Stability analysis of fractional order systems based on T-S fuzzy model with the fractional order \(\alpha:0<\alpha<1\). (English) Zbl 1331.93131
Summary: This paper addresses the problems of the robust stability and stabilization for fractional order systems based on uncertain Takagi-Sugeno fuzzy model. A sufficient condition of asymptotical stability for fractional order uncertain T-S fuzzy model is given, and a parallel distributed compensating fuzzy controller is designed to asymptotically stabilize the model. The sufficient conditions are formulated in the format of linear matrix inequalities. The fractional order T-S fuzzy model of a chaotic system, which has complex nonlinearity, is developed as a test bed. The effectiveness of the approach is tested on fractional order Rössler system and fractional order uncertain Lorenz system.
MSC:
| 93C42 | Fuzzy control/observation systems |
| 93D09 | Robust stability |
| 93D20 | Asymptotic stability in control theory |
| 34K37 | Functional-differential equations with fractional derivatives |
| 34K36 | Fuzzy functional-differential equations |
| 34K35 | Control problems for functional-differential equations |
| 37M05 | Simulation of dynamical systems |
| 37N35 | Dynamical systems in control |
Keywords:
fractional order uncertain T-S fuzzy model; robust stability; parallel distributed compensation (PDC); state feedback control; fractional order Rössler system; fractional order uncertain Lorenz systemReferences:
| [1] | Asemani, M.H., Majd, V.J.: A robust observer-based controller design for uncertain TCS fuzzy systems with unknown premise variables via lmi. Fuzzy Sets Syst. 212, 21-40 (2013) · Zbl 1285.93051 · doi:10.1016/j.fss.2012.07.008 |
| [2] | Bagley, R., Calico, R.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 14, 304-311 (1991) · doi:10.2514/3.20641 |
| [3] | Boyd S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear matrix inequalities in system and control theory. In: SIAM Studies in Applied Mathematics, vol. 15, Philadelphia, PA: Society for Industrial and Applied Mathematics (1994) · Zbl 0816.93004 |
| [4] | Caponetto, R., Dongola, G., Fortuna, L., Petráš, I.: fractional Order System: Modeling and Control Applications. World Scientific, Singapore (2010) · Zbl 1207.82058 |
| [5] | Chen, Y.Q., Ahn, H.S., Podlubny, I.: Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Process. 86(10), 2611-2618 (2006) · Zbl 1172.94385 · doi:10.1016/j.sigpro.2006.02.011 |
| [6] | Demirci, E., Ozalp, N.: A method for solving differential equations of fractional order. J. Comput. Appl. Math. 236(11), 2754-2762 (2012) · Zbl 1243.34003 · doi:10.1016/j.cam.2012.01.005 |
| [7] | Farges, C., Moze, M., Sabatier, J.: Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica 46(10), 1730-1734 (2010) · Zbl 1204.93094 · doi:10.1016/j.automatica.2010.06.038 |
| [8] | Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2001) · Zbl 0998.26002 |
| [9] | Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of non-integer order transfer functions for analysis of electrode processes. J. Electroanal. Chem. Interfacial Electrochem. 33(2), 253-265 (1971) · doi:10.1016/S0022-0728(71)80115-8 |
| [10] | Lan, Y.H., Zhou, Y.: LMI-based robust control of fractional-order uncertain linear systems. Comput. Math. Appl. 62(3), 1460-1471 (2011) · Zbl 1228.93087 · doi:10.1016/j.camwa.2011.03.028 |
| [11] | Lee, H.J., Park, J.B., Chen, G.R.: Robust fuzzy control of nonlinear systems with parametric uncertainties. IEEE Trans. Fuzzy Syst. 9(2), 369-379 (2001) · doi:10.1109/91.919258 |
| [12] | Li, J.M., Li, Y.T.: Robust stability and stabilization of fractional order systems based on uncertain Takagi-Sugeno fuzzy model with the fractional order \[1\,{\le }\,v\,<\,21\]≤v<2. J. Comput. Nonlinear Dyn. 8(4), (2013). doi:10.1115/1.4023,739 · Zbl 1228.93087 |
| [13] | Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59(5), 1810-1821 (2010) · Zbl 1189.34015 · doi:10.1016/j.camwa.2009.08.019 |
| [14] | Liu, Y., Zhao, S.W.: TCS fuzzy model-based impulsive control for chaotic systems and its application. Math. Comput. Simul. 81(11), 2507-2516 (2011) · Zbl 1219.93059 · doi:10.1016/j.matcom.2011.02.012 |
| [15] | Lu, J.G., Chen, G.R.: Robust stability and stabilization of fractional-order interval systems: an LMI approach. IEEE Trans. Autom. Control 54(6), 1294-1299 (2009) · Zbl 1367.93472 · doi:10.1109/TAC.2009.2013056 |
| [16] | Lu, J.G., Chen, Y.Q.: Robust stability and stabilization of fractional-order interval systems with the fractional order \[\alpha\] α: the \[0 < \alpha < 10\]<α<1 case. IEEE Trans. Autom. Control 158(1), 152-158 (2010) · Zbl 1368.93506 |
| [17] | Luo, Y., Chao, H., Di, L., Chen, Y.Q.: Lateral directional fractional order (pi) control of a small fixed-wing unmanned aerial vehicles: controller designs and flight tests. IET Control Theory Appl. 5(18), 2156-2167 (2011) · doi:10.1049/iet-cta.2010.0314 |
| [18] | Manai, Y., Benrejeb, M.: Robust stabilization for continuous Takagi-Sugeno fuzzy system based on observer design. Math. Prob. Eng. 2012, 1-18 (2012) · Zbl 1264.93131 |
| [19] | Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Proceedings of Computational Engineering in Systems and Application Multi Conference, vol. 2, pp. 963-968. IMACS. IEEE-SMC (1996) |
| [20] | N’Doye, I., Darouach, M., Zasadzinski, M., Radhy, N.E.: Robust stabilization of uncertain descriptor fractional-order systems. Automatica 49(6), 1907-1913 (2013) · Zbl 1360.93620 · doi:10.1016/j.automatica.2013.02.066 |
| [21] | Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008 |
| [22] | Qiang, G., Harris, C.J.: Fuzzy local linearization and local basis function expansion in nonlinear system modeling. IEEE Trans. Syst. Man Cybern. Part B Cybern. 29(4), 559-565 (1999) · doi:10.1109/3477.775275 |
| [23] | Rossikhin, Y., Shitikova, M.: Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass system. Acta Mech. 120, 109-125 (1997) · Zbl 0901.73030 · doi:10.1007/BF01174319 |
| [24] | Sabatier, J., Moze, M., Farges, C.: LMI stability conditions for fractional order systems. Comput. Math. Appl. 59(5), 1594-1609 (2010) · Zbl 1189.34020 · doi:10.1016/j.camwa.2009.08.003 |
| [25] | Song, X., Liu, L., Wang, Z.: Stabilization of singular fractional-order systems: a linear matrix inequality approach. In: Proceedings of the IEEE International Conference on Automation and Logistics (2012) · Zbl 0841.93014 |
| [26] | Sun, H., Abdelwahad, A., Onaral, B.: Linear approximation of transfer function with a pole of fractional power. IEEE Trans. Autom. Control 29, 441-444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551 |
| [27] | Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. 15(1), 116-132 (1985) · Zbl 0576.93021 · doi:10.1109/TSMC.1985.6313399 |
| [28] | Tanaka, K., Sugeno, M.: Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst. 45(2), 135-156 (1992) · Zbl 0758.93042 · doi:10.1016/0165-0114(92)90113-I |
| [29] | Tian, E., Peng, C.: Delay-dependent stability analysis and synthesis of uncertain TCS fuzzy systems with time-varying delay. Fuzzy Sets Syst. 157(4), 544-559 (2006) · Zbl 1082.93031 · doi:10.1016/j.fss.2005.06.022 |
| [30] | Wei, Y., Karimi, H.R., Liang, S., Gao, Q., Wang, Y.: General output feedback stabilization for fractional order systems: an LMI approach. Abstr. Appl. Anal. 2014, 1-9 (2014) · Zbl 1406.93271 |
| [31] | Xie, L.H.: Output feedback \[{H}_\infty H\]∞ control of systems with parameter uncertainty. Int. J. Control 63(4), 741-750 (1996) · Zbl 0841.93014 · doi:10.1080/00207179608921866 |
| [32] | Yoneyama, J.: New delay-dependent approach to robust stability and stabilization for Takagi-Sugeno fuzzy time-delay systems. Fuzzy Sets Syst. 158(20), 2225-2237 (2007) · Zbl 1122.93050 · doi:10.1016/j.fss.2007.03.022 |
| [33] | Yu, J.M., Hu, H., Zhou, S.B., Lin, X.R.: Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems. Automatica 49(6), 1798-1803 (2013) · Zbl 1360.93513 · doi:10.1016/j.automatica.2013.02.041 |
| [34] | Zheng, Y.G., Chen, G.R.: Fuzzy impulsive control of chaotic systems based on T-S fuzzy model. Chaos Solitons Fract. 39(4), 2002-2011 (2009) · Zbl 1197.93109 · doi:10.1016/j.chaos.2007.06.061 |
| [35] | Zheng, Y.G., Nian, Y.B., Wang, D.J.: Controlling fractional order chaotic systems based on Takagi-Sugeno fuzzy model and adaptive adjustment mechanism. Phys. Lett. A 375(2), 125-129 (2010) · Zbl 1241.34088 · doi:10.1016/j.physleta.2010.10.038 |
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.
