logo

SCIENTIA SINICA Informationis, Volume 49, Issue 8: 1031-1049(2019) https://doi.org/10.1360/N112018-00095

Pole assignment of two-dimensional switched linear time- invariant systems with multiple equilibria

More info
  • ReceivedApr 19, 2018
  • AcceptedAug 22, 2018
  • PublishedAug 9, 2019

Abstract

This paper discusses the pole assignment issues of two-dimensional switched linear time-invariant (LTI) systems with multiple equilibria. For the case in which each subsystem has a unique equilibrium point, a necessary and sufficient condition of arbitrary pole assignments for such switched LTI systems with multiple equilibria is proposed. A numerical example shows that even when all the poles of all the closed-loop subsystems are assigned to only two locations on the left-half side of the complex plane, the overall switched LTI systems may not be stable under arbitrary switching. For switched systems in which all the subsystems have a common single equilibrium point or different multiple equilibria, several sufficient criteria of stabilizing pole assignments and corresponding algorithms are proposed. The results imply that to stabilize switched LTI systems via the pole assignment method, all or some of the poles of some or all the subsystems can be assigned to suitable locations on the right-half side of the complex plane. An illustrative example shows that our new results are very effective and practical.


Funded by

山东省自然科学基金项目(ZR2017MF071,ZR2017MA018))

国家自然科学基金项目(61873311))

哈尔滨工业大学机器人技术与系统国家重点实验室项目(: SKLRS201801A03))

山东工商学院博士启动基金项目(: BS201617))


References

[1] Zheng D Z. Linear System Theory. 2nd ed. Beijing: Tsinghua University Press, 2002 [郑大钟 .线性系统理论 .第 2版.北京:清华大学出版社, 2002]. Google Scholar

[2] Antsaklis P J, Michel A N. Linear Systems. Englewood Cliffs: McGraw-Hill, 1997. Google Scholar

[3] Kautsky J, Nichols N K, Van Dooren P. Robust pole assignment in linear state feedback. Int J Control, 1985, 41: 1129-1155 CrossRef Google Scholar

[4] Rissanen J. Control system synthesis by analogue computer based on the generalized linear feedback concept. In: Proceedings of Symposium on Analog Comput Applied to the Study of Chemistry Processes, Brussels, 1960. 1–13. Google Scholar

[5] 5 Wonham W M. On pole assignment in multi-input controllable linear systems. IEEE Trans Automat Contr, 1967, 12:660–665. Google Scholar

[6] Morse A S, Wonham W M. Decoupling and pole assignment by dynamic compensation. SIAM J Control, 1970, 8: 317-337 CrossRef Google Scholar

[7] Feng G, Zhang C, Palaniswami M. Stability of input amplitude constrained adaptive pole placement control systems. Automatica, 1994, 30: 1065-1070 CrossRef Google Scholar

[8] Daizhan Cheng , Lei Guo , Yuandan Lin , et al. Stabilization of switched linear systems. IEEE Trans Automat Contr, 2005, 50: 661-666 CrossRef Google Scholar

[9] Sun Z, Ge S S. Analysis and synthesis of switched linear control systems. Automatica, 2005, 41: 181-195 CrossRef Google Scholar

[10] 0 Zhu L Y, Wang Y Z. Necessary and sufficient conditions for uniform controllability and observability of switched linear systems. J Shandong Univ (Eng Sci ), 2007, 37: 43–46 [朱礼营 ,王玉振 .线性切换系统一致能控性和能观性的充要条件.山东大学学报 (工学版), 2007, 37: 43–46]. Google Scholar

[11] Lin H, Antsaklis P J. Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans Automat Contr, 2009, 54: 308-322 CrossRef Google Scholar

[12] Li Q K, Lin H. Effects of mixed-modes on the stability analysis of switched time-varying delay systems. IEEE Trans Automat Contr, 2016, 61: 3038-3044 CrossRef Google Scholar

[13] Branicky M S. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Automat Contr, 1998, 43: 475-482 CrossRef Google Scholar

[14] Zhu L Y, Xiang Z R. Aggregation analysis for competitive multiagent systems with saddle points via switching strategies. IEEE Trans Neural Netw Learn Syst, 2018, 29: 2931–2943. Google Scholar

[15] Zhu L. Stability and stabilization of switched linear time-invariant systems with saddle points and switching delays. Inf Sci, 2016, 326: 146-159 CrossRef Google Scholar

[16] Zhu L. Stability and stabilization of two-dimensional lti switched systems with potentially unstable focus. Asian J Control, 2015, 17: 892-907 CrossRef Google Scholar

[17] Sun X M, Zhao J, Hill D J. Stability and L2-gain analysis for switched delay systems: a delay-dependent method. Google Scholar

[18] Sun X M, Zhao J, Hill D J. Stability and L2-gain analysis for switched delay systems: A delay-dependent method. Automatica, 2006, 42: 1769-1774 CrossRef Google Scholar

[19] Li Z G, Wen C Y, Soh Y C. Observer-based stabilization of switching linear systems. Automatica, 2003, 39: 517-524 CrossRef Google Scholar

[20] Sun Y G, Wang L, Xie G, et al. Improved overshoot estimation in pole placements and its application in observer-based stabilization for switched systems. IEEE Trans Automat Contr, 2006, 51: 1962-1966 CrossRef Google Scholar

[21] Mansouri B, Manamanni N, Guelton K, et al. Robust pole placement controller design in LMI region for uncertain and disturbed switched systems. NOnlinear Anal-Hybrid Syst, 2008, 2: 1136-1143 CrossRef Google Scholar

[22] Zhu L Y, Wang Y Z. Stability analysis of switched dissipative Hamiltonian systems. Sci China Ser F-Inf Sci, 2006, 36: 617–630 [朱礼营,王玉振.切换耗散 Hamilton系统的稳定性研究.中国科学 E辑:信息科学, 2006, 36: 617–630]. Google Scholar

[23] Zhao J, Hill D J. Passivity and stability of switched systems: a multiple storage function method. Syst Control Lett, 2008, 57: 158-164 CrossRef Google Scholar

[24] Bao G, Zeng Z. Region stability analysis for switched discrete-time recurrent neural network with multiple equilibria. Neurocomputing, 2017, 249: 182-190 CrossRef Google Scholar

[25] Zhu L. Dynamics of switching van der pol circuits. NOnlinear Dyn, 2017, 87: 1217-1234 CrossRef Google Scholar

[26] Zhu L, Qiu J, Chadli M. Modelling and stability analysis of switching impulsive power systems with multiple equilibria. Int J Syst Sci, 2017, 48: 3470-3490 CrossRef Google Scholar

[27] Zhu L, Qiu J. Region stability and stabilisation of switched linear systems with multiple equilibria. Int J Control, 2019, 92: 1061-1083 CrossRef Google Scholar

[28] Narendra K S, Balakrishnan J. A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Trans Automat Contr, 1994, 39: 2469-2471 CrossRef Google Scholar

[29] Alpcan T, Basar T. A stability result for switched systems with multiple equilibria. Dyn Contin Discret Impuls Syst Ser A Math Anal, 2010, 17: 949–958. Google Scholar

[30] Guo R W, Wang Y Z. Estimation of stability region for a class of switched linear systems with multiple equilibrium points. Control Theor Appl, 2012, 29: 409–414 [郭荣伟 ,王玉振 .一类多平衡点线性切换系统稳定区域的估计 .控制理论与应用, 2012, 29: 409–414]. Google Scholar

[31] Navarro-Lopez E M, Laila D S. Group and total dissipativity and stability of multi-equilibria hybrid automata. IEEE Trans Automat Contr, 2013, 58: 3196-3202 CrossRef Google Scholar

[32] Wu W, Duan G, Tan F. Switching signal design for asymptotic stability of uncertain multiple equilibrium switched systems with actuator saturation. Int J Robust NOnlinear Control, 2016, 26: 1705-1717 CrossRef Google Scholar

[33] Guo R W, Wang Y Z. Region stability analysis for switched nonlinear systems with multiple equilibria. Int J Control Autom Syst, 2017, 15: 567-574 CrossRef Google Scholar

[34] Liu Z, Zhang X F, Wang Y Z. Stability and stabilization for discrete-time positive switched multiple equilibria systems on finite time intervals. Control Theor Appl, 2017, 34: 433–440 [刘志 ,张宪福 ,王玉振 .离散多平衡点正切换系统有限区间稳定与镇定.控制理论与应用, 2017, 34: 433–440]. Google Scholar

[35] Liu Z, Wang Y Z. Regional stability of positive switched linear systems with multi-equilibrium points. Int J Autom Comput, 2017, 14: 213-220 CrossRef Google Scholar

[36] Li Y, Tong S, Liu L, et al. Adaptive output-feedback control design with prescribed performance for switched nonlinear systems. Automatica, 2017, 80: 225-231 CrossRef Google Scholar

[37] Li Y M, Tong S C. Adaptive neural networks prescribed performance control design for switched interconnected uncertain nonlinear systems. IEEE Trans Neural Netw Learn Syst, 2018, 29: 3059–3068. Google Scholar

[38] Chen M, Ren B B, Wu Q X, et al. Anti-disturbance control of hypersonic flight vehicles with input saturation using disturbance observer. Sci China Inf Sci, 2015, 58: 070202. Google Scholar

[39] Chen M, Tao G. Adaptive fault-tolerant control of uncertain nonlinear large-scale systems with unknown dead zone. IEEE Trans Cybern, 2016, 46: 1851-1862 CrossRef PubMed Google Scholar

[40] Zhu L, Feng G. Necessary and sufficient conditions for stability of switched nonlinear systems. J Franklin Institute, 2015, 352: 117-137 CrossRef Google Scholar

[41] Zhu L. Region aggregation analysis for multi-agent networks with multi-equilibria in multi-dimensional coordinate systems via switching strategies. Neurocomputing, 2016, 171: 991-1002 CrossRef Google Scholar

[42] 1 Zhai G, Hu B, Yasuda K, et al. Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. Int J Syst Sci, 2001, 32: 1055–1061 42 Zhu L Y. Stability analysis of two-dimensional LTI switched systems with multiple equilibria. In: Proceedings of the 33rd Chinese Conference (CCC), Nanjing, 2014. 4131–4136. Google Scholar

[43] Zhu L Y, Fang Y Y. Stability and stabilization of two-dimensional linear time-invariant switched systems with multi-equilibria. Control Decis, 2015, 30: 599–604 [朱礼营 ,方盈盈 .多平衡点二维线性时不变切换系统的稳定性及镇定性.控制与决策, 2015, 30: 599–604]. Google Scholar

[44] Zhu L. A condition for boundedness of solutions of bidimensional switched affine systems with multiple foci and centers. Asian J Control, 2018, 20: 585-594 CrossRef Google Scholar

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1       京公网安备11010102003388号