# Analysis of Lyapunovmatrices’application Methods for Optimization of Stationary Dynamic Systems

2021;
: pp. 1 - 7
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University
4
Lviv Polytechnic National University, Ukraine

In this article there has been conducted analysis of Lyapunov matrix application in order to form control inputs under different dynamic systems’ optimization methods oriented by quadratic integral criterion. For this purpose, the methods of finding the Lyapunov matrix and optimization based on the Bellman functional equation with subsequent application of the Riccati equation, optimization taking into account the initial values of state variables, optimization based on the Bellman equation using linear matrix inequalities and Lyapunov equation are considered. Despite the complexity of solving the Riccati equation, the problem of finding the Lyapunov matrix is unambiguous only in the case of application of optimization methods based on dynamic Bellman programming and representation of the Bellman function by the Lyapunov function. Optimization based on the application of the linear matrix inequality condition is not unambiguous, as it requires the choice of the inequality solution. The optimization of the system by the integral quadratic criterion and the initial values of the state variables is also ambiguous because there is a problem of solving nonlinear interconnected optimization equations.

1. An operator theoretical approach to the sequence entropy of dynamical Systems / Dynamical Systems // Taylor & Francis, 2021. Vol. 36. No. 4. P. 1468-9367 1. 10.1080/14689367.2021.1999907.
2. Kim D. Automatic Control Theory. Vol. 1. Linear Systems [Teoriya avtomaticheskogo upravleniya. T. 1: Linejnye sistemy]. Moscow: Phismatlit, 2003. 288 p. (in Russian).
3. The Interconnection and Damping Assignment Passivity-Based Control Synthesis via the Optimal Control Method for Electric Vehicle Subsystems / A. Lozynskyy, T. Perzyński, J. Kozyra, Y. Biletskyi, L. Kasha // Energies - 2021. Vol. 14. No. 12. P. 3711. 10.3390/en14123711.
https://doi.org/10.3390/en14123711
4. Synthesis of the Intelligent Position Controller of an Electromechanical System / Y. Paranchuk, O. Kuznyetsov // 2020 IEEE Problems of Automated Electrodrive. Theory and Practice (PAEP), 2020, pp. 1-4 10.1109/PAEP49887.2020.9240889.
5. Aghababa M. Adaptive control for electromechanical systems considering dead-zone phenomenon // Nonlinear Dynamics, 2014. Vol. 75 10.1007/s11071-013-1056-8.
https://doi.org/10.1007/s11071-013-1056-8
6. Synthesis of Combine Feedback Control of Electromechanical System by Feedback Linearization Method /A. Lozynskyy, Y. Marushchak, O. Lozynskyy, L. Kasha // 2020 IEEE Problems of Automated Electro.
7. Lozynskyj O., Lozynskyj A., Marushhak Ya., Paranchuk Ya., Tcyapa V., Synthesis of linear optimal dynamic systems: Tutorial, Lviv, Ukraine: Publishing House of Lviv Polytechnic National University, 2016. 392 p. (in Ukrainian).
8. Kaleniuk P., Rudavskyi Y., Tatsiyi R., Differential equations: Tutorial, Lviv, Ukraine: Publishing House of Lviv Polytechnic National University, 2014. 380 p. (in Ukrainian).
9. Optimization of the Electromechanical System by Formation of a Feedback Matrix Based on State Variables /A. Lozynskyy, L. Demkiv, O. Lozynskyy, Y. Biletskyi // Electrical Power and Electromechanical Systems. Lviv : Lviv Politechnic Publishing House, 2020. Vol. 3. No. 1(s). P. 18-26. doi.org/ 10.23939/sepes2020.01s.018.
10. Analytical design of dynamic systems regulators taking into account the effect of disturbing factors / Orest Lozynskyi, Volodymyr Moroz, Roman Biletskyi, Yurii Biletskyi // Computational Problems of Electrical Engineering. Lviv : Lviv Politechnic Publishing House, 2019. Vol. 9. No. 1. P. 21-26.
11. Dorf R. and Bishop R. Modern control systems.Translated by V. Kopylov, Moscow, Russia: Binom. Laboratory of knowledge, 2002. (in Russian).