Reaction-diffusion systems are widely used to describe pattern formation,and various control strategies have been applied to reaction-diffusion systems to achieve control objectives such as boundary control,output fee...Reaction-diffusion systems are widely used to describe pattern formation,and various control strategies have been applied to reaction-diffusion systems to achieve control objectives such as boundary control,output feedback stabilization,and synchronization.However,controlling pattern dynamics in reaction-diffusion systems with fractional-order diffusion remains an unresolved problem.This paper presents a proportional-derivative(PD)control strategy for the Schnakenberg system with fractional-order diffusion and cross-diffusion.Theoretical analysis explores the amplitude equation near the Turing bifurcation threshold,determining the selection and stability of pattern formations.Numerical simulations demonstrate that the PD controller accomplishes the modification of pattern structures and suppression of Turing instability by adjusting only two control parameters.Additionally,it is found that for smaller fractional diffusion order,the region can accommodate more hexagonal and stripe patterns in space.This work contributes to the control of complex pattern dynamics and offers a new approach to enhancing stability in fractional reaction-diffusion systems.展开更多
A novel repetitive control strategy for the output waveform of single-phase CVCF inverters is presented. In this scheme, the inverse transfer function of inverter is used as a compensator to obtain stable and satisfy ...A novel repetitive control strategy for the output waveform of single-phase CVCF inverters is presented. In this scheme, the inverse transfer function of inverter is used as a compensator to obtain stable and satisfy harmonic rejection. Besides, PD controller is adopted to improve transient performance. Simulation and experimental results, which are gotten from a DSP-based 400Hz, 5.5KW inverter, indicate that the proposed control scheme can achieve not only low THD during steady-state operation but also fast transient response during load step change.展开更多
This paper proposes an adaptive nonlinear proportional-derivative(ANPD)controller for a two-wheeled self-balancing robot(TWSB)modeled by the Lagrange equation with external forces.The proposed control scheme is design...This paper proposes an adaptive nonlinear proportional-derivative(ANPD)controller for a two-wheeled self-balancing robot(TWSB)modeled by the Lagrange equation with external forces.The proposed control scheme is designed based on the combination of a nonlinear proportional-derivative(NPD)controller and a genetic algorithm,in which the proportional-derivative(PD)parameters are updated online based on the tracking error and the preset error threshold.In addition,the genetic algorithm is employed to adaptively select initial controller parameters,contributing to system stability and improved control accuracy.The proposed controller is basic in design yet simple to implement.The ANPD controller has the advantage of being computationally lightweight and providing high robustness against external forces.The stability of the closed-loop system is rigorously analyzed and verified using Lyapunov theory,providing theoretical assurance of its robustness.Simulations and experimental results show that the TWSB robot with the proposed ANPD controller achieves quick balance and tracks target values with very small errors,demonstrating the effectiveness and performance of the proposed controller.The proposed ANPD controller demonstrates significant improvements in balancing and tracking performance for two-wheeled self-balancing robots,which has great applicability in the field of robot control systems.This represents a promising solution for applications requiring precise and stable motion control under varying external conditions.展开更多
In this paper, a decentralized proportional-derivative （PD） controller design for non-uniform motion of a Hamiltonian hybrid system is considered. A Hamiltonian hybrid system with the capability of producing a non-u...In this paper, a decentralized proportional-derivative （PD） controller design for non-uniform motion of a Hamiltonian hybrid system is considered. A Hamiltonian hybrid system with the capability of producing a non-uniform motion is developed. The structural properties of the system are investigated by means of the theory of Hamiltonian systems. A relationship between the parameters of the system and the parameters of the proposed decentralized PD controller is shown to ensure local stability and tracking performance. Simulation results are included to show the obtained non-uniform motion.展开更多
A novel type of control law was adopted to reduce the vertical acceleration of a fast ferry as well as the motion sickness incidence suffered by the passengers onboard by means of a submerged T-foil.Considering the sy...A novel type of control law was adopted to reduce the vertical acceleration of a fast ferry as well as the motion sickness incidence suffered by the passengers onboard by means of a submerged T-foil.Considering the system changing characteristics under high disturbances,a model-free approach was adopted.In addition,an upgraded proportional-derivative(PD)controller with correction terms resulting from a fast-online estimation of the system dynamics was designed.The overall controller,known as intelligent PD(i-PD)controller,was tested,and the obtained results were compared with those of a classic PD controller.The controllers were also tested in a changing environment and at different operating velocities.The results confirmed the effectiveness of the i-PD controller to smooth the motions with low computational cost control schemes.Furthermore,thanks to ability of the i-PD controller to continually update the estimated dynamics of the system,it showed a better reduction in both vertical motions and the seasickness level of the passengers with the needed robustness under external disturbances and system changing parameters.展开更多
In conjunction with a second order uncertain nonlinear system, this paper makes some comparisons between PID control and general-integral-proportional-derivative (GI-PD) control;that is, by Routh’s stability criterio...In conjunction with a second order uncertain nonlinear system, this paper makes some comparisons between PID control and general-integral-proportional-derivative (GI-PD) control;that is, by Routh’s stability criterion, we demonstrate that the system matrix under GI-PD control can be stabilized more easily;by linear system theory and Lyapunov method, we demonstrate that GI-PD control can deal with the uncertain nonlinearity more effectively;by analyzing and comparing the integral control action, we demonstrate that GI-PD control has the better control performance. Design example and simulation results verify the justification of our conclusions again. All these mean that GI-PD control has the stronger robustness and higher control performance than PID control. Consequently, GI-PD control has broader application prospects than PID control.展开更多
This paper presents the formulation of novel implementation method based on parameter varying PD controller for fuzzy servo controllers. This formulation uses the approximation of fuzzy nonlinear function including er...This paper presents the formulation of novel implementation method based on parameter varying PD controller for fuzzy servo controllers. This formulation uses the approximation of fuzzy nonlinear function including error and error derivation in operation point. Obtained fuzzy control law has been employed to control angular position of servo using digital control technique applied to a typical microcontroller like AVR. The performance and robustness of modified fuzzy controller in comparison with PID controller evaluated in no load, applied external disturbance with different magnitude conditions has been studied. The simulation results showed that the proposed fuzzy controller has a considerable advantage in rise time, settling time and overshoot respect to PID controller when the servo system encounters with nonlinear features like saturation and friction.展开更多
This paper presents the model of a SVC （Static VAR Compensator） which is controlled externally by a PI （Proportional Integral） ＆ PD （Proportional Differential） controllers for the improvements of voltage stabil...This paper presents the model of a SVC （Static VAR Compensator） which is controlled externally by a PI （Proportional Integral） ＆ PD （Proportional Differential） controllers for the improvements of voltage stability and damping effect of an on line power system. Both controller parameters has been optimized by using Ziegler-Nichols close loop tuning method. Both single phase and three phase （L-L） faults have been considered in the research. In this paper, a power system network is considered which is simulated in the phasor simulation method ＆ the network is simulated in four steps; without SVC, With SVC but no externally controlled, SVC with PI controller ＆ SVC with PD controller. Simulation result shows that without SVC, the system parameters become unstable during faults. When SVC is imposed in the network, then system parameters become stable. Again, when SVC is controlled externally by PI ＆ PD controllers, then system parameters becomes stable in faster way then without controller. It has been observed that the SVC ratings are only 50 MVA with controllers and 200 MVA without controllers. So, SVC with PI ＆ PD controllers are more effective to enhance the voltage stability and increases power transmission capacity of a power system. The power system oscillations are also reduced with controllers in compared to that of without controllers. So with both controllers the system performance is greatly enhanced.展开更多
目的系统评价程序性死亡蛋白-1(PD-1)抑制剂联合化疗对比单纯化疗治疗晚期食管鳞状细胞癌的有效性与安全性。方法计算机检索截至2025年4月,PubMed、Web of Science、EMBASE、Cochrane图书馆、中国学术期刊全文数据库(CNKI)、万方数据库(...目的系统评价程序性死亡蛋白-1(PD-1)抑制剂联合化疗对比单纯化疗治疗晚期食管鳞状细胞癌的有效性与安全性。方法计算机检索截至2025年4月,PubMed、Web of Science、EMBASE、Cochrane图书馆、中国学术期刊全文数据库(CNKI)、万方数据库(Wanfang Data)中评估PD-1抑制剂联合化疗对比单纯化疗治疗晚期食管鳞状细胞癌的随机对照试验(RCTs)。采用RevMan和Stata软件计算总生存期(OS)、无进展生存期(PFS)的合并风险比(HR),以及客观缓解率(ORR)、疾病控制率(DCR)和≥3级不良事件的相对风险比(RR)。结果共纳入15项RCTs。与单纯化疗相比,PD-1抑制剂联合化疗显著改善OS(HR=0.70,95%CI:0.65~0.75,P<0.00001)、PFS(HR=0.63,95%CI:0.58~0.67,P<0.00001)、ORR(RR=1.48,95%CI:1.33~1.65,P<0.00001)和DCR(RR=1.10,95%CI:1.07~1.13,P<0.00001)。但联合治疗的≥3级不良事件发生率更高(RR=1.06,95%CI:1.01~1.10,P=0.01)。结论在晚期食管鳞状细胞癌治疗中,PD-1抑制剂联合化疗较单纯化疗可显著提升生存获益和应答效果,但会增加严重不良事件。支持将PD-1抑制剂联合化疗方案作为一线选择,但需注意毒性风险的管理。展开更多
In this paper, we address the problem of bifurcation control for a delayed neuron system. By introducing a new fractional-order Proportional-Derivative(PD) feedback controller, this paper aims to control the stability...In this paper, we address the problem of bifurcation control for a delayed neuron system. By introducing a new fractional-order Proportional-Derivative(PD) feedback controller, this paper aims to control the stability and Hopf bifurcation through adjusting the control gain parameters. The order chosen in PD controller is different with that of the integer-order neuron system. Sufficient conditions for guaranteeing the stability and generating Hopf bifurcation are constructed for the controlled neuron system. Finally,numerical simulation results are illustrated to verify our theoretical derivations and the relationships between the onset of the Hopf bifurcation and the gain parameters are obtained.展开更多
In order to improve the trajectory tracking precision and reduce the synchronization error of a 6-DOF lightweight robot, nonlinear proportion-deviation （N-PD） cross-coupling synchronization control strategy based on...In order to improve the trajectory tracking precision and reduce the synchronization error of a 6-DOF lightweight robot, nonlinear proportion-deviation （N-PD） cross-coupling synchronization control strategy based on adjacent coupling error analysis is presented. The mathematical models of the robot, including kinematic model, dynamic model and spline trajectory planing, are established and verified. Since it is difficult to describe the real-time contour error of the robot for complex trajectory, the adjacent coupling error is analyzed to solve the problem. Combined with nonlinear control and coupling performance of the robot, N-PD cross-coupling synchronization controller is designed and validated by simulation analysis. A servo control experimental system which mainly consists of laser tracking system, the robot mechanical system and EtherCAT based servo control system is constructed. The synchronization error is significantly decreased and the maximum trajectory error is reduced from 0.33 mm to 0.1 mm. The effectiveness of the control algorithm is validated by the experimental results, thus the control strategy can improve the robot＇s trajectory tracking precision significantly.展开更多
Long-period pulses in near-field earthquakes lead to large displacements in the base of isolated structures.To dissipate energy in isolated structures using semi-active control,piezoelectric friction dampers（PFD） ca...Long-period pulses in near-field earthquakes lead to large displacements in the base of isolated structures.To dissipate energy in isolated structures using semi-active control,piezoelectric friction dampers（PFD） can be employed.The performance of a PFD is highly dependent on the strategy applied to adjust its contact force.In this paper,the seismic control of a benchmark isolated building equipped with PFD using PD/PID controllers is developed.Using genetic algorithms,these controllers are optimized to create a balance between the performance and robustness of the closed-loop structural system.One advantage of this technique is that the controller forces can easily be estimated.In addition,the structure is equipped with only a single sensor at the base floor to measure the base displacement.Considering seven pairs of earthquakes and nine performance indices,the performance of the closed-loop system is evaluated.Then,the results are compared with those given by two well-known methods：the maximum possive operation of piezoelectric friction dampers and LQG controllers.The simulation results show that the proposed controllers perform better than the others in terms of simultaneous reduction of floor acceleration and maximum displacement of the isolator.Moreover,they are able to reduce the displacement of the isolator systems for different earthquakes without losing the advantages of isolation.展开更多
A stochastic averaging technique is proposed to study the randomly excited single-degree-of-freedom(SDOF) strongly nonlinear systems with delayed feedback fractional-order proportional-derivative(PD) controller. The d...A stochastic averaging technique is proposed to study the randomly excited single-degree-of-freedom(SDOF) strongly nonlinear systems with delayed feedback fractional-order proportional-derivative(PD) controller. The delayed feedback fractional-order PD control force is approximated by an equivalent non-delay feedback control force combining with a quasi-linear elastic force and a quasi-linear damping force. The averaged It? stochastic differential equation for amplitude of the equivalent nonlinear system is derived by the generalized harmonic functions. The analytical stationary probability density function(PDF) is obtained with solving the reduced Fokker-Planck-Kolmogorov(FPK) equation. Two examples of van der Pol oscillator and RayleighDuffing oscillator are studied to illustrate the application and effectiveness of the proposed method. Numerical results display that the proposed method can yield to the high precision, and the time delay could ruin the control effectiveness, but also even amplifies the response of the system more than that of uncontrolled system. Furthermore, the study finds that the parameters of fractional-order α and time delay may cause the stochastic P-bifurcation. It is indicated that the delayed feedback fractional-order PD controller can offer a potentially effective tool for anti-control of stochastic bifurcation展开更多
基金supported by the National Natural Science Foundation of China(62073172)the Natural Science Foundation of Jiangsu Province of China(BK20221329)。
文摘Reaction-diffusion systems are widely used to describe pattern formation,and various control strategies have been applied to reaction-diffusion systems to achieve control objectives such as boundary control,output feedback stabilization,and synchronization.However,controlling pattern dynamics in reaction-diffusion systems with fractional-order diffusion remains an unresolved problem.This paper presents a proportional-derivative(PD)control strategy for the Schnakenberg system with fractional-order diffusion and cross-diffusion.Theoretical analysis explores the amplitude equation near the Turing bifurcation threshold,determining the selection and stability of pattern formations.Numerical simulations demonstrate that the PD controller accomplishes the modification of pattern structures and suppression of Turing instability by adjusting only two control parameters.Additionally,it is found that for smaller fractional diffusion order,the region can accommodate more hexagonal and stripe patterns in space.This work contributes to the control of complex pattern dynamics and offers a new approach to enhancing stability in fractional reaction-diffusion systems.
基金This work was supported by the National Natural Science Foundation of China (No. 50007004)
文摘A novel repetitive control strategy for the output waveform of single-phase CVCF inverters is presented. In this scheme, the inverse transfer function of inverter is used as a compensator to obtain stable and satisfy harmonic rejection. Besides, PD controller is adopted to improve transient performance. Simulation and experimental results, which are gotten from a DSP-based 400Hz, 5.5KW inverter, indicate that the proposed control scheme can achieve not only low THD during steady-state operation but also fast transient response during load step change.
文摘This paper proposes an adaptive nonlinear proportional-derivative(ANPD)controller for a two-wheeled self-balancing robot(TWSB)modeled by the Lagrange equation with external forces.The proposed control scheme is designed based on the combination of a nonlinear proportional-derivative(NPD)controller and a genetic algorithm,in which the proportional-derivative(PD)parameters are updated online based on the tracking error and the preset error threshold.In addition,the genetic algorithm is employed to adaptively select initial controller parameters,contributing to system stability and improved control accuracy.The proposed controller is basic in design yet simple to implement.The ANPD controller has the advantage of being computationally lightweight and providing high robustness against external forces.The stability of the closed-loop system is rigorously analyzed and verified using Lyapunov theory,providing theoretical assurance of its robustness.Simulations and experimental results show that the TWSB robot with the proposed ANPD controller achieves quick balance and tracks target values with very small errors,demonstrating the effectiveness and performance of the proposed controller.The proposed ANPD controller demonstrates significant improvements in balancing and tracking performance for two-wheeled self-balancing robots,which has great applicability in the field of robot control systems.This represents a promising solution for applications requiring precise and stable motion control under varying external conditions.
文摘In this paper, a decentralized proportional-derivative （PD） controller design for non-uniform motion of a Hamiltonian hybrid system is considered. A Hamiltonian hybrid system with the capability of producing a non-uniform motion is developed. The structural properties of the system are investigated by means of the theory of Hamiltonian systems. A relationship between the parameters of the system and the parameters of the proposed decentralized PD controller is shown to ensure local stability and tracking performance. Simulation results are included to show the obtained non-uniform motion.
文摘A novel type of control law was adopted to reduce the vertical acceleration of a fast ferry as well as the motion sickness incidence suffered by the passengers onboard by means of a submerged T-foil.Considering the system changing characteristics under high disturbances,a model-free approach was adopted.In addition,an upgraded proportional-derivative(PD)controller with correction terms resulting from a fast-online estimation of the system dynamics was designed.The overall controller,known as intelligent PD(i-PD)controller,was tested,and the obtained results were compared with those of a classic PD controller.The controllers were also tested in a changing environment and at different operating velocities.The results confirmed the effectiveness of the i-PD controller to smooth the motions with low computational cost control schemes.Furthermore,thanks to ability of the i-PD controller to continually update the estimated dynamics of the system,it showed a better reduction in both vertical motions and the seasickness level of the passengers with the needed robustness under external disturbances and system changing parameters.
文摘In conjunction with a second order uncertain nonlinear system, this paper makes some comparisons between PID control and general-integral-proportional-derivative (GI-PD) control;that is, by Routh’s stability criterion, we demonstrate that the system matrix under GI-PD control can be stabilized more easily;by linear system theory and Lyapunov method, we demonstrate that GI-PD control can deal with the uncertain nonlinearity more effectively;by analyzing and comparing the integral control action, we demonstrate that GI-PD control has the better control performance. Design example and simulation results verify the justification of our conclusions again. All these mean that GI-PD control has the stronger robustness and higher control performance than PID control. Consequently, GI-PD control has broader application prospects than PID control.
文摘This paper presents the formulation of novel implementation method based on parameter varying PD controller for fuzzy servo controllers. This formulation uses the approximation of fuzzy nonlinear function including error and error derivation in operation point. Obtained fuzzy control law has been employed to control angular position of servo using digital control technique applied to a typical microcontroller like AVR. The performance and robustness of modified fuzzy controller in comparison with PID controller evaluated in no load, applied external disturbance with different magnitude conditions has been studied. The simulation results showed that the proposed fuzzy controller has a considerable advantage in rise time, settling time and overshoot respect to PID controller when the servo system encounters with nonlinear features like saturation and friction.
文摘This paper presents the model of a SVC （Static VAR Compensator） which is controlled externally by a PI （Proportional Integral） ＆ PD （Proportional Differential） controllers for the improvements of voltage stability and damping effect of an on line power system. Both controller parameters has been optimized by using Ziegler-Nichols close loop tuning method. Both single phase and three phase （L-L） faults have been considered in the research. In this paper, a power system network is considered which is simulated in the phasor simulation method ＆ the network is simulated in four steps; without SVC, With SVC but no externally controlled, SVC with PI controller ＆ SVC with PD controller. Simulation result shows that without SVC, the system parameters become unstable during faults. When SVC is imposed in the network, then system parameters become stable. Again, when SVC is controlled externally by PI ＆ PD controllers, then system parameters becomes stable in faster way then without controller. It has been observed that the SVC ratings are only 50 MVA with controllers and 200 MVA without controllers. So, SVC with PI ＆ PD controllers are more effective to enhance the voltage stability and increases power transmission capacity of a power system. The power system oscillations are also reduced with controllers in compared to that of without controllers. So with both controllers the system performance is greatly enhanced.
文摘目的系统评价程序性死亡蛋白-1(PD-1)抑制剂联合化疗对比单纯化疗治疗晚期食管鳞状细胞癌的有效性与安全性。方法计算机检索截至2025年4月,PubMed、Web of Science、EMBASE、Cochrane图书馆、中国学术期刊全文数据库(CNKI)、万方数据库(Wanfang Data)中评估PD-1抑制剂联合化疗对比单纯化疗治疗晚期食管鳞状细胞癌的随机对照试验(RCTs)。采用RevMan和Stata软件计算总生存期(OS)、无进展生存期(PFS)的合并风险比(HR),以及客观缓解率(ORR)、疾病控制率(DCR)和≥3级不良事件的相对风险比(RR)。结果共纳入15项RCTs。与单纯化疗相比,PD-1抑制剂联合化疗显著改善OS(HR=0.70,95%CI:0.65~0.75,P<0.00001)、PFS(HR=0.63,95%CI:0.58~0.67,P<0.00001)、ORR(RR=1.48,95%CI:1.33~1.65,P<0.00001)和DCR(RR=1.10,95%CI:1.07~1.13,P<0.00001)。但联合治疗的≥3级不良事件发生率更高(RR=1.06,95%CI:1.01~1.10,P=0.01)。结论在晚期食管鳞状细胞癌治疗中,PD-1抑制剂联合化疗较单纯化疗可显著提升生存获益和应答效果,但会增加严重不良事件。支持将PD-1抑制剂联合化疗方案作为一线选择,但需注意毒性风险的管理。
基金supported by the National Natural Science Foundation of China(Grant Nos. 61573194, 51775284)the Natural Science Foundation of Jiangsu Province of China(Grant Nos. BK20181389, BK20171441)+1 种基金the Key Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu Province(Grant No. 2018SJZDI142)the Australian Research Council(Grant No. DP120104986)
文摘In this paper, we address the problem of bifurcation control for a delayed neuron system. By introducing a new fractional-order Proportional-Derivative(PD) feedback controller, this paper aims to control the stability and Hopf bifurcation through adjusting the control gain parameters. The order chosen in PD controller is different with that of the integer-order neuron system. Sufficient conditions for guaranteeing the stability and generating Hopf bifurcation are constructed for the controlled neuron system. Finally,numerical simulation results are illustrated to verify our theoretical derivations and the relationships between the onset of the Hopf bifurcation and the gain parameters are obtained.
基金Project(2015AA043003)supported by National High-technology Research and Development Program of ChinaProject(GY2016ZB0068)supported by Application Technology Research and Development Program of Heilongjiang Province,ChinaProject(SKLR201301A03)supported by Self-planned Task of State Key Laboratory of Robotics and System(Harbin Institute of Technology),China
文摘In order to improve the trajectory tracking precision and reduce the synchronization error of a 6-DOF lightweight robot, nonlinear proportion-deviation （N-PD） cross-coupling synchronization control strategy based on adjacent coupling error analysis is presented. The mathematical models of the robot, including kinematic model, dynamic model and spline trajectory planing, are established and verified. Since it is difficult to describe the real-time contour error of the robot for complex trajectory, the adjacent coupling error is analyzed to solve the problem. Combined with nonlinear control and coupling performance of the robot, N-PD cross-coupling synchronization controller is designed and validated by simulation analysis. A servo control experimental system which mainly consists of laser tracking system, the robot mechanical system and EtherCAT based servo control system is constructed. The synchronization error is significantly decreased and the maximum trajectory error is reduced from 0.33 mm to 0.1 mm. The effectiveness of the control algorithm is validated by the experimental results, thus the control strategy can improve the robot＇s trajectory tracking precision significantly.
文摘Long-period pulses in near-field earthquakes lead to large displacements in the base of isolated structures.To dissipate energy in isolated structures using semi-active control,piezoelectric friction dampers（PFD） can be employed.The performance of a PFD is highly dependent on the strategy applied to adjust its contact force.In this paper,the seismic control of a benchmark isolated building equipped with PFD using PD/PID controllers is developed.Using genetic algorithms,these controllers are optimized to create a balance between the performance and robustness of the closed-loop structural system.One advantage of this technique is that the controller forces can easily be estimated.In addition,the structure is equipped with only a single sensor at the base floor to measure the base displacement.Considering seven pairs of earthquakes and nine performance indices,the performance of the closed-loop system is evaluated.Then,the results are compared with those given by two well-known methods：the maximum possive operation of piezoelectric friction dampers and LQG controllers.The simulation results show that the proposed controllers perform better than the others in terms of simultaneous reduction of floor acceleration and maximum displacement of the isolator.Moreover,they are able to reduce the displacement of the isolator systems for different earthquakes without losing the advantages of isolation.
基金supported by the National Natural Science Foundation of China(Grant Nos.11672111,11432012,11602089)the Program for New Century Excellent Talents in Fujian Province University,and the Technological Project of Huaqiao University(Grant Nos.ZQN-YX307,ZQNYX505)
文摘A stochastic averaging technique is proposed to study the randomly excited single-degree-of-freedom(SDOF) strongly nonlinear systems with delayed feedback fractional-order proportional-derivative(PD) controller. The delayed feedback fractional-order PD control force is approximated by an equivalent non-delay feedback control force combining with a quasi-linear elastic force and a quasi-linear damping force. The averaged It? stochastic differential equation for amplitude of the equivalent nonlinear system is derived by the generalized harmonic functions. The analytical stationary probability density function(PDF) is obtained with solving the reduced Fokker-Planck-Kolmogorov(FPK) equation. Two examples of van der Pol oscillator and RayleighDuffing oscillator are studied to illustrate the application and effectiveness of the proposed method. Numerical results display that the proposed method can yield to the high precision, and the time delay could ruin the control effectiveness, but also even amplifies the response of the system more than that of uncontrolled system. Furthermore, the study finds that the parameters of fractional-order α and time delay may cause the stochastic P-bifurcation. It is indicated that the delayed feedback fractional-order PD controller can offer a potentially effective tool for anti-control of stochastic bifurcation
