Abstract
Takagi–Sugeno (T–S) fuzzy model is an effective technology for describing complex nonlinear industrial processes and dynamic systems with unmeasurable parameters. However, to stabilize this type of system, feedback linearization techniques and complex adaptive schemes usually need to be adopted. This paper focuses on the stabilization of uncertain fractional-order (FO) systems with unmeasurable states, external disturbances, and time delays, where certain “IF-THEN” rules based on an FO T–S model are proposed to describe FO systems. A new FO \(H_\infty\) performance model is established to provide stabilization sufficient conditions. A fuzzy observer is implemented to reconstruct system states, and a feedback controller is established to stabilize the integrated system. Benefiting from a continuous frequency distribution model, the stabilization can be discussed through integer-order stabilization theories, and several stabilization conditions are obtained. Moreover, the proposed control approach avoids the usage of feedback linearization techniques, which effectively reduces the computational complexity. Apart from theoretical analysis, through a numerical experiment, the feasibility of derived results is verified.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article since no associated the date.
References
Magin, R.L.: Fractional calculus in bioengineering: a tool to model complex dynamics. In: Proceedings of the 13th International Carpathian Control Conference (ICCC), pp. 464–469. IEEE (2012)
Kumar D, Baleanu D.: Fractional calculus and its applications in physics. Frontiers physics. 7, 81 (2019)
Machado, J.T., Mata, M.E.: Pseudo phase plane and fractional calculus modeling of western global economic downturn. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 396–406 (2015)
Li, Y., Chen, Y., Podlubny, I.: Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8), 1965–1969 (2009)
Precup, R.-E., Angelov, P., Costa, B.S.J., Sayed-Mouchaweh, M.: An overview on fault diagnosis and nature-inspired optimal control of industrial process applications. Comput. Ind. 74, 75–94 (2015)
Efe, M.Ö.: Fractional fuzzy adaptive sliding-mode control of a 2-DOF direct-drive robot arm. IEEE Trans. Syst, Man Cybern. B (Cybern.) 38(6), 1561–1570 (2008)
Zhou, Y., Wang, H., Liu, H.: Generalized function projective synchronization of incommensurate fractional-order chaotic systems with inputs saturation. Int. J. Fuzzy Syst. 21, 823–836 (2019)
Gegov, A.E., Frank, P.M.: Hierarchical fuzzy control of multivariable systems. Fuzzy Sets Syst. 72(3), 299–310 (1995)
Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. 1, 116–132 (1985)
Mirzajani, S., Aghababa, M.P., Heydari, A.: Adaptive T–S fuzzy control design for fractional-order systems with parametric uncertainty and input constraint. Fuzzy Sets Syst. 365, 22–39 (2019)
Kavikumar, R., Ma, Y.K., Ren, Y., Anthoni, S.M., et al.: Observer-based \({H}_\infty\) repetitive control for fractional-order interval type-2 TS fuzzy systems. IEEE Access 6, 49828–49837 (2018)
Bai, J., Wen, G., Rahmani, A., Yu, Y.: Distributed consensus tracking for the fractional-order multi-agent systems based on the sliding mode control method. Neurocomputing 235, 210–216 (2017)
Lin, T.C., Lee, T.Y.: Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control. IEEE Trans. Fuzzy Syst. 19(4), 623–635 (2011)
Zhang, H., Zeng, Z.: Synchronization of nonidentical neural networks with unknown parameters and diffusion effects via robust adaptive control techniques. IEEE Trans. Cybern. 51(2), 660–672 (2019)
Tan, L.N., Cong, T.P., Cong, D.P.: Neural network observers and sensorless robust optimal control for partially unknown PMSM with disturbances and saturating voltages. IEEE Trans. Power Electron. 36(10), 12045–12056 (2021)
Liu, H., Pan, Y., Cao, J.: Composite learning adaptive dynamic surface control of fractional-order nonlinear systems. IEEE Trans. Cybern. 50(6), 2557–2567 (2019)
Li, Z., Gao, L., Chen, W., Xu, Y.: Distributed adaptive cooperative tracking of uncertain nonlinear fractional-order multi-agent systems. IEEE/CAA J. Autom. Sin. 7(1), 292–300 (2019)
Seuret, A.: A novel stability analysis of linear systems under asynchronous samplings. Automatica 48(1), 177–182 (2012)
Zeng, H.-B., Teo, K.L., He, Y.: A new looped-functional for stability analysis of sampled-data systems. Automatica 82, 328–331 (2017)
Liu, H., Pan, Y., Cao, J., Wang, H., Zhou, Y.: Adaptive neural network backstepping control of fractional-order nonlinear systems with actuator faults. IEEE Trans. Neural Netw. Learn. Syst. 31(12), 5166–5177 (2020)
Sakthivel, R., Raajananthini, K., Kwon, O., Mohanapriya, S.: Estimation and disturbance rejection performance for fractional order fuzzy systems. ISA Trans. 92, 65–74 (2019)
Zheng, Y., Nian, Y., Wang, D.: Controlling fractional order chaotic systems based on Takagi–Sugeno fuzzy model and adaptive adjustment mechanism. Phys. Lett. A 375(2), 125–129 (2010)
Wang, X., Park, J.H., She, K., Zhong, S., Shi, L.: Stabilization of chaotic systems with T–S fuzzy model and nonuniform sampling: a switched fuzzy control approach. IEEE Trans. Fuzzy Syst. 27(6), 1263–1271 (2018)
Tseng, C.-S., Chen, B.-S., Uang, H.-J.: Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model. IEEE Trans. Fuzzy Syst. 9(3), 381–392 (2001)
Djennoune, S., Bettayeb, M., Al Saggaf, U.M.: Impulsive observer with predetermined finite convergence time for synchronization of fractional-order chaotic systems based on Takagi–Sugeno fuzzy model. Nonlinear Dyn. 98, 1331–1354 (2019)
Sajewski, Ł.: Decentralized stabilization of descriptor fractional positive continuous-time linear systems with delays. In: 2017 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), pp. 482–487. IEEE (2017)
Kaczorek, T.: Stabilization of fractional positive continuous-time linear systems with delays in sectors of left half complex plane by state-feedbacks. Control Cybern. 39(3), 783–795 (2010)
Mahmoudabadi, P., Tavakoli-Kakhki, M.: Improved stability criteria for nonlinear fractional order fuzzy systems with time-varying delay. Soft Comput. 26(9), 4215–4226 (2022)
Shen, J., Lam, J.: Stability and performance analysis for positive fractional-order systems with time-varying delays. IEEE Trans. Autom. Control 61(9), 2676–2681 (2015)
Li, Y., Li, J.: Decentralized stabilization of fractional order TS fuzzy interconnected systems with multiple time delays. J. Intell. Fuzzy Syst. 30(1), 319–331 (2016)
Liu, H., Pan, Y., Cao, J., Zhou, Y., Wang, H.: Positivity and stability analysis for fractional-order delayed systems: a T–S fuzzy model approach. IEEE Trans. Fuzzy Syst. 29(4), 927–939 (2020)
Lee, C.-C.: Fuzzy logic in control systems: fuzzy logic controller. I. IEEE Trans. Syst. Man Cybern. 20(2), 404–418 (1990)
Palm, R., Driankov, D., Hellendoorn, H.: Model Based Fuzzy Control: Fuzzy Gain Schedulers and Sliding Mode Fuzzy Controllers. Springer Science & Business Media, Berlin (1997)
Bai, Z., Li, S., Liu, H.: Composite observer-based adaptive event-triggered backstepping control for fractional-order nonlinear systems with input constraints. Math. Methods Appl. Sci. 46, 16415–16433 (2022)
Trigeassou, J.-C., Maamri, N., Sabatier, J., Oustaloup, A.: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91(3), 437–445 (2011)
Ma, X.-J., Sun, Z.-Q., He, Y.-Y.: Analysis and design of fuzzy controller and fuzzy observer. IEEE Trans. Fuzzy Syst. 6(1), 41–51 (1998)
Tanaka, K., Ikeda, T., Wang, H.O.: Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE Trans. Fuzzy Syst. 6(2), 250–265 (1998)
Cao, Y.-Y., Frank, P.M.: Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach. IEEE Trans. Fuzzy Syst. 8(2), 200–211 (2000)
Hua, C., Wu, S., Guan, X.: Stabilization of T–S fuzzy system with time delay under sampled-data control using a new looped-functional. IEEE Trans. Fuzzy Syst. 28(2), 400–407 (2019)
Gassara, H., Hajjaji, A.E., Chaabane, M.: Robust \({H}\infty\) control for T–S fuzzy systems with time-varying delay. Int. J. Syst. Sci. 41(12), 1481–1491 (2010)
Gahinet, P., Nemirovskii, A., Laub, A.J., Chilali, M.: The LMI control toolbox. In: Proceedings of 1994 33rd IEEE Conference on Decision and Control, vol. 3, pp. 2038–2041. IEEE (1994)
Acknowledgements
This work was supported by the Innovation Project of Guangxi Graduate Education (YCSW2023253) and the National Natural Science Foundation of China (12261009).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Liu, Y., Zhang, X. Stabilization of Fractional-Order T–S Fuzzy Systems with Time Delays via an \(H_\infty\) Performance Model. Int. J. Fuzzy Syst. (2024). https://doi.org/10.1007/s40815-023-01667-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40815-023-01667-y