# A Line-integral Fuzzy Lyapunov Functional Approach to Sampled-data Tracking Control of Takagi-Sugeno Fuzzy Systems

• Kim, Han Sol (Dept. of Electrical and Electronic Engineering, Yonsei University) ;
• Joo, Young Hoon (School of IT Information and Control Eng., Kunsan National University)
• Accepted : 2018.08.19
• Published : 2018.11.01

#### Abstract

This paper deals with a sampled-data tracking control problem for the Takagi-Sugeno fuzzy system with external disturbances. We derive a stability condition guaranteeing both asymptotic stability and H-infinity tracking performance by employing a newly proposed time-dependent line-integral fuzzy Lyapunov-Krasovskii functional. A new integral inequality is also introduced, by which the proposed stability condition is formulated in terms of linear matrix inequalities. Finally, the effectiveness of the proposed method is demonstrated through a simulation example.

#### Acknowledgement

Supported by : Korea Electric Power Corporation, National Research Foundation of Korea (NRF)

#### References

1. K. Tanaka and H. O. Wang, "Fuzzy control systems design and analysis: a LMI approach," Wiely, 2001.
2. D. H. Lee, J. B. Park, Y. H. Joo, K. C. Lin, and C. H. Ham, "Robust $H_{\infty}$ control for uncertain nonlinear active magnetic bearing systems via Takagi-Sugeno fuzzy models," Int. J. Control, Autom. Syst., vol. 8, no. 3, pp. 636-646, 2010. https://doi.org/10.1007/s12555-010-0317-2
3. Y. Y. Cao and P. M. Frank, "Robust $H_{\infty}$ disturbance attenuation for a class of uncertain discrete-time fuzzy systems," IEEE Trans. Fuzzy Syst., vol. 8, no. 4, pp. 406-415, 2000. https://doi.org/10.1109/91.868947
4. H. K. Lam, F. H. F. Leung, and P. K. S. Tam, "A switching controller for uncertain nonlinear systems," IEEE Trans. Control Syst. Mag., vol. 22, no. 1, pp. 7-14, 2002. https://doi.org/10.1109/37.980242
5. G. B. Koo, J. B. Park, and Y. H. Joo, "Robust fuzzy controller for large-scale nonlinear systems using decentralized static output-feedback," Int. J. Control, Autm, Systm., vol. 9, no. 4, pp. 649-658, 2011. https://doi.org/10.1007/s12555-011-0405-y
6. H. J. Kim, G. B. Koo, J. B. Park, and Y. H. Joo, "Decentralized sampled-data $H_{\infty}$ fuzzy filter for nonlinear large-scale systems," Fuzzy Sets and Syst. vol. 273, pp. 68-86, 2015. https://doi.org/10.1016/j.fss.2014.11.024
7. G. B. Koo, J. B. Park, and Y. H. Joo, "Intelligent digital redesign for nonlinear systems using a guaranteed cost control method," Int. J. Control, Autom. Syst., vol. 11, no.6, pp. 1075-1086, 2013. https://doi.org/10.1007/s12555-013-0093-x
8. D. H. Lee, J. B. Park, and Y. H. Joo, "Further improvement of periodic control approach for relaxed stabilization condition of discrete-time Takagi-Sugeno fuzzy systems," Fuzzy Sets and Syst., vol. 174, pp. 50-65, 2011. https://doi.org/10.1016/j.fss.2011.02.013
9. X. L. Zhu, B. Chen, D. Yue, and Y. Wang, "An improved input delay approach to stabilization of fuzzy systems under variable sampling," IEEE Trans. Fuzzy Systs., vol. 20, no. 2, pp. 330-341, 2012. https://doi.org/10.1109/TFUZZ.2011.2174242
10. J. Yoneyama, "Robust sampled-data stabilization of uncertain fuzzy systems via input delay approach," Inform. Sci., vol. 198, pp. 169-176, 2012. https://doi.org/10.1016/j.ins.2012.02.007
11. Z. G. Wu, P. Shi, H. Su, and J. Chu, "Sampled-data fuzzy control of chaotic systems based on a T-S fuzzy model," IEEE Trans. Fuzzy Systs., vol. 22, no. 1, pp. 153-163, March, 2013.
12. F. Yang, H. Zhang, and Y. Wang, "An enhanced input-delay approach to sampled-data stabilization of T-S fuzzy systems via mixed convex combination," Nonlinear Dyn., vol. 75, pp. 501-512, 2014. https://doi.org/10.1007/s11071-013-1080-8
13. G. B. Koo, J. B. Park, and Y. H. Joo, "Guaranteed cost sampled-data fuzzy control for non-linear systems: a continuous-time Lyapunov approach," IET Cont. Theory, Appl., vol. 7, no. 13, 1745-175, 2013. https://doi.org/10.1049/iet-cta.2013.0186
14. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, "Linear matrix inequalities in system and control theory," SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, 1994.
15. T. Takagi and M. Sugeno, "Fuzzy identification of systems and its applications to modeling and control," IEEE Trans. Syst., Man, Cybern., vol. SMC-15, no. 1, pp. 116-132, 1985. https://doi.org/10.1109/TSMC.1985.6313399
16. E. Fridman, "A refined input delay approach to sampled-data control," Automatica, vol. 46, pp. 421-427, 2010. https://doi.org/10.1016/j.automatica.2009.11.017
17. K. Tanaka, T. Hori, and H. O. Wang, "A multiple Lyapunov function approach to stabilization of fuzzy control systems," IEEE Trans. Fuzzy Syst., vol. 11, no. 4, pp. 582-589, 2003. https://doi.org/10.1109/TFUZZ.2003.814861
18. D. H. Lee, J. B. Park, and Y. H. Joo, "A new fuzzy Lyapunov function for relaxed stability condition of continuous-time Takagi-Sugeno fuzzy systems," IEEE Trans. Fuzzy Syst., vol. 19, no. 4, pp. 785-791, 2011. https://doi.org/10.1109/TFUZZ.2011.2142315
19. H. S. Kim, J. B. Park, and Y. H. Joo, "Relaxed stability conditions for the Takagi-Sugeno fuzzy system using a polynomial non-quadratic Lyapunov function," to appear in IET Cont. Theory, Appl., vol. 10, iss. 13, pp. 1590-1599, 2016. https://doi.org/10.1049/iet-cta.2015.1273
20. L. A. Mozelli, R. M. Palhares, and G. S. C. Avellar, "A systematic approach to improve multiple Lyapunov function stability and stabilization conditions for fuzzy systems," Inform. Sci., vol. 179, pp. 1149-1162, 2009. https://doi.org/10.1016/j.ins.2008.12.002
21. B. J. Rhee and S. Won, "A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design," Fuzzy Sets Syst., vol. 157, pp. 1211-1228, 2006. https://doi.org/10.1016/j.fss.2005.12.020
22. Z. Zhang, C. Lin, and B. Chen, "New stability and stabilization conditions for T-S fuzzy systems with time delay," Fuzzy Sets Syst., vol. 263, pp. 82-91, 2015. https://doi.org/10.1016/j.fss.2014.09.012
23. C. S. Tseng, B. S. Chen, and H. J. Uang, "Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model," IEEE Trans. Fuzzy Syst., vol. 9, no. 3, pp. 381-392, 2001. https://doi.org/10.1109/91.928735
24. B. Mansouri, N. Manamanni, K. Guelton, A. Kruszewski, and T. M. Guerra, "Output feedback LMI tracking control conditions with $H_{\infty}$ criterion for uncertain and disturbed T-S models," Inform. Sci., vol. 179, pp. 446-457, 2009.
25. H. K. Lam and L. D. Seneviratne, "Tracking control of sampled-data fuzzy-model-based control systems," IET Cont. Theory Appl., vol. 3, no. 1, pp. 56-67, 2009. https://doi.org/10.1049/iet-cta:20070466
26. D. G. Zill and, M. R. Cullen, "Advanced engineering mathematics," Jones and Bartlett, 2000.
27. E. Kreyszig, "Advanced engineering mathematics," Wiely, 2011.
28. S. K. Nguang, "Comments on "Fuzzy $H_{\infty}$ tracking control for nonlinear networked control systems in T-S fuzzy model"," IEEE Trans. Syst. Man, Cybern., vol. 40, no. 3, pp. 957, 2010. https://doi.org/10.1109/TSMCB.2009.2030883
29. J. Lfberg, "YALMIP: a toolbox for modeling and optimization in MATLAB," in Proc. CACSD Conf., pp. 284-289, 2004.
30. H. S. Kim, J. B. Park, and Y. H. Joo, "Decentralized sampled-data tracking control of large-scale fuzzy systems: An exact discretization approach," IEEE Access, vol. 5, pp. 12668-12681, 2017. https://doi.org/10.1109/ACCESS.2017.2723982