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

Citation:

Simos, T. E., Katsikis, V. N., Mourtas, S. D., & Stanimirović, P. 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, 202, 164-180. Copy at http://www.tinyurl.com/2a865ts8

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.

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National and Kapodistrian University of Athens
Office: 601 (Floor 6)
Phone number: (+30) 210-3689489
Address: Sofokleous 1, 10559, Athens, Greece
Email: vaskatsikis@econ.uoa.gr
URL: http://scholar.uoa.gr/vaskatsikis
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Recent Publications

Cao, X., Li, S., Katsikis, V., Khan, A. T., He, H., Liu, Z., Zhang, L., et al. (2024). Empowering financial futures: Large language models in the modern financial landscape. EAI Endorsed Transactions on AI and Robotics, 3. presented at the Jul. Website

Stanimirović, P. S., Mourtas, S. D., Mosić, D., Katsikis, V. N., Cao, X., & Li, S. (2024). A Zeroing Neural Network Approach for Calculating Time-Varying G-Outer Inverse of Arbitrary Matrix. IEEE Transactions on Neural Networks and Learning Systems, 1-10.

Balios, D., Katsikis, V. N., Naoum, V. C., & Zaroulea, T. (2024). The effect of industry level characteristics and cross-country differences on earnings management: A European comparative perspective. Journal of Governance & Regulation, 13(2), 403-418. Publisher's Version

Jerbi, H., Al-Darraji, I., Albadran, S., Aoun, S. B., Simos, T. E., Mourtas, S. D., & Katsikis, V. N. (2024). Solving quaternion nonsymmetric algebraic Riccati equations through zeroing neural networks. AIMS Mathematics, 9, 5794-5809. Website

Mourtas, S. D., Katsikis, V. N., Stanimirović, P. S., & Kazakovtsev, L. A. (2024). Credit and Loan Approval Classification Using a Bio-Inspired Neural Network. Biomimetics, 9. Website Abstract

Numerous people are applying for bank loans as a result of the banking industry’s expansion, but because banks only have a certain amount of assets to lend to, they can only do so to a certain number of applicants. Therefore, the banking industry is very interested in finding ways to reduce the risk factor involved in choosing the safe applicant in order to save lots of bank resources. These days, machine learning greatly reduces the amount of work needed to choose the safe applicant. Taking this into account, a novel weights and structure determination (WASD) neural network has been built to meet the aforementioned two challenges of credit approval and loan approval, as well as to handle the unique characteristics of each. Motivated by the observation that WASD neural networks outperform conventional back-propagation neural networks in terms of sluggish training speed and being stuck in local minima, we created a bio-inspired WASD algorithm for binary classification problems (BWASD) for best adapting to the credit or loan approval model by utilizing the metaheuristic beetle antennae search (BAS) algorithm to improve the learning procedure of the WASD algorithm. Theoretical and experimental study demonstrate superior performance and problem adaptability. Furthermore, we provide a complete MATLAB package to support our experiments together with full implementation and extensive installation instructions.

He, Y., Dong, X., Simos, T. E., Mourtas, S. D., Katsikis, V. N., Lagios, D., Zervas, P., et al. (2024). A bio-inspired weights and structure determination neural network for multiclass classification: Applications in occupational classification systems. AIMS Mathematics, 9, 2411-2434. Website

Jerbi, H., Alshammari, O., Aoun, S. B., Kchaou, M., Simos, T. E., Mourtas, S. D., & Katsikis, V. N. (2024). Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control. Mathematics, 12. Website Abstract

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called ’ZQ-ARE’ that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor’s flight control system faster than the traditional differential-algebraic Riccati equation solution.

Gerontitis, D., Mo, C., Stanimirović, P. S., & Katsikis, V. N. (2024). Improved zeroing neural models based on two novel activation functions with exponential behavior. Theoretical Computer Science, 986, 114328. Website Abstract

A family of zeroing neural networks based on new nonlinear activation functions is proposed for solving various time-varying linear matrix equations (TVLME). The proposed neural network dynamical systems, symbolized as Li-VPZNN1 and Li-VPZNN2, include an exponential parameter in nonlinear activation function (AF) that leads to faster convergence to the theoretical result compared to previous categories of nonlinearly activated neural networks. Theoretical analysis as well as numerical tests in MATLAB's environment confirm the efficiency and accelerated convergence property of the novel dynamics.

Stanimirović, P. S., Mourtas, S. D., Mosić, D., Katsikis, V. N., Cao, X., & Li, S. (2024). Zeroing neural network approaches for computing time-varying minimal rank outer inverse. Applied Mathematics and Computation, 465, 128412. Website Abstract

Generalized inverses are extremely effective in many areas of mathematics and engineering. The zeroing neural network (ZNN) technique, which is currently recognized as the state-of-the-art approach for calculating the time-varying Moore-Penrose matrix inverse, is investigated in this study as a solution to the problem of calculating the time-varying minimum rank outer inverse (TV-MROI) with prescribed range and/or TV-MROI with prescribed kernel. As a result, four novel ZNN models are introduced for computing the TV-MROI, and their efficiency is examined. Numerical tests examine and validate the effectiveness of the introduced ZNN models for calculating TV-MROI with prescribed range and/or prescribed kernel.

Cao, X., Peng, C., Zheng, Y., Li, S., Ha, T. T., Shutyaev, V., Katsikis, V. N., et al. (2024). Neural Networks for Portfolio Analysis in High-Frequency Trading. IEEE Transactions on Neural Networks and Learning Systems, 1-10.

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Vasilios Katsikis

Professor, Economics

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

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