BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, , FINITE-DIMENSIONAL APPROXIMATION OF CONTROLS IN OPTIMIZATION PROBLEMS FOR LINEAR SYSTEMS // BSU Bulletin. Mathematics, Informatics. - 2020. №3. . - С. 19-31.
Title:
FINITE-DIMENSIONAL APPROXIMATION OF CONTROLS IN OPTIMIZATION PROBLEMS FOR LINEAR SYSTEMS
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The article studies extremum problems of the final state norm of a linear
dynamical system using methods of parameterization of admissible controls. Piecewise
continuous controls are approximated in the class of piecewise constant functions on
a uniform grid of nodes of the time interval by linear combinations of special support
functions. In this case, the restriction of a control of the original problem to the interval
induces the same restrictions for the variables of the finite-dimensional problems.
The finite-dimensional version of a minimum norm problem can effectively be resolved with the help of modern convex optimization programs. In the case of two variables, we propose an analytical method of resolution that uses a one-dimensional
minimization problem for a parabola over a segment.
For a non-convex norm maximization problem, the finite-dimensional version is resolved globally by exhaustive search over the vertices of a hypercube. The proposed
approach provides further insights into global resolution of non-convex optimal control
problems and is exemplified by some illustrative problems.
dynamical system using methods of parameterization of admissible controls. Piecewise
continuous controls are approximated in the class of piecewise constant functions on
a uniform grid of nodes of the time interval by linear combinations of special support
functions. In this case, the restriction of a control of the original problem to the interval
induces the same restrictions for the variables of the finite-dimensional problems.
The finite-dimensional version of a minimum norm problem can effectively be resolved with the help of modern convex optimization programs. In the case of two variables, we propose an analytical method of resolution that uses a one-dimensional
minimization problem for a parabola over a segment.
For a non-convex norm maximization problem, the finite-dimensional version is resolved globally by exhaustive search over the vertices of a hypercube. The proposed
approach provides further insights into global resolution of non-convex optimal control
problems and is exemplified by some illustrative problems.
Keywords:
linear control system; extremum problems of the final state norm; piecewise constant approximation; finite-dimensional problems.
List of references:
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3. Galyaev A. A., Lysenko P. V. Optimalnoe po energii upravlenie garmonicheskim ostsillyatorom [Energy-Optimal Control of Harmonic Oscillator]. Automation and Remote Control. 2019. Vol. 80, Pp. 16–29. htts://doi.org/10.1134/S0005231019010021
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В. А. Срочко, Е. В. Аксенюшкина, В. Г. Антоник. Конечномерная аппроксимация управлений в задачах оптимизации линейных систем
31
5. Srochko V. A. Iteratsionnye metody resheniya zadach optimalnogo upravleniya [Iterative Methods for Solving Optimal Control Problems]. Moscow, Fizmatlit Publ., 2000. 160 p.
6. Srochko V. A., Aksenyushkina E. V. Parametrizatsiya nekotorykh zadach upravleniya lineinymi sistemami [Parameterization of Some Control Problems by Linear Systems]. Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika. 2019. Vol. 30. Pp. 83–98. htts://doi.org/10.26516/1997-7670.2019.30.83
7. Strekalovsky A. S. Elementy nevypukloi optimizatsii [Elements of Nonconvex Optimization]. Novosibirsk: Nauka Publ., 2003. 356 p.
8. Strekalovsky A. S., Sharankhaeva E. V. Globalnyi poisk v nevypukloi zadache optimalnogo upravleniya [Global Search in a Nonconvex Optimal Control Problem]. Computational Mathematics and Mathematical Physics. 2005. Vol. 45, no. 10. Pp. 1719–1734.
9. Sukharev A. G., Timokhov V. V., Fedorov V. V. Kurs metodov optimizatsii [A Course of Optimization Methods]. Moscow: Nauka Publ., 1986. 328 p.
10. Tyatyushkin A. I. Mnogometodnaya tekhnologiya optimizatsii upravlyaemykh sistem [Multi-Method Technology for Optimization of Control Systems]. Novosibirsk: Nauka Publ., 2006. 343 p.
11. Chernov A. V. O primenenii funktsii Gaussa dlya chislennogo resheniya zadach optimalnogo upravleniya [On Application of Gaussian Functions for Numerical Solution of Optimal Control Problems]. Automation and Remote Control. 2019. Vol. 80. Pp. 1026–1040. htts://doi.org/10.1134/S0005231019060035
2. Antonik V. G., Srochko V. A. Metody nelokalnogo uluchsheniya ekstremalnykh upravlenii v zadache na maksimum normy konechnogo sostoyaniya [Methods for Nonlocal Improvement of Extreme Controls in the Maximization of the Terminal State Norm]. Computational Mathematics and Mathematical Physics. 2009. Vol. 49, no. 5. Pp. 762–775.
3. Galyaev A. A., Lysenko P. V. Optimalnoe po energii upravlenie garmonicheskim ostsillyatorom [Energy-Optimal Control of Harmonic Oscillator]. Automation and Remote Control. 2019. Vol. 80, Pp. 16–29. htts://doi.org/10.1134/S0005231019010021
4. Gorbunov V. K. O svedenii zadach optimalnogo upravleniya k konechnomernym [The Reduction of Optimal Control Problems for Finite-Dimensional]. Computational Mathematics and Mathematical Physics. 1978. Vol. 18, no. 5. Pp. 1083–1095.
В. А. Срочко, Е. В. Аксенюшкина, В. Г. Антоник. Конечномерная аппроксимация управлений в задачах оптимизации линейных систем
31
5. Srochko V. A. Iteratsionnye metody resheniya zadach optimalnogo upravleniya [Iterative Methods for Solving Optimal Control Problems]. Moscow, Fizmatlit Publ., 2000. 160 p.
6. Srochko V. A., Aksenyushkina E. V. Parametrizatsiya nekotorykh zadach upravleniya lineinymi sistemami [Parameterization of Some Control Problems by Linear Systems]. Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika. 2019. Vol. 30. Pp. 83–98. htts://doi.org/10.26516/1997-7670.2019.30.83
7. Strekalovsky A. S. Elementy nevypukloi optimizatsii [Elements of Nonconvex Optimization]. Novosibirsk: Nauka Publ., 2003. 356 p.
8. Strekalovsky A. S., Sharankhaeva E. V. Globalnyi poisk v nevypukloi zadache optimalnogo upravleniya [Global Search in a Nonconvex Optimal Control Problem]. Computational Mathematics and Mathematical Physics. 2005. Vol. 45, no. 10. Pp. 1719–1734.
9. Sukharev A. G., Timokhov V. V., Fedorov V. V. Kurs metodov optimizatsii [A Course of Optimization Methods]. Moscow: Nauka Publ., 1986. 328 p.
10. Tyatyushkin A. I. Mnogometodnaya tekhnologiya optimizatsii upravlyaemykh sistem [Multi-Method Technology for Optimization of Control Systems]. Novosibirsk: Nauka Publ., 2006. 343 p.
11. Chernov A. V. O primenenii funktsii Gaussa dlya chislennogo resheniya zadach optimalnogo upravleniya [On Application of Gaussian Functions for Numerical Solution of Optimal Control Problems]. Automation and Remote Control. 2019. Vol. 80. Pp. 1026–1040. htts://doi.org/10.1134/S0005231019060035
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