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The main purpose of this paper is to reveal the evolution mechanism of the bursting oscillation and suppress the bursting oscillation. The permanent magnet synchronous motor (PMSM) system is taken as a research object, and the case of the PMSM with periodic external load perturbation is considered. The first part in this paper is for the analysis of bursting oscillation. First, a mathematical model of the non-autonomous PMSM system with external load perturbation is established, and the frequency of the external load perturbation is set to be far less than the natural frequency of the PMSM system, so that the PMSM system has a fast-slow coupling effect. Then, the non-autonomous PMSM system with external load perturbation is transformed into a generalized autonomous PMSM system by taking the external load perturbation as a slow-varying parameter of the PMSM system. In order to obtain the bifurcation behaviors and different equilibrium types of the PMSM system, the time series diagram, the equilibrium point distribution curve that changes with slow-varying parameter, and the transformed phase portrait are analyzed. Finally, the evolution mechanism of bursting oscillation is revealed by analyzing the overlay of the equilibrium point distribution curve and the transformed phase portrait, and it is found that the change of the equilibrium type and the corresponding bifurcation behavior will cause the PMSM system to exhibit “periodic symmetrical subcritical Hopf bursting oscillation”. The second part focuses on the control of the bursting oscillation. First, a macro-variable is defined by using the synergetic control strategy, which is a linear combination of all state variables of the PMSM system. Then, the synergetic controller is designed based on the constraint that the macro-variable converges to the invariant manifold. When the macro-variable converges to the invariant manifold, the PMSM system is also stabilized to the equilibrium. In addition, in order to explore the influence of controller parameters, a large number of simulation experiments are carried out, and the relationship between the control parameters with the response speed of the PMSM system is obtained. Finally, the effectiveness of the synergetic control strategy is verified by changing the amplitude of the external load perturbation. The simulation results show that the synergetic control strategy has a continuous control law when the system has external load perturbations, and can effectively suppress the bursting oscillation phenomenon of the PMSM system, so that the PMSM system runs stably.
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Keywords:
- permanent magnet synchronous motor /
- bursting oscillation /
- synergetic control /
- bursting suppression
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图 1
$\tilde \omega $ 的时间历程图Figure 1. Time history of
$\tilde \omega $ .
图 2
$\tilde \omega $ 关于慢变参数${\tilde T_{\rm{L}}}$ 的平衡点分布曲线Figure 2. Equilibrium point distribution curve of
$\tilde \omega $ with respect to the slow-varying parameter${\tilde T_{\rm{L}}}$ .
图 3 转子机械角速度
$\tilde \omega $ 的簇发振荡示意图 (a) (${\tilde T_{\rm{L}}}$ ,$\tilde \omega $ )平面上的转换相图与平衡点分布曲线的叠加图; (b) 图3(a)的局部放大图Figure 3. Bursting Oscillation of
$\tilde \omega $ : (a) Overlay of the transformed phase portrait and equilibrium point distribution curve on the (${\tilde T_L}$ ,$\tilde \omega $ ) plane; (b) locally enlarged diagram of Fig. 3(a).
图 4 PMSM系统的状态响应曲线
Figure 4. State response curves of PMSM system.
图 5 PMSM系统状态的相图
Figure 5. Phase portrait of PMSM system state.
图 6 宏变量
$\varphi $ 的状态响应曲线Figure 6. State response curve of macro-variable
$\varphi $ .
图 7 PMSM系统和宏变量
$\varphi $ 随参数T变化的响应曲线 (a)$\tilde \omega $ 的响应曲线; (b) 宏变量$\varphi $ 的响应曲线Figure 7. Response curves of PMSM system and macro-variable
$\varphi $ as T changes: (a) Response curves of$\tilde \omega $ ; (b) response curves of macro-variable$\varphi $ .
图 8 PMSM系统和宏变量
$\varphi $ 随参数${k_3}$ 变化的响应曲线 (a)$\tilde \omega $ 的响应曲线; (b) 宏变量$\varphi $ 的响应曲线Figure 8. Response curves of PMSM system and macro-variable
$\varphi $ as${k_3}$ changes: (a) Response curves of$\tilde \omega $ ; (b) response curves of macro-variable$\varphi $ .
图 9 不同扰动下的PMSM系统的状态响应
Figure 9. State response of PMSM system under different perturbations.
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[1] Ananthamoorthy N P, Baskaran K 2015 J. Vib. Control 21 181
Google Scholar
[2] Wang L B, Fan J, Wang Z C, Zhan B S, Li J 2016 J. Dyn. Syst. Meas. Contr. 138 011003
Google Scholar
[3] Lu S K, Wang X C, Li Y N 2019 AIP Adv. 9 055105
Google Scholar
[4] Zhang F C, Liao X F, Mu C L 2017 Adv. Differ. Equations 2017 76
Google Scholar
[5] Li Z, Park J B, Joo Y H, Zhang B, Chen G R 2002 IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49 383
Google Scholar
[6] Singh J P, Roy B K, Kuznetsov N V 2019 Int. J. Bifurcation Chaos 29 1950056
Google Scholar
[7] 罗晓曙, 张波, 丘东元, 韦笃取 2009 物理学报 58 6026
Google Scholar
Luo X S, Zhang B, Qiu D Y, Wei D Q 2009 Acta Phys. Sin. 58 6026
Google Scholar
[8] 唐传胜, 戴跃洪 2013 物理学报 62 180504
Google Scholar
Tang C S, Dai Y H 2013 Acta Phys. Sin. 62 180504
Google Scholar
[9] 邢雅清, 陈小可, 张正娣, 毕勤胜 2016 物理学报 65 090501
Google Scholar
Xing Y Q, Chen X K, Zhang Z D, Bi Q S 2016 Acta Phys. Sin. 65 090501
Google Scholar
[10] 张正娣, 刘杨, 张苏珍, 毕勤胜 2017 物理学报 66 020501
Google Scholar
Zhang Z D, Liu Y, Zhang S Z, Bi Q S 2017 Acta Phys. Sin. 66 020501
Google Scholar
[11] Hodgkin A L, Huxley A F 1952 J. Physiol. 117 500
Google Scholar
[12] Izhikevich E M 2000 Int. J. Bifurcation Chaos 10 1171
Google Scholar
[13] 吴天一, 陈小可, 张正娣, 张晓芳, 毕勤胜 2017 物理学报 66 110501
Google Scholar
Wu T Y, Chen X K, Zhang Z D, Zhang X F, Bi Q S 2017 Acta Phys. Sin. 66 110501
Google Scholar
[14] Wu H G, Bao B C, Liu Z, Xu Q, Jing P 2016 Nonlinear Dyn. 83 893
Google Scholar
[15] Wen Z H, Li Z J, Li X 2019 Chaos, Solitons Fractals 128 58
Google Scholar
[16] 李向红, 毕勤胜 2012 物理学报 61 020504
Google Scholar
Li X H, Bi Q S 2012 Acta Phys. Sin. 61 020504
Google Scholar
[17] Razvan M R, Yasaman S 2020 Chaos, Solitons Fractals 132 109482
Google Scholar
[18] Ryashko L, Slepukhina E 2020 Commun. Nonlinear Sci. Numer. Simul. 82 105071
Google Scholar
[19] Han X J, Zhang Y, Bi Q S, Kurths J 2018 Chaos: An Interdiscip. J. Nonlinear Sci. 28 043111
Google Scholar
[20] Peng M, Zhang Z D, Qu Z F, Bi Q S 2020 Pramana - J. Phys. 94 14
Google Scholar
[21] Yu Y, Zhang Z D, Han X J 2018 Commun. Nonlinear Sci. Numer. Simul. 56 380
Google Scholar
[22] Zhang Z D, Chen Z Y, Bi Q S 2019 Theor. Appl. Mech. Lett. 9 358
Google Scholar
[23] Han X J, Xia F B, Zhang C, Yu Y 2017 Nonlinear Dyn. 88 2693
Google Scholar
[24] Bi Q S, Ma R, Zhang Z D 2015 Nonlinear Dyn. 79 101
Google Scholar
[25] Wei D Q, Zhang B, Luo X S 2012 Chin. Phys. B 21 030504
Google Scholar
[26] Han X J, Bi Q S 2011 Commun. Nonlinear Sci. Numer. Simul. 16 4146
Google Scholar
[27] 王江彬, 刘崇新 2020 西安交通大学学报 54 26
Google Scholar
Wang J B, Liu C X 2020 J. Xi'an Jiaotong Univ. 54 26
Google Scholar
[28] Wang J B, Liu L, Liu C X 2019 Int. J. Bifurcation Chaos 29 1950130
Google Scholar
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