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 . .

TY  - JOUR
AU  - Li, Wei
AU  - Huang, Dongmei
AU  - Zhang, Meiting
AU  - Trišović, Nataša
AU  - Zhao, Junfeng
PY  - 2019
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3145
AB  - Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker-Planck-Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods.
PB  - Pergamon-Elsevier Science Ltd, Oxford
T2  - Chaos Solitons & Fractals
T1  - Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller
EP  - 38
SP  - 30
VL  - 121
DO  - 10.1016/j.chaos.2019.01.026
ER  - 

@article{
author = "Li, Wei and Huang, Dongmei and Zhang, Meiting and Trišović, Nataša and Zhao, Junfeng",
year = "2019",
abstract = "Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker-Planck-Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods.",
publisher = "Pergamon-Elsevier Science Ltd, Oxford",
journal = "Chaos Solitons & Fractals",
title = "Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller",
pages = "38-30",
volume = "121",
doi = "10.1016/j.chaos.2019.01.026"
}

Li, W., Huang, D., Zhang, M., Trišović, N.,& Zhao, J.. (2019). Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller. in Chaos Solitons & Fractals
Pergamon-Elsevier Science Ltd, Oxford., 121, 30-38.
https://doi.org/10.1016/j.chaos.2019.01.026

Li W, Huang D, Zhang M, Trišović N, Zhao J. Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller. in Chaos Solitons & Fractals. 2019;121:30-38.
doi:10.1016/j.chaos.2019.01.026 .

Li, Wei, Huang, Dongmei, Zhang, Meiting, Trišović, Nataša, Zhao, Junfeng, "Bifurcation control of a generalized VDP system driven by color-noise excitation via FOPID controller" in Chaos Solitons & Fractals, 121 (2019):30-38,
https://doi.org/10.1016/j.chaos.2019.01.026 . .