Sliding mode control of a class of uncertain nonlinear fractional-order time-varying delayed systems based on Razumikhin approach

Document Type : Research Paper


Department of Mathematics, Payame Noor University (PNU), Tehran, Iran.


Within the current paper, we design a sliding-based control law to stabilize a set of systems that are nonlinear, fractional order involve delay, perturbation, and uncertainty. A control law-based sliding mode is considered in such a way that the variables of the closed loop system reach the sliding surface in a limited time and stay on it for later times. Then, using the Razomokhin stability theorem, the stability of the systems is proved and in the end, a calculation is found to search for useful methods. 


  • [1]          A. Si. Ammour, S. Djennoune, and M. Bettayeb, A sliding mode control for linear fractional systems with input and state delays, Communications in Nonlinear Science and Numerical Simulation, 14(5) (2009), 2310-2318.
  • [2]          A. Ashyralyev, D. Agirseven, and R. P. Agarwal, Stability Estimates for Delay Parabolic Differential and Differ- ence Equations, Applied and computational mathematics, 19(2) (2020), 175-204
  • [3]          V. Badri and M. S. Tavazoei, Stability analysis of fractional order time delay systems: constructing new Lyapunov functions from those of integer order counterparts, IET Control Theory and Applications, 13(15) (2019), 2476- 2481.
  • [4]          T. Binazadeh and M. Yousefi, Asymptotic stabilization of a class of uncertain nonlinear time-delay fractional-order systems via a robust delay-independent controller, Journal of Vibration and Control, 24(19) (2018), 4541-4550.
  • [5]          B. Chen and J. Chen, Razumikhin type stability theorems for functional fractional order differential systems and applications, Applied Mathematics and Computation, 254 (2015), 63-69.
  • [6]          Y. Chen, I. Petr, and D. Xue, Fractional order control a tutorial, American Control Conference ACC09, (2009).
  • [7]          L. Chen, R. Wu, Y. Cheng, and Y. Chen, Delay dependent and order dependent stability and stabilization of fractional order linear systems with time varying delay, IEEE Transactions on Circuits and Systems II: Express Briefs, (2019).
  • [8]          R. M. Colorado, Finite-time sliding mode controller for perturbed second-order systems, ISA Transactions, 95 (2019), 82–92.
  • [9]          S. Dadras and H. R. Momeni, Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty, Communications in Nonlinear Science and Numerical Simulation, 17(1) (2012), 367-377.
  • [10]        N. Djeghali, S. Djennoune, M. Bettayeb, M. Ghanes, and J. P. Barbot, Observation and sliding mode observer for nonlinear fractional-order system with unknown input, ISA Transactions, 63 (2016), 1–10.
  • [11]        M. O. Efe, Fractional order sliding mode control with reaching law approach, Turkish Journal of Electrical Engi- neering and Computer Sciences, 18 (2010), 731–747.
  • [12]        I. Eker, Second-order sliding mode control with experimental application, ISA Trans, (2010), 394–405 .
  • [13]        F. Gao, M. Wu, J. She, and W. Cao, Disturbance rejection in nonlinear systems based on equivalent input disturbance approach, Asian Control Conference (ASCC), (2017).
  • [14]        T. Haghi and K. Ghanbari, Positive solutions for discrete fractional initial value problem, ComputationalMethods for Differential Equations, 4(4) (2016), 285-297.
  • [15]        S. He and J. Song, Finite-time Sliding Mode Control Design for a Class of Uncertain Conic NonlinearSystems, Journal of Automatic Sinica, 4(4) (2017), 809-816.
  • [16]        B. B. He, H. C. Zhou, C. H. Kou, and Y. Chen, Stabilization of uncertain fractional order system with time varying delay using BMI approach, Asian Journal of Control, (2019), 1-9.
  • [17]        D. T. Hong and N. H. Sau, Output feedback finite-time dissipative control for uncertain nonlinear fractional-order systems, Asian Journal of Control, (2021), 1–10.
  • [18]        J. B. Hu, G. P. Lu, S. B. Zhang, and L. D. Zhao, Lyapunov stability theorem about fractional system without and with delay, Communications in Nonlinear Science and Numerical Simulation, 20(3) (2015), 905-913.
  • [19]        D. C. Huong and M. V. Thuan, Mixed H∞ and Passive Control for Fractional-Order Nonlinear Systems Via LMI Approach, Acta Applicandae Mathematicae, 170 (2020), 37–52.
  • [20]        F. C. Kong and R. Sakthivel, Uncertain External Perturbation and Mixed Time Delay Impact on Fixed-Time Synchronization of Discontinuous Neutral-Type Neural Networks, Applied and computational mathematics, 20(2) (2021), 290-312.
  • [21]        Y. Ji, M. Du, and Y. Guo, Stabilization of nonlinear fractional order uncertain systems, Asian Journal of Control, 20(2) (2018), 669-677.
  • [22]        Y. Ji and J. Qiu, Stabilization of fractional order singular uncertain systems, ISA Transactions, 56 (2015), 53-64.
  • [23]        M. Li and J. Wang, Finite time stability of fractional delay differential equations, Applied Mathematics Letters, 64 (2017), 170-176.
  • [24]        M. Li and J. Wang, Exploring delayed mittaglefer type matrix functions to study finite time stability of fractional delay differential equations, Applied Mathematics and Computation, 324 (2018), 254-265.
  • [25]        J. Li, X. Guo, C. Chen, and Q. Su, Robust fault diagnosis for switched systems based on sliding mode observer, Applied Mathematics and Computation, 341 (2019), 193–203.
  • [26]        S. Liu, W. Jiang, and X. Zhou, Lyapunov stability analysis of fractional nonlinear systems, Applied Mathematics Letters, 51 (2015) , 13-19.
  • [27]        K. Mathiyalagan and G. Sangeetha, Finite-time stabilization of nonlinear time delay systems using LQR based sliding mode control, Journal of the Franklin Institute, 356(7) (2019), 3948-3964.
  • [28]        D. Matignon, Stability results on fractional differential equations to control processing, Proceedings Computational Engineering in Systems and Application multiconference, (1996), 963-968.
  • [29]        S. M. Mirhosseini-Alizamani, Solving linear optimal control problems of the time-delayed systems by Adomian decomposition method, Iranian Journal of Numerical Analysis of Optimization, 9(2) (2019),165-185.
  • [30]        S. M. Mirhosseini-Alizamani, S. Effati, and A. Heydari, An iterative method for suboptimal control of linear time-delayed systems, Systems and Control Letters, 82 (2015), 40-50.
  • [31]        S. M. Mirhosseini-Alizamani, S. Effati, and A. Heydari, Solution of linear time-varying multi-delay systems via variational iteration method, Journal of Mathematics and Computer Science, 16(2) (2016), 282-297.
  • [32]        S. M. Mirhosseini-Alizamani, S. Effati, and A. Heydari, An iterative method for suboptimal control of a class of nonlinear time-delayed systems, International Journal of Control, 92(12) (2019), 2869-2885.
  • [33]        O. Naifar, A. B. Makhlouf, and M. A. Hammami, Comments on ‘Lyapunov stability theorem about fractional system without and with delay’, Communications in Nonlinear Science and Numerical Simulation, 30 (2016), 360-361.
  • [34]        A. Neirameh, S. Shokooh, and M. Eslami, Solutions structure of integrable families of Riccati equations and their applications to the perturbed nonlinear fractional Schrodinger equation, Computational Methods for Differential Equations, 4(4) (2016), 261-275.
  • [35]        V. Phat, P. Niamsup, and M. V. Thuan, A new design method for observer-based control of nonlinear fractional- order systems with time-variable delay, European Control Association, 56 (2020), 124-131.
  • [36]        L. Podlubny, R. Magin, and L. Trymorush, Niels Henrik Abel and the birth of fractional calculus , Fractional Calculus and Applied Analysis, 20(5) (2017), 1068-1075.
  • [37]        P. Rahimkhani, Y. Ordokhani, and E. Babolian, Fractional-order Legendre wavelets and their applications for solving fractional-order differential equations with initial/boundary conditions, Computational Methods for Dif- ferential Equations, 5(2) (2017), 117-140.
  • [38]        N. H. Sau , D. T. Hong, N. T. T. Huyen, B. V. Huong, and M. V. Thuan, Delay-Dependent and Order-Dependent H Control for Fractional-Order Neural Networks with Time-Varying Delay, Differential Equations and Dynam- ical Systems, (2021).
  • [39]        Y. B. Shtessel, I. A. Shkolnikov, and M. D. J. Brown, An asymptotic second order smooth sliding mode control, Asian Journal of Control, 5 (2003), 498–504.
  • [40]        M. V. Thuan, N. H. Sau, and N. T. T. Huyen, Finite-time H control of uncertain fractional-order neural Networks, Computational and Applied Mathematics, 59 (2020), 39-59.
  • [41]        M. Thuan and D. C. Huong, New Results on Stabilization of Fractional-Order Nonlinear Systems via an LMI Approach , Asian Journal of Control, 20(4) (2018), 1541-1550.
  • [42]        Y.Wen, X. F. Zhou, Z. Zhan, and S. Liu, Lyapunov method for nonlinear fractional differential systems with delay, Nonlinear Dynamics, 82 (2015), 1015-1025.
  • [43]        S. Yaghoobi and B. Parsa Moghaddam, A numerical approach for variable-order fractional unified chaotic systems with time-delay, Computational Methods for Differential Equations, 6(4) (2018), 396-410.
  • [44]        C. Yin, Y. Chen, and S. Zhong, LMI based design of a sliding mode controller for a class of uncertain fractional order nonlinear systems, American Control Conference (ACC), (2013), 6526-6531.
  • [45]        H. Zhang, R. Ye, J. Cao, A. Ahmed, X. Li, and Y. Wan, Lyapunov functional approach to stability analysis  of Riemann-Liouville fractional neural networks with time-varying delays, Asian Journal of Control, 20 (2018), 1938-1951.
Volume 10, Issue 4
October 2022
Pages 860-875
  • Receive Date: 11 May 2021
  • Revise Date: 26 November 2021
  • Accept Date: 06 December 2021
  • First Publish Date: 08 December 2021