×

Finite-time stability and controller design of continuous-time polynomial fuzzy systems. (English) Zbl 1470.93090

Summary: Finite-time stability and stabilization problem is first investigated for continuous-time polynomial fuzzy systems. The concept of finite-time stability and stabilization is given for polynomial fuzzy systems based on the idea of classical references. A sum-of-squares- (SOS-) based approach is used to obtain the finite-time stability and stabilization conditions, which include some classical results as special cases. The proposed conditions can be solved with the help of powerful Matlab toolbox SOSTOOLS and a semidefinite-program (SDP) solver. Finally, two numerical examples and one practical example are employed to illustrate the validity and effectiveness of the provided conditions.

MSC:

93C42 Fuzzy control/observation systems
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D15 Stabilization of systems by feedback

Software:

Sostools; Matlab
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Tanaka, K.; Wang, H. O., Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, (2001), Hoboken, NJ, USA: Wiley, Hoboken, NJ, USA
[2] Zhang, H.; Liu, D., Fuzzy Modeling and Fuzzy Control. Fuzzy Modeling and Fuzzy Control, Control Engineering, (2006), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 1203.93115
[3] Zhang, H.; Lun, S.; Liu, D., Fuzzy H_{∞} filter design for a class of nonlinear discrete-time systems with multiple time delays, IEEE Transactions on Fuzzy Systems, 15, 3, 453-469, (2007) · doi:10.1109/TFUZZ.2006.889841
[4] Yang, F.; Zhang, H., T-S model-based relaxed reliable stabilization of networked control systems with time-varying delays under variable sampling, International Journal of Fuzzy Systems, 13, 4, 260-269, (2011)
[5] Zhao, X.; Zhang, L.; Shi, P.; Karimi, H. R., Novel stability criteria for T - S fuzzy systems, IEEE Transactions on Fuzzy Systems, 22, 2, 313-323, (2014) · doi:10.1109/TFUZZ.2013.2254491
[6] Wu, L.; Su, X.; Shi, P.; Qiu, J., A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 41, 1, 273-286, (2011) · doi:10.1109/TSMCB.2010.2051541
[7] Mozelli, L. A.; Palhares, R. M.; Souza, F. O.; Mendes, E. M. A. M., Reducing conservativeness in recent stability conditions of TS fuzzy systems, Automatica, 45, 6, 1580-1583, (2009) · Zbl 1166.93344 · doi:10.1016/j.automatica.2009.02.023
[8] Choi, H. D.; Ahn, C. K.; Shi, P.; Wu, L.; Lim, M. T., Dynamic Output-Feedback Dissipative Control for T-S Fuzzy Systems With Time-Varying Input Delay and Output Constraints, IEEE Transactions on Fuzzy Systems, 25, 3, 511-526, (2017) · doi:10.1109/TFUZZ.2016.2566800
[9] Zhou, Q.; Li, H.; Wu, C.; Wang, L.; Ahn, C. K., Adaptive fuzzy control of nonlinear systems with unmodeled dynamics and input saturation using small-gain approach, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47, 8, 1979-1989, (2017) · doi:10.1109/TSMC.2016.2586108
[10] Tanaka, K.; Yoshida, H.; Ohtake, H.; Wang, H. O., Stabilization of polynomial fuzzy systems via a sum of squares approach, Proceedings of the 2007 IEEE 22nd International Symposium on Intelligent Control, ISIC 2007 · doi:10.1109/ISIC.2007.4450878
[11] Tanaka, K.; Ohtake, H.; Seo, T.; Tanaka, M.; Wang, H. O., Polynomial fuzzy observer designs: A sum-of-squares approach, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 42, 5, 1330-1342, (2012) · doi:10.1109/TSMCB.2012.2190277
[12] Tanaka, K.; Yoshida, H.; Ohtake, H.; Wang, H. O., A sum of squares approach to stability analysis of polynomial fuzzy systems, Proceedings of the 2007 American Control Conference, ACC · doi:10.1109/ACC.2007.4282579
[13] Tanaka, K.; Ohtake, H.; Wang, H. O., Guaranteed cost control of polynomial fuzzy systems via a sum of squares approach, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 39, 2, 561-567, (2009) · doi:10.1109/TSMCB.2008.2006639
[14] Tanaka, K.; Yoshida, H.; Ohtake, H.; Wang, H. O., A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems, IEEE Transactions on Fuzzy Systems, 17, 4, 911-922, (2009) · doi:10.1109/TFUZZ.2008.924341
[15] 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, International Journal of Fuzzy Systems, 15, 1, 1-8, (2013)
[16] Prajna, S.; Papachristodoulou, A.; Seiler, P.; Parrilo, P., SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, version 3.01, (2016), Pasadena, CA, USA: California Institute of Technology, Pasadena, CA, USA
[17] Tanaka, K.; Tanaka, M.; Chen, Y.-J.; Wang, H. O., A new sum-of-squares design framework for robust control of polynomial fuzzy systems with uncertainties, IEEE Transactions on Fuzzy Systems, 24, 1, 94-110, (2016) · doi:10.1109/TFUZZ.2015.2426719
[18] Narimani, M.; Lam, H. K., SOS-based stability analysis of polynomial fuzzy-model-based control systems via polynomial membership functions, IEEE Transactions on Fuzzy Systems, 18, 5, 862-871, (2010) · doi:10.1109/TFUZZ.2010.2050890
[19] Lam, H. K., Polynomial fuzzy-model-based control systems: stability analysis via piecewise-linear membership functions, IEEE Transactions on Fuzzy Systems, 19, 3, 588-593, (2011) · doi:10.1109/tfuzz.2011.2118215
[20] Lam, H. K.; Narimani, M.; Li, H.; Liu, H., Stability analysis of polynomial-fuzzy-model-based control systems using switching polynomial lyapunov function, IEEE Transactions on Fuzzy Systems, 21, 5, 800-813, (2013) · doi:10.1109/TFUZZ.2012.2230005
[21] Pitarch, J. L.; Sala, A.; Ariño, C. V., Closed-form estimates of the domain of attraction for nonlinear systems via fuzzy-polynomial models, IEEE Transactions on Cybernetics, 44, 4, 526-538, (2014) · doi:10.1109/TCYB.2013.2258910
[22] Chen, Z.; Zhang, B.; Li, H.; Yu, J., Tracking control for polynomial fuzzy networked systems with repeated scalar nonlinearities, Neurocomputing, 171, 185-193, (2016) · doi:10.1016/j.neucom.2015.06.030
[23] Wang, Y.; Zhang, H.; Wang, Y.; Zhang, J., Stability analysis and controller design of discrete-time polynomial fuzzy time-varying delay systems, Journal of The Franklin Institute, 352, 12, 5661-5685, (2015) · Zbl 1395.93334 · doi:10.1016/j.jfranklin.2015.09.015
[24] Amato, F.; Ariola, M.; Cosentino, C., Finite-time stabilization via dynamic output feedback, Automatica, 42, 2, 337-342, (2006) · Zbl 1099.93042 · doi:10.1016/j.automatica.2005.09.007
[25] Amato, F.; Ariola, M.; Dorato, P., Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37, 9, 1459-1463, (2001) · Zbl 0983.93060 · doi:10.1016/S0005-1098(01)00087-5
[26] He, S.; Liu, F., Finite-time H_{∞} fuzzy control of nonlinear jump systems with time delays via dynamic observer-based state feedback, IEEE Transactions on Fuzzy Systems, 20, 4, 605-614, (2012) · doi:10.1109/tfuzz.2011.2177842
[27] Yang, D.; Cai, K.-Y., Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems, Expert Systems with Applications, 37, 10, 6963-6967, (2010) · doi:10.1016/j.eswa.2010.03.024
[28] Shen, H.; Park, J. H.; Wu, Z.-G., Finite-time reliable control for Takagi-Sugeno fuzzy systems with actuator faults, IET Control Theory & Applications, 8, 9, 688-696, (2014) · doi:10.1049/iet-cta.2013.0486
[29] Zheng, C.; Cao, J.; Hu, M.; Fan, X., Finite-time stabilisation for discrete-time T–S fuzzy model system with channel fading and two types of parametric uncertainty, International Journal of Systems Science, 48, 1, 34-42, (2017) · Zbl 1358.93185 · doi:10.1080/00207721.2016.1146972
[30] Prajna, S.; Papachristodoulou, A.; Wu, F., Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach, Proceedings of the 2004 5th Asian Control Conference
[31] Tanaka, K.; Komatsu, T.; Ohtake, H.; Wang, H. O., Micro helicopter control: LMI approach vs SOS approach, Proceedings of the 2008 IEEE International Conference on Fuzzy Systems, FUZZ 2008 · doi:10.1109/FUZZY.2008.4630389
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.