Solution of generalized matrix Riccati differential equation for indefinite stochastic linear quadratic singular fuzzy system with cross-term using neural networks

Kumaresan, N.

doi:10.1007/s00521-010-0431-3

Cited by 3 publications

(3 citation statements)

References 28 publications

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“…In other words, the authors used the residual of the MDRE as the loss function. Following the work of Reference [16], more complex MDREs from singular, stochastic, and fuzzy systems were quickly solved [17][18][19][20][21]. Each neural network in these studies was trained with the Levenberg-Marquardt algorithm and validated by comparisons with the Runge-Kutta algorithm.…”

Section: Related Workmentioning

confidence: 99%

Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation

Drozd,

Furfaro,

Schiassi

et al. 2023

Mathematics

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In this manuscript, we explore how the solution of the matrix differential Riccati equation (MDRE) can be computed with the Extreme Theory of Functional Connections (X-TFC). X-TFC is a physics-informed neural network that uses functional interpolation to analytically satisfy linear constraints, such as the MDRE’s terminal constraint. We utilize two approaches for solving the MDRE with X-TFC: direct and indirect implementation. The first approach involves solving the MDRE directly with X-TFC, where the matrix equations are vectorized to form a system of first order differential equations and solved with iterative least squares. In the latter approach, the MDRE is first transformed into a matrix differential Lyapunov equation (MDLE) based on the anti-stabilizing solution of the algebraic Riccati equation. The MDLE is easier to solve with X-TFC because it is linear, while the MDRE is nonlinear. Furthermore, the MDLE solution can easily be transformed back into the MDRE solution. Both approaches are validated by solving a fluid catalytic reactor problem and comparing the results with several state-of-the-art methods. Our work demonstrates that the first approach should be performed if a highly accurate solution is desired, while the second approach should be used if a quicker computation time is needed.

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Section: Related Workmentioning

confidence: 99%

Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation

Drozd,

Furfaro,

Schiassi

et al. 2023

Mathematics

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show abstract

“…By mapping the output of the neural network to the expected value and using the characteristics of NM (19), the numerical integration can be acquired [11]:…”

Section: Neural Networkmentioning

confidence: 99%

“…Another advantage is its ability to estimate unseen or untrained points. Previous studies of the first order differential equation (FODE) and second order differential equation (SODE) solution via neural networks [10][11][12] have been promising. It has also been shown in [10] that neural network methods are more stable and accurate than Euler method, first order implicit method, and second order implicit method.…”

Section: Introductionmentioning

confidence: 99%

A Multiagent Transfer Function Neuroapproach to Solve Fuzzy Riccati Differential Equations

Shahrir

Kumaresan

Ratnavelu

et al. 2014

Journal of Applied Mathematics

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A numerical solution of fuzzy quadratic Riccati differential equation is estimated using a proposed new approach for neural networks (NN). This proposed new approach provides different degrees of polynomial subspaces for each of the transfer function. This multitude of transfer functions creates unique “agents” in the structure of the NN. Hence it is named as multiagent neuroapproach (multiagent NN). Previous works have shown that results using Runge-Kutta 4th order (RK4) are reliable. The results can be achieved by solving the 1st order nonlinear differential equation (ODE) that is found commonly in Riccati differential equation. Multiagent NN shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over Mabood et al. (2013), RK-4, and the existing neuromethod (NM). Numerical examples are discussed to illustrate the proposed method.

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Legendre Neural Network Method for Several Classes of Singularly Perturbed Differential Equations Based on Mapping and Piecewise Optimization Technology

Liu

Xing

Wang

et al. 2020

Neural Process Lett

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Solution of generalized matrix Riccati differential equation for indefinite stochastic linear quadratic singular fuzzy system with cross-term using neural networks

Cited by 3 publications

References 28 publications

Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation

Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation

A Multiagent Transfer Function Neuroapproach to Solve Fuzzy Riccati Differential Equations

Legendre Neural Network Method for Several Classes of Singularly Perturbed Differential Equations Based on Mapping and Piecewise Optimization Technology

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