Abstract
This paper investigates the polynomial fuzzy observer design for discrete-time uncertain polynomial systems. Three classes of discrete-time polynomial fuzzy systems are studied via a sum of squares (SOS) approach. A polynomial fuzzy system is a more general representation of the well-known Takagi–Sugeno (T–S) fuzzy system. The conditions in the proposed approach are derived in terms of SOS, which is the extension of the LMI method. Hence, the conditions obtained in this paper are more general than the corresponding LMI approaches for T–S fuzzy systems. All the design conditions in the proposed approach can be symbolically and numerically solved via the recently developed SOSTOOLS and a semidefinite-program solver, respectively. Numerical examples are provided to demonstrate the validity and applicability of the proposed SOS-based design approach.
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Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. 15(2), 116–132 (1985)
Tong, S.C., Li, H.X.: Fuzzy adaptive sliding-mode control for MIMO nonlinear systems. IEEE Trans. Fuzzy Syst. 11(3), 354–360 (2003)
Zhang, H.G., Liu, D.R.: Fuzzy Modeling and Fuzzy Control. Birkhauser, Boston (2006)
Wang, Y.C., Zhang, H.G., Wang, Y.Z.: Fuzzy adaptive control of stochastic nonlinear systems with unknown virtual control gain function. Acta Autom. Sinica 32(2), 170–178 (2006)
Tong, S.C., Li, Y., Li, Y.M., Liu, Y.J.: Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems. IEEE Trans. Syst. Man Cybern. Part B 41(6), 1693–1704 (2011)
Zhang, H.G., Zhang, J.L., Yang, G.H., Luo, Y.H.: Leader-based optimal coordination control for the consensus problem of multi-agent differential games via fuzzy adaptive dynamic programming (published on-line on 11th, March). IEEE Trans. Fuzzy Syst. (2014). doi:10.1109/TFUZZ.2014.2310238
Yang, F.S., Zhang, H.G.: T–S model-based relaxed reliable stabilization of networked control systems with time-varying delays under variable sampling. Int. J. Fuzzy Syst. 13(4), 260–269 (2011)
Zhang, H.G., Li, M., Yang, J., Yang, D.D.: Fuzzy model-based robust networked control for a class of nonlinear systems. IEEE Trans. Syst. Man Cybern. Part A 39(2), 437–447 (2009)
Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. Wiley, Hoboken (2001)
Zhang, H.G., Yang, D.D.: Guaranteed cost networked control for T–S fuzzy systems with time delays. IEEE Trans. Syst. Man Cybern. Part C 37(2), 250–265 (2007)
Feng, G.: A survey on analysis and design of model-based fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(5), 676–697 (2006)
Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
Baumann, W., Rugh, W.: Feedback control of nonlinear systems by extended linearization. IEEE Trans. Autom. Control 31(1), 40–46 (1986)
Tanaka, K., Yoshida, H., Ohtake, H., Wang, H. O,: Stabilization of polynomial fuzzy systems via a sum of squares approach. In: Proceeding of the 22nd IEEE International Symposium on Intelligent Control, 160–165 (2007)
Tanaka, K., Yoshida, H., Ohtake, H., Wang, H.O.: Polynomial fuzzy observer designs: A sum-of-squares approach. IEEE Trans. Syst. Man Cybern. Part B 14(5), 1330–1342 (2012)
Tanaka, K., Ohtake, H., Wada, M., Wang, H. O. Chen, Y.-J.: Polynomial fuzzy observer design: a sum of squares approach, In: 48th IEEE Conference on Decision and Control, 7771–7776 (2009)
Seo, T., Ohtake, H., Chen, Y.J., Tanaka, K., Wang, H.O.: A polynomial observer design for a wider class of polynomial fuzzy systems. Int. Conf. Fuzzy Syst. 2011, 1305–1311 (2011)
Tanaka, K., Ohtake, H., Seo, T., Wang, H.O.: An SOS-based observer design for polynomial fuzzy systems. Am. Control Conf. 2011, 4953–4958 (2011)
Guelton, K., Manamanni, N., Duong, C.C., Koumba-Emianiwe, D.L.: Sum-of-squares stability analysis of Takagi-Sugeno systems based on multiple polynomial lyapunov functions. Int. J. Fuzzy Syst. 15(1), 1–8 (2013)
Prajna, S., Papachristodoulou, A., Seiler, P., Parrilo, P.: SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, Version 2.00. California Institute Technology, Pasadena (2004)
Sturm, J.: Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones. Optim. Methods Softw 11(4), 625–653 (1999)
Tong, S.C., Li, Y.M.: Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones. IEEE Trans. Fuzzy Syst. 20(1), 168–180 (2012)
Lee, C.H., Hsueh, H.Y.: Observer-based adaptive control for a class of nonlinear non-affine systems using recurrent-type fuzzy logic systems. Int. J Fuzzy Syst. 15(1), 55–65 (2013)
Shen, Q.K., Jiang, B., Cocquempot, V.: Adaptive fuzzy observer-based active fault-tolerant dynamic surface control for a class of nonlinear systems with actuator faults. IEEE Trans. Fuzzy Syst 22(2), 338–349 (2014)
Wang, Y.C., Chien, C.J.: An observer-based model reference adaptive iterative learning controller for nonlinear systems. Int. J.Fuzzy Syst. 16(1), 73–85 (2014)
Zhang, L.L., Tong, S.C., Li, Y.M.: Adaptive fuzzy output-feedback control with prescribed performance for uncertain nonlinear systems. Int. J. Fuzzy Syst. 16(2), 212–221 (2014)
Kim, S.H.: Nonquadratic H ∞ stabilization conditions for observer-based T–S fuzzy control systems. IEEE Trans. Fuzzy Syst. 22(3), 699–706 (2014)
Xie, L.: Output feedback H ∞ control of systems with parameter uncertainty. Int. J. Control 63, 741–750 (1996)
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61433004, 61273027), Science and Technology planning project of Liaoning Province, China (2013219005), and IAPI Fundamental Research Funds 2013ZCX14. This work was supported also by the development project of key laboratory of Liaoning province.
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Wang, Y., Zhang, H., Zhang, J. et al. An SOS-Based Observer Design for Discrete-Time Polynomial Fuzzy Systems. Int. J. Fuzzy Syst. 17, 94–104 (2015). https://doi.org/10.1007/s40815-015-0003-x
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DOI: https://doi.org/10.1007/s40815-015-0003-x