Volume 13, Issue 1 (Journal of Control, V.13, N.1 Spring 2019)
JoC 2019, 13(1): 21-33 |
Back to browse issues page
1- Shahrood University of Technology
2- Ferdowsi University of Mashhad
2- Ferdowsi University of Mashhad
Abstract: (5566 Views)
Stability and stabilization of impulsive switched system have been considered in recent decades, but there are some issues that are not yet fully addressed such as actuator saturation. This paper deals with expo-nential stabilization for a class of nonlinear impulsive switched systems with different types of non-vanishing uncertainties under the norm-bounded control input. Due to the constrained control signal, the local stabilization is here considered. To establish local stabilization criteria, at first, based on multiple Lyapunov functions technique and minimum dwell-time approach, sufficient conditions for a more gen-eral model of impulsive switched systems are proposed. These conditions are also local which increases the chances of achieving the goals compared to existing global techniques. Also, unlike conventional mul-tiple Lyapunov function techniques, this paper considers converging to a sufficient small ultimate bound because of non-vanishing property of the uncertainties. Secondly, the proposed conditions for the general model are applied to the given system and the sufficient stability conditions are formed into linear and bi-linear matrix inequalities. After these, to achieve the parameters of stabilizing control signal along with the largest convergence area and smallest ultimate bound, an optimization problem is proposed. Finally, some illustrative numerical examples are presented to demonstrate the proposed approach.
Keywords: Exponential stability, Nonlinear impulsive switched systems, Non-vanishing uncertainty, Norm-bounded input, State feedback.
Type of Article: Research paper |
Subject:
Special
Received: 2017/04/5 | Accepted: 2018/07/23 | Published: 2019/07/15
Received: 2017/04/5 | Accepted: 2018/07/23 | Published: 2019/07/15
References
1. [1] R. Shi, X. Jiang, L. Chen, The effect of impulsive vaccination on an SIR epidemic model, Appl. Math. Comput. 212 (2009) 305-311. [DOI:10.1016/j.amc.2009.02.017]
2. [2] X.-M. Sun, W. Wang, Integral input-to-state stability for hybrid delayed systems with unstable continuous dynamics, Automatica. 48 (2012) 2359-2364. [DOI:10.1016/j.automatica.2012.06.056]
3. [3] G. Pang, Z. Liang, W. Xu, L. Li, G. Fu, A Pest Management Model with Stage Structure and Impulsive State Feedback Control, Discret. Dyn. Nat. Soc. 2015 (2015) 1-12. [DOI:10.1155/2015/617379]
4. [4] J. Jiao, S. Cai, L. Chen, Dynamics of a plankton-nutrient chemostat model with hibernation and it described by impulsive switched systems, J. Appl. Math. Comput. 53 (2017) 583-598. [DOI:10.1007/s12190-015-0983-6]
5. [5] A.D. Ames, K. Galloway, K. Sreenath, J.W. Grizzle, Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics, IEEE Trans. Automat. Contr. 59 (2014) 876-891. [DOI:10.1109/TAC.2014.2299335]
6. [6] M. Posa, M. Tobenkin, R. Tedrake, Stability Analysis and Control of Rigid-Body Systems with Impacts and Friction, IEEE Trans. Automat. Contr. (2015) 1423-1437. [DOI:10.1109/TAC.2015.2459151]
7. [7] Z.-G. Wu, P. Shi, H. Su, J. Chu, Sampled-Data Fuzzy Control of Chaotic Systems Based on a T-S Fuzzy Model, IEEE Trans. Fuzzy Syst. 22 (2014) 153-163. [DOI:10.1109/TFUZZ.2013.2249520]
8. [8] X. Wan, J. Sun, Adaptive-impulsive synchronization of chaotic systems, Math. Comput. Simul. 81 (2011) 1609-1617. [DOI:10.1016/j.matcom.2010.11.012]
9. [9] T. Fang, J. Sun, Stability of complex-valued impulsive and switching system and application to the Lü system, Nonlinear Anal. Hybrid Syst. 14 (2014) 38-46. [DOI:10.1016/j.nahs.2014.04.004]
10. [10] ملا احمدیان کاسب حامد، کریم پور علی، پریز ناصر، سیستم های تکه ای خطی تبار مستقیم: کلاس جدیدی از سیستم های هایبرید با دینامیک‌های خطی تبار و مرزهای کلیدزنی قابل تنظیم. مجله کنترل. ۱۳۹۱; ۶ (۱) :۲۱-۲۹.
11. [11] Zhi-Hong Guan, D.J. Hill, Xuemin Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Automat. Contr. 50 (2005) 1058-1062. [DOI:10.1109/TAC.2005.851462]
12. [12] W.M. Haddad, V. Chellaboina, S.G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, 2006. [DOI:10.1515/9781400865246]
13. [13] H. Lin, P.J. Antsaklis, Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results, IEEE Trans. Automat. Contr. 54 (2009) 308-322. [DOI:10.1109/TAC.2008.2012009]
14. [14] F. Xu, L. Dong, D. Wang, X. Li, R. Rakkiyappan, Globally exponential stability of nonlinear impulsive switched systems, Math. Notes. 97 (2015) 803-810. [DOI:10.1134/S0001434615050156]
15. [15] H. Xu, K.L. Teo, Exponential Stability With $L_{2}$-Gain Condition of Nonlinear Impulsive Switched Systems, IEEE Trans. Automat. Contr. 55 (2010) 2429-2433. [DOI:10.1109/TAC.2010.2060173]
16. [16] Y. Chen, S. Fei, K. Zhang, Stabilization of impulsive switched linear systems with saturated control input, Nonlinear Dyn. 69 (2012) 793-804. [DOI:10.1007/s11071-011-0305-y]
17. [17] اقرزاده کوهبنانی محمد علی، قیصری جعفر، عسکری جواد، مجیری محسن. پایدارسازی مقاوم سیستم‌های سوئیچینگ خطی با استفاده از فیدبک حالت مبتنی بر رویتگر و سیگنال سوئیچ با حداقل زمان اقامت مشخص. مجله کنترل. ۱۳۹۳; ۸ (۴) :۵۵-۶۴.
18. [18] Y. Tian, Y. Cai, Y. Sun, H. Gao, Finite-time stability for impulsive switched delay systems with nonlinear disturbances, J. Franklin Inst. 353 (2016) 3578-3594. [DOI:10.1016/j.jfranklin.2016.06.021]
19. [19] S. Li, Z. Xiang, Stability and L1-gain control for positive impulsive switched systems with mixed time-varying delays, IMA J. Math. Control Inf. (2016) dnw030. [DOI:10.1093/imamci/dnw030]
20. [20] Y. Yang, G. Chen, Finite-time stability of fractional order impulsive switched systems, Int. J. Robust Nonlinear Control. 25 (2014) 2207-2222. [DOI:10.1002/rnc.3202]
21. [21] G. Zong, Q. Wang, Robust resilient control for impulsive switched systems under asynchronous switching, Int. J. Comput. Math. 92 (2015) 1143-1159. [DOI:10.1080/00207160.2014.946413]
22. [22] G. Feng, J. Cao, Stability analysis of impulsive switched singular systems, IET Control Theory Appl. 9 (2015) 863-870. [DOI:10.1049/iet-cta.2013.1142]
23. [23] اعظم بالغی نصراله، شفیعی محمد حسین. تحلیل پایداری سیستم‌های سوئیچ‌شوندۀ خطی گسسته‌زمان با در نظر گرفتن تاخیر زمانی و عدم قطعیت پارامتری. مجله کنترل. ۱۳۹۴; ۹ (۴) :۷۷-۸۵.
24. [24] X. Zhao, P. Shi, Y. Yin, S.K. Nguang, New Results on Stability of Slowly Switched Systems: A Multiple Discontinuous Lyapunov Function Approach, IEEE Trans. Automat. Contr. 62 (2017) 3502-3509. [DOI:10.1109/TAC.2016.2614911]
25. [25] B. Wang, H. Zhang, G. Wang, C. Dang, S. Zhong, Asynchronous control of discrete-time impulsive switched systems with mode-dependent average dwell time, ISA Trans. 53 (2014) 367-372. [DOI:10.1016/j.isatra.2013.11.019]
26. [26] X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and Stabilization of Switched Linear Systems With Mode-Dependent Average Dwell Time, IEEE Trans. Automat. Contr. 57 (2012) 1809-1815. [DOI:10.1109/TAC.2011.2178629]
27. [27] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Contr. 43 (1998) 475-482. [DOI:10.1109/9.664150]
28. [28] Hui Ye, A.N. Michel, Ling Hou, Stability theory for hybrid dynamical systems, IEEE Trans. Automat. Contr. 43 (1998) 461-474. [DOI:10.1109/9.664149]
29. [29] W. Xiang, J. Xiao, Stabilization of switched continuous-time systems with all modes unstable via dwell time switching, Automatica. 50 (2014) 940-945. [DOI:10.1016/j.automatica.2013.12.028]
30. [30] X. Xie, H. Xu, R. Zhang, Exponential Stabilization of Impulsive Switched Systems with Time Delays Using Guaranteed Cost Control, Abstr. Appl. Anal. 2014 (2014) 1-8. [DOI:10.1155/2014/126836]
31. [31] L. Gao, D. Wang, Input-to-state stability and integral input-to-state stability for impulsive switched systems with time-delay under asynchronous switching, Nonlinear Anal. Hybrid Syst. 20 (2016) 55-71. [DOI:10.1016/j.nahs.2015.12.002]
32. [32] P. Li, J. Lam, K.C. Cheung, Stability, stabilization and L2-gain analysis of periodic piecewise linear systems, Automatica. 61 (2015) 218-226. [DOI:10.1016/j.automatica.2015.08.024]
33. [33] L. Lu, Z. Lin, Design of Switched Linear Systems in the Presence of Actuator Saturation, IEEE Trans. Automat. Contr. 53 (2008) 1536-1542. [DOI:10.1109/TAC.2008.921021]
34. [34] A. Benzaouia, O. Akhrif, L. Saydy, Stabilisation and control synthesis of switching systems subject to actuator saturation, Int. J. Syst. Sci. 41 (2010) 397-409. [DOI:10.1080/00207720903045791]
35. [35] W. Ni, D. Cheng, Control of switched linear systems with input saturation, Int. J. Syst. Sci. 41 (2010) 1057-1065. [DOI:10.1080/00207720903201865]
36. [36] A. Poznyak, A. Polyakov, V. Azhmyakov, Attractive Ellipsoids in Robust Control, Springer International Publishing, Cham, 2014. [DOI:10.1007/978-3-319-09210-2]
37. [37] H. Yang, B. Jiang, J. Zhao, On Finite-Time Stability of Cyclic Switched Nonlinear Systems, IEEE Trans. Automat. Contr. 60 (2015) 2201-2206. [DOI:10.1109/TAC.2014.2366856]
38. [38] M. Kocvara, M. Stingl, PENNON: Software for Linear and Nonlinear Matrix Inequalities, in: M.F. Anjos, J.B. Lasserre (Eds.), Handb. Semidefinite, Conic Polynomial Optim., Springer US, 2012: pp. 755-791. [DOI:10.1007/978-1-4614-0769-0_26]
39. [39] X. Liao, G. Chen, E.N. Sanchez, Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach, Neural Networks. 15 (2002) 855-866. [DOI:10.1016/S0893-6080(02)00041-2]
40. [40] L. V. Hien, V.N. Phat, Exponential stabilization for a class of hybrid systems with mixed delays in state and control, Nonlinear Anal. Hybrid Syst. 3 (2009) 259-265. [DOI:10.1016/j.nahs.2009.01.009]
41. [41] K. Derinkuyu, M.Ç. Pınar, On the S-procedure and Some Variants, Math. Methods Oper. Res. 64 (2006) 55-77. [DOI:10.1007/s00186-006-0070-8]
| Rights and permissions | |
 |
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |