Investigation of one linear non-smooth two-parameter discrete optimal control problem

Authors

DOI:

https://doi.org/10.15587/2312-8372.2019.180548

Keywords:

optimal control problem for discrete systems, necessary optimality condition, special control, non-smooth functional

Abstract

The object of research is the linear optimal control problem described by discrete two-parameter systems under the assumption that the controlled process is stepwise.

The work is aimed at deriving the necessary first-order optimality conditions in the case of a non-smooth quality function. And also to establish the necessary conditions of second-order optimality in stepwise control problems for discrete two-parameter systems. The paper investigates one linear two-parameter discrete optimal control problem with a non-smooth quality criterion. A special increment of the quality functional is calculated. Cases under the condition of a convex set are considered. The concept of special control in the problem under study is given. A number of necessary optimality conditions for the first and second orders are established. And also the necessary second-order optimality conditions are obtained in terms of directional derivatives. In the case of a linear quality criterion, the necessary and sufficient optimality condition is proved using the increment formula by analogous arguments. Under the assumption that the set is convex, a special increment of the quality criterion for admissible control is defined.

The methods of calculus of variations and optimal control, the theory of difference equations are used. The result is obtained for the optimality of a special, first-order control, in the case of convexity of the set. The case when the minimized functional is linear is considered. In this case, a necessary and sufficient condition is obtained for the optimality of the admissible control.

Thanks to the research results, it is possible to obtain the necessary first-order optimality conditions in terms of directional derivatives in the stepwise problem of optimal control of discrete two-parameter systems. As well as the necessary conditions of optimality of the second order in the case of convexity of the control domain and the necessary optimality conditions of special controls.

The theoretical results obtained in the work are of interest in the theory of optimal control of step systems and can be used in the further development of the theory of necessary optimality conditions for step control problems.

Author Biography

Mehriban Nasiyyati, Institute of Control Systems of Azerbaijan National Academy of Sciences, 9, F. Agaeva str., Baku, Azerbaijan, Az 1141

PhD, Researcher

Laboratory of Control in Complex Dynamic Systems

References

  1. Gaishun, I. V., Hoang, V. Q. (1991). Conditions for complete controllability and observability of discrete two-parameter systems. Differ. Equ., 27 (2), 125–130.
  2. Kaczorek, T. (1985). Two-Dimensional Linear Systems. Springer, 399. doi: https://doi.org/10.1007/bfb0005617
  3. Fornasini, E., Marchesini, G. (1976). State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control, 21 (4), 484–492. doi: https://doi.org/10.1109/tac.1976.1101305
  4. Afkhami, H., Argha, A., Roopaei, M., Ahrari Nouri, M. (2011). Optimal Iterative Learning Control Method for 2-D Systems using 1-D Model (WAM) of 2-D Systems. World Applied Sciences Journal, 13 (11), 2410–2419.
  5. Nyman, P.-O. (2009). Hierarchical least squares optimal control of 2-D systems. 2009 American Control Conference. doi: https://doi.org/10.1109/acc.2009.5160019
  6. Rogers, E., Galkowski, K., Owens, D. H. (2007). Control Systems Theory and Applications for Linear Repetitive Processes. Springer. doi: https://doi.org/10.1007/978-3-540-71537-5
  7. Kaczorek, T. (2005). Controllability and minimum energy control of positive 2D systems with delays. Control and Cybernetics, 34 (2), 411–423.
  8. Kaczorek, T. (1991). Some recent results in singular 2-D systems theory. Kybernetika, 27 (3), 253–262.
  9. Przyborowski, P., Kaczorek, T. (2009). Positive 2D Discrete-Time Linear Lyapunov Systems. International Journal of Applied Mathematics and Computer Science, 19 (1), 95–106. doi: https://doi.org/10.2478/v10006-009-0009-3
  10. Gayshun, I. V. (1996). Mnogoparametricheskie sistemy upravleniya. Minsk: Nauka i tehnika, 200.
  11. Bisiacco, M., Fornasini, E. (1990). Optimal Control of Two-Dimensional Systems. SIAM Journal on Control and Optimization, 28 (3), 582–601. doi: https://doi.org/10.1137/0328035
  12. Fornasini, E., Marchesini, G. (1976). State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control, 21 (4), 484–492. doi: https://doi.org/10.1109/tac.1976.1101305
  13. Prett, U. (1982). Tsifrovaya obrabotka izobrazheniy. Vol. I, II. Moscow: Nauka.
  14. Mansimov, K. B. (2013). Diskretnye sistemy. Baku: Izd-vo BGU, 151.
  15. Dem'yanov, V. F. (1974). Minimaks: Differentsiruemost' po napravleniyam. Leningrad: Izd-vo LGU, 120.
  16. Dem'yanov, V. F., Rubinov, A. M. (1990). Osnovy negladkogo analiza i kvazi-differentsial'noe ischislenie. Moscow: Nauka, 432.

Downloads

Published

2019-07-12

How to Cite

Nasiyyati, M. (2019). Investigation of one linear non-smooth two-parameter discrete optimal control problem. Technology Audit and Production Reserves, 4(2(48), 38–41. https://doi.org/10.15587/2312-8372.2019.180548

Issue

Vol. 4 No. 2(48) (2019): Information and control systems

Section

Reports on research projects

License

Copyright (c) 2019 Mehriban Nasiyyati

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.