Volume 13, Issue 3 (Journal of Control, V.13, N.3 Fall 2019)                   JoC 2019, 13(3): 15-27 | Back to browse issues page

XML Persian Abstract Print

Download citation:
BibTeX | RIS | EndNote | Medlars | ProCite | Reference Manager | RefWorks
Send citation to:

Cheraghi Shami F, Gharaveisi A A, M. Farsangi M, Mohammadian M. Enlarging Domain of Attraction for a Special Class of Continuous-time Quadratic Lyapunov Function Piecewise Affine Systems based on Discontinuous Piecewise. JoC. 2019; 13 (3) :15-27
URL: http://joc.kntu.ac.ir/article-1-514-en.html
1- Shahid Bahonar University of Kerman
Abstract:   (4095 Views)
This paper presents a new approach to estimate and to enlarge the domain of attraction for a planar continuous-time piecewise affine system. Various continuous Lyapunov functions have been proposed to estimate and to enlarge the system’s domain of attraction. In the proposed method with a new vision and with the aids of a discontinuous piecewise quadratic Lyapunov function, the domain of attraction at the origin is enlarged by designing a state feedback controller. This paper shows that the continuity of the Lyapunov function on the boundaries, increases the conservativeness in estimating the domain of attraction, and gives more powerful search ability to the domain of attraction estimation algorithm by relaxing this continuity condition. The simulation results show the superiority of the proposed method so that using this method the larger estimation of the domain of attraction is obtained than continuous one.
Full-Text [PDF 1134 kb]   (1397 Downloads)    
Type of Article: Research paper | Subject: Special
Received: 2017/08/8 | Accepted: 2018/06/11 | Published: 2019/12/31

1. [1] P. Siewniak and B. Grzesik, "A generalized geometrical piecewise‐affine model of DC‐DC power electronic converters," International Journal of Circuit Theory and Applications, vol. 43, pp. 342-373, 2015. [DOI:10.1002/cta.1945]
2. [2] P. Siewniak and B. Grzesik, "The piecewise‐affine model of buck converter suitable for practical stability analysis," International Journal of Circuit Theory and Applications, vol. 43, pp. 3-21, 2015. [DOI:10.1002/cta.1915]
3. [3] کاردهی مقدم ریحانه، پریز ناصر، مدیر شانه چی حسن، وحیدیان کامیاد علی. افزایش زمان بحرانی سیستمهای غیر خطی بوسیله گسترش جهت‌دار ناحیه جذب. مجله کنترل. ۱۳۸۹; ۴ (۲) :۱-۱۰.
4. [4] S. Anbu and N. Jaya, "Design of gain scheduling adaptive control for continuous stirred tank reactor," International Journal of Automation and Control, vol. 8, pp. 141-157, 2014. [DOI:10.1504/IJAAC.2014.063360]
5. [5] A. Chakraborty, P. Seiler, and G. J. Balas, "Nonlinear region of attraction analysis for flight control verification and validation," Control Engineering Practice, vol. 19, pp. 335-345, 2011. [DOI:10.1016/j.conengprac.2010.12.001]
6. [6] M. L. Matthews and C. M. Williams, "Region of attraction estimation of biological continuous Boolean models," in Systems, Man, and Cybernetics (SMC), 2012 IEEE International Conference on, 2012, pp. 1700-1705. [DOI:10.1109/ICSMC.2012.6377982]
7. [7] S. Sundar, "Effect of Elevated Carbon Dioxide Concentration on Plant Growth: A Mathematical Model," American Journal of Applied Mathematics and Statistics, vol. 3, pp. 59-67, 2015.
8. [8] J. Haddad and N. Geroliminis, "On the stability of traffic perimeter control in two-region urban cities," Transportation Research Part B: Methodological, vol. 46, pp. 1159-1176, 2012. [DOI:10.1016/j.trb.2012.04.004]
9. [9] A. Bemporad, "Efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form," IEEE Transactions on Automatic Control, vol. 49, pp. 832-838, 2004. [DOI:10.1109/TAC.2004.828315]
10. [10] ملااحمدیان کاسب حامد، کریم پور علی، پریز ناصر. سیستم های تکه‌ای خطی تبار مستقیم: کلاس جدیدی از سیستم‌های هایبرید با دینامیک‌های خطی تبار و مرزهای کلیدزنی قابل تنظیم. مجله کنترل. ۱۳۹۱; ۶ (۱) :۲۱-۲۹.
11. [11] N. Eghbal, N. Pariz, and A. Karimpour, "Uniform modeling of parameter dependent nonlinear systems," Journal of Zhejiang University SCIENCE C, vol. 13, pp. 850-858, 2012. [DOI:10.1631/jzus.C1200096]
12. [12] کشوری خور هادی، کریم پور علی، پریز ناصر. شناسائی سیستم های سوئیچ شونده خطی با استفاده از نگاشت معادلات خطی همزمان. مجله کنترل. ۱۳۹۳; ۸ (۱) :۲۱-۳۰.
13. [13] J. H. Richter, W. Heemels, N. van de Wouw, and J. Lunze, "Reconfigurable control of piecewise affine systems with actuator and sensor faults: stability and tracking," Automatica, vol. 47, pp. 678-691, 2011. [DOI:10.1016/j.automatica.2011.01.048]
14. [14] L. Khodadadi, B. Samadi, and H. Khaloozadeh, "Estimation of region of attraction for polynomial nonlinear systems: A numerical method," ISA transactions, 2013. [DOI:10.1016/j.isatra.2013.08.005]
15. [15] H. K. Khalil and J. Grizzle, Nonlinear systems vol. 3: Prentice hall Upper Saddle River, 2002.
16. [16] Y. Chen, Y. Sun, C.-S. Tang, Y.-G. Su, and A. P. Hu, "Characterizing regions of attraction for piecewise affine systems by continuity of discrete transition functions," Nonlinear Dynamics, vol. 90, pp. 2093-2110, 2017. [DOI:10.1007/s11071-017-3786-5]
17. [17] Y. Chen, Y. Sun, C. Tang, Y. Su, and A. P. Hu, "Computing Regions of Stability for Limit Cycles of Piecewise Affine Systems," Information Technology And Control, vol. 46, pp. 459-469, 2017. [DOI:10.5755/j01.itc.46.4.16072]
18. [18] M. Johansson and A. Rantzer, "Computation of piecewise quadratic Lyapunov functions for hybrid systems," IEEE transactions on automatic control, vol. 43, pp. 555-559, 1998. [DOI:10.1109/9.664157]
19. [19] B. Samadi and L. Rodrigues, "A unified dissipativity approach for stability analysis of piecewise smooth systems," Automatica, vol. 47, pp. 2735-2742, 2011. [DOI:10.1016/j.automatica.2011.09.018]
20. [20] H. Nakada and K. Takaba, "Local stability analysis of piecewise affine systems," Rn, vol. 10, p. 1, 2003. [DOI:10.23919/ECC.2003.7084935]
21. [21] R. Iervolino, F. Vasca, and L. Iannelli, "Cone-copositive piecewise quadratic lyapunov functions for conewise linear systems," IEEE Transactions on Automatic Control, vol. 60, pp. 3077-3082, 2015. [DOI:10.1109/TAC.2015.2409933]
22. [22] M. Johansson, "Analysis of piecewise linear system via convex optimization-a unifying approach," in Proceedings of the 1999 IFAC World Congress, 1999, pp. 521-526.
23. [23] M. K.-J. Johansson, Piecewise linear control systems: a computational approach vol. 284: Springer, 2003.
24. [24] J. Xu and L. Xie, "Homogeneous polynomial Lyapunov functions for piecewise affine systems," in American Control Conference, 2005. Proceedings of the 2005, 2005, pp. 581-586.
25. [25] N. Eghbal, N. Pariz, and A. Karimpour, "Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems," Journal of Mathematical Analysis and Applications, vol. 399, pp. 586-593, 2013. [DOI:10.1016/j.jmaa.2012.09.054]
26. [26] T. González and M. Bernal, "Progressively better estimates of the domain of attraction for nonlinear systems via piecewise Takagi-Sugeno models: Stability and stabilization issues," Fuzzy Sets and Systems, 2015. [DOI:10.1016/j.fss.2015.11.010]
27. [27] S. Gering, L. Eciolaza, J. Adamy, and M. Sugeno, "A piecewise approximation approach to nonlinear systems: Stability and region of attraction," IEEE Transactions on Fuzzy Systems, vol. 23, pp. 2231-2244, 2015. [DOI:10.1109/TFUZZ.2015.2417870]
28. [28] R. Iervolino, D. Tangredi, and F. Vasca, "Lyapunov stability for piecewise affine systems via cone-copositivity," Automatica, vol. 81, pp. 22-29, 2017. [DOI:10.1016/j.automatica.2017.03.011]
29. [29] A.-T. Nguyen, M. Sugeno, V. Campos, and M. Dambrine, "LMI-based stability analysis for piecewise multi-affine systems," IEEE Transactions on Fuzzy Systems, vol. 25, pp. 707-714, 2017. [DOI:10.1109/TFUZZ.2016.2566798]
30. [30] L. Rodrigues and S. Boyd, "Piecewise-affine state feedback for piecewise-affine slab systems using convex optimization," Systems & Control Letters, vol. 54, pp. 835-853, 2005. [DOI:10.1016/j.sysconle.2005.01.002]
31. [31] K. Liu, Y. Yao, D. Sun, and V. Balakrishnan, "Improved state feedback controller synthesis for piecewise-linear systems," International Journal of Innovative Computing, Information and Control, vol. 8, pp. 6945-6957, 2012.
32. [32] A. Benine-Neto, S. Mammar, B. Lusetti, and S. Scalzi, "Piecewise affine control for lane departure avoidance," Vehicle System Dynamics, vol. 51, pp. 1121-1150, 2013. [DOI:10.1080/00423114.2013.783220]
33. [33] N. Dadkhah and L. Rodrigues, "Non-fragile state-feedback control of uncertain piecewise-affine slab systems with input constraints: a convex optimisation approach," IET Control Theory & Applications, vol. 8, pp. 626-632, 2014. [DOI:10.1049/iet-cta.2013.0202]
34. [34] J. Raouf and L. Rodrigues, "Stability and stabilization of piecewise‐affine slab systems subject to Wiener process noise," International Journal of Robust and Nonlinear Control, vol. 25, pp. 949-960, 2015. [DOI:10.1002/rnc.3117]
35. [35] H. Razavi, K. Merat, H. Salarieh, A. Alasty, and A. Meghdari, "Observer based minimum variance control of uncertain piecewise affine systems subject to additive noise," Nonlinear Analysis: Hybrid Systems, vol. 19, pp. 153-167, 2016. [DOI:10.1016/j.nahs.2015.09.002]
36. [36] Y. Eren, J. Shen, and K. Camlibel, "Quadratic stability and stabilization of bimodal piecewise linear systems," Automatica, vol. 50, pp. 1444-1450, 2014. [DOI:10.1016/j.automatica.2014.03.009]
37. [37] M. di Bernardo, U. Montanaro, R. Ortega, and S. Santini, "Extended hybrid model reference adaptive control of piecewise affine systems," Nonlinear Analysis: Hybrid Systems, vol. 21, pp. 11-21, 2016. [DOI:10.1016/j.nahs.2015.12.003]
38. [38] M. Rubagotti, L. Zaccarian, and A. Bemporad, "A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains," International Journal of Control, vol. 89, pp. 950-959, 2016. [DOI:10.1080/00207179.2015.1108456]
39. [39] P. Li, J. Lam, and K. C. Cheung, "Stability, stabilization and L2-gain analysis of periodic piecewise linear systems," Automatica, vol. 61, pp. 218-226, 2015. [DOI:10.1016/j.automatica.2015.08.024]
40. [40] L. Rodrigues, Dynamic output feedback controller synthesis for piecewise-affine systems: Stanford University, 2002.
41. [41] S. Pettersson and B. Lennartson, "Exponential stability of hybrid systems using piecewise quadratic Lyapunov functions resulting in LMIs," in IFAC, 14th Triennial World Congress, Beijing, PR China, 1999. [DOI:10.1016/S1474-6670(17)56820-2]
42. [42] J. P. LaSalle and S. Lefschetz, Stability by Liapunov's direct method: with applications vol. 4: Academic Press New York, 1961.
43. [43] O. Hachicho, "A novel LMI-based optimization algorithm for the guaranteed estimation of the domain of attraction using rational Lyapunov functions," Journal of the Franklin Institute, vol. 344, pp. 535-552, 2007. [DOI:10.1016/j.jfranklin.2006.02.032]
44. [44] G. Chesi, Domain of attraction: analysis and control via SOS programming vol. 415: Springer Science & Business Media, 2011.
45. [45] M. Johansson, "Piecewise quadratic estimates of domains of attraction for linear systems with saturation," in CD-ROM of 15th IFAC World Congress, 2002. [DOI:10.3182/20020721-6-ES-1901.00281]
46. [46] T. González and M. Bernal, "Progressively better estimates of the domain of attraction for nonlinear systems via piecewise Takagi-Sugeno models: Stability and stabilization issues," Fuzzy Sets and Systems, vol. 297, pp. 73-95, 2016. [DOI:10.1016/j.fss.2015.11.010]
47. [47] J. Lofberg, "YALMIP: A toolbox for modeling and optimization in MATLAB," in Computer Aided Control Systems Design, 2004 IEEE International Symposium on, 2004, pp. 284-289.
48. [48] D. Henrion, J. Lofberg, M. Kocvara, and M. Stingl, "Solving polynomial static output feedback problems with PENBMI," in Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC'05. 44th IEEE Conference on, 2005, pp. 7581-7586.¬

Add your comments about this article : Your username or Email:

Send email to the article author

Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

© 2022 CC BY-NC 4.0 | Journal of Control

Designed & Developed by : Yektaweb