July  2022, 15(7): 1615-1631. doi: 10.3934/dcdss.2021144

Fuzzy adaptive asymptotic tracking of fractional order nonlinear systems with uncertain disturbances

College of Science, Liaoning University of Technology, Jinzhou, 121001, China

*Corresponding author: Yuan-Xin Li

Received  August 2021 Revised  September 2021 Published  July 2022 Early access  December 2021

Fund Project: The first author is supported part by the Funds of National Science of China (Grant Nos. 61973146, 61773188), in part by the Doctoral Research Initiation of Foundation of Liaoning Province (No. 20180540047), and in part by the Distinguished Young Scientific Research Talents Plan in Liaoning Province (Nos. XLYC1907077, JQL201915402)

In this article, an adaptive asymptotic tracking control scheme is proposed for fractional order nonlinear systems (FONSs) with time-varying disturbance. By introducing some well defined smooth functions and the bounded estimation approach, the effects caused by the unknown virtual control coefficients (UVCC) and unknown nonlinear functions are counteracted. For the UVCC, we only need to assume that their lower bounds are positive constants. Fuzzy logic systems (FLSs) are applied to approximate unknown nonlinear functions. Moreover, the fractional directed Lyapunov method is used to prove that the tracking error asymptotically converges to zero. Finally, an illustrative simulation example is applied to verify the superior performance of the presented control algorithms.

Citation: Jin-Zi Yang, Yuan-Xin Li, Ming Wei. Fuzzy adaptive asymptotic tracking of fractional order nonlinear systems with uncertain disturbances. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1615-1631. doi: 10.3934/dcdss.2021144
References:
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S. Djennoune and M. Bettayeb, Optimal synergetic control for fractional-order systems, Automatica, 49 (2013), 2243-2249.  doi: 10.1016/j.automatica.2013.04.007.

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M. A. Duart-MermoudN. Aguila-CamachoJ. A. Gallegos and R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 650-659.  doi: 10.1016/j.cnsns.2014.10.008.

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MÖ. Efe, Fractional order systems in industrial automation-a survey, IEEE Transactions on Industrial Informatics, 7 (2011), 582-591.  doi: 10.1109/TII.2011.2166775.

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P. GongW. Lan and Q. L. Han, Robust adaptive fault-tolerant consensus control for uncertain nonlinear fractional-order multi-agent systems with directed topologies, Automatica, 117 (2020), 109011.  doi: 10.1016/j.automatica.2020.109011.

[9]

X. Li, J. He, C. Wen and X. K. Liu, Backstepping based adaptive control of a class of uncertain incommensurate fractional-order nonlinear systems with external disturbance, IEEE Transactions on Industrial Electronics, (2021), 1–1. doi: 10.1109/TIE.2021.3070513.

[10]

X. Li, C. Wen and Y. Zou, Adaptive backstepping control for fractional-order nonlinear systems with external disturbance and uncertain parameters using smooth control, IEEE Transactions on Systems, Man, and Cybernetics: Systems., (2020), 1–10. doi: 10.1109/TSMC.2020.2987335.

[11]

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[12]

X. D. LiJ. H. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[13]

X. D. LiX. Y. Yang and T. W. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[14]

Y. M. LiS. C. Tong and T. S. Li, Composite adaptive fuzzy output feedback control design for uncertain nonlinear strict-feedback systems with input saturation, IEEE Transactions on Cybernetics, 45 (2014), 2299-2308.  doi: 10.1109/TCYB.2014.2370645.

[15]

Y. X. Li, M. Wei and S. C. Tong, Event-triggered adaptive neural control for fractional-order nonlinear systems based on finite-time scheme, IEEE Transactions on Cybernetics, (2021), 1–9. doi: 10.1109/TCYB.2021.3056990.

[16]

H. LiuS. LiJ. CaoG. LiA. Alsaedi and F. E. Alsaadi, Adaptive fuzzy prescribed performance controller design for a class of uncertain fractional-order nonlinear systems with external disturbances, Neurocomputing, 219 (2017), 422-430.  doi: 10.1016/j.neucom.2016.09.050.

[17]

H. Liu, Y. Pan, S. Li and Y. Chen, Adaptive fuzzy backstepping control of fractional-order nonlinear systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2209-2217. doi: 10.1109/TSMC.2016.2640950.

[18]

S. LuoF. L. LewisY. D. Song and M. Hassen, Accelerated adaptive fuzzy optimal control of three coupled fractional-order chaotic electromechanical transducers, IEEE Transactions on Fuzzy Systems, 29 (2021), 1701-1714.  doi: 10.1109/TFUZZ.2020.2984998.

[19]

Z. Y. Ma and H. Ma, Adaptive fuzzy backstepping dynamic surface control of strict-feedback fractional-order uncertain nonlinear systems, IEEE Transactions on Fuzzy Systems, 28 (2019), 122-133.  doi: 10.1109/TFUZZ.2019.2900602.

[20]

R. E. PrecupM. L. Tomescu and C. A. Dragos, Stabilization of Rossler chaotic dynamical system using fuzzy logic control algorithm, Int. J. Gen. Syst., 43 (2014), 413-433.  doi: 10.1080/03081079.2014.893299.

[21]

M. Radgolchin and H. Moeenfard, Development of a multi-level adaptive fuzzy controller for beyond pull-in stabilization of electrostatically actuated microplates, J. Vib. Control, 24 (2018), 860-878.  doi: 10.1177/1077546316653040.

[22]

J. Shen and J. Lam, Non-existence of finite-time stable equilibria in fractional-order nonlinear systems, Automatica, 50 (2014), 547-551.  doi: 10.1016/j.automatica.2013.11.018.

[23]

Q. ShenP. Shi and Y. Shi, Distributed adaptive fuzzy control for nonlinear multiagent systems via sliding mode observers, IEEE Transactions on Cybernetics, 46 (2016), 3086-3097.  doi: 10.1109/TCYB.2015.2496963.

[24]

S. SongJ. H. ParkB. Zhang and X. Song, Observer-based adaptive hybrid fuzzy resilient control for fractional-order nonlinear systems with time-varying delays and actuator failures, IEEE Transactions on Fuzzy Systems, 29 (2021), 471-485.  doi: 10.1109/TFUZZ.2019.2955051.

[25]

S. SongB. ZhangJ. Xia and Z. Q. Zhang, Adaptive backstepping hybrid fuzzy sliding mode control for uncertain fractional-order nonlinear systems based on finite-time scheme, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 50 (2018), 1559-1569.  doi: 10.1109/TSMC.2018.2877042.

[26]

S. C. Tong and Y. M. Li, Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones, IEEE Transactions on Fuzzy Systems, 20 (2011), 168-180.  doi: 10.1109/TFUZZ.2011.2171189.

[27]

S. C. TongX. Min and Y. X. Li, Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions, IEEE Transactions on Cybernetics, 50 (2020), 3903-3913.  doi: 10.1109/TCYB.2020.2977175.

[28]

S. C. TongY. M. Li and S. Sui, Adaptive fuzzy tracking control design for SISO uncertain nonstrict feedback nonlinear systems, IEEE Transactions on Fuzzy Systems, 24 (2016), 1441-1454.  doi: 10.1109/TFUZZ.2016.2540058.

[29]

L. X. Wang and J. M. Mendel, Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Transactions on Neural Networks, 3 (1992), 807-814.  doi: 10.1109/72.159070.

[30]

T. D. WeiX. Xie and X. D. Li, Persistence and periodicity of survival red blood cells model with time-varying delays and impulses, Mathematical Modelling and Control, 1 (2021), 12-25.  doi: 10.3934/mmc.2021002.

[31]

Y. WeiY. ChenS. Liang and Y. Wang, A novel algorithm on adaptive backstepping control of fractional order systems, Neurocomputing, 165 (2015), 395-402.  doi: 10.1016/j.neucom.2015.03.029.

[32]

Y. WeiW. T. PeterZ. Yao and Y. Wang, Adaptive backstepping output feedback control for a class of nonlinear fractional order systems, Nonlinear Dynamics, 86 (2016), 1047-1056.  doi: 10.1007/s11071-016-2945-4.

[33]

D. YangX. D. Li and J. L. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. Hybrid Syst., 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[34]

W. YangW. YuY. LvL. Zhu and T. Hayat, Tracking control design for a class of uncertain nonstrict-feedback fractional-order nonlinear SISO systems, IEEE Transactions on Cybernetics, 51 (2021), 3039-3053.  doi: 10.1109/TCYB.2019.2931401.

show all references

References:
[1]

N. Aguila-CamachoM. A. Duarte-Mermoud and J. A. Gallegos, Lyapunov functions for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2951-2957.  doi: 10.1016/j.cnsns.2014.01.022.

[2]

H. B. Bao and J. D. Cao, Projective synchronization of fractional-order memristor-based neural networks, Neural Networks, 63 (2016), 1-9.  doi: 10.1016/j.neunet.2014.10.007.

[3]

A. ChatterjeeR. ChatterjeeF. Matsuno and T. Endo, Augmented stable fuzzy control for flexible robotic arm using LMI approach and neuro-fuzzy state space modeling, IEEE Transactions on Industrial Electronics, 55 (2008), 1256-1270.  doi: 10.1109/TIE.2007.896439.

[4]

B. ChenX. P. LiuS. S. Ge and C. Lin, Adaptive fuzzy control of a class of nonlinear systems by fuzzy approximation approach, IEEE Transactions on Fuzzy Systems, 20 (2012), 1012-1021.  doi: 10.1109/TFUZZ.2012.2190048.

[5]

S. Djennoune and M. Bettayeb, Optimal synergetic control for fractional-order systems, Automatica, 49 (2013), 2243-2249.  doi: 10.1016/j.automatica.2013.04.007.

[6]

M. A. Duart-MermoudN. Aguila-CamachoJ. A. Gallegos and R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 650-659.  doi: 10.1016/j.cnsns.2014.10.008.

[7]

MÖ. Efe, Fractional order systems in industrial automation-a survey, IEEE Transactions on Industrial Informatics, 7 (2011), 582-591.  doi: 10.1109/TII.2011.2166775.

[8]

P. GongW. Lan and Q. L. Han, Robust adaptive fault-tolerant consensus control for uncertain nonlinear fractional-order multi-agent systems with directed topologies, Automatica, 117 (2020), 109011.  doi: 10.1016/j.automatica.2020.109011.

[9]

X. Li, J. He, C. Wen and X. K. Liu, Backstepping based adaptive control of a class of uncertain incommensurate fractional-order nonlinear systems with external disturbance, IEEE Transactions on Industrial Electronics, (2021), 1–1. doi: 10.1109/TIE.2021.3070513.

[10]

X. Li, C. Wen and Y. Zou, Adaptive backstepping control for fractional-order nonlinear systems with external disturbance and uncertain parameters using smooth control, IEEE Transactions on Systems, Man, and Cybernetics: Systems., (2020), 1–10. doi: 10.1109/TSMC.2020.2987335.

[11]

Y. LiY. Q. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965-1969.  doi: 10.1016/j.automatica.2009.04.003.

[12]

X. D. LiJ. H. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.

[13]

X. D. LiX. Y. Yang and T. W. Huang, Persistence of delayed cooperative models: Impulsive control method, Applied Mathematics and Computation, 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.

[14]

Y. M. LiS. C. Tong and T. S. Li, Composite adaptive fuzzy output feedback control design for uncertain nonlinear strict-feedback systems with input saturation, IEEE Transactions on Cybernetics, 45 (2014), 2299-2308.  doi: 10.1109/TCYB.2014.2370645.

[15]

Y. X. Li, M. Wei and S. C. Tong, Event-triggered adaptive neural control for fractional-order nonlinear systems based on finite-time scheme, IEEE Transactions on Cybernetics, (2021), 1–9. doi: 10.1109/TCYB.2021.3056990.

[16]

H. LiuS. LiJ. CaoG. LiA. Alsaedi and F. E. Alsaadi, Adaptive fuzzy prescribed performance controller design for a class of uncertain fractional-order nonlinear systems with external disturbances, Neurocomputing, 219 (2017), 422-430.  doi: 10.1016/j.neucom.2016.09.050.

[17]

H. Liu, Y. Pan, S. Li and Y. Chen, Adaptive fuzzy backstepping control of fractional-order nonlinear systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2209-2217. doi: 10.1109/TSMC.2016.2640950.

[18]

S. LuoF. L. LewisY. D. Song and M. Hassen, Accelerated adaptive fuzzy optimal control of three coupled fractional-order chaotic electromechanical transducers, IEEE Transactions on Fuzzy Systems, 29 (2021), 1701-1714.  doi: 10.1109/TFUZZ.2020.2984998.

[19]

Z. Y. Ma and H. Ma, Adaptive fuzzy backstepping dynamic surface control of strict-feedback fractional-order uncertain nonlinear systems, IEEE Transactions on Fuzzy Systems, 28 (2019), 122-133.  doi: 10.1109/TFUZZ.2019.2900602.

[20]

R. E. PrecupM. L. Tomescu and C. A. Dragos, Stabilization of Rossler chaotic dynamical system using fuzzy logic control algorithm, Int. J. Gen. Syst., 43 (2014), 413-433.  doi: 10.1080/03081079.2014.893299.

[21]

M. Radgolchin and H. Moeenfard, Development of a multi-level adaptive fuzzy controller for beyond pull-in stabilization of electrostatically actuated microplates, J. Vib. Control, 24 (2018), 860-878.  doi: 10.1177/1077546316653040.

[22]

J. Shen and J. Lam, Non-existence of finite-time stable equilibria in fractional-order nonlinear systems, Automatica, 50 (2014), 547-551.  doi: 10.1016/j.automatica.2013.11.018.

[23]

Q. ShenP. Shi and Y. Shi, Distributed adaptive fuzzy control for nonlinear multiagent systems via sliding mode observers, IEEE Transactions on Cybernetics, 46 (2016), 3086-3097.  doi: 10.1109/TCYB.2015.2496963.

[24]

S. SongJ. H. ParkB. Zhang and X. Song, Observer-based adaptive hybrid fuzzy resilient control for fractional-order nonlinear systems with time-varying delays and actuator failures, IEEE Transactions on Fuzzy Systems, 29 (2021), 471-485.  doi: 10.1109/TFUZZ.2019.2955051.

[25]

S. SongB. ZhangJ. Xia and Z. Q. Zhang, Adaptive backstepping hybrid fuzzy sliding mode control for uncertain fractional-order nonlinear systems based on finite-time scheme, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 50 (2018), 1559-1569.  doi: 10.1109/TSMC.2018.2877042.

[26]

S. C. Tong and Y. M. Li, Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones, IEEE Transactions on Fuzzy Systems, 20 (2011), 168-180.  doi: 10.1109/TFUZZ.2011.2171189.

[27]

S. C. TongX. Min and Y. X. Li, Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions, IEEE Transactions on Cybernetics, 50 (2020), 3903-3913.  doi: 10.1109/TCYB.2020.2977175.

[28]

S. C. TongY. M. Li and S. Sui, Adaptive fuzzy tracking control design for SISO uncertain nonstrict feedback nonlinear systems, IEEE Transactions on Fuzzy Systems, 24 (2016), 1441-1454.  doi: 10.1109/TFUZZ.2016.2540058.

[29]

L. X. Wang and J. M. Mendel, Fuzzy basis functions, universal approximation, and orthogonal least-squares learning, IEEE Transactions on Neural Networks, 3 (1992), 807-814.  doi: 10.1109/72.159070.

[30]

T. D. WeiX. Xie and X. D. Li, Persistence and periodicity of survival red blood cells model with time-varying delays and impulses, Mathematical Modelling and Control, 1 (2021), 12-25.  doi: 10.3934/mmc.2021002.

[31]

Y. WeiY. ChenS. Liang and Y. Wang, A novel algorithm on adaptive backstepping control of fractional order systems, Neurocomputing, 165 (2015), 395-402.  doi: 10.1016/j.neucom.2015.03.029.

[32]

Y. WeiW. T. PeterZ. Yao and Y. Wang, Adaptive backstepping output feedback control for a class of nonlinear fractional order systems, Nonlinear Dynamics, 86 (2016), 1047-1056.  doi: 10.1007/s11071-016-2945-4.

[33]

D. YangX. D. Li and J. L. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. Hybrid Syst., 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.

[34]

W. YangW. YuY. LvL. Zhu and T. Hayat, Tracking control design for a class of uncertain nonstrict-feedback fractional-order nonlinear SISO systems, IEEE Transactions on Cybernetics, 51 (2021), 3039-3053.  doi: 10.1109/TCYB.2019.2931401.

Figure 1.  Trajectories of $ y $ and $ y_{r} $
Figure 2.  State trajectories of $ x_1 $ and $ x_2 $
Figure 3.  Adaptive parameters
Figure 4.  Control input
Figure 5.  Tracking error $ z_1 $
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