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## General robust stability controller design method for a class of hopf bifurcation systems

Lu Jin-Bo, Hou Xiao-Rong, Luo Min
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• #### 摘要

针对一类多项式形式的Hopf分岔系统, 提出了一种鲁棒稳定的控制器设计方法. 使用该方法设计控制器时不需要求解出系统在分岔点处的分岔参数值, 只需要估算出分岔参数的上下界, 然后设计一个参数化的控制器, 并通过Hurwitz判据和柱形代数剖分技术求解出满足上下界条件的控制器参数区域, 最后在得到的这个区域内确定出满足鲁棒稳定的控制器参数值. 该方法设计的控制器是由包含系统状态的多项式构成, 形式简单, 具有通用性, 且添加控制器后不会改变原系统平衡点的位置. 本文首先以Lorenz系统为例说明了控制器的推导和设计过程, 然后以van der Pol振荡系统为例, 进行了工程应用. 通过对这两个系统的控制器设计和仿真, 说明了文中提出的控制器设计方法能够有效地应用于这类Hopf分岔系统的鲁棒稳定控制, 并且具有通用性.

#### Abstract

For the nonlinear Hopf bifurcation system, the change of bifurcation parameter has an important influence on the state of the system. In order to control the Hopf bifurcations of the nonlinear dynamic system, the parameter values of bifurcation points in the system need to be found out before controller designing. However, due to uncertainties of the system structure and parameters in the nonlinear system, or disturbance, it is difficult to determine the bifurcation point precisely. So it is a good way of designing a robust controller near the bifurcation point. Although, lots of works have discussed the robust control of a Hopf bifurcation in a nonlinear dynamic system, the solutions are not satisfactory and there are still many problems. The controller is always designed for some special system. Its structure is usually complex, not general, and the design process is complicated. And before controller design, the value of bifurcation point must be solved accurately.In this paper, a parametric robust stability controller design method is proposed for a class of polynomial form Hopf bifurcation systems. Using this method, it is not necessary to solve the exact values of the bifurcation parameter, it is only needed to determine the bifurcation parameter range. The designed controller includes a system state polynomial; its structure is general, simple and keeps the equilibrium of the original system unchanged. By using the Hurwitz criterion, the system stability constraints for bifurcation parameter boundaries are obtained at equilibrium, and they are described by algebraic inequalities. Cylindrical algebraic decomposition is applied to calculate the stability region of the controller parameters. And then, in the region, parameters of the robust controller can be calculated to make the dynamic system stable. In this paper, the Lorenz system without disturbance is used as an example to show the designing process of the method, and then the controller of the van der Pol oscillator system with disturbance is designed by this method as an engineering application. Simulations of the two systems are given to demonstrate that the proposed controller designing method can be effectively applied to the robust stability control of the Hopf bifurcation systems.

#### 作者及机构信息

###### 通信作者: 侯晓荣, houxr@uestc.edu.cn
• 基金项目: 国家自然科学基金(批准号: 61374001)资助的课题.

#### Authors and contacts

###### Corresponding author: Hou Xiao-Rong, houxr@uestc.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61374001).

#### 参考文献

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#### 施引文献

•  [1] Wu Z Q, Sun L M 2011 Acta Phys. Sin. 60 050504 (in Chinese) [吴志强, 孙立明 2011 物理学报 60 050504] [2] Zhu L H, Zhao H Y 2014 Acta Phys. Sin. 63 090203 (in Chinese) [朱霖河, 赵洪涌 2014 物理学报 63 090203] [3] Wang J S, Yuan R X, Gao Z W, Wang D J 2011 Chin. Phys. B 20 090506 [4] Pei Y, Chen G R 2004 Int. J. Bifurcat. Chaos 14 1683 [5] Nguyen L, Hong K 2012 Phys. Lett. A 376 442 [6] Wang F Q, Ma X K 2013 Chin. Phys. B 22 120504 [7] Yfoulis C, Giaouris D, Stergiopoulos F, Ziogou C, Voutetakis S, Papadopoulou S 2015 Control Eng. Pract. 35 67 [8] Zhao H Y, Xie W 2011 Nonlinear Dyn. 63 345 [9] Wu Z, Yu P 2006 IEEE Trans. Automat. Control 51 1019 [10] Mohammadi A, Marvdasht I 2012 Problems of Cybernetics and Informatics (PCI), 2012 IV International Conference Baku, September 12-14, 2012 p1 [11] Kishida M, Braatz R 2014 American Control Conference (ACC), Portland, OR, June 4-6, 2014 p5085 [12] Cao G Y, Hill D J 2010 IET Gener. Transm. Dis. 4 873 [13] Inoue M, Imura J, Kashima K, Aihara K 2015 Analysis and Control of Complex Dynamical Systems 7 3 [14] Hu J, Liu L, Ma D W 2014 J. Korean Phys. Soc. 65 2132 [15] Liang J S, Chen Y S, Leung A Y T 2004 Appl. Math. Mech. 25 263 [16] Yue M, Schlueter R 2005 IEEE Trans. Power Syst. 20 301 [17] Chen D W, Gu H B, Liu H 2010 J. Vib. and Shock 29 30 (in Chinese) [陈大伟, 顾宏斌, 刘晖 2010 振动与冲击 29 30] [18] He C M, Jin L, Chen D G, Geiger R 2007 IEEE Trans. Circ. Syst. 54 964 [19] Liu S, Zhao S S, Wang Z L, Li H B 2015 Chin. Phys. B 24 014501 [20] Li H Y, Che Yan Q, Wang J, Jin Q T, Deng B, Wei X L, Dong F 2012 The 10th World Congress on Intelligent Control and Automation (WCICA) Beijing, July 6-8, 2012 p4953 [21] Yang T B, Chen X 2008 Proceedings of the 47th IEEE Conference on Decision and Control(CDC) Cancun, Mexico, December 9-11, 2008 p4103 [22] LaValle S M 2006 Planning Algorithms (Cambridge: Cambridge University Press) pp 280-290 [23] Gandy S, Kanno M, Anai H, Yokoyama K 2011 Math. Comput. Sci. 5 209 [24] England M, Bradford R, Chen C, Davenport J H, Maza M M, Wilson D 2014 Intellig. Comput. Math. 8543 45
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##### 出版历程
• 收稿日期:  2015-10-24
• 修回日期:  2015-12-10
• 刊出日期:  2016-03-05

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