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Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

Author

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  • Simos, Theodore E.
  • Katsikis, Vasilios N.
  • Mourtas, Spyridon D.
  • Stanimirović, Predrag S.

Abstract

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.

Suggested Citation

  • Simos, Theodore E. & Katsikis, Vasilios N. & Mourtas, Spyridon D. & Stanimirović, Predrag S., 2022. "Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 164-180.
    • Handle: RePEc:eee:matcom:v:202:y:2022:i:c:p:164-180
    DOI: 10.1016/j.matcom.2022.05.033
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    References listed on IDEAS

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    1. Li Wang, 2020. "Numerical Algorithms of the Discrete Coupled Algebraic Riccati Equation Arising in Optimal Control Systems," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-8, August.
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    Cited by:

    1. Vladislav N. Kovalnogov & Ruslan V. Fedorov & Dmitry A. Generalov & Andrey V. Chukalin & Vasilios N. Katsikis & Spyridon D. Mourtas & Theodore E. Simos, 2022. "Portfolio Insurance through Error-Correction Neural Networks," Mathematics, MDPI, vol. 10(18), pages 1-14, September.
    2. Houssem Jerbi & Hadeel Alharbi & Mohamed Omri & Lotfi Ladhar & Theodore E. Simos & Spyridon D. Mourtas & Vasilios N. Katsikis, 2022. "Towards Higher-Order Zeroing Neural Network Dynamics for Solving Time-Varying Algebraic Riccati Equations," Mathematics, MDPI, vol. 10(23), pages 1-16, November.
    3. Liu, Yiqun & Zhuang, Guangming & Zhao, Junsheng & Lu, Junwei & Wang, Zekun, 2023. "H∞.. admissibilization for time-varying delayed nonlinear singular impulsive jump systems based on memory state-feedback control," Applied Mathematics and Computation, Elsevier, vol. 447(C).
    4. Spyridon D. Mourtas & Chrysostomos Kasimis, 2022. "Exploiting Mean-Variance Portfolio Optimization Problems through Zeroing Neural Networks," Mathematics, MDPI, vol. 10(17), pages 1-20, August.

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