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Local H ∞ controller design for continuous-time T-S fuzzy systems

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  • Intelligent and Information Systems
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Abstract

This paper addresses local H ∞ controller design problems for continuous-time Takagi–Sugeno (T–S) systems with magnitude- and energy-bounded disturbances. The design procedure is formulated as optimizations subject to linear matrix inequalities (LMIs) which can be solved by means of convex optimization techniques. The designed controllers not only guarantee the H ∞ performance but also ensure the state not to escape an invariant set that is included by the region where the T–S fuzzy model is defined. Finally, an example is given to illustrate the proposed method.

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References

  1. P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox, MathWorks, Natick, MA, 1995.

    Google Scholar 

  2. J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cons,” Optim. Meth. Software, vol. 11-12, pp. 625–653, 1999.

    Article  MathSciNet  Google Scholar 

  3. J. Löfberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” Proc. IEEE CCA/ISIC/CACSD Multiconf., pp. 284–289, http://control.ee.ethz.ch/joloef/yalmip.php, 2004.

    Google Scholar 

  4. T. Taniguchi, K. Tanaka, and H. O. Wang, “Model construction, rule reduction and robust compensation for generalized form of Takagi–Sugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 525–537, 2001.

    Article  Google Scholar 

  5. K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Syst., vol. 6, no. 2, pp. 250–265, 1998.

    Article  Google Scholar 

  6. E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 5, pp. 523–534, 2000.

    Article  Google Scholar 

  7. H. D. Tuan, P. Apkarian, T. Narikiyo, and Y. Yamamoto, “Parameterized linear matrix inequality techniques in fuzzy control system design,” IEEE Trans. Fuzzy Syst., vol. 9, no. 2, pp. 324–332, 2001.

    Article  Google Scholar 

  8. X. Liu and Q. Zhang, “New approaches to H8 controller designs based on fuzzy observers for T–S fuzzy systems via LMI,” Automatica, vol. 39, no. 9, pp. 1571–1582, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  9. C. H. Fang, Y. S. Liu, S. W. Kau, L. Hong, and C. H. Lee, “A new LMI-based approach to relaxed quadratic stabilization of T–S fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 14, no. 3, pp. 386–397, 2006.

    Article  Google Scholar 

  10. A. Sala and C. Ariño, “Relaxed stability and performance conditions for Takagi–Sugeno fuzzy systems with knowledge on membership function overlap,” IEEE Trans. Syst., Man Cybern., vol. 37, no. 3, pp. 727–732, 2007.

    Article  Google Scholar 

  11. M. Narimani and H. K. Lam, “Relaxed LMI-based stability conditions for Takagi–Sugeno fuzzy control systems using regional-membership-functionshape-dependent analysis approach,” IEEE Trans. Fuzzy Syst., vol. 17, no. 5, pp. 1221–1228, 2009.

    Article  Google Scholar 

  12. A. Sala and C. Ariño, “Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: applications of Polya’s theorem,” Fuzzy Sets Syst., vol. 158, no. 24, pp. 2671–2686, 2007.

    Article  MATH  Google Scholar 

  13. Y. J. Chen, H. Ohtake, K. Tanaka, W. J. Wang, and H. O. Wang, “Relaxed stabilization criterion for T–S fuzzy systems by minimum-type piecewise Lyapunov function based switching fuzzy controller,” IEEE Trans. Fuzzy Syst., vol. 20, no. 6, pp. 1166–1173, 2012.

    Article  Google Scholar 

  14. V. C. S. Campos, F. O. Souza, L. A. B. Tôrres, and R. M. Palhares, “New stability conditions based on piecewise fuzzy Lyapunov functions and tensor product transformations,” IEEE Trans. Fuzzy Syst., vol. 21, no. 4, pp. 748–760, 2013.

    Article  Google Scholar 

  15. K. Tanaka, T. Hori, and H. O. Wang, “A fuzzy Lyapunov approach to fuzzy control system design,” Proc. Amer. Control Conf., Arlington, VA, pp. 4790–4795, 2001.

    Google Scholar 

  16. 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.

    Article  Google Scholar 

  17. L. A. Mozelli, R. M. Palhares, F. O. Souza, and E. M. A. M. Mendes, “Reducing conservativeness in recent stability conditions of T–S fuzzy systems,” Automatica, vol. 45, no. 6, pp. 1580–1583, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  18. 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, no. 9, pp. 1211–1228, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Sala and C. Ariño, “Polynomial fuzzy models for nonlinear control: a Taylor series approach,” IEEE Trans. Fuzzy Syst., vol. 17, no. 6, pp. 1284–1295, 2009.

    Article  Google Scholar 

  20. K. Tanaka, H. Yoshida, H. Ohtake, and H. O. Wang, “A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 17, no. 4, pp. 911–922, 2009.

    Article  Google Scholar 

  21. H. K. Lam, M. Narimani, H. Li, and H. Liu “Stability analysis of polynomial-fuzzy-modelbased control systems using switching polynomial Lyapunov function,” IEEE Trans. Fuzzy Syst., vol. 21, no. 5, pp. 800–813, 2013.

    Article  Google Scholar 

  22. B. Ding, “Homogeneous polynomially nonquadratic stabilization of discrete-time Takagi–Sugeno systems via nonparallel distributed compensation law,” IEEE Trans. Fuzzy Syst., vol. 18, no. 5, pp. 994–1000, 2010.

    Article  Google Scholar 

  23. H. Zhang and X. Xie, “Relaxed stability conditions for continuous-time T–S fuzzy-control systems via augmented multiindexed matrix approach,” IEEE Trans. Fuzzy Syst., vol. 19, no. 3, pp 478–492, 2011.

    Article  Google Scholar 

  24. E. S. Tognetti, R. C. L. F. Oliveira, and P. L. D. Peres, “Selective H2 and H1 stabilization of Takagi–Sugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 890–900, 2011.

    Article  Google Scholar 

  25. X. Xie, H. Ma, Y. Zhao, D. W. Ding, and Y. Wang, “Control synthesis of discrete-time T–S fuzzy systems based on a novel non-PDC control scheme,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 147–157, 2013.

    Article  Google Scholar 

  26. A. Kruszewski, R. Wang, and T. M. Guerra, “Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time T–S fuzzy models: a new approach,” IEEE Trans. Autom. Control, vol. 53, no. 2, pp. 606–611, 2008.

    Article  MathSciNet  Google Scholar 

  27. T. M. Guerra, A. Kruszewski, and M. Bernal, “Control law proposition for the stabilization of discrete Takagi–Sugeno models,” IEEE Trans. Fuzzy Syst., vol. 17, no. 3, pp. 724–731, 2009.

    Article  Google Scholar 

  28. 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 Syst., vol. 174, no. 1, pp. 50–65, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  29. H. K. Lam and F. H. F. Leung, “LMI-based stability and performance conditions for continuous-time nonlinear systems in Takagi–Sugeno’s form,” IEEE Trans. Syst., Man Cybern., vol. 37, no. 5, pp. 1396–1406, 2007.

    Article  Google Scholar 

  30. X.-H. Chang and G.-H. Yang, “Relaxed stabilization conditions for continuous-time Takagi–Sugeno fuzzy control systems,” Inf. Sci., vol. 180, pp. 32733287, 2010.

    Google Scholar 

  31. S. H. Kim, “Relaxation technique for a T-S fuzzy control design based on a continuous-time fuzzy weighting-dependent Lyapunov function,” IEEE Trans. Fuzzy Syst., vol. 21, no. 4, pp. 761–766, 2013.

    Article  Google Scholar 

  32. M. Bernal and T. M. Guerra, “Generalized nonquadratic stability of continuous-time Takagi–Sugeno models,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 815–822, 2010.

    Article  Google Scholar 

  33. J. T. Pan, T. M. Guerra, S. M. Fei, and A. Jaadari, “Nonquadratic stabilization of continuous T–S fuzzy models: LMI solution for a local approach,” IEEE Trans. Fuzzy Syst., vol. 20, no. 3, pp. 594–602, 2012.

    Article  Google Scholar 

  34. A. Jaadari, T. M. Guerra, and M. Bernal, “New controllers and new designs for continuous-time Takagi–Sugeno models,” Proc. IEEE Would Congress on Computational Intelligence, Brisbane, Australia, 2012, pp. 1–7.

    Google Scholar 

  35. D. H. Lee, J. B. Park, and Y. H. Joo, “A fuzzy Lyapunov function approach to estimating the domain of attraction for continuous-time Takagi–Sugeno fuzzy systems,” Inf. Sci., vol. 185, no. 1, pp. 230–248, 2012.

    Article  MATH  MathSciNet  Google Scholar 

  36. D. H. Lee, “Domain of attraction analysis for continuous-time Takagi–Sugeno fuzzy systems: an LMI approach,” Proc. 51st IEEE Conf. Decision Control, Maui, HI, USA, 2012, pp. 6187–6192.

    Google Scholar 

  37. D. H. Lee, “Local stability analysis of continuoustime Takagi–Sugeno fuzzy systems: an LMI approach,” Proc. Amer. Control Conf., Washington, DC, 2013, pp. 5625–5630.

    Google Scholar 

  38. D. H. Lee, Y. H. Joo, and M. H. Tak, “Local stability analysis of continuous-time Takagi–Sugeno fuzzy systems: a fuzzy Lyapunov function approach,” Inf. Sci., vol. 257, no. 1, pp. 163–175, 2014.

    Article  MathSciNet  Google Scholar 

  39. M. Bernal, A. Soto-Cota, and J. Cortez, “Local non-quadratic H-infinity control for continuoustime Takagi–Sugeno models,” Proc. IEEE Int. Conf. Fuzzy Syst., Taipei, Taiwan, pp. 1615–1620, 2011.

    Google Scholar 

  40. A. Jaadari, T. M. Guerra, A. Sala, and M. Bernal, “Finsler’s relaxation for local H-infinity controller design of continuous time Takagi–Sugeno models via non-quadratic Lyapunov functions,” Proc. Amer. Control Conf., Washington, DC, pp. 5648–5653, 2013.

    Google Scholar 

  41. L. Wang and X. Liu, “Local analysis of continuoustime Takagi–Sugeno fuzzy system with disturbances bounded by magnitude or energy: a Lagrange multiplier method,” Inf. Sci., vol. 248, no. 1, pp. 89–102, 2013.

    Article  Google Scholar 

  42. D. H. Lee, Y. H. Joo, and M. H. Tak, “Local H8 control and invariant set analysis for continuoustime T-S fuzzy systems with magnitude- and energy-bounded disturbances,” Proc. IEEE Int. Conf. Fuzzy Syst., China, Beijing, pp. 1990–1997, 2014.

    Google Scholar 

  43. H. K. Khalil, Nonlinear Systems, 3rd ed., Prentice-Hall, Upper Saddle River, NJ, 2001.

    Google Scholar 

  44. M. Herceg, M. Kvasnica, C. N. Jones, and M. Morari, “Multi-Parametric Toolbox 3.0,” Proc. Eur. Control Conf., Zurich, Switzerland, pp. 502–510, http://control.ee.ethz.ch/mpt, 2013.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47906, USA

    Dong Hwan Lee

  2. Department of Electrical and Electronic Engineering, Yonsei University, Seodaemun-gu, Seoul, 120-749, Korea

    Jin Bae Park

  3. Department of Control and Robotics Engineering, Kunsan National University, Kunsan, Chonbuk, 573-701, Korea

    Young Hoon Joo & Sung Kwan Kim

Authors
  1. Dong Hwan Lee
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  2. Jin Bae Park
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Corresponding author

Correspondence to Jin Bae Park.

Additional information

Recommended by Associate Editor Ho Jae Lee under the direction of Editor Euntai Kim.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2015R1A2A2A05001610) and the Human Resources Development program (No. 20144030200590) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy.

Dong Hwan Lee received his B.S. degree in Electronic Engineering from Konkuk University, Seoul, Korea, in 2008 and M.S. degree in Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, in 2010. From 2014, he is working toward a Ph.D. degree in the Department of Electrical and Computer Engineering, Purdue University, USA. He was the recipient of the Outstanding Paper Award in Graduate School of Yonsei University Thesis Award Fall 2010 and the recipient of the Student Paper Award in ICCAS 2010. He is an Associate Editor of IEEE Transactions on Fuzzy Systems. His current research interests include stability analysis in fuzzy systems, fuzzy-model-based control, and robust control of uncertain linear systems.

Jin Bae Park received his B.E. degree in Electrical Engineering from Yonsei University, Seoul, Korea, and his M.S. and Ph.D. degrees in Electrical Engineering from Kansas State University, Manhattan, in 1977, 1985, and 1990, respectively. Since 1992, he has been with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea, where he is currently a professor. His research interests include robust control and filtering, nonlinear control, mobile robot, fuzzy logic control, neural networks, genetic algorithms, and Hadamard transform spectroscopy. He served as the President for the Institute of Control, Robotics and Systems (ICROS) (2013) and Editor-in-Chief for the International Journal of Control, Automation, and Systems (IJCAS) (2006-2010) and is serving as Vice-President of Yonsei University, Seoul, Korea (2014-present).

Young Hoon Joo received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from Yonsei University, Seoul, Korea, in 1982, 1984, and 1995, respectively. He worked with Samsung Electronics Company, Seoul, Korea, from 1986 to 1995, as a project manager. He was with the University of Houston, Houston, TX, from 1998 to 1999, as a visiting professor in the Department of Electrical and Computer Engineering. He is currently a professor in the Department of Control and Robotics Engineering, Kunsan National University, Korea. His major interest is mainly in the field of intelligent robot, robot vision, intelligent control, human-robot interaction, wind-farm control, and intelligent surveillance systems. He served as the President for Korea Institute of Intelligent Systems (KIIS) (2008-2009) and the Vice-President for the Korean Institute of Electrical Engineers (KIEE) (2013-2014); and is serving as Editor-in-Chief for the International Journal of Control, Automation, and Systems (IJCAS) (2014-present).

Sung Kwan Kim received the B.S. and M.S. degree in the School of Electronics and Information Engineering from Kunsan National University, Korea, in 2013 and 2015, respectively. He is currently working toward a Ph. D. degree in the Department of Control and Robotics Engineering from Kunsan National University, Korea. His research interests include intelligent surveillance system, robot vision, and human-robot interaction.

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Lee, D.H., Park, J.B., Joo, Y.H. et al. Local H ∞ controller design for continuous-time T-S fuzzy systems. Int. J. Control Autom. Syst. 13, 1499–1507 (2015). https://doi.org/10.1007/s12555-014-0439-z

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  • DOI: https://doi.org/10.1007/s12555-014-0439-z

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