一类具时滞反馈控制的摩擦模型的稳定性和Hopf分支分析
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摘要
摩擦是一种复杂的非线性物理现象,产生于具有相对运动的接触面之间。在机械机电系统的摩擦模型的研究方面,以往的学者们比较注重系统的控制策略及其实现方法的研究,而对系统动力学方面的探讨尚不多见。具时滞反馈控制的Stribeck摩擦模型是一个非线性控制系统,反馈控制力是被控对象位移的线性函数,Stribeck摩擦力则为速度的非线性函数,当系统参数位于某些区域时,反馈控制力会使滑块产生相当大的振动,所以分析系统的振动特性、稳定性等动力学性质是具时滞反馈控制的摩擦模型的重要研究课题。另外,反馈控制系统中不可避免地带有时滞,考虑时滞的影响,一方面可以更客观、更准确地反应系统的实际工作情况,另一方面可以通过时滞反馈去控制动力学系统,以实现理想的控制策略。通过对具有时滞反馈控制的摩擦模型的研究,可以最大程度地消除和减少摩擦引起的不利因素,提高系统的动力学性能。因此,对具时滞反馈控制的Stribeck摩擦模型的研究有重要的理论和实际意义。
在本文中,我们首先引入了一类具时滞反馈控制的Stribeck摩擦模型系统。然后利用线性稳定性方法对此系统的平衡点进行稳定性分析,当系统的线性部分对应的特征方程的特征根为纯虚根时,算出相应的时滞τ,得到了平衡点的稳定性在时滞τ取某些值时发生翻转,并且在平衡点处系统经历Hopf分支,而且我们发现当时滞量发生变化时系统产生了“稳定性开关”的现象。接着利用规范型理论和中心流形定理讨论了系统Hopf分支的分支方向和分支周期解的稳定性,给出了关于分支方向和分支周期解稳定性的计算公式。最后利用MATLAB软件进行相应的数值模拟,数值模拟结果与所得理论分析结果具有一致性。
Friction is one complicated kind of non-linear physics phenomena, which is produced between the interfaces with rightabout movement. In the study of mechanism system of friction model, former researchers have paid more attention to the controlling tactics and the realization methods, but few explorations were made in terms of the dynamical behavior of the system. As time-delayed feedback control of friction system is a non-linear control system, the feedback control force is a linear function for the displacement of the object and the Stribeck friction is a non-linear function for the velocity, the effects of the control will make rotor produce considerable vibrations when the parameters belong to certain domain. Taking the effect of time delay into consideration, we can understand the practical performance of the system more objectively and accurately; on the other hand we can control the dynamical system by delayed feedback and realize an ideal strategy. Thus investigating the Stribeck friction system with time-delayed feedback control is of great theoretical and practical significance.
In this thesis, in the first, we introduce one kind of Stribeck friction system with time-delayed feedback control. Secondly, we analyze the stability of the equilibrium by using linearizing stability method. When the engenvalues of the linear part are pure imaginary numbers, we obtain the corresponding delay value. The stability of the steady state is lost when the delay passes through the critical value, as well as there will be a family periodic solutions bifurcate from the steady states, and we find that the stability switch occurs when delay varies. Then, we derive the explicit formulae for determining the direction of the Hopf bifurcation and the stability of these periodic solutions bifurcating from the steady states, by using the normal form method and center manifold theorem. Finally, some numerical simulations are carried out by using MATLAB, and the results are consistent with our analysis results.
在本文中,我们首先引入了一类具时滞反馈控制的Stribeck摩擦模型系统。然后利用线性稳定性方法对此系统的平衡点进行稳定性分析,当系统的线性部分对应的特征方程的特征根为纯虚根时,算出相应的时滞τ,得到了平衡点的稳定性在时滞τ取某些值时发生翻转,并且在平衡点处系统经历Hopf分支,而且我们发现当时滞量发生变化时系统产生了“稳定性开关”的现象。接着利用规范型理论和中心流形定理讨论了系统Hopf分支的分支方向和分支周期解的稳定性,给出了关于分支方向和分支周期解稳定性的计算公式。最后利用MATLAB软件进行相应的数值模拟,数值模拟结果与所得理论分析结果具有一致性。
Friction is one complicated kind of non-linear physics phenomena, which is produced between the interfaces with rightabout movement. In the study of mechanism system of friction model, former researchers have paid more attention to the controlling tactics and the realization methods, but few explorations were made in terms of the dynamical behavior of the system. As time-delayed feedback control of friction system is a non-linear control system, the feedback control force is a linear function for the displacement of the object and the Stribeck friction is a non-linear function for the velocity, the effects of the control will make rotor produce considerable vibrations when the parameters belong to certain domain. Taking the effect of time delay into consideration, we can understand the practical performance of the system more objectively and accurately; on the other hand we can control the dynamical system by delayed feedback and realize an ideal strategy. Thus investigating the Stribeck friction system with time-delayed feedback control is of great theoretical and practical significance.
In this thesis, in the first, we introduce one kind of Stribeck friction system with time-delayed feedback control. Secondly, we analyze the stability of the equilibrium by using linearizing stability method. When the engenvalues of the linear part are pure imaginary numbers, we obtain the corresponding delay value. The stability of the steady state is lost when the delay passes through the critical value, as well as there will be a family periodic solutions bifurcate from the steady states, and we find that the stability switch occurs when delay varies. Then, we derive the explicit formulae for determining the direction of the Hopf bifurcation and the stability of these periodic solutions bifurcating from the steady states, by using the normal form method and center manifold theorem. Finally, some numerical simulations are carried out by using MATLAB, and the results are consistent with our analysis results.
引文
1刘维民.摩擦学研究及发展趋势.中国机械工程. 2000, 11(2):77~79
2刘丽兰,刘宏昭.机械系统中摩擦模型的研究进展.力学进展. 2008, 38(2):201~202
3黄毅,王太勇.干摩擦系统的自激振动数值研究.机械强度. 2007, 3:539~543
4张涛,路长厚,李泉.基于扭矩测量的交流伺服工作台摩擦建模与仿真.研究润滑与密封. 2004:32～35
5刘国平.机械系统中的摩擦模型及仿真.西安理工大学硕士学位论文. 2007
6 B. Armstrong, D. Neevel and T. Kusik. New Result in NPID Control: Tracking, Integral, Control, Friction Compensation and Experimental Result. IEEE Transactions on Control Systems Technology. 2001, 3:339~406
7 W. Gharieb, G. Nagib. Fuzzy intervention in PID controller design. Industrial Electronics, 2001, Proceedings. ISIE 2001. IEEE International Symposium on Pusan, South Korea. 2001:1639~1643
8 L. Horowitz, S. Oidak and A. Shapiro. Extensions of Dithered Feedback Systems. International Journal of Contro1. 1991, 54(1):83~109
9 S. Yang, M. Tomizuka. Adaptive Pulse Width Control for Piece Positioning under Influence of Stiction and Coulomb Friction. Journal of Dynamic Systems. 1988, 110(3):221~227
10 G. Morcl, K. Iagnemma and S. Dubowsky. The Precise Control of Manipulators with High Joint-Friction Using Base Force Torque Sensing. Automatic. 2000, 36:931~941
11 Shih Jung Huang, Jia Yush Yen, Shui Shong Lu. Dual Mode Control of a System with Friction. IEEE Transactions on Control Systems Technology. 1999, 7(3):306~314
12 H. Kwatny, C. Teolic and M. Matlice. Variable Structure Control of System with Uncertain Nonlinear Friction. Automatic. 2002, 38(7):1251~1256
13刘强,刘金琨.摩擦非线性环节的特性、建模与控制补偿综述.系统工程与电子技术. 2002, 11:45～52
14 B. Friedland, Y. J.Park. On Adaptive Friction Compensation. IEEE Trans on Automatic Control. 1992, 37(10):1605~1612
15 L. Weiping, X.Cheng. Adaptive High-Precision Control of Positioning Tables-Theory and Experiments. IEEE Trans on Control Systems Technology. 1994, 9:265~270
16 C. Canudas, P. Noel and B.Brogliato. Adaptive Friction Compensation in Robot Manipulators:Low-Velocities. International Journal of Robotics Research. 1993, 10(3):189~199
17 K. Menon, K. Krishnamurthy. Control of Low Velocity Friction and Gear Backlash in a Machine Tool Feed Drive System. Mechatronies. 1999, 9:33~51
18 K. C. Cheok, Hu Hongxing and N. K. Loh. Modeling and Identification of a Class of Servomechanism Systems with Stick-Slip Friction. Journal of Dynamic Systems. 1988, 110:324~328
19 J. Y. Yen, S. J. Huang and S. S.Lu. A New Compensation for Servo Systems with Position Dependent Friction. Journal of Dynamic Systems. 1999, 121: 612~618
20 R. M. Hirschorn, G. Miller. Control of Nonlinear Systems with Friction. Trans on Control Systems Technology. 1999, 7(5):588~595
21 K. Pyragas. Control of Chaos by Self-Controlling Feedback.Phys. Lett. A. 1992, 170:421~428
22 Y. Song, J. Wei. Bifurcation Analysis for Chen’s System with Delayed Feedback and Its Application to Control of Chaos. Chaos, Solitons&Fractals. 2004, 22(1):75~91
23 H. Wang, J. Liu. Stability and Bifurcation Analysis in a Magnetic Bearing System with Time Delays. Chaos, Solitons&Fractals. 2005, 26:813~825
24 J. Wei, W. Jiang. Stability and Bifurcation Analysis in Van der Pol’s Oscillator with Delayed Feedback. Journal of Sound and Vibration. 2005, 283(4):801~819
25梁建术.高维非线性系统的分岔和混沌控制.天津大学博士学位论文. 2002, 1~13
26张伟,陈予恕.非线性振子的主共振—基本参数共振.非线性动力学学报. 1995, 2(4):361~365
27 S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990:253~283
28季进臣,陈予恕.两自由度非线性振动系统主参数激励下的分岔分析.应用数学与力学. 2001, 33(2):215~223
29张芷芬,李承治,郑志明,李伟固.向量场的分岔理论基础.高等教育出版社, 1997:58~157
30 B. Hassard, N. Kazarionff and Y. H. Wan. Theory and Application of Hopf Bifuecation. Cambrige University Press, Cambrige, 1981:14~35
31 J. K. Hale. Theory of Functional Differential Equations. Springer-Verlag, New York, 1977
32 E. Hopf. Abzweigung Einer Priodischen Losung Von einer stationaren Losung Einer Differential Systems. Ber. Math. Phys. Sachs Akad Wiss Leipzig. 1942, 1-22
33 N. Chafee. A Bifurcation Problem for Functional Differential Equations of Finitely Retarded Type. J. Math. 1971, 35(4):312-348
34 M. A. Heckl, D. Abrahams. Active Control of Friction Driven Oscillations. J. Sound Vib. 1996, 193(1):417~426
35 K. Popp, M. Rudolph. Vibration Control to Avoid Stick-slip Motion. J. Vib Control. 2004, 10:1585~1600
36 S. Chatterjee, T. K. Singha and S. K. Karmakar. Effect of High-frequency Excition on a Class of Mechanical Systems with Dynamic Friction. J. Sound Vib. 2004, 269:61~89
37 S. Chatterjee. Non-linear Control of Friction-induced Self-excited Vibration. Int. J. Nonlinear Mech. 2007, 42(3):459~469
38 A. Maccari. Vibration Control for the Primary Resonance of a Cantilever Beam by a Time Delay State Feedback. J. Sound Vib. 2003,259(2): 241~251
39 A. Maccari. Vibration Control of Parametrically Excited Lienard System. Nonlinear Mech. 2006, 41:146~155
40 X. Li, J. C. Li, C. H. Hansen, C. Tan. Response of a Duffing-Van der Pol Oscillator under Delayed Feedback Control. J. Sound Vib. 2006, 291: 644~655
41 J. Das, A. K. Mallik. Control of Friction Driven Oscillation by Time-delayed State Feedback. J. Sound Vib. 2006, 297(3-5)578~594
42 S. Chatterjee. Time-delayed Feedback Control of Friction-induced Instability. J. Nonlinear Mech. 2007, 42:1127~1143
43 C. Canudas, J. Carillo. A Modified EW-RLS Algorithm for System with Bounded Isturbance. Automatica. 1990, 26(2):599~606
44 N. Hinrichs, M. Oestreich, K. Popp, On the Modeling of Friction Oscillators, J. Sound Vib. 1998, 216(3):435~459
45 S. Ruan, J. Wei, On the Zeros of Transcendental Functions with Applications to Stability of Delay Differential Equations with Two Delays. Dynamics of Discrete, Continuous and Impulsive Systems. 2003, 10: 863~874
46 S. Ruan, J. Wei. On the Zeros of a Third Degree Exponential Polynomial with Applications to a Delayed Model for the Control of Testosterone Secretion. Journal of Mathematics. Applied in Medicine and Biology. 2001, 18:41~52
47沈启宏,魏俊杰.具时滞的人类呼吸系统模型的稳定性与分支.应用数学和力学. 2004, 25:1169~1181
2刘丽兰,刘宏昭.机械系统中摩擦模型的研究进展.力学进展. 2008, 38(2):201~202
3黄毅,王太勇.干摩擦系统的自激振动数值研究.机械强度. 2007, 3:539~543
4张涛,路长厚,李泉.基于扭矩测量的交流伺服工作台摩擦建模与仿真.研究润滑与密封. 2004:32～35
5刘国平.机械系统中的摩擦模型及仿真.西安理工大学硕士学位论文. 2007
6 B. Armstrong, D. Neevel and T. Kusik. New Result in NPID Control: Tracking, Integral, Control, Friction Compensation and Experimental Result. IEEE Transactions on Control Systems Technology. 2001, 3:339~406
7 W. Gharieb, G. Nagib. Fuzzy intervention in PID controller design. Industrial Electronics, 2001, Proceedings. ISIE 2001. IEEE International Symposium on Pusan, South Korea. 2001:1639~1643
8 L. Horowitz, S. Oidak and A. Shapiro. Extensions of Dithered Feedback Systems. International Journal of Contro1. 1991, 54(1):83~109
9 S. Yang, M. Tomizuka. Adaptive Pulse Width Control for Piece Positioning under Influence of Stiction and Coulomb Friction. Journal of Dynamic Systems. 1988, 110(3):221~227
10 G. Morcl, K. Iagnemma and S. Dubowsky. The Precise Control of Manipulators with High Joint-Friction Using Base Force Torque Sensing. Automatic. 2000, 36:931~941
11 Shih Jung Huang, Jia Yush Yen, Shui Shong Lu. Dual Mode Control of a System with Friction. IEEE Transactions on Control Systems Technology. 1999, 7(3):306~314
12 H. Kwatny, C. Teolic and M. Matlice. Variable Structure Control of System with Uncertain Nonlinear Friction. Automatic. 2002, 38(7):1251~1256
13刘强,刘金琨.摩擦非线性环节的特性、建模与控制补偿综述.系统工程与电子技术. 2002, 11:45～52
14 B. Friedland, Y. J.Park. On Adaptive Friction Compensation. IEEE Trans on Automatic Control. 1992, 37(10):1605~1612
15 L. Weiping, X.Cheng. Adaptive High-Precision Control of Positioning Tables-Theory and Experiments. IEEE Trans on Control Systems Technology. 1994, 9:265~270
16 C. Canudas, P. Noel and B.Brogliato. Adaptive Friction Compensation in Robot Manipulators:Low-Velocities. International Journal of Robotics Research. 1993, 10(3):189~199
17 K. Menon, K. Krishnamurthy. Control of Low Velocity Friction and Gear Backlash in a Machine Tool Feed Drive System. Mechatronies. 1999, 9:33~51
18 K. C. Cheok, Hu Hongxing and N. K. Loh. Modeling and Identification of a Class of Servomechanism Systems with Stick-Slip Friction. Journal of Dynamic Systems. 1988, 110:324~328
19 J. Y. Yen, S. J. Huang and S. S.Lu. A New Compensation for Servo Systems with Position Dependent Friction. Journal of Dynamic Systems. 1999, 121: 612~618
20 R. M. Hirschorn, G. Miller. Control of Nonlinear Systems with Friction. Trans on Control Systems Technology. 1999, 7(5):588~595
21 K. Pyragas. Control of Chaos by Self-Controlling Feedback.Phys. Lett. A. 1992, 170:421~428
22 Y. Song, J. Wei. Bifurcation Analysis for Chen’s System with Delayed Feedback and Its Application to Control of Chaos. Chaos, Solitons&Fractals. 2004, 22(1):75~91
23 H. Wang, J. Liu. Stability and Bifurcation Analysis in a Magnetic Bearing System with Time Delays. Chaos, Solitons&Fractals. 2005, 26:813~825
24 J. Wei, W. Jiang. Stability and Bifurcation Analysis in Van der Pol’s Oscillator with Delayed Feedback. Journal of Sound and Vibration. 2005, 283(4):801~819
25梁建术.高维非线性系统的分岔和混沌控制.天津大学博士学位论文. 2002, 1~13
26张伟,陈予恕.非线性振子的主共振—基本参数共振.非线性动力学学报. 1995, 2(4):361~365
27 S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990:253~283
28季进臣,陈予恕.两自由度非线性振动系统主参数激励下的分岔分析.应用数学与力学. 2001, 33(2):215~223
29张芷芬,李承治,郑志明,李伟固.向量场的分岔理论基础.高等教育出版社, 1997:58~157
30 B. Hassard, N. Kazarionff and Y. H. Wan. Theory and Application of Hopf Bifuecation. Cambrige University Press, Cambrige, 1981:14~35
31 J. K. Hale. Theory of Functional Differential Equations. Springer-Verlag, New York, 1977
32 E. Hopf. Abzweigung Einer Priodischen Losung Von einer stationaren Losung Einer Differential Systems. Ber. Math. Phys. Sachs Akad Wiss Leipzig. 1942, 1-22
33 N. Chafee. A Bifurcation Problem for Functional Differential Equations of Finitely Retarded Type. J. Math. 1971, 35(4):312-348
34 M. A. Heckl, D. Abrahams. Active Control of Friction Driven Oscillations. J. Sound Vib. 1996, 193(1):417~426
35 K. Popp, M. Rudolph. Vibration Control to Avoid Stick-slip Motion. J. Vib Control. 2004, 10:1585~1600
36 S. Chatterjee, T. K. Singha and S. K. Karmakar. Effect of High-frequency Excition on a Class of Mechanical Systems with Dynamic Friction. J. Sound Vib. 2004, 269:61~89
37 S. Chatterjee. Non-linear Control of Friction-induced Self-excited Vibration. Int. J. Nonlinear Mech. 2007, 42(3):459~469
38 A. Maccari. Vibration Control for the Primary Resonance of a Cantilever Beam by a Time Delay State Feedback. J. Sound Vib. 2003,259(2): 241~251
39 A. Maccari. Vibration Control of Parametrically Excited Lienard System. Nonlinear Mech. 2006, 41:146~155
40 X. Li, J. C. Li, C. H. Hansen, C. Tan. Response of a Duffing-Van der Pol Oscillator under Delayed Feedback Control. J. Sound Vib. 2006, 291: 644~655
41 J. Das, A. K. Mallik. Control of Friction Driven Oscillation by Time-delayed State Feedback. J. Sound Vib. 2006, 297(3-5)578~594
42 S. Chatterjee. Time-delayed Feedback Control of Friction-induced Instability. J. Nonlinear Mech. 2007, 42:1127~1143
43 C. Canudas, J. Carillo. A Modified EW-RLS Algorithm for System with Bounded Isturbance. Automatica. 1990, 26(2):599~606
44 N. Hinrichs, M. Oestreich, K. Popp, On the Modeling of Friction Oscillators, J. Sound Vib. 1998, 216(3):435~459
45 S. Ruan, J. Wei, On the Zeros of Transcendental Functions with Applications to Stability of Delay Differential Equations with Two Delays. Dynamics of Discrete, Continuous and Impulsive Systems. 2003, 10: 863~874
46 S. Ruan, J. Wei. On the Zeros of a Third Degree Exponential Polynomial with Applications to a Delayed Model for the Control of Testosterone Secretion. Journal of Mathematics. Applied in Medicine and Biology. 2001, 18:41~52
47沈启宏,魏俊杰.具时滞的人类呼吸系统模型的稳定性与分支.应用数学和力学. 2004, 25:1169~1181