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Improved stability criteria for nonlinear fractional order fuzzy systems with time-varying delay

  • Fuzzy systems and their mathematics
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Abstract

This paper addresses the issue of the stability analysis and stabilization of nonlinear time-delayed systems with fractional order \(\alpha \in \left( {0,1} \right)\) modeled by Takagi–Sugeno fuzzy model. A new Lyapunov-Krasovskii functional is proposed based on Caputo derivative which leads to obtaining less conservative stability conditions are proposed. Both approaches state feedback and delayed state feedback are applied in the form of a parallel-distributed compensation scheme for stabilization of the closed-loop system. Each of the stability conditions is written as a linear matrix inequality. Finally, some examples are presented to illustrate the effectiveness of the proposed approach.

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References

  • Aguila-Camacho N, Duarte-Mermoud MA, Gallegos JA (2014) Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 19:2951–2957

    Article  MathSciNet  Google Scholar 

  • Binazadeh T, Yousefi M (2018) Asymptotic stabilization of a class of uncertain nonlinear time-delay fractional-order systems via a robust delay-independent controller. J Vib Control 24:4541–4550

    Article  MathSciNet  Google Scholar 

  • Boubellouta A, Boulkroune A (2019) Intelligent fractional-order control-based projective synchronization for chaotic optical systems. Soft Comput 23:5367–5384

    Article  Google Scholar 

  • Bushnaq S, Saeed T, Torres D et al (2021) Control of COVID-19 dynamics through a fractional-order model. Alexandria Eng J 60:3587–3592

    Article  Google Scholar 

  • Cai X, Shi K, Zhong S et al (2021a) Dissipative analysis for high speed train systems via looped-functional and relaxed condition methods. Appl Math Model 96:570–583

    Article  MathSciNet  Google Scholar 

  • Cai X, Wang J, Zhong S et al (2021b) Fuzzy quantized sampled-data control for extended dissipative analysis of T-S fuzzy system and its application to WPGSs. J Franklin Inst 358:1350–1375

    Article  MathSciNet  Google Scholar 

  • Datta R, Dey R, Bhattacharya B, Chakrabarti A (2019) Delayed state feedback controller design for inverted pendulum using T–S fuzzy modeling: An LMI approach. In: Advances in Intelligent Systems and Computing. Springer Verlag, pp 67–79

  • Dong S, Zhu H, Zhong S et al (2021) New study on fixed-time synchronization control of delayed inertial memristive neural networks. Appl Math Comput 399:126035

    MathSciNet  MATH  Google Scholar 

  • Duan R, Li J (2018) Observer-based non- PDC controller design for t–S fuzzy systems with the fractional-order a : 0 < a < 1. IET Control Theory Appl 12:661–668

    Article  MathSciNet  Google Scholar 

  • Gholami H, Binazadeh T (2019) Sliding-mode observer design and finite-time control of one-sided Lipschitz nonlinear systems with time-delay. Soft Comput 23:6429–6440

    Article  Google Scholar 

  • Gu K, Chen J, Kharitonov V (2003) Stability of time-delay systems. Springer

  • Harjule P, Bansal MK (2020) Fractional order models for viscoelasticity in lung tissues with power, exponential and Mittag-Leffler memories. Int J Appl Comput Math 6:1–10

    Article  MathSciNet  Google Scholar 

  • Hartley TT, Lorenzo CF (2002) Dynamics and control of initialized fractional-order systems. Nonlinear Dyn 29:201–233

    Article  MathSciNet  Google Scholar 

  • He BB, Zhou HC, Kou CH, Chen YQ (2019) Stabilization of uncertain fractional order system with time-varying delay using BMI approach. Asian J Control 1:1–9

    Google Scholar 

  • Hu JB, Lu GP, Zhang SB, Zhao LD (2015) Lyapunov stability theorem about fractional system without and with delay. Commun Nonlinear Sci Numer Simul 20:905–913

    Article  MathSciNet  Google Scholar 

  • Hua C, Zhang T, Li Y, Guan X (2016) Robust output feedback control for fractional order nonlinear systems with time-varying delays. IEEE/CAA J Autom Sin 3:477–482

    Article  MathSciNet  Google Scholar 

  • Jafari P, Teshnehlab M, Tavakoli-Kakhki M (2017) Synchronization and stabilization of fractional order nonlinear systems with adaptive fuzzy controller and compensation signal. Nonlinear Dyn 90:1037–1052

    Article  MathSciNet  Google Scholar 

  • Jafari P, Teshnehlab M, Tavakoli-Kakhki M (2018) Adaptive type-2 fuzzy system for synchronisation and stabilisation of chaotic non-linear fractional order systems. IET Control Theory Appl 12:183–193

    Article  MathSciNet  Google Scholar 

  • Kilbas AA, Marzan SA (2005) Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differ Equations 41:84–89

    Article  MathSciNet  Google Scholar 

  • Li P, Chen L, Wu R et al (2018) Robust asymptotic stability of interval fractional-order nonlinear systems with time-delay. J Franklin Inst 355:7749–7763

    Article  MathSciNet  Google Scholar 

  • Liu H, Pan Y, Cao J, et al (2020) Positivity and stability analysis for fractional-order delayed systems: a T-S fuzzy model approach. IEEE Trans Fuzzy Syst 1–1

  • Mahmoudabadi P, Shasadeghi M, Zarei J (2017) New stability and stabilization conditions for nonlinear systems with time-varying delay based on delay-partitioning approach. ISA Trans 70:46–52

    Article  Google Scholar 

  • Mahmoudabadi P, Tavakoli-Kakhki M (2021a) Fuzzy observer–based disturbance rejection control for nonlinear fractional-order systems with time-varying delay. J Vib Control. https://doi.org/10.1177/10775463211006958

    Article  Google Scholar 

  • Mahmoudabadi P, Tavakoli-Kakhki M (2021b) Tracking control with disturbance rejection of nonlinear fractional order fuzzy systems: Modified repetitive control approach. Chaos, Solitons Fractals. https://doi.org/10.1016/j.chaos.2021.111142

    Article  MathSciNet  Google Scholar 

  • Mirzajani S, Aghababa MP, Heydari A (2019) Adaptive T-S fuzzy control design for fractional-order systems with parametric uncertainty and input constraint. Fuzzy Sets Syst 365:22–39

    Article  MathSciNet  Google Scholar 

  • Pang H, Wang Y, Zhang X, Xu Z (2019) Robust state-feedback control design for active suspension system with time-varying input delay and wheelbase preview information. J Franklin Inst 356:1899–1923

    Article  MathSciNet  Google Scholar 

  • Parvizian M, Khandani K, Majd VJ (2019) An H∞non-fragile observer-based adaptive sliding mode controller design for uncertain fractional-order nonlinear systems with time delay and input nonlinearity. Asian J Control 1:1–9

    Google Scholar 

  • Phat V, Niamsup P, Thuan MV (2020) A new design method for observer-based control of nonlinear fractional-order systems with time-variable delay. Eur J Control 56:124–131

    Article  MathSciNet  Google Scholar 

  • Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their. Elsevier

  • Rahmanipour P, Ghadiri H (2020) Stability analysis for a class of fractional-order nonlinear systems with time-varying delays. Soft Comput 24:17445–17453

    Article  Google Scholar 

  • Sakthivel R, Raajananthini K, Kwon OM, Mohanapriya S (2019) Estimation and disturbance rejection performance for fractional order fuzzy systems. ISA Trans 92:65–74

    Article  Google Scholar 

  • Sene N (2018a) Exponential form for Lyapunov function and stability analysis of the fractional differential equations. J Math Comput Sci 18:388–397

    Article  Google Scholar 

  • Sene N (2018b) Lyapunov characterization of the fractional nonlinear systems with exogenous input. Fractal Fract 2:1–10

    Article  Google Scholar 

  • Sene N (2019) Global asymptotic stability of the fractional differential equations. J Nonlinear Sci Appl 13:171–175

    Article  MathSciNet  Google Scholar 

  • Shi K, Wang J, Tang Y, Zhong S (2020) Reliable asynchronous sampled-data filtering of T-S fuzzy uncertain delayed neural networks with stochastic switched topologies. Fuzzy Sets Syst 381:1–25

    Article  MathSciNet  Google Scholar 

  • Stamov G, Stamova I (2018) Uncertain impulsive differential systems of fractional order: almost periodic solutions. Int J Syst Sci 49:631–638

    Article  MathSciNet  Google Scholar 

  • Thanh NT, Trinh H, Phat VN (2017) Stability analysis of fractional differential time-delay equations. IET Control Theory Appl 11:1006–1015

    Article  MathSciNet  Google Scholar 

  • Trigeassou JC, Maamri N, Sabatier J, Oustaloup A (2011) A Lyapunov approach to the stability of fractional differential equations. Signal Process 91:437–445

    Article  Google Scholar 

  • Wang B, Xue J, Chen D (2016) Takagi-Sugeno fuzzy control for a wide class of fractional-order chaotic systems with uncertain parameters via linear matrix inequality. J Vib Control 22:2356–2369

    Article  MathSciNet  Google Scholar 

  • Wang L, Lam HK (2018) A new approach to stability and stabilization analysis for continuous-time Takagi-Sugeno fuzzy systems with time delay. IEEE Trans Fuzzy Syst 26:2460–2465

    Article  Google Scholar 

  • Wang Y, Gao G, Li X, Chen Z (2020) A fractional-order model-based state estimation approach for lithium-ion battery and ultra-capacitor hybrid power source system considering load trajectory. J Power Sources 449:227543

    Article  Google Scholar 

  • Xie W, Zhang R, Zeng D et al (2020) Strictly dissipative stabilization of multiple-memory Markov jump systems with general transition rates: A novel event-triggered control strategy. Int J Robust Nonlinear Control 30:1956–1978

    Article  MathSciNet  Google Scholar 

  • Yousefi M, Binazadeh T (2018) Delay-independent sliding mode control of time-delay linear fractional order systems. Trans Inst Meas Control 40:1212–1222

    Article  Google Scholar 

  • Zhang X, Ma Y (2020) LMIs conditions to robust pinning synchronization of uncertain fractional-order neural networks with discontinuous activations. Soft Comput 24:15927–15935

    Article  Google Scholar 

  • Zhao Y, Gao H, Lam J, Du B (2009) Stability and stabilization of delayed T-S fuzzy systems: a delay partitioning approach. IEEE Trans Fuzzy Syst 17:750–762

    Article  Google Scholar 

  • Zheng Y, Nian Y, Wang D (2010) Controlling fractional order chaotic systems based on Takagi-Sugeno fuzzy model and adaptive adjustment mechanism. Phys Lett Sect A Gen at Solid State Phys 375:125–129

    MATH  Google Scholar 

Download references

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This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Faculty of Electrical Engineering, K.N. Toosi University of Technology, Seyed-Khandan bridge, Shariati Ave., 163171419, Tehran, Iran

    Parvin Mahmoudabadi & Mahsan Tavakoli-Kakhki

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  1. Parvin Mahmoudabadi
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  2. Mahsan Tavakoli-Kakhki
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Correspondence to Mahsan Tavakoli-Kakhki.

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Mahmoudabadi, P., Tavakoli-Kakhki, M. Improved stability criteria for nonlinear fractional order fuzzy systems with time-varying delay. Soft Comput 26, 4215–4226 (2022). https://doi.org/10.1007/s00500-022-06893-4

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