Application of the interval explosion method

Mathematics. Physics. Mechanics


Аuthors

Panovskiy V. N.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: panovskiy.v@gmail.com

Abstract

In the given work the interval explosion method, which was created by the author, for the solution of the problem of determination of optimum program control of the discrete and continuous dynamic systems is considered (the problems about spacecraft orientation and stabilization and «smooth landing» are solved as the example).
In the modern mathematics a great attention is given to a solution of problems of global optimization and synthesis of optimum control of dynamic system. These problems arise during designing of designs of planes, helicopters, spacecrafts when arises the necessity of optimization of characteristic parameters and of creation of control systems.
Existing numerical methods use various approaches, but their use is connected with various difficulties: the big computing loadings, requirements to problem statement, difficulties in reaching of convergence. Thus, working out of new methods of the optimization, which combine the newest mathematic approaches, is the extremely important.
Besides, it is necessary to notice that it is the extremely important to use and develop heuristic methods. Despite the lack of its strict substantiation, these methods give an acceptable solution of a problem in the majority of almost significant cases. Heuristic algorithms do not guarantee finding the solution and can give an incorrect solution in certain cases. However an essential advantage of such algorithms is their lowest computing complexity that allows to apply them to a solution of problems of the raised difficulty (for example, the problems belonging to NP class). In aggregate with key singularities of the interval analysis (handling of ranges instead of isolated points, low insistence to problem statement) working out of heuristic interval algorithms is the extremely perspective direction.
The main feature of the developed method is combination of interval analysis and explosion heuristics. Method has two search procedures: global and clarifying. This helps to find areas where the global minima can be situated and then to locate its position.
In the given work the algorithm and the software of the interval explosion method for a solution of a problem of determination of optimum programmed control by the discrete and continuous determined dynamic systems were created, applied examples on which efficiency of the developed method is shown were solved. Besides, some recommendations about possible improvements of the created method and its further development were made.

Keywords:

interval analysis, global optimization, heuristic algorithm, explosion method, optimal control, dynamic system

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