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The tracking control of chaotic system has been one of the research focus areas of nonlinear control in recent years, in which the vital problem is to enable chaotic system to stabilize to an equilibrium point or to track a deterministic trajectory quickly. The conventional chaos control methods make the control power unnecessarily large and generate the phenomenon of chattering easily, resulting in the instabilities of the systems.The problems above can be transformed into the solutions of differential algebraic equations effectively. Considering that the group preserving scheme not only approximates the original system, but also preserve as much as possible the geometric structure and invariants of the original system, this paper takes advantage of the group preserving method to study the control method in chaotic system from two different perspectives.A new group preserving scheme based on the fast descending control method is presented, which enables chaotic system to stabilize to an equilibrium point quickly. Firstly, we introduce a novel approach to replace the optimal control problem of nonlinear system by directly specifying a time-decaying Lagrangian function, which helps us to transform the optimal control problem into a system of differential algebraic equations. Then we derive a modified group preserving scheme for the system.Similarly, we propose a new group preserving scheme based on the sliding mode control method for chaotic system to track a deterministic trajectory quickly. Owing to numerical discretization errors, signal noises and structural uncertainties in dynamical systems, the conventional sliding mode control method cannot guarantee to maintain the trajectories on the sliding surface, unless the numerical integration method is designed to do so. On the other hand, the conventional sliding mode control method easily induces high frequency chattering of the control force. Therefore, we modify the conventional sliding mode control method and use the modified group preserving scheme to find the control force.The above two methods are the combination of traditional control method and the Lie-group method. An invariant manifold is properly designed, and the original system is transformed into the differential algebraic system, in which the modified group preserving scheme can be used to find the control force. The resulting controlled system is stable.Finally, the proposed methods are applied to the classic Lorenz system and Duffing system correspondingly. Numerical experimental results show that the new approaches are very accurate and stable. Since the two controlled methods are fast in convergence and chattering-free, each of them has a good application prospect in the tracking control of chaotic systems.
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Keywords:
- tracking control of chaos /
- group preserving method /
- Lorenz system /
- Duffing system
[1] Chen G R, Ueta T 1999 Int. J. Bifurcat. Chaos 9 1465
[2] L J H, Chen G R 2002 Int. J. Bifurcat. Chaos 12 659
[3] Zhang H G, Wang Z L,Huang W 2008 Control Theory of Chaotic Systems (Vol. 1) (Shenyang: Northeast Univesity Press) pp1-4 (in Chinese) [张化光, 王智良, 黄伟 2008 混沌系统的控制理论 (沈阳: 东北大学出版社) 第1-4页]
[4] Zhang X H, Shen J, Mei L, Wang D M 2011 Syst. Eng. Elect. 33 603 (in Chinese) [张兴华, 沈捷, 梅磊, 王德明 2011 系统工程与电子技术 33 603]
[5] Davies M J 1972 J. Inst. Math. Appl. 9 357
[6] Van D R, Vlassenbroeck J 1982 J. Comput. Phys. 47 321
[7] El-Kady M, Elbarbary E M E 2002 Appl. Math. Comput. 129 171
[8] Razzaghi M, Elnagar G 1994 J. Comput. Appl. Math. 56 253
[9] Lakestani M, Razzaghi M, Dehghan M 2006 Phys. Scr. 74 362
[10] Song R Z, Xiao W D, Sun C Y, Wei Q L 2013 Chin. Phys. B 22 090502
[11] Wei Q L, Liu D R, Xu Y C 2015 Chin. Phys. B 24 030502
[12] Wei Q L, Song R Z, Sun Q Y, Xiao W D 2015 Chin. Phys. B 24 090504
[13] Liu C S {2012 CMES: Comput. Model. Eng. Sci. 86 171
[14] Liu C S 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2012
[15] Utkin V I 1977 IEEE Trans. Autom. Contr. 22 212
[16] Utkin V I 1992 Sliding Modes in Control and Optimization (Vol. 1) (New York: Springer-Verlag) pp7-11
[17] Slotine J J E, Sastry S S 1983 Int. J. Control 38 465
[18] Levant A 1993 Int. J. Control 58 1247
[19] Wong L K, Leung F H F, Tam P K S 1998 Mechatronics 8 765
[20] Lee H, Utkin V I 2007 Ann. Rev. Control 31 179
[21] Yang G L, Li H G 2009 Acta Phys. Sin. 58 7552 (in Chinese) [杨国良, 李惠光 2009 物理学报 58 7552]
[22] Falahpoor M, Ataei M, Kiyoumarsi A 2009 Chaos Solitons Fract. 42 1755
[23] Qi L 2013 Ph. D. Dissertation (Shanghai: East China University of Science and Technology) (in Chinese) [齐亮 2013 博士学位论文 (上海: 华东理工大学)]
[24] Dong K W, Zhang X 2007 Electrotechnical Application 2 6 (in Chinese) [董克文, 张兴 2007 电气应用 2 6]
[25] Liu C S 2001 Int. J. Non-Linear Mech. 36 1047
[26] Liu C S {2006 CMES: Comput. Model. Eng. Sci. 12 83
[27] Wu W G, Gu T X 2000 Acta Phys. Sin. 49 1922 (in Chinese) [伍维根, 古天祥 2000 物理学报 49 1922]
[28] Cai G L, Tan Z M, Zhou W H, Tu W T 2007 Acta Phys. Sin. 56 6230 (in Chinese) [蔡国梁, 谭振梅, 周维怀, 涂文桃 2007 物理学报 56 6230]
[29] Li G Y 2008 Optimal Control Theory and Application (Vol. 1) (Beijing: National Defence Industry Press) p4 (in Chinese) [李国勇 2008 最优化控制理论与应用 (北京: 国防工业出版社) 第4页]
[30] Wang W 2009 M. S. Dissertation (Dalian: Dalian Jiaotong University) (in Chinese) [王文 2009 硕士学位论文 (大连: 大连交通大学)]
[31] Roopaei M, Sahraei B R, Lin T C 2010 Commun. Nonlinear Sci. Numer. Simul. 15 4158
[32] Shen C W, Yu S M, L J H, Chen G R 2014 IEEE Trans. Circuits Syst. I 61 854
[33] Shen C W, Yu S M, L J H, Chen G R 2014 IEEE Trans. Circuits Syst. I 61 2380
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[1] Chen G R, Ueta T 1999 Int. J. Bifurcat. Chaos 9 1465
[2] L J H, Chen G R 2002 Int. J. Bifurcat. Chaos 12 659
[3] Zhang H G, Wang Z L,Huang W 2008 Control Theory of Chaotic Systems (Vol. 1) (Shenyang: Northeast Univesity Press) pp1-4 (in Chinese) [张化光, 王智良, 黄伟 2008 混沌系统的控制理论 (沈阳: 东北大学出版社) 第1-4页]
[4] Zhang X H, Shen J, Mei L, Wang D M 2011 Syst. Eng. Elect. 33 603 (in Chinese) [张兴华, 沈捷, 梅磊, 王德明 2011 系统工程与电子技术 33 603]
[5] Davies M J 1972 J. Inst. Math. Appl. 9 357
[6] Van D R, Vlassenbroeck J 1982 J. Comput. Phys. 47 321
[7] El-Kady M, Elbarbary E M E 2002 Appl. Math. Comput. 129 171
[8] Razzaghi M, Elnagar G 1994 J. Comput. Appl. Math. 56 253
[9] Lakestani M, Razzaghi M, Dehghan M 2006 Phys. Scr. 74 362
[10] Song R Z, Xiao W D, Sun C Y, Wei Q L 2013 Chin. Phys. B 22 090502
[11] Wei Q L, Liu D R, Xu Y C 2015 Chin. Phys. B 24 030502
[12] Wei Q L, Song R Z, Sun Q Y, Xiao W D 2015 Chin. Phys. B 24 090504
[13] Liu C S {2012 CMES: Comput. Model. Eng. Sci. 86 171
[14] Liu C S 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2012
[15] Utkin V I 1977 IEEE Trans. Autom. Contr. 22 212
[16] Utkin V I 1992 Sliding Modes in Control and Optimization (Vol. 1) (New York: Springer-Verlag) pp7-11
[17] Slotine J J E, Sastry S S 1983 Int. J. Control 38 465
[18] Levant A 1993 Int. J. Control 58 1247
[19] Wong L K, Leung F H F, Tam P K S 1998 Mechatronics 8 765
[20] Lee H, Utkin V I 2007 Ann. Rev. Control 31 179
[21] Yang G L, Li H G 2009 Acta Phys. Sin. 58 7552 (in Chinese) [杨国良, 李惠光 2009 物理学报 58 7552]
[22] Falahpoor M, Ataei M, Kiyoumarsi A 2009 Chaos Solitons Fract. 42 1755
[23] Qi L 2013 Ph. D. Dissertation (Shanghai: East China University of Science and Technology) (in Chinese) [齐亮 2013 博士学位论文 (上海: 华东理工大学)]
[24] Dong K W, Zhang X 2007 Electrotechnical Application 2 6 (in Chinese) [董克文, 张兴 2007 电气应用 2 6]
[25] Liu C S 2001 Int. J. Non-Linear Mech. 36 1047
[26] Liu C S {2006 CMES: Comput. Model. Eng. Sci. 12 83
[27] Wu W G, Gu T X 2000 Acta Phys. Sin. 49 1922 (in Chinese) [伍维根, 古天祥 2000 物理学报 49 1922]
[28] Cai G L, Tan Z M, Zhou W H, Tu W T 2007 Acta Phys. Sin. 56 6230 (in Chinese) [蔡国梁, 谭振梅, 周维怀, 涂文桃 2007 物理学报 56 6230]
[29] Li G Y 2008 Optimal Control Theory and Application (Vol. 1) (Beijing: National Defence Industry Press) p4 (in Chinese) [李国勇 2008 最优化控制理论与应用 (北京: 国防工业出版社) 第4页]
[30] Wang W 2009 M. S. Dissertation (Dalian: Dalian Jiaotong University) (in Chinese) [王文 2009 硕士学位论文 (大连: 大连交通大学)]
[31] Roopaei M, Sahraei B R, Lin T C 2010 Commun. Nonlinear Sci. Numer. Simul. 15 4158
[32] Shen C W, Yu S M, L J H, Chen G R 2014 IEEE Trans. Circuits Syst. I 61 854
[33] Shen C W, Yu S M, L J H, Chen G R 2014 IEEE Trans. Circuits Syst. I 61 2380
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