基于速度反馈分数阶PID控制的达芬振子的主共振
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摘要
与传统整数阶比例-积分-微分(PID)控制器相比,分数阶比例-积分-微分控制器由于增加了两个控制参数,因此能够更灵活地控制受控对象.研究了基于速度反馈分数阶比例-积分-微分控制的达芬振子的主共振,利用平均法获得了系统的近似解析解.研究发现分数阶比例-积分-微分控制器的比例环节以等效线性阻尼的形式影响系统的共振振幅,积分环节以等效线性阻尼和等效线性刚度的形式影响系统的动力学特性,微分环节以等效线性阻尼和等效质量的形式影响系统的动力学特性.建立了主共振幅频响应方程的解析表达式和稳定性判断准则,并对主共振幅频响应的近似解析解和数值解进行了比较,二者吻合良好,验证了求解过程和近似解析解的正确性.分析了分数阶比例-积分-微分控制器的比例环节系数、积分环节系数、微分环节系数以及分数阶阶次变化时,对系统主共振幅频响应的影响.对分数阶比例-积分-微分控制器与传统整数阶比例-积分-微分控制器的控制效果进行了对比,发现当控制器各个环节的系数相同时,基于速度反馈的分数阶比例-积分-微分控制对达芬振子主共振的控制效果要优于传统整数阶比例-积分-微分控制.
Compared with the traditional integer-order PID controller, the fractional-order PID(FOPID) controller may present much better control performance due to its two surplus adjustable parameters. The primary resonance of Duffing oscillator with FOPID controller of velocity feedback is investigated by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters in FOPID controller on the dynamical properties are studied and characterized by some equivalent parameters. The proportional component of FOPID controller is characterized in the form of equivalent linear damping. The integral component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent linear stiffness. The differential component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent mass. Those equivalent parameters could distinctly illustrate the effects of the parameters in FOPID controller on the dynamical response. The amplitude-frequency equation for steady-state solution and associated with the stability condition are also presented. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximatelyanalytical results. Finally, the effects on the amplitude-frequency performance of the coefficients and the orders in FOPID controller are analyzed, and the control performances of fractional-order and integer-order PID controller are compared.The results show that the control performance of FOPID controller is better than the traditional integer-order counterpart for controlling the vibration of the primary resonance of Duffing oscillator, when the coefficients of the two controllers are the same.
Compared with the traditional integer-order PID controller, the fractional-order PID(FOPID) controller may present much better control performance due to its two surplus adjustable parameters. The primary resonance of Duffing oscillator with FOPID controller of velocity feedback is investigated by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters in FOPID controller on the dynamical properties are studied and characterized by some equivalent parameters. The proportional component of FOPID controller is characterized in the form of equivalent linear damping. The integral component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent linear stiffness. The differential component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent mass. Those equivalent parameters could distinctly illustrate the effects of the parameters in FOPID controller on the dynamical response. The amplitude-frequency equation for steady-state solution and associated with the stability condition are also presented. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximatelyanalytical results. Finally, the effects on the amplitude-frequency performance of the coefficients and the orders in FOPID controller are analyzed, and the control performances of fractional-order and integer-order PID controller are compared.The results show that the control performance of FOPID controller is better than the traditional integer-order counterpart for controlling the vibration of the primary resonance of Duffing oscillator, when the coefficients of the two controllers are the same.
引文
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11 Li CP,Deng WH.Remarks on fractional derivatives.Applied Mathematics and Computation,2007,187(1):777-784
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14 Agrawal OP.A quadratic numerical scheme for fractional optimal control problems.Journal of Dynamic Systems Measurement and Control,Transactions of the ASME,2008,130(1):011010.1-011010.6
15 Chen JH,Chen WC.Chaotic dynamics of the fractionally damped van der Pol equation.Chaos,Solitons and Fractals,2008,35(1):188-198
16 吴光强,黄焕军,叶光湖.基于分数阶微积分的汽车空气悬架半主动控制.农业机械学报,2014,45(7):19-24(Wu Guangqiang,Huang Huanjun,Ye Guanghu.Semi-active control of automotive air suspension based on fractional calculus.Transactions of the Chinese Society for Agricultural Machinery,2014,45(7):19-24(in Chinese))
17 李媛萍,张卫,欧阳东.具有分数导数本构关系的粘弹性浅拱的非线性动力学行为.振动工程学报,2012,25(3):342-350(Li Yuanping,Zhang Wei,Ouyang Dong.Nonlinear dynamic behaviors of viscoelastic shallow arch with fractional derivative constitutive relationship.Journal of Vibration Engineering,2012,25(3):342-350(in Chinese))
18 Podlubny I.Fractional-order systems and PIλDμ–controllers.IEEE Transactions on Automatic Control,1999,44(1):208-214
19 陈宁,陈南,王乃洲等.基于分数阶参考模型的车辆悬架自适应控制.南京林业大学学报,2009,33(3):247-253(Chen Ning,Chen Nan,Wang Naizhou,et al.Adaptive control of vehicle suspension using fractional order reference models.Journal of Nanjing Forestry University,2009,33(3):116-120(in Chinese))
20 赵春娜,赵雨,张祥德等.分数阶控制器与整数阶控制器仿真研究.系统仿真学报,2009,21(3):768-775(Zhao Chunna,Zhao Yu,Zhang Xiangde,et al.Simulation research on fractional order controllers with integer order controllers.Journal of System Simulation,2009,21(3):768-775(in Chinese))
21 Chen YQ,Petras I,Xue DY.Fractional order control—a tutorial.2009 Conference on American Control Conference,2009 American Control Conference,St.Louis,MO,USA,2009-6-10-12.Piscataway:IEEE Press,2009.1397-1411
22 Ostalczyk PW,Duch P,Brzeziski DW,et al.Order functions selection in the variable-,fractional-order PID controller.Advances in Modelling and Control of Non-integer Order Systems,6th Conference on Non-integer Order Calculus and Its Applications,Opole,Poland,2014-11-6-6.New York:Springer Verlag,2015.159-170
23 薛定宇,赵春娜.分数阶系统的分数阶PID控制器设计.控制理论与应用,2007,24(5):771-775(Xue Dingyu,Zhao Chunna.Fractional order PID controller design for fractional order system.Control Theory&Applications,2007,24(5):771-775(in Chinese))
24 曹军义,谢航,蒋庄德.分数阶阻尼Duffing系统的非线性动力学特性.西安交通大学学报,2009,43(3):50-53(Cao Junyi,Xie Hang,Jiang Zhuangde.Nonlinear dynamics of Duffing system with fractional order damping.Journal of Xi’An Jiaotong University,2009,43(3):50-53(in Chinese))
25 黄建亮,黄惠仪,陈恒等.梁的强非线性超、次谐波共振.振动与冲击,2004,23(1):1-3(Huang Jianliang,Huang Huiyi,Chen Heng,et al.Superharmonic and subharmonic resonances of strongly nonlinear vibration of beams.Journal of Shock and Vibration,2004,23 (1):1-3(in Chinese))
26 刘灿昌,裘进浩,孙慧玉等.悬臂梁智能结构主共振响应的最优化控制.中国机械工程,2013,24(12):1600-1604(Liu Canchang,Qiu Jinhao,Sun Huiyu,et al.Optimal control of primary resonance of smart structures of cantilever beams.China Mechanical Engineering,2013,24(12):1600-1604(in Chinese))
27 Zillettia M,Elliotta SJ,Gardoniob P,et al.Experimental implementation of a self-tuning control system for decentralised velocity feedback.Journal of Sound and Vibration,2012,331(1-2):1-14
28 石秀东,钱林方,陈龙淼.基于加速度和速度反馈的半主动减振系统控制策略研究.南京理工大学学报(自然科学版),2005,29(5):556 -559(Shi Xiudong,Qian Linfang,Chen Longmiao.Semi-active vibration control strategy based on acceleration and velocity feedback.Journal of Nanjing University of Science and Technology,2005,29(5):556-559(in Chinese))
29 Zhang T,Li HG,Cai GP,et al.Experimental verifications of vibration suppression for a smart cantilever beam with a modified velocity feedback controller.Shock and Vibration,2014:172570
30 申永军,杨绍普,邢海军.含分数阶微分的线性单自由度振子的动力学分析.物理学报,2012,61(11):110505(Shen Yongjun,Yang Shaopu,Xing Haijun.Dynamical analysis of linear single degreeof-freedom oscillator with fractional-order derivative.Acta Physica Sinica,2012,61(11):110505(in Chinese))
31 Shen YJ,Yang SP,Sui CY.Analysis on limit cycle of fractionalorder van der Pol oscillator.Chaos,Solitons&Fractals,2014,67:94-102
2 曹青松,洪芸芸,周继惠等.基于PSO自整定PID控制器的柔性臂振动控制.振动、测试与诊断,2014,34(6):1045-1049(Cao Qingsong,Hong Yunyun,Zhou Jihui,et al.Vibration control of flexible manipulator based on self-tuning PID controller by PSO.Journal of Vibration,Measurement&Diagnosis,2014,34(6):1045-1049(in Chinese))
3 Thenozhi S,Yu W.Stability analysis of active vibration control ofbuilding structures using PD/PID control.Engineering Structures,2014,81:208-218
4 Zhou LW,Chen GP.Intelligent vibration control for high-speed spinning beam based on fuzzy self-tuning PID controller.Shock and Vibration,2015,ID:617038
5 周兵,赵保华.汽车主动悬架自适应模糊PID控制仿真研究.湖南大学学报(自然科学版),2009,32(16):27-30(Zhou Bing,Zhao Baohua.Simulation study of self-adaptive fuzzy-PID control of active suspension.Journal of Hunan University,2009,32(16):27-30(in Chinese))
6 Petras I.Fractional-Order Nonlinear Systems.Beijing:Higher Education Press,2011
7 Podlubny I.Fractional Differential Equations,Mathematics in Science and Engineering.New York:Academic Press,1999
8 Shen YJ,Yang SP,Xing HJ,et al.Primary resonance of Duffing oscillator with fractional-order derivative.Communication in Nonlinear Science and Numerical Simulation,2012,17(7):3092-3100
9 申永军,杨绍普,邢海军.分数阶Duffing振子的超谐共振.力学学报,2012,44(4):762-767(Shen Yongjun,Yang Shaopu,Xing Haijun.Super-harmonic response of fractional-order Duffing oscillator.Chinese Journal of Theoretical and Applied Mechanics,2012,44 (4):762-767(in Chinese))
10 Yang SP,Shen YJ.Recent advances in dynamics and control of hysteretic nonlinear systems.Chaos Solitons and Fractals,2009,40(4):1808-1822
11 Li CP,Deng WH.Remarks on fractional derivatives.Applied Mathematics and Computation,2007,187(1):777-784
12 Chen LC,Zhu WQ.Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations.International Journal of NonLinear Mechanics,2011,46(10):1324-1329
13 曹建雄,丁恒飞,李常品.分数阶扩散方程的隐差分格式.应用数学与计算数学学报,2013,27(1):61-74(Cao Jianxiong,Ding Hengfei,Li Changpin.Implicit difference schemes for fractional diffusion equations.Communication on Applied Mathematics and Computation,2013,27(1):61-74(in Chinese))
14 Agrawal OP.A quadratic numerical scheme for fractional optimal control problems.Journal of Dynamic Systems Measurement and Control,Transactions of the ASME,2008,130(1):011010.1-011010.6
15 Chen JH,Chen WC.Chaotic dynamics of the fractionally damped van der Pol equation.Chaos,Solitons and Fractals,2008,35(1):188-198
16 吴光强,黄焕军,叶光湖.基于分数阶微积分的汽车空气悬架半主动控制.农业机械学报,2014,45(7):19-24(Wu Guangqiang,Huang Huanjun,Ye Guanghu.Semi-active control of automotive air suspension based on fractional calculus.Transactions of the Chinese Society for Agricultural Machinery,2014,45(7):19-24(in Chinese))
17 李媛萍,张卫,欧阳东.具有分数导数本构关系的粘弹性浅拱的非线性动力学行为.振动工程学报,2012,25(3):342-350(Li Yuanping,Zhang Wei,Ouyang Dong.Nonlinear dynamic behaviors of viscoelastic shallow arch with fractional derivative constitutive relationship.Journal of Vibration Engineering,2012,25(3):342-350(in Chinese))
18 Podlubny I.Fractional-order systems and PIλDμ–controllers.IEEE Transactions on Automatic Control,1999,44(1):208-214
19 陈宁,陈南,王乃洲等.基于分数阶参考模型的车辆悬架自适应控制.南京林业大学学报,2009,33(3):247-253(Chen Ning,Chen Nan,Wang Naizhou,et al.Adaptive control of vehicle suspension using fractional order reference models.Journal of Nanjing Forestry University,2009,33(3):116-120(in Chinese))
20 赵春娜,赵雨,张祥德等.分数阶控制器与整数阶控制器仿真研究.系统仿真学报,2009,21(3):768-775(Zhao Chunna,Zhao Yu,Zhang Xiangde,et al.Simulation research on fractional order controllers with integer order controllers.Journal of System Simulation,2009,21(3):768-775(in Chinese))
21 Chen YQ,Petras I,Xue DY.Fractional order control—a tutorial.2009 Conference on American Control Conference,2009 American Control Conference,St.Louis,MO,USA,2009-6-10-12.Piscataway:IEEE Press,2009.1397-1411
22 Ostalczyk PW,Duch P,Brzeziski DW,et al.Order functions selection in the variable-,fractional-order PID controller.Advances in Modelling and Control of Non-integer Order Systems,6th Conference on Non-integer Order Calculus and Its Applications,Opole,Poland,2014-11-6-6.New York:Springer Verlag,2015.159-170
23 薛定宇,赵春娜.分数阶系统的分数阶PID控制器设计.控制理论与应用,2007,24(5):771-775(Xue Dingyu,Zhao Chunna.Fractional order PID controller design for fractional order system.Control Theory&Applications,2007,24(5):771-775(in Chinese))
24 曹军义,谢航,蒋庄德.分数阶阻尼Duffing系统的非线性动力学特性.西安交通大学学报,2009,43(3):50-53(Cao Junyi,Xie Hang,Jiang Zhuangde.Nonlinear dynamics of Duffing system with fractional order damping.Journal of Xi’An Jiaotong University,2009,43(3):50-53(in Chinese))
25 黄建亮,黄惠仪,陈恒等.梁的强非线性超、次谐波共振.振动与冲击,2004,23(1):1-3(Huang Jianliang,Huang Huiyi,Chen Heng,et al.Superharmonic and subharmonic resonances of strongly nonlinear vibration of beams.Journal of Shock and Vibration,2004,23 (1):1-3(in Chinese))
26 刘灿昌,裘进浩,孙慧玉等.悬臂梁智能结构主共振响应的最优化控制.中国机械工程,2013,24(12):1600-1604(Liu Canchang,Qiu Jinhao,Sun Huiyu,et al.Optimal control of primary resonance of smart structures of cantilever beams.China Mechanical Engineering,2013,24(12):1600-1604(in Chinese))
27 Zillettia M,Elliotta SJ,Gardoniob P,et al.Experimental implementation of a self-tuning control system for decentralised velocity feedback.Journal of Sound and Vibration,2012,331(1-2):1-14
28 石秀东,钱林方,陈龙淼.基于加速度和速度反馈的半主动减振系统控制策略研究.南京理工大学学报(自然科学版),2005,29(5):556 -559(Shi Xiudong,Qian Linfang,Chen Longmiao.Semi-active vibration control strategy based on acceleration and velocity feedback.Journal of Nanjing University of Science and Technology,2005,29(5):556-559(in Chinese))
29 Zhang T,Li HG,Cai GP,et al.Experimental verifications of vibration suppression for a smart cantilever beam with a modified velocity feedback controller.Shock and Vibration,2014:172570
30 申永军,杨绍普,邢海军.含分数阶微分的线性单自由度振子的动力学分析.物理学报,2012,61(11):110505(Shen Yongjun,Yang Shaopu,Xing Haijun.Dynamical analysis of linear single degreeof-freedom oscillator with fractional-order derivative.Acta Physica Sinica,2012,61(11):110505(in Chinese))
31 Shen YJ,Yang SP,Sui CY.Analysis on limit cycle of fractionalorder van der Pol oscillator.Chaos,Solitons&Fractals,2014,67:94-102