to the
limited speed and memory size of computers.
Dynamic
programming serves as an effective method of the solution of problems of
optimization of discrete multistage processes for which the criterion of an optimality
is set as additive function of criteria of an optimality of separate stages.
Without special difficulties the specified method can be extended and to a case
when the criterion of an optimality is set in other form, however thus
dimension of separate stages usually increases.
In
essence the method of dynamic programming represents algorithm of definition of
optimum strategy of management at all stages of process. Thus the management
law at each stage find a solution of private problems of optimization
consistently for all stages of process by means of methods of research of
functions of the classical analysis or methods of nonlinear programming.
Results of the decision can't be usually expressed in an analytical form, and
turn out in the form of tables. Restrictions on variable tasks have no impact
on the general algorithm of the decision, and are considered at the solution of
private problems of optimization at each stage of process. In the presence of
restrictions like equalities sometimes even it is possible to reduce dimension
of these private tasks due to use of multipliers of Lagrange. Application of a
method of dynamic programming for optimization of processes with the
distributed parameters or in problems of dynamic optimization leads to the solution
of the differential equations in private derivatives. Instead of the solution
of such equations often it is much simpler to present continuous process as
discrete with rather large number of stages. Similar reception is justified
especially when there are restrictions on variable tasks and the direct
solution of the differential equations becomes complicated need of the
accounting of the specified restrictions.
As the
best way at a choice of a method of the optimization which is most suitable for
the solution of the corresponding task, it is necessary to recognize research
of opportunities and experience of application of various methods of
optimization.
Literature:
1. 1 . A. G. Trifonov.
"Problem definition of optimization and numerical methods of its
decision", 2012.
2.
2 . The automated design of technical systems: Manual. Baby seals V.N.,
Lanshakov V. L. Natural Sciences Academy publishing house, 2009, ISBN
978-5-91327-056-6.