Abstract
In this paper, a design method of control for Polynomial Fuzzy Models (PFM) with time delay is developed. By using a Polynomial Lyapunov Krasovskii Functional (PLKF) with double integral and by imposing bounds on the derivatives of each state, less conservative sufficient conditions are established to ensure the local stability of the closed loop system. Furthermore, a Domain Of Attraction (DOA) in which the initial states are ensured to converge asymptotically to the origin is estimated. The resulting conditions are formulated in terms of Sum-Of- Squares (SOS) which can be numerically (partially symbolically) solved via the recently developed SOSTOOLS. Some examples are provided to show the effectiveness and the merit of the design procedure.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Fuzzy Systems, vol. 15, no. 1, pp. 116–132, 1985.
P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, “LMI control toolbox for use with Matlab,” Natick, MA: Math Works, 1995.
K. Tanaka, H. Yosihida, H. Ohtake, and H. Wang, “A sumof-squares approach to stability analaysis of polynomial fuzzy systems,” Proceedings of the American Control Conference, New York City, USA, 2007.
S. Prajna, A. Papachristodoulou, P. Seiler, P. A. Parrilo, J. Anderson1, and G. Valmorbida1, SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, Version 3.00, 3rd October.
K. Guelton, N. Manamanni, C.-C. Duong, and D. L. Koumba-Emianiwe, “Sum-of-squares stability analysis of Takagi-Sugeno systems based on multiple polynomial Lyapunov functions,” International Journal of Fuzzy Systems, vol. 15, no. 1, pp. 1–7, 2013.
H. K. Lam, M. Narimani, H. Li, and H. Liu, “Stability analysis of polynomial-fuzzy-model-based control systems using switching polynomial Lyapunov function,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 5, pp. 800–813, 2013.
K. Cao, X. Z. Gao, T. Vasilakos, and W. Pedrycz, “Analysis of stability and robust stability of polynomial fuzzy modelbased control systems using a sum-of-squares approach,” Soft Comput, vol. 18, no. 3, pp. 433–442, 2014.
Y.-J. Chen, M. Tanaka, K. Tanaka, and H. O. Wang, “Stability analysis and region-of-attraction estimation using piecewise polynomial Lyapunov functions: polynomial fuzzy model approach,” IEEE Transactions on Fuzzy Systems, vol. 23, no. 4, pp. 1314–1322, 2015.
K. Tanaka, H. Yosihida, H. Ohtake, and H. Wang, “Stabilization of polynomial fuzzy systems via a sum of squares approach,” Proc. of 22nd IEEE International Symposium on Intelligent Control Part of IEEE Multi-conference on Systems and Control, Singapore, 2007.
K. Tanaka, H. Yosihida, H. Ohtake, and H. Wang, “A sumof-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 4, pp. 911–922, 2009.
T. Seo, H. Ohtake, K. Tanaka, Y. Chen, and H. Wang, “A polynomial observer design for a wider class of polynomial fuzzy systems,” Proc. of IEEE International Conference on Fuzzy Systems, pp. 1305–1311, Taipei, 2011.
H. K. Lam and J. C. Lo, “Output regulation of polynomial-Fuzzy-Model-Based control systems,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 2, pp. 262–274, 2013.
H.-J. Ma and G.-H. Yang, “Fault-tolerant control synthesis for a class of nonlinear systems: Sum of squares optimization approach,” International Journal of Robust and Nonlinear Control, vol. 19, no. 5, pp. 591–610, 2009.
W. Li and W. Wang, “Guaranteed cost control for polynomial fuzzy time delay systems by sum-of-squares approach,” Proceedings of the 10th World Congress on Intelligent Control and Automation, pp. 1806–1811, Beijing, 2012.
S. Tarbouriech, J. M. Gomes, Jr. Da Silva, and G. Garcia, “Delay-depend anti-windup strategy for linear systems with saturating inputs and delayed outputs,” International Journal of Robust and Nonlinear Control, vol. 14, no. 7, pp. 665–682, 2004.
B. Chen and X. Liu, “Delay-dependent robust H ∞ control for T-S fuzzy systems with time delay,” IEEE Transactions on Fuzzy Systems, vol. 13, no. 9, pp. 238–249, 2005.
H. Gassara, A. El Hajjaji, and M. Chaabane, “Robust control of T-S fuzzy systems with time-varying delay using new approach,” International Journal of Robust and Nonlinear Control, vol. 20, no. 14, pp. 1566–1578, 2010.
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, “Linear matrix inequalities in systems and control theory,” SIAM Studies in Appl. Mathematics, Philadelphia, 1994.
T. Hu, Z. Lin, and B. M. Chen, “An analysis and design method for linear systems subject to actuator saturation and disturbance,” Automatica, vol. 28, no. 2, pp. 351–359, 2002.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Ho Jae Lee under the direction of Editor Euntai Kim. This journal was supported by the Korean Federation of Science and Technology Societies Grant.
Hamdi Gassara received the Ph.D degree in automatic control from the University of Picardie Jules Vernes (UPJV), in 2011. Prior to his Ph.D., he received his Master degree from UPJV in 2008. His teaching experience started when he was a Ph.D. student in UPJV France from 2008 to 2011. He is currently an assistant professor in Electrical department at National School of Engineering of Sfax, Tunisia. His research focuses on analysis and control for fuzzy model with time delay, fault tolerant control, diagnostics, saturations, polynomial fuzzy model.
Fatma Siala received her Master degree in 2011 from the National School of Engineering of Sfax, Tunisia, where she is currently working toward a Ph.D. degree. Her current research interests include polynomial fuzzy systems, sum of squares approach and delay systems.
Ahmed El Hajjaji received the Ph.D. degree in automatic control and HDR degree from the University of Picardie Jules Verne (UPJV), France in 1993 and 2000, respectively. He is currently a full professor and head of the research team of control in MIS Lab (Modeling Information Systems Laboratory) of UPJV. He has been the director of the Professional Institute of Electrical Engineering and Industrial Computing from 2006 to 2012. Since 1994, he has published more than 259 Journal and conference papers in the areas of advanced fuzzy control, fault detection and diagnosis and fault tolerant control and their applications to vehicle dynamics, engine control, power systems, renewable energy conversion systems and to industrial processes. His research interests include fuzzy control, vehicle dynamics, fault-tolerant control, neural networks, maglev systems, and renewable energy conversion systems.
Mohamed Chaabane was born in Sfax, Tunisia, on August 26, 1961. He received the Ph.D. degree in electrical engineering from the University of Nancy, Nancy, France, in 1991. He is currently a Professor with the National School of Engineering, University of Sfax, where he has been a Researcher with the Laboratory of Sciences and Techniques of Automatic Control and Computer Engineering (Lab-STA) since 1997. The main research interests are in the filed of robust and optimal control, fault tolerant control, delay systems, descriptor systems, fuzzy logic systems and applications of these techniques to fed-batch processes, asynchronous machines, agriculture systems and renewable energy. Currently, he is an associate editor of the International Journal on Sciences and Techniques of Automatic Control and Computer Engineering (www.sta-tn.com).
Rights and permissions
About this article
Cite this article
Gassara, H., Siala, F., El Hajjaji, A. et al. Local stabilization of Polynomial Fuzzy Model with time delay: SOS approach. Int. J. Control Autom. Syst. 15, 385–393 (2017). https://doi.org/10.1007/s12555-014-0575-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-014-0575-5