Abstract
By using Takagi–Sugeno (T–S) fuzzy set approach, this paper proposes a robust dynamic output feedback (DOF) control for nonlinear fractional-order systems satisfying \(0<\alpha <1\). First, using a Fractional Lyapunov function, the novel DOF controller guarantees the stability of the closed-loop system. The proposed approach allows avoiding appearance of crossing terms between the controller’s and the T–S system’s input matrices leading to easier LMI formulation. Second, a new controller is developed by combining a fuzzy dependent Lyapunov function and some special derivations on the controller parameters. This leads to some sufficient conditions in the form of strict linear matrix inequalities (LMIs). When compared with previous work, the proposed method not only has abilities to handle the fuzzy system with the time-derivatives of the membership functions but also can deal with the parametric uncertainties effectively. Simulation examples are provided to demonstrate the validity of the proposed conditions.
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References
Dong K, Wu F (2008) Almost output regulation for parameter-dependent linear fractional transformation systems. IET Control Theory Appl 2(3):200–209
Badri V, Tavazoei MS (2014) Fractional order control of thermal systems: achievability of frequency-domain requirements. Nonlinear Dyn 80(4):1773–1783
Gabano JD, Poinot T, Kanoun H (2011) Identification of a thermal system using continuous linear parameter-varying fractional modelling. IET Control Theory Appl 5(7):889–899
Ahn H, Chen Y (2008) Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica 44(11):2985–2988
Dzielinski A, Sierociuk D (2008) Stability of discrete fractional order state-space systems. J Vib Control 14(9–10):1543–1556
Lu J, Chen Y (2010) Robust stability and stabilization of fractional-order interval systems with the fractional order: the case \(0 < \alpha < 1\). IEEE Transactions Automatic Control 55(1):152–158
Chilali M, Gahinet P, Apkarian P (1999) Robust pole placement in LMI regions. IEEE Transactions on Automatic Control 44(12):2257–2270
Alaviyan Shahri ES, Alfi A, Tenreiro Machado JA (2018) Stability analysis of a class of nonlinear fractional-order systems under control input saturation. Int J Robust Nonlinear Control 28(7):2887–2905
Marir S, Chadli M, Bouagada D (2017) New admissibility conditions for singular linear continuous-time fractional-order systems. J Franklin Inst 354(2):752–766
Takagi T, Sugeno M (1985) Fuzzy identification of systems and its application to modeling and control, IEEE Transactions Syst Man Cybern pp 116–132
Mamdani EH (1974) Application of fuzzy algorithms for control of simple dynamic plant In Proceedings of the Institution of Electrical Engineers, IET 121(12):1585–1588
Chaibi R, El Aiss H, El Hajjaji A, Hmamed A (2020) Stability analysis and robust \(H_{\infty }\) controller synthesis with derivatives of membership functions for T-S fuzzy systems with time-varying delay: Input-output stability approach. Int J Control Automation Syst 18:1–13
Chaibi R, Er Rachid R, Tissir El H, Hmamed A (2019) Finite frequency \(H_{\infty }\) control for 2-D T–S fuzzy FM II model with stochastic perturbation, Int J Syst Sci
Esfahani SH, Sichani AK (2011) Improvement on the problem of optimal fuzzy \(H_{\infty }\) tracking control design for non-linear systems. IET Control Theory Appl 5(18):2179–2190
Esfahani SH (2016) Improvement on the problem of output feedback fuzzy \(H_{\infty }\) tracking control design for non-linear discrete-time systems with state and input delay. IET Control Theory Appl 10(1):24–34
Esfahani SH (2020) Further Improvements on the problem of optimal fuzzy \(H_{\infty }\) tracking control design for T-S fuzzy systems. J Control Automation Electric Syst 31(4):874–884
Li Y, Li J (2014) Stability analysis of fractional order systems based on T-S fuzzy model with the fractional order \(\alpha \) : \(0<\alpha <1\). Nonlinear Dyn 78:2909–2919
Mozelli LA, Palhares RM, Souza FO, Mendes EM (2009) Reducing conservativeness in recent stability conditions of T-S fuzzy systems. Automatica 45(6):1580–1583
Feng G (2004) Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions. IEEE Transactions Fuzzy Syst 12(1):22–28
Guelton K, Bouarar T, Manamanni N (2009) Robust dynamic output feedback fuzzy Lyapunov stabilization of Takagi-Sugeno systems -a descriptor redundancy approach. Fuzzy Sets Syst 160(19):2796–2811
Choi HD, Ki Ahn C, Lim MT, Song MK (2016) Dynamic output-feedback \(H_{\infty }\) control for active half-vehicle suspension systems with time-varying input delay. Int J Control Automation Syst 14(1):59–68
Hu J, Ding B (2018) Dynamic output feedback predictive control with one free control move for the takagi-sugeno model with bounded disturbance. IEEE Transaction Fuzzy Syst 27(3):462–473
Guo Y, Lin C, Chen B, Wang Q (2019) Necessary and sufficient conditions for the dynamic output feedback stabilization of fractional-order systems with order \(0 < \alpha < 1\) Sci China Inf Sci 62(9): 199201:1–199201:3
Podlubny I (1999) Fract Differ Equ. Academic Press, San Diego, Boston
Badri P, Sojoodi M (2018) Robust fixed-order dynamic output feedback controller design for fractional-order systems. IET Control Theory Appl 12(9):1236–1243
Demirci E, Ozalp N (2012) A method for solving differential equations of fractional order. J Comput Appl Math 236(11):2754–2762
Chang XH, Zhang L, Park JH (2015) Robust static output feedback \(H_{\infty }\) control for uncertain fuzzy systems. Fuzzy Sets Syst 273:87–104
Kau SW, Lee HJ, Yang CM, Lee CH, Hong L, Fang CH (2007) Robust \(H_{\infty }\) fuzzy static output feedback control of T-S fuzzy systems with parametric uncertainties. Fuzzy Sets Syst 158:135–146
Ji Y, Su L, Qiu J (2015) Design of fuzzy output feedback stabilization for uncertain fractional-order systems. Neurocomputing 173:1683–1693
Chang XH, Yang GH (2011) Non-fragile \(H_{\infty }\) filtering of continuous-time fuzzy systems. IEEE Transactions Signal Process 59:1528–1538
Tanaka K, Hori T, Wang HO (2003) A multiple Lyapunov function approach to stabilization of fuzzy control systems. IEEE Transactions Fuzzy Syst 11(4):582–589
Li J, Li Y (2013) Robust stability and stabilization of fractional order systems based on uncertain Takagi-Sugeno fuzzy model with the fractional order \(1<\alpha <2\). J Comput Nonlinear Dyn 8:041005
Cao Y, Samidurai R, Sriraman R (2019) Robust passivity analysis for uncertain neural networks with leakage delay and additive time-varying delays by using general activation function. Math Computers Simul 155:57–77
Cao Y, Sriraman R, Samidurai R (2020) Stability and stabilization analysis of nonlinear time-delay systems with randomly occurring controller gain fluctuation. Math Computers Simul 171:36–51
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El Bachtiri, R., Yagoubi, M. & Chaibi, R. A new T–S fuzzy model based robust output-feedback stabilizing controller for fractional-order systems. Int. J. Dynam. Control 10, 1217–1227 (2022). https://doi.org/10.1007/s40435-021-00874-5
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DOI: https://doi.org/10.1007/s40435-021-00874-5