考虑间隙反馈控制时滞的磁浮车辆稳定性研究
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摘要
常导磁吸型(EMS)磁悬浮列车在悬浮控制中的每个环节,时滞是不可避免的,当时滞超过一定程度后,系统有可能失稳.本文针对EMS磁浮列车控制环节的临界时滞与车辆参数(如运行速度、反馈控制增益、导轨参数和悬挂参数)的关系开展研究.建立了磁浮车辆/导轨耦合动力学模型,车辆包含1节车辆和4个磁浮架,考虑车辆的10个自由度,每个磁浮架上包含4个悬浮电磁铁.导轨模拟为一系列简支Bernoulli-Euler梁,采用模态叠加法对导轨振动方程进行求解.采用传统线性电磁力模型实现车辆和轨道的耦合.采用比例–微分控制算法对电磁铁电流进行反馈控制,实现车辆稳定悬浮,并假设时滞均发生在控制环节,且只考虑间隙反馈控制环节的时滞.采用四阶龙格库塔法对耦合系统动力学方程进行求解,编写了数值仿真程序,计算得到车辆导轨耦合系统在考虑间隙反馈控制时滞时的响应.将系统运动发散时的时滞大小视为临界时滞,开展了参数规律影响分析.通过分析,给出了提高时滞条件下车辆稳定性的方法,包括增大导轨的弯曲刚度和阻尼比,减小间隙反馈控制增益并增大速度反馈控制增益,以及增大二系悬挂阻尼.
Because of the inherent instability, EMS maglev train requires the application of active control to achieve a stable suspension. In every suspension control loop, there is inevitable time-delay, which may affect the performance of the system. When the time-delay exceeds a critical value, the maglev train will become unstable. Previous studies have proved the existence of the critical time-delay. The relationship between critical time delay in control loop and vehicle parameters(including vehicle speed, position feedback control gains, guideway parameters, and suspension parameters) is studied in this paper. A dynamic model of a maglev vehicle/guideway coupling system is established. The 10 DOF vehicle includes a carbody and four maglev frames. Each maglev frame contains four electromagnets. The guideway is modelled as a series of continuous simply supported Bernoulli-Euler beams. The vibration equation of the guideway is obtained by a modal superposition method. In order to achieve vehicle/guideway coupling, a conventional electromagnetic force model which is linearized in the neighborhood of stable suspension position is adopted. The fourth-order Runge-Kutta method is used to obtain the dynamic response of the vehicle/guideway coupling system. A proportional derivative(PD) control algorithm is used for the feedback control of the electromagnet current. To facilitate the analysis, this paper assumes that all the time-delay occurs in the control loop, and that only the position feedback control loop exists. In order to analyze the motion property when critical delay occurs, we write a simulation program. Using the program, the dynamic responses of maglev vehicle considering different position feedback control delay are calculated. The delay value which results in motion divergence is defined as critical time-delay. Based on the calculations and analysis, following suggestions to promote the stability and weaken the effect of time-delay in position feedback control loop are provided: Enlarge bending rigidity and damping ratio of the guideway; Reduce position feedback control gain; Enlarge velocity feedback control gain; Enlarge second suspension damping. In addition, in view of that the critical time-delay of a stationary vehicle is always smaller than that of a running vehicle, hence, the critical time-delay of stationary vehicle shall be considered as safety limit value.
Because of the inherent instability, EMS maglev train requires the application of active control to achieve a stable suspension. In every suspension control loop, there is inevitable time-delay, which may affect the performance of the system. When the time-delay exceeds a critical value, the maglev train will become unstable. Previous studies have proved the existence of the critical time-delay. The relationship between critical time delay in control loop and vehicle parameters(including vehicle speed, position feedback control gains, guideway parameters, and suspension parameters) is studied in this paper. A dynamic model of a maglev vehicle/guideway coupling system is established. The 10 DOF vehicle includes a carbody and four maglev frames. Each maglev frame contains four electromagnets. The guideway is modelled as a series of continuous simply supported Bernoulli-Euler beams. The vibration equation of the guideway is obtained by a modal superposition method. In order to achieve vehicle/guideway coupling, a conventional electromagnetic force model which is linearized in the neighborhood of stable suspension position is adopted. The fourth-order Runge-Kutta method is used to obtain the dynamic response of the vehicle/guideway coupling system. A proportional derivative(PD) control algorithm is used for the feedback control of the electromagnet current. To facilitate the analysis, this paper assumes that all the time-delay occurs in the control loop, and that only the position feedback control loop exists. In order to analyze the motion property when critical delay occurs, we write a simulation program. Using the program, the dynamic responses of maglev vehicle considering different position feedback control delay are calculated. The delay value which results in motion divergence is defined as critical time-delay. Based on the calculations and analysis, following suggestions to promote the stability and weaken the effect of time-delay in position feedback control loop are provided: Enlarge bending rigidity and damping ratio of the guideway; Reduce position feedback control gain; Enlarge velocity feedback control gain; Enlarge second suspension damping. In addition, in view of that the critical time-delay of a stationary vehicle is always smaller than that of a running vehicle, hence, the critical time-delay of stationary vehicle shall be considered as safety limit value.
引文
1 Lee HW,Kim KC,Lee J.Review of maglev train technologies.IEEE Transactions on Magnetics,2006,42(7):1917-1925
2翟婉明,赵春发.磁浮车辆/轨道系统动力学(I)--磁轨相互作用及稳定性.机械工程学报,2005,41(7):1-10(Zhai Wanming,Zhao Chunfa.Dynamics of maglev vehicle/guideway systems(I)-magnet/rail interaction and system stability.Chinese Journal of Mechanical Engineering,2005,41(7):1-10(in Chinese))
3时瑾,魏庆朝,万传风等.随机不平顺激励下磁浮车辆轨道梁动力响应.力学学报,2006,38(6):850-857(Shi Jin,Wei Qingchao,Wan Chuanfeng,et al.Dynamic response of maglev vehicle track beam under random irregularity.Chinese Journal of Theoretical and Applied Mechanics,2006,38(6):850-857(in Chinese))
4 Zhao CF,Zhai WM.Maglev vehicle guideway vertical random response and ride quality.Vehicle System Dynamics,2002,38(3):185-210
5 Wu H,Zeng XH,Yu Y.Motion stability of high-speed maglev systems in consideration of aerodynamic effect:A study of a single magnetic suspension system.Acta Mechanica Sinica,2017,33(6):1084-1094
6 Han HS,Yim BH,Lee JK,et al.Effects of guideway’s vibration characteristics on the dynamics of a maglev vehicle.Vehicle System Dynamics,2009,47(3):309-324
7 Han HS,Yim BH,Lee NJ,et al.Prediction of ride quality of a Maglev vehicle using a full vehicle multi-body dynamic model.Vehicle System Dynamics,2009,47(10):1271-1286
8 Yim BH,Han HS,Lee JK,et al.Curving performance simulation of an EMS-type maglev vehicle.Vehicle System Dynamics,2009,47(10):1287-1304
9 Lee JS,Kwon SD,Kim MY,et al.A parametric study on the dynamics of urban transit maglev vehicle running on flexible guideway bridges.Journal of Sound and Vibration,2009,328(3):301-317
10 Kim KJ,Han JB,Han HS,et al.Coupled vibration analysis of Maglev vehicle-guideway while standing still or moving at low speeds.Vehicle System Dynamics,2015,53(4):587-601
11 Zhou DF,Hansen CH,Li J.Suppression of maglev vehicle-girder self-excited vibration using a virtual tuned mass damper.Journal of Sound and Vibration,2011,330(5):883-901
12 Zhou DF,Li J,Hansen CH.Application of least mean square algorithm to suppression of maglev track-induced self-excited vibration.Journal of Sound and Vibration,2011,330(24):5791-5811
13 Zhou DF,Li J,Zhang K.Amplitude control of the track-induced self-excited vibration for a maglev system.ISA Transactions,2014,53(5):1463-1469
14 Li JH,Li J,Zhou DF,et al.The active control of maglev stationary self-excited vibration with a virtual energy harvester.IEEE Transactions on Industrial Electronics,2015,62(5):2942-2951
15王洪坡.EMS型低速词素列车/轨道系统的动力相互作用问题研究.[博士论文].长沙:国防科技大学,2007(Wang Hongpo.Vehicle-Guideway Dynamic Interaction of the EMS Low Speed Maglev Vehicle.[PhD Thesis].Changsha:National University of Defense Technology,2007(in Chinese))
16 Xu J Q,Chen C,Gao D G,et al.Nonlinear dynamic analysis on maglev train system with flexible guideway and double time-delay feedback control.Journal of Vibroengineering,2017,19(8):6346-6362
17张舒,徐鉴.时滞耦合系统非线性动力学的研究进展.力学学报,2017,49(3):565-587(Zhang Shu,Xu Jian.Review on nonlinear dynamics in system with coupling delays.Chinese Journal of Theoretical Applied Mechnaics,2017,49(3):565-587(in Chinese))
18李帅,周继磊,任传波等.时变参数时滞减振控制研究.力学学报,2018,50(1):99-108.(Li Shuai,Zhou Jilei,Ren Chuanbo,et al.The research of time delay vibration control with time-varying parameters.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):99-108(in Chinese))
19公徐路,许鹏飞.含时滞反馈与涨落质量的记忆阻尼系统的随机共振.力学学报,2018,50(4):880-889(Gong Xulu,Xu Pengfei.Stochastic resonance of a memorial-damped system with time delay feedback and fluctuating mass.Chinese Journal of Theoretical and Applied Mechanics,2018,50(4):880-889(in Chinese))
20 Li JH,Li J,Zhou DF,et al.Self-excited vibration problems of maglev vehicle-bridge interaction system.Journal of Central South University,2014,21(11):4184-4192
21 Wang HP,Li J,Zhang K.Stability and Hopf bifurcation of the maglev system with delayed speed feedback control.Acta Automatica Sinica,2007,33(8):829-834
22 Wang HP,Li J,Zhang K.Non-resonant response,bifurcation and oscillation suppression of a non-autonomous system with delayed position feedback control.Nonlinear Dynamics,2008,51(3):447-464
23 Wang HP,Li J,Zhang K.Sup-resonant response of a nonautonomous maglev system with delayed acceleration feedback control.IEEE Transactions on Magnetics,2008,44(10):2338-2350
24 Zhang LL,Campbell SA,Huang LH.Nonlinear analysis of a maglev system with time-delayed feedback control.Physica D-Nonlinear Phenomena,2011,240(21):1761-1770
25 Zhang LL,Huang LH,Zhang ZZ.Stability and Hopf bifurcation of the maglev system with delayed position and speed feedback control.Nonlinear Dynamics,2009,57(1-2):197-207
26 Zhang LL,Zhang ZZ,Huang LH.Double Hopf bifurcation of timedelayed feedback control for maglev system.Nonlinear Dynamics,2012,69(3):961-967
27 Zhang ZZ,Zhang LL.Hopf bifurcation of time-delayed feedback control for maglev system with flexible guideway.Applied Mathematics&Computation,2013,219(11):6106-6112
28 Zhang LL,Huang JH,Huang LH,et al.Stability and bifurcation analysis in a maglev system with multiple delays.International Journal of Bifurcation&Chaos,2015,25(5):1550074-1-1550074-9
29 Zhang LL,Huang LH,Zhang ZZ.Hopf bifurcation of the maglev time-delay feedback system via pseudo-oscillator analysis.Mathematical and Computer Modelling,2010,52(5):667-673
2翟婉明,赵春发.磁浮车辆/轨道系统动力学(I)--磁轨相互作用及稳定性.机械工程学报,2005,41(7):1-10(Zhai Wanming,Zhao Chunfa.Dynamics of maglev vehicle/guideway systems(I)-magnet/rail interaction and system stability.Chinese Journal of Mechanical Engineering,2005,41(7):1-10(in Chinese))
3时瑾,魏庆朝,万传风等.随机不平顺激励下磁浮车辆轨道梁动力响应.力学学报,2006,38(6):850-857(Shi Jin,Wei Qingchao,Wan Chuanfeng,et al.Dynamic response of maglev vehicle track beam under random irregularity.Chinese Journal of Theoretical and Applied Mechanics,2006,38(6):850-857(in Chinese))
4 Zhao CF,Zhai WM.Maglev vehicle guideway vertical random response and ride quality.Vehicle System Dynamics,2002,38(3):185-210
5 Wu H,Zeng XH,Yu Y.Motion stability of high-speed maglev systems in consideration of aerodynamic effect:A study of a single magnetic suspension system.Acta Mechanica Sinica,2017,33(6):1084-1094
6 Han HS,Yim BH,Lee JK,et al.Effects of guideway’s vibration characteristics on the dynamics of a maglev vehicle.Vehicle System Dynamics,2009,47(3):309-324
7 Han HS,Yim BH,Lee NJ,et al.Prediction of ride quality of a Maglev vehicle using a full vehicle multi-body dynamic model.Vehicle System Dynamics,2009,47(10):1271-1286
8 Yim BH,Han HS,Lee JK,et al.Curving performance simulation of an EMS-type maglev vehicle.Vehicle System Dynamics,2009,47(10):1287-1304
9 Lee JS,Kwon SD,Kim MY,et al.A parametric study on the dynamics of urban transit maglev vehicle running on flexible guideway bridges.Journal of Sound and Vibration,2009,328(3):301-317
10 Kim KJ,Han JB,Han HS,et al.Coupled vibration analysis of Maglev vehicle-guideway while standing still or moving at low speeds.Vehicle System Dynamics,2015,53(4):587-601
11 Zhou DF,Hansen CH,Li J.Suppression of maglev vehicle-girder self-excited vibration using a virtual tuned mass damper.Journal of Sound and Vibration,2011,330(5):883-901
12 Zhou DF,Li J,Hansen CH.Application of least mean square algorithm to suppression of maglev track-induced self-excited vibration.Journal of Sound and Vibration,2011,330(24):5791-5811
13 Zhou DF,Li J,Zhang K.Amplitude control of the track-induced self-excited vibration for a maglev system.ISA Transactions,2014,53(5):1463-1469
14 Li JH,Li J,Zhou DF,et al.The active control of maglev stationary self-excited vibration with a virtual energy harvester.IEEE Transactions on Industrial Electronics,2015,62(5):2942-2951
15王洪坡.EMS型低速词素列车/轨道系统的动力相互作用问题研究.[博士论文].长沙:国防科技大学,2007(Wang Hongpo.Vehicle-Guideway Dynamic Interaction of the EMS Low Speed Maglev Vehicle.[PhD Thesis].Changsha:National University of Defense Technology,2007(in Chinese))
16 Xu J Q,Chen C,Gao D G,et al.Nonlinear dynamic analysis on maglev train system with flexible guideway and double time-delay feedback control.Journal of Vibroengineering,2017,19(8):6346-6362
17张舒,徐鉴.时滞耦合系统非线性动力学的研究进展.力学学报,2017,49(3):565-587(Zhang Shu,Xu Jian.Review on nonlinear dynamics in system with coupling delays.Chinese Journal of Theoretical Applied Mechnaics,2017,49(3):565-587(in Chinese))
18李帅,周继磊,任传波等.时变参数时滞减振控制研究.力学学报,2018,50(1):99-108.(Li Shuai,Zhou Jilei,Ren Chuanbo,et al.The research of time delay vibration control with time-varying parameters.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):99-108(in Chinese))
19公徐路,许鹏飞.含时滞反馈与涨落质量的记忆阻尼系统的随机共振.力学学报,2018,50(4):880-889(Gong Xulu,Xu Pengfei.Stochastic resonance of a memorial-damped system with time delay feedback and fluctuating mass.Chinese Journal of Theoretical and Applied Mechanics,2018,50(4):880-889(in Chinese))
20 Li JH,Li J,Zhou DF,et al.Self-excited vibration problems of maglev vehicle-bridge interaction system.Journal of Central South University,2014,21(11):4184-4192
21 Wang HP,Li J,Zhang K.Stability and Hopf bifurcation of the maglev system with delayed speed feedback control.Acta Automatica Sinica,2007,33(8):829-834
22 Wang HP,Li J,Zhang K.Non-resonant response,bifurcation and oscillation suppression of a non-autonomous system with delayed position feedback control.Nonlinear Dynamics,2008,51(3):447-464
23 Wang HP,Li J,Zhang K.Sup-resonant response of a nonautonomous maglev system with delayed acceleration feedback control.IEEE Transactions on Magnetics,2008,44(10):2338-2350
24 Zhang LL,Campbell SA,Huang LH.Nonlinear analysis of a maglev system with time-delayed feedback control.Physica D-Nonlinear Phenomena,2011,240(21):1761-1770
25 Zhang LL,Huang LH,Zhang ZZ.Stability and Hopf bifurcation of the maglev system with delayed position and speed feedback control.Nonlinear Dynamics,2009,57(1-2):197-207
26 Zhang LL,Zhang ZZ,Huang LH.Double Hopf bifurcation of timedelayed feedback control for maglev system.Nonlinear Dynamics,2012,69(3):961-967
27 Zhang ZZ,Zhang LL.Hopf bifurcation of time-delayed feedback control for maglev system with flexible guideway.Applied Mathematics&Computation,2013,219(11):6106-6112
28 Zhang LL,Huang JH,Huang LH,et al.Stability and bifurcation analysis in a maglev system with multiple delays.International Journal of Bifurcation&Chaos,2015,25(5):1550074-1-1550074-9
29 Zhang LL,Huang LH,Zhang ZZ.Hopf bifurcation of the maglev time-delay feedback system via pseudo-oscillator analysis.Mathematical and Computer Modelling,2010,52(5):667-673