一类非光滑系统的无模型自适应混沌控制
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摘要
以一类单自由度非光滑系统为研究对象。分析了系统n-1周期运动经周期倍化分岔通向混沌运动的动力学行为。研究了系统混沌运动的控制问题,提出了一种无模型自适应参数反馈混沌控制方法。该混沌控制方法基于无模型自适应控制方法的思想设计混沌控制器:仅利用被控系统的输入/输出数据在线估计伪偏导数以建立系统的非参数时变动态线性化模型,进而基于该非参数时变动态线性化模型进行混沌控制器设计。当系统进入混沌运动状态后,利用混沌控制器输出微幅调整量施加于系统的可控参数,对混沌系统进行参数反馈控制,将系统的混沌运动控制为期望的周期运动。该混沌控制方法不依赖被控系统的数学模型,可适用于系统模型未知而仅获得系统输入/输出数据的情况,且具有控制器设计简单、计算量小等优点。数学证明了控制系统的稳定性和收敛性,仿真结果验证了该控制方法的可行性和有效性。
A single-degree-of-freedom non-smooth system is considered.The dynamic behaviors of the system from n-1periodic motion to chaotic motion via period-doubling bifurcation are analyzed by numerical simulation.By studying the chaotic motion control of the system,aparameter feedback chaos control method based on model-free adaptive control is proposed in this paper.The chaos controller is designed based on model-free adaptive control method:the pseudo-partial-derivative is estimated on-line by using the input/output data of the controlled system so that the non-parametric time-varying dynamic linear model of the controlled system can be established,and on this basis the chaos controller is designed.When the motion is chaotic,the chaos controller will be used to output a small perturbation to adjust the controllable parameter of the system so that the chaotic system can be controlled by using the parameter feedback chaos control method based on model-free adaptive control,thus the chaotic motion is controlled to the expected periodic motion.The proposed method in this paper does not need the exact mathematical model of the controlled system,so it is suitable for the cases where the exact mathematical model of the controlled system is unknown and only the I/O data can be obtained,and it has advantages of simpler controller design and smaller computing load,and so on.The stability and the convergence of the control system are proved mathematically,and the effectiveness and feasibility of the proposed method in this paper are verified by simulation results.
A single-degree-of-freedom non-smooth system is considered.The dynamic behaviors of the system from n-1periodic motion to chaotic motion via period-doubling bifurcation are analyzed by numerical simulation.By studying the chaotic motion control of the system,aparameter feedback chaos control method based on model-free adaptive control is proposed in this paper.The chaos controller is designed based on model-free adaptive control method:the pseudo-partial-derivative is estimated on-line by using the input/output data of the controlled system so that the non-parametric time-varying dynamic linear model of the controlled system can be established,and on this basis the chaos controller is designed.When the motion is chaotic,the chaos controller will be used to output a small perturbation to adjust the controllable parameter of the system so that the chaotic system can be controlled by using the parameter feedback chaos control method based on model-free adaptive control,thus the chaotic motion is controlled to the expected periodic motion.The proposed method in this paper does not need the exact mathematical model of the controlled system,so it is suitable for the cases where the exact mathematical model of the controlled system is unknown and only the I/O data can be obtained,and it has advantages of simpler controller design and smaller computing load,and so on.The stability and the convergence of the control system are proved mathematically,and the effectiveness and feasibility of the proposed method in this paper are verified by simulation results.
引文
[1]丁旺才,谢建华.碰撞振动系统分岔与混沌的研究进展[J].力学进展,2005,35(4):513-524.DING Wangcai,XIE Jianhua.Advances of research on bifurcations and chaos in vibro-impact system[J].Advances in Mechanics,2005,35(4):513-524.
[2]Cai Y,Chen S S.Nonlinear dynamics of loosely supported tubes in crossflow[J].Journal of Sound and Vibration,1993,168(3):449-468.
[3]Laggiard E,Runkel J,Stegemann D.One-dimensional bimodal model of vibration and impacting of instrument tubes in a boiling water reactor[J].Nuclear Science&Engineering,1993,115(1):62-70.
[4]Ko P L.Wear of power plant components due to impact and sliding[J].Applied Mechanics Reviews,1997,50(7):387-411.
[5]徐惠东,谢建华.一类单自由度分段线性系统的分岔和混沌控制[J].振动与冲击,2008,27(6):20-24.XU Huidong,XIE Jianhua.Bifurcation and chaos control of a single-degree-of-freedom system with piecewise-linearity[J].Journal of Vibration and Shock,2008,27(6):20-24.
[6]张惠,丁旺才,褚衍东,等.非光滑动力系统局部奇异性及擦边条件分析[J].兰州交通大学学报,2015,34(4):150-156.ZHANG Hui,DING Wangcai,CHU Yandong,et al.Local singularity and grazing condition analysis of nonsmooth dynamical system[J].Journal of Lanzhou Jiaotong University,2015,34(4):150-156.
[7]Leine R I,Nijmeijer H.Dynamics and Bifurcations of Non-Smooth Mechanical Systems[M].Berlin:Springer,2004.
[8]Ding W C,Li G F,Luo G W,et al.Torus T2and its locking,doubling,chaos of a vibro-impact system[J].Journal of the Franklin Institute,2012,349(1):337-348.
[9]Lee J Y,Yan J J.Control of impact oscillator[J].Chaos Solitons and Fractals,2006,28(1):136-142.
[10]王子俊.利用OGY方法控制单自由度碰撞振动系统的混沌行为[D].成都:西南交通大学,2013.WANG Zijun.Controlling chaos in a one-degree-offreedom vibro-impact system by the OGY method[D].Chengdu:Southwest Jiaotong University,2013.
[11]Souza S L T D,Caldas I L,Viana R L.Damping control law for a chaotic impact oscillator[J].Chaos,Solitons&Fractals,2007,32(2):745-750.
[12]丁旺才,马永靖,王靖岳.碰撞振动系统的状态预测反馈控制[J].振动工程学报,2007,20(6):589-593.DING Wangcai,MA Yongjing,WANG Jingyue.Feedback control of a vibro-impact system by states prediction[J].Journal of Vibration Engineering,2007,20(6):589-593.
[13]卫晓娟,李宁洲,张惠,等.一类含间隙碰撞振动系统混沌运动的RBF神经网络控制[J].振动工程学报,2018,31(2):336-342.WEI Xiaojuan,LI Ningzhou,ZHANG Hui,et al.Chaos control of a vibro-impact system with clearance based on RBF neural network[J].Journal of Vibration Engineering,2018,31(2):336-342.
[14]张庆爽,丁旺才,孙闯.一类单自由度非光滑系统混沌运动的延迟反馈控制[J].振动与冲击,2008,27(1):155-158.ZHANG Qingshuang,DING Wangcai,SUN Chuang.Delayed feedback control of chaos in a single DOF nonsmooth system[J].Journal of Vibration and Shock,2008,27(1):155-158.
[15]马永靖,丁旺才,杨小刚.碰撞振动系统的参数自调节混沌控制[J].振动与冲击,2007,26(1):24-26.MA Yongjing,DING Wangcai,YANG Xiaogang.Chaos control of a vibro-impact system with parameter adjustment[J].Journal of Vibration and Shock,2007,26(1):24-26.
[16]乐源.一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学[J].力学学报,2016,48(1):163-172.YUE YUAN.Local dynamical behavior of two-parameter family near the Neimark-Sacker-pitchfork bifurcation point in a vibro-impact system[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(1):163-172.
[17]吴少培,李国芳,丁旺才.含间隙运动副模型的机械动力学分析[J].兰州交通大学学报,2016,35(4):111-116.Wu Shaopei,Li Guofang,Ding Wangcai.Dynamics analysis of mechanisms with joint clearance[J].Journal of Lanzhou Jiaotong University,2016,35(4):111-116.
[18]Luo G W,Zhu X F,Shi Y Q.Dynamics of a two-degree-of freedom periodically-forced system with a rigid stop:Diversity and evolution of periodic-impact motions[J].Journal of Sound and Vibration,2015,334(334):338-362.
[19]朱喜锋,罗冠炜.两自由度含间隙弹性碰撞系统的颤碰运动分析[J].振动与冲击,2015,34(15):195-200.Zhu Xifeng,Luo Guanwei.Chattering-impact motion of a 2-DOF system with clearance and soft impacts[J].Journal of Vibration and shock,2015,34(15):195-200.
[20]张惠,丁旺才,李飞.两自由度含间隙和预紧弹簧碰撞振动系统动力学分析[J].工程力学,2011,28(3):209-217.Zhang Hui,Ding Wangcai,Li Fei.Dynamics of a twodegree-of-freedom impact system with clearance and Pre-compressed spring[J].Engineering Mechanics,2011,28(3):209-217.
[21]Wen G L,Xie J H.Period-doubling bifurcation and non-typical route to chaos of a two-degree-of-freedom vibro-impact system[J].ASME,J.of Appl.Mech.,2001,68(4):670-674.
[22]柴林,吴晓明.机械碰撞振动系统分岔与混沌的参数演化[J].厦门大学学报(自然科学版),2014,53(4):508-513.CHAI Lin,WU Xiaoming.Evolution of bifurcation and chaos in mechanical vibro-impact system with parameters[J].Journal of Xiamen University(Natural Sciences),2014,53(4):508-513.
[23]吕小红.阻尼控制策略抑制碰撞振动系统的分岔与混沌[J].机械科学与技术,2016,35(9):1337-1342.LüXiao-hong.Controlling bifurcation and chaos of vibro-impact system by damping control law[J].Mechanical Science and Technology for Aerospace Engineering,2016,35(9):1337-1342.
[24]刘庆丰.分段线性系统的动力学分析及混沌控制[D].兰州:兰州交通大学,2015.LIU Qingfeng.Dynamic analysis of piecewise-linear system and chaos control[D].Lanzhou:Lanzhou Jiaotong University,2015.
[25]李群宏,魏艳辉,谭洁燕.两自由度碰撞振动系统的混沌控制[J].广西大学学报(自然科学版),2009,34(2):205-210.LI Qunhong,WEI Yanhui,TAN Jieyan.Chaos control of a vibro-impact dynamical system with two degrees of freedom[J].Journal of Guangxi University(Nat Sci Ed),2009,34(2):206-210.
[26]侯忠生.非参数模型及其自适应控制理论[M].北京:科学出版社,1999.HOU Zhongsheng.Nonparametric Models and Its A-daptive Control Theory[M].Beijing:Science Press,1999.
[27]侯忠生,董航瑞,金尚泰.基于坐标补偿的自动泊车系统无模型自适应控制[J].自动化学报,2015,41(4):823-831.HOU ZHongsheng,DONG Hangrui,JIN Shangtai.Model-free adaptive control with coordinates compensation for automatic car parking systems[J].Acta Automatica Sinica,2015,41(4):823-831.
[2]Cai Y,Chen S S.Nonlinear dynamics of loosely supported tubes in crossflow[J].Journal of Sound and Vibration,1993,168(3):449-468.
[3]Laggiard E,Runkel J,Stegemann D.One-dimensional bimodal model of vibration and impacting of instrument tubes in a boiling water reactor[J].Nuclear Science&Engineering,1993,115(1):62-70.
[4]Ko P L.Wear of power plant components due to impact and sliding[J].Applied Mechanics Reviews,1997,50(7):387-411.
[5]徐惠东,谢建华.一类单自由度分段线性系统的分岔和混沌控制[J].振动与冲击,2008,27(6):20-24.XU Huidong,XIE Jianhua.Bifurcation and chaos control of a single-degree-of-freedom system with piecewise-linearity[J].Journal of Vibration and Shock,2008,27(6):20-24.
[6]张惠,丁旺才,褚衍东,等.非光滑动力系统局部奇异性及擦边条件分析[J].兰州交通大学学报,2015,34(4):150-156.ZHANG Hui,DING Wangcai,CHU Yandong,et al.Local singularity and grazing condition analysis of nonsmooth dynamical system[J].Journal of Lanzhou Jiaotong University,2015,34(4):150-156.
[7]Leine R I,Nijmeijer H.Dynamics and Bifurcations of Non-Smooth Mechanical Systems[M].Berlin:Springer,2004.
[8]Ding W C,Li G F,Luo G W,et al.Torus T2and its locking,doubling,chaos of a vibro-impact system[J].Journal of the Franklin Institute,2012,349(1):337-348.
[9]Lee J Y,Yan J J.Control of impact oscillator[J].Chaos Solitons and Fractals,2006,28(1):136-142.
[10]王子俊.利用OGY方法控制单自由度碰撞振动系统的混沌行为[D].成都:西南交通大学,2013.WANG Zijun.Controlling chaos in a one-degree-offreedom vibro-impact system by the OGY method[D].Chengdu:Southwest Jiaotong University,2013.
[11]Souza S L T D,Caldas I L,Viana R L.Damping control law for a chaotic impact oscillator[J].Chaos,Solitons&Fractals,2007,32(2):745-750.
[12]丁旺才,马永靖,王靖岳.碰撞振动系统的状态预测反馈控制[J].振动工程学报,2007,20(6):589-593.DING Wangcai,MA Yongjing,WANG Jingyue.Feedback control of a vibro-impact system by states prediction[J].Journal of Vibration Engineering,2007,20(6):589-593.
[13]卫晓娟,李宁洲,张惠,等.一类含间隙碰撞振动系统混沌运动的RBF神经网络控制[J].振动工程学报,2018,31(2):336-342.WEI Xiaojuan,LI Ningzhou,ZHANG Hui,et al.Chaos control of a vibro-impact system with clearance based on RBF neural network[J].Journal of Vibration Engineering,2018,31(2):336-342.
[14]张庆爽,丁旺才,孙闯.一类单自由度非光滑系统混沌运动的延迟反馈控制[J].振动与冲击,2008,27(1):155-158.ZHANG Qingshuang,DING Wangcai,SUN Chuang.Delayed feedback control of chaos in a single DOF nonsmooth system[J].Journal of Vibration and Shock,2008,27(1):155-158.
[15]马永靖,丁旺才,杨小刚.碰撞振动系统的参数自调节混沌控制[J].振动与冲击,2007,26(1):24-26.MA Yongjing,DING Wangcai,YANG Xiaogang.Chaos control of a vibro-impact system with parameter adjustment[J].Journal of Vibration and Shock,2007,26(1):24-26.
[16]乐源.一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学[J].力学学报,2016,48(1):163-172.YUE YUAN.Local dynamical behavior of two-parameter family near the Neimark-Sacker-pitchfork bifurcation point in a vibro-impact system[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(1):163-172.
[17]吴少培,李国芳,丁旺才.含间隙运动副模型的机械动力学分析[J].兰州交通大学学报,2016,35(4):111-116.Wu Shaopei,Li Guofang,Ding Wangcai.Dynamics analysis of mechanisms with joint clearance[J].Journal of Lanzhou Jiaotong University,2016,35(4):111-116.
[18]Luo G W,Zhu X F,Shi Y Q.Dynamics of a two-degree-of freedom periodically-forced system with a rigid stop:Diversity and evolution of periodic-impact motions[J].Journal of Sound and Vibration,2015,334(334):338-362.
[19]朱喜锋,罗冠炜.两自由度含间隙弹性碰撞系统的颤碰运动分析[J].振动与冲击,2015,34(15):195-200.Zhu Xifeng,Luo Guanwei.Chattering-impact motion of a 2-DOF system with clearance and soft impacts[J].Journal of Vibration and shock,2015,34(15):195-200.
[20]张惠,丁旺才,李飞.两自由度含间隙和预紧弹簧碰撞振动系统动力学分析[J].工程力学,2011,28(3):209-217.Zhang Hui,Ding Wangcai,Li Fei.Dynamics of a twodegree-of-freedom impact system with clearance and Pre-compressed spring[J].Engineering Mechanics,2011,28(3):209-217.
[21]Wen G L,Xie J H.Period-doubling bifurcation and non-typical route to chaos of a two-degree-of-freedom vibro-impact system[J].ASME,J.of Appl.Mech.,2001,68(4):670-674.
[22]柴林,吴晓明.机械碰撞振动系统分岔与混沌的参数演化[J].厦门大学学报(自然科学版),2014,53(4):508-513.CHAI Lin,WU Xiaoming.Evolution of bifurcation and chaos in mechanical vibro-impact system with parameters[J].Journal of Xiamen University(Natural Sciences),2014,53(4):508-513.
[23]吕小红.阻尼控制策略抑制碰撞振动系统的分岔与混沌[J].机械科学与技术,2016,35(9):1337-1342.LüXiao-hong.Controlling bifurcation and chaos of vibro-impact system by damping control law[J].Mechanical Science and Technology for Aerospace Engineering,2016,35(9):1337-1342.
[24]刘庆丰.分段线性系统的动力学分析及混沌控制[D].兰州:兰州交通大学,2015.LIU Qingfeng.Dynamic analysis of piecewise-linear system and chaos control[D].Lanzhou:Lanzhou Jiaotong University,2015.
[25]李群宏,魏艳辉,谭洁燕.两自由度碰撞振动系统的混沌控制[J].广西大学学报(自然科学版),2009,34(2):205-210.LI Qunhong,WEI Yanhui,TAN Jieyan.Chaos control of a vibro-impact dynamical system with two degrees of freedom[J].Journal of Guangxi University(Nat Sci Ed),2009,34(2):206-210.
[26]侯忠生.非参数模型及其自适应控制理论[M].北京:科学出版社,1999.HOU Zhongsheng.Nonparametric Models and Its A-daptive Control Theory[M].Beijing:Science Press,1999.
[27]侯忠生,董航瑞,金尚泰.基于坐标补偿的自动泊车系统无模型自适应控制[J].自动化学报,2015,41(4):823-831.HOU ZHongsheng,DONG Hangrui,JIN Shangtai.Model-free adaptive control with coordinates compensation for automatic car parking systems[J].Acta Automatica Sinica,2015,41(4):823-831.