TY  - JOUR
AU  - Li, Wei
AU  - Guan, Yu
AU  - Huang, Dongmei
AU  - Trišović, Nataša
PY  - 2023
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/6880
AB  - Solving the Fokker–Planck–Kolmogorov (FPK) equation and the Backward-Kolmogorov (BK) equation is a crucial task to obtain the transient response of stochastic dynamical systems. Fractional order PID (FOPID) is a new efficient controller to change the system response to be the expected one. Therefore, in this paper, the Gaussian Radial Basis Functions Neural Network (RBFNN) is proposed to solve FPK and BK equations, to obtain the transient probability density function and the reliability function for a generalized Van der Pol system under a FOPID controller. The values of the different fractional orders are analyzed to discuss the performance of the FOPID controller. A data collection strategy is adopted to deal with the associated boundary conditions by way of a one-time Monte-Carlo simulation and uniform distribution in our Gaussian RBFNN method. The advantage of this method is that the solution process of FPK and BK equations is converted into solving algebraic equations. Numerical results with regard to the transient system response prove that the Gaussian RBFNN is efficient and accurate in getting the solutions of FPK and BK equations. The order of the fractional integration and the fractional derivative are critical parameters to control the system response. Moreover, we conclude that the fractional order parameters in a FOPID controller can indeed enhance the system’s response to a certain extent and lead to bifurcation.
PB  - Elsevier
T2  - International Journal of Non-Linear Mechanics
T1  - Gaussian RBFNN method for solving FPK and BK equations in stochastic dynamical system with FOPID controller
SP  - 104403
VL  - 153
DO  - 10.1016/j.ijnonlinmec.2023.104403
ER  - 

@article{
author = "Li, Wei and Guan, Yu and Huang, Dongmei and Trišović, Nataša",
year = "2023",
abstract = "Solving the Fokker–Planck–Kolmogorov (FPK) equation and the Backward-Kolmogorov (BK) equation is a crucial task to obtain the transient response of stochastic dynamical systems. Fractional order PID (FOPID) is a new efficient controller to change the system response to be the expected one. Therefore, in this paper, the Gaussian Radial Basis Functions Neural Network (RBFNN) is proposed to solve FPK and BK equations, to obtain the transient probability density function and the reliability function for a generalized Van der Pol system under a FOPID controller. The values of the different fractional orders are analyzed to discuss the performance of the FOPID controller. A data collection strategy is adopted to deal with the associated boundary conditions by way of a one-time Monte-Carlo simulation and uniform distribution in our Gaussian RBFNN method. The advantage of this method is that the solution process of FPK and BK equations is converted into solving algebraic equations. Numerical results with regard to the transient system response prove that the Gaussian RBFNN is efficient and accurate in getting the solutions of FPK and BK equations. The order of the fractional integration and the fractional derivative are critical parameters to control the system response. Moreover, we conclude that the fractional order parameters in a FOPID controller can indeed enhance the system’s response to a certain extent and lead to bifurcation.",
publisher = "Elsevier",
journal = "International Journal of Non-Linear Mechanics",
title = "Gaussian RBFNN method for solving FPK and BK equations in stochastic dynamical system with FOPID controller",
pages = "104403",
volume = "153",
doi = "10.1016/j.ijnonlinmec.2023.104403"
}

Li, W., Guan, Y., Huang, D.,& Trišović, N.. (2023). Gaussian RBFNN method for solving FPK and BK equations in stochastic dynamical system with FOPID controller. in International Journal of Non-Linear Mechanics
Elsevier., 153, 104403.
https://doi.org/10.1016/j.ijnonlinmec.2023.104403

Li W, Guan Y, Huang D, Trišović N. Gaussian RBFNN method for solving FPK and BK equations in stochastic dynamical system with FOPID controller. in International Journal of Non-Linear Mechanics. 2023;153:104403.
doi:10.1016/j.ijnonlinmec.2023.104403 .

Li, Wei, Guan, Yu, Huang, Dongmei, Trišović, Nataša, "Gaussian RBFNN method for solving FPK and BK equations in stochastic dynamical system with FOPID controller" in International Journal of Non-Linear Mechanics, 153 (2023):104403,
https://doi.org/10.1016/j.ijnonlinmec.2023.104403 . .

TY  - JOUR
AU  - Li, Wei
AU  - Guan, Yu
AU  - Huang, Dongmei
AU  - Trišović, Nataša
PY  - 2023
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/6881
AB  - Solving the Fokker–Planck–Kolmogorov (FPK) equation and the Backward-Kolmogorov (BK) equation is a crucial task to obtain the transient response of stochastic dynamical systems. Fractional order PID (FOPID) is a new efficient controller to change the system response to be the expected one. Therefore, in this paper, the Gaussian Radial Basis Functions Neural Network (RBFNN) is proposed to solve FPK and BK equations, to obtain the transient probability density function and the reliability function for a generalized Van der Pol system under a FOPID controller. The values of the different fractional orders are analyzed to discuss the performance of the FOPID controller. A data collection strategy is adopted to deal with the associated boundary conditions by way of a one-time Monte-Carlo simulation and uniform distribution in our Gaussian RBFNN method. The advantage of this method is that the solution process of FPK and BK equations is converted into solving algebraic equations. Numerical results with regard to the transient system response prove that the Gaussian RBFNN is efficient and accurate in getting the solutions of FPK and BK equations. The order of the fractional integration and the fractional derivative are critical parameters to control the system response. Moreover, we conclude that the fractional order parameters in a FOPID controller can indeed enhance the system’s response to a certain extent and lead to bifurcation.
PB  - Elsevier
T2  - Communications in Nonlinear Science and Numerical Simulation
T1  - Two methods for studying the response and the reliability of a fractional stochastic dynamical system
SP  - 107144
VL  - 120
DO  - 10.1016/j.cnsns.2023.107144
ER  - 

@article{
author = "Li, Wei and Guan, Yu and Huang, Dongmei and Trišović, Nataša",
year = "2023",
abstract = "Solving the Fokker–Planck–Kolmogorov (FPK) equation and the Backward-Kolmogorov (BK) equation is a crucial task to obtain the transient response of stochastic dynamical systems. Fractional order PID (FOPID) is a new efficient controller to change the system response to be the expected one. Therefore, in this paper, the Gaussian Radial Basis Functions Neural Network (RBFNN) is proposed to solve FPK and BK equations, to obtain the transient probability density function and the reliability function for a generalized Van der Pol system under a FOPID controller. The values of the different fractional orders are analyzed to discuss the performance of the FOPID controller. A data collection strategy is adopted to deal with the associated boundary conditions by way of a one-time Monte-Carlo simulation and uniform distribution in our Gaussian RBFNN method. The advantage of this method is that the solution process of FPK and BK equations is converted into solving algebraic equations. Numerical results with regard to the transient system response prove that the Gaussian RBFNN is efficient and accurate in getting the solutions of FPK and BK equations. The order of the fractional integration and the fractional derivative are critical parameters to control the system response. Moreover, we conclude that the fractional order parameters in a FOPID controller can indeed enhance the system’s response to a certain extent and lead to bifurcation.",
publisher = "Elsevier",
journal = "Communications in Nonlinear Science and Numerical Simulation",
title = "Two methods for studying the response and the reliability of a fractional stochastic dynamical system",
pages = "107144",
volume = "120",
doi = "10.1016/j.cnsns.2023.107144"
}

Li, W., Guan, Y., Huang, D.,& Trišović, N.. (2023). Two methods for studying the response and the reliability of a fractional stochastic dynamical system. in Communications in Nonlinear Science and Numerical Simulation
Elsevier., 120, 107144.
https://doi.org/10.1016/j.cnsns.2023.107144

Li W, Guan Y, Huang D, Trišović N. Two methods for studying the response and the reliability of a fractional stochastic dynamical system. in Communications in Nonlinear Science and Numerical Simulation. 2023;120:107144.
doi:10.1016/j.cnsns.2023.107144 .

Li, Wei, Guan, Yu, Huang, Dongmei, Trišović, Nataša, "Two methods for studying the response and the reliability of a fractional stochastic dynamical system" in Communications in Nonlinear Science and Numerical Simulation, 120 (2023):107144,
https://doi.org/10.1016/j.cnsns.2023.107144 . .