OPTIMIZATION OF NONLINEAR STOCHASTIC SYSTEMS IN THE SPECTRAL CHARACTERISTICS OF CONTROLS
The author presents the spectral method of determining relatively optimal control in case of incomplete infor- mation about the state vector for multidimensional nonlinear continuous stochastic systems, which are governed by Itô’s stochastic differential equations. The quality criterion is given as...
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Moscow State Technical University of Civil Aviation
2017-05-01
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| Series: | Научный вестник МГТУ ГА |
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| Online Access: | https://avia.mstuca.ru/jour/article/view/1053 |
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| author | K. A. Rybakov |
| author_facet | K. A. Rybakov |
| author_sort | K. A. Rybakov |
| collection | DOAJ |
| description | The author presents the spectral method of determining relatively optimal control in case of incomplete infor- mation about the state vector for multidimensional nonlinear continuous stochastic systems, which are governed by Itô’s stochastic differential equations. The quality criterion is given as the mean of the function determined on the system tracks. One should find the equation that depends on state vector time and coordinates, of which there is exact information from measuring system. Solving the problem of finding optimal control is based upon the actual sufficient optimum condition and the ratios derived from them. These ratios, which determine nonlinear continuous stochastic systems optimal control in case of incomplete state vector information (Fokker-Planck-Kolmogorov and Bellman equation systems and the tying ratios that allow to determine control structure) with the help of a spectral transformation usually lead to the system of nonlinear equations for the coefficients of optimal control and optimal state vector probability density coordinates expansion into a basic system functions series. This nonlinear equations system solving method does not depend on the chosen basis, it is solved either with iterative methods or with reducing it to the equivalent method of unconditional optimization with the following usage of zero-order method, including metaheuristic methods global extremum search. In this article, determin- ing optimal control goes down to improving control spectral characteristics in space (in the coefficient space of dividing control according to the orthonormal system functions). The author dwells upon the issue of taking so called geometrical control constraints into account as a special case. Using the spectral form of the mathematical description it is necessary to reduce spectral characteristics of functions, operators and functionals to some chosen orders, and therefore moving to finite-dimensional problems of optimization. Reduction order choice and basis system choice determine the approximate solution accuracy for optimal control problem. |
| format | Article |
| id | doaj-art-fd8581c0bd4f4b54a3a714ce83d882e7 |
| institution | DOAJ |
| issn | 2079-0619 2542-0119 |
| language | Russian |
| publishDate | 2017-05-01 |
| publisher | Moscow State Technical University of Civil Aviation |
| record_format | Article |
| series | Научный вестник МГТУ ГА |
| spelling | doaj-art-fd8581c0bd4f4b54a3a714ce83d882e72025-08-20T03:23:19ZrusMoscow State Technical University of Civil AviationНаучный вестник МГТУ ГА2079-06192542-01192017-05-0120216261053OPTIMIZATION OF NONLINEAR STOCHASTIC SYSTEMS IN THE SPECTRAL CHARACTERISTICS OF CONTROLSK. A. Rybakov0Moscow Aviation Institute (National Research University)The author presents the spectral method of determining relatively optimal control in case of incomplete infor- mation about the state vector for multidimensional nonlinear continuous stochastic systems, which are governed by Itô’s stochastic differential equations. The quality criterion is given as the mean of the function determined on the system tracks. One should find the equation that depends on state vector time and coordinates, of which there is exact information from measuring system. Solving the problem of finding optimal control is based upon the actual sufficient optimum condition and the ratios derived from them. These ratios, which determine nonlinear continuous stochastic systems optimal control in case of incomplete state vector information (Fokker-Planck-Kolmogorov and Bellman equation systems and the tying ratios that allow to determine control structure) with the help of a spectral transformation usually lead to the system of nonlinear equations for the coefficients of optimal control and optimal state vector probability density coordinates expansion into a basic system functions series. This nonlinear equations system solving method does not depend on the chosen basis, it is solved either with iterative methods or with reducing it to the equivalent method of unconditional optimization with the following usage of zero-order method, including metaheuristic methods global extremum search. In this article, determin- ing optimal control goes down to improving control spectral characteristics in space (in the coefficient space of dividing control according to the orthonormal system functions). The author dwells upon the issue of taking so called geometrical control constraints into account as a special case. Using the spectral form of the mathematical description it is necessary to reduce spectral characteristics of functions, operators and functionals to some chosen orders, and therefore moving to finite-dimensional problems of optimization. Reduction order choice and basis system choice determine the approximate solution accuracy for optimal control problem.https://avia.mstuca.ru/jour/article/view/1053optimal controloptimizationincomplete informationspectral methodspectral transformationspectral characteristicspectral form of mathematical descriptionstochastic system |
| spellingShingle | K. A. Rybakov OPTIMIZATION OF NONLINEAR STOCHASTIC SYSTEMS IN THE SPECTRAL CHARACTERISTICS OF CONTROLS Научный вестник МГТУ ГА optimal control optimization incomplete information spectral method spectral transformation spectral characteristic spectral form of mathematical description stochastic system |
| title | OPTIMIZATION OF NONLINEAR STOCHASTIC SYSTEMS IN THE SPECTRAL CHARACTERISTICS OF CONTROLS |
| title_full | OPTIMIZATION OF NONLINEAR STOCHASTIC SYSTEMS IN THE SPECTRAL CHARACTERISTICS OF CONTROLS |
| title_fullStr | OPTIMIZATION OF NONLINEAR STOCHASTIC SYSTEMS IN THE SPECTRAL CHARACTERISTICS OF CONTROLS |
| title_full_unstemmed | OPTIMIZATION OF NONLINEAR STOCHASTIC SYSTEMS IN THE SPECTRAL CHARACTERISTICS OF CONTROLS |
| title_short | OPTIMIZATION OF NONLINEAR STOCHASTIC SYSTEMS IN THE SPECTRAL CHARACTERISTICS OF CONTROLS |
| title_sort | optimization of nonlinear stochastic systems in the spectral characteristics of controls |
| topic | optimal control optimization incomplete information spectral method spectral transformation spectral characteristic spectral form of mathematical description stochastic system |
| url | https://avia.mstuca.ru/jour/article/view/1053 |
| work_keys_str_mv | AT karybakov optimizationofnonlinearstochasticsystemsinthespectralcharacteristicsofcontrols |