Abstract
This study presents a novel fractional-order nonlinear sliding mode controller (FONSMC) based on an extended nonlinear disturbance observer (ENDOB) for a class of fractional order systems with matched and mismatched disturbances. Firstly, an ENDOB is introduced to estimate both the matched and mismatched disturbances. Then, the fractional-order nonlinear sliding surface is designed to satisfy the sliding condition in finite time. Accordingly, the corresponding FONSMC is proposed using the Lyapunov stability theorem. The proposed method shows an impressive disturbances rejection and also guarantees finite-time stability of closed-loop systems. Finally, the effectiveness of the proposed FONSMC-ENDOB structure is illustrated via numerical simulation. The simulation results exhibit the superiority of the proposed controlling method.
Similar content being viewed by others
References
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, vol 204. Elsevier, Amsterdam
Yang X-J, Srivastava HM, Machado JA (2015) A new fractional derivative without singular kernel: application to the modelling of the steady heat flow. ArXiv preprint arXiv:1601.01623
Yang X-J, Abdel-Aty M, Cattani C (2019) A new general fractional-order derivataive with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer. Therm Sci 23(3 Part A):1677–1681
Yang A-M et al (2016) On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel. Therm Sci 20(Suppl 3):717–721
Yang X-J (2019) New non-conventional methods for quantitative concepts of anomalous rheology. Therm Sci 00:427
Yang X-J (2019) New general calculi with respect to another functions applied to describe the Newton-like dashpot models in anomalous viscoelasticity. Therm Sci 00:260
Yang X-J, Tenreiro Machado JA (2019) A new fractal nonlinear Burgers’ equation arising in the acoustic signals propagation. Math Methods Appl Sci 42(18):7539–7544
Monje CA et al (2010) Fractional-order systems and controls: fundamentals and applications. Springer, Berlin
Gómez-Aguilar JF et al (2016) Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl Math Model 40(21-22):9079–9094
Gómez-Aguilar JF et al (2016) Analytical solutions of the electrical RLC circuit via Liouville–Caputo operators with local and non-local kernels. Entropy 18(8):402
Aguilar JFG (2016) Behavior characteristics of a cap-resistor, memcapacitor, and a memristor from the response obtained of RC and RL electrical circuits described by fractional differential equations. Turk J Electr Eng Comput Sci 24(3):1421–1433
Gómez-Aguilar JF, Atangana A, Morales-Delgado VF (2017) Electrical circuits RC, LC, and RL described by Atangana–Baleanu fractional derivatives. Int J Circuit Theory Appl 45(11):1514–1533
Morales-Delgado VF et al (2018) Fractional operator without singular kernel: applications to linear electrical circuits. Int J Circuit Theory Appl 46(12):2394–2419
Xiong R, Tian J, Shen W, Sun F (2019) A novel fractional order model for state of charge estimation in lithium ion batteries. IEEE Trans Veh Technol 68(5):4130–4139
Liu RJ, Nie ZY, Wu M, She J (2018) Robust disturbance rejection for uncertain fractional-order systems. Appl Math Comput 322:79–88
Gómez-Aguilar JF, Dumitru B (2014) Fractional transmission line with losses. Zeitschrift für Naturforschung A 69(10-11):539–546
Aguilar JG, Baleanu D (2014) Solutions of the telegraph equations using a fractional calculus approach. Proc Romanian Acad A 15:27–34
Tavazoei MS (2020) Fractional order chaotic systems: history, achievements, applications, and future challenges. Eur Phys J Special Topics 229:887–904
Čermák J, Nechvátal L (2019) Stability and chaos in the fractional Chen system. Chaos Solitons Fractals 125:24–33
Wang X, Kingni ST, Volos C, Pham VT, Hoang DV, Jafari S (2019) A fractional system with five terms: analysis, circuit, chaos control and synchronization. Int J Electron 106(1):109–120
Shahri ES, Alaviyan AA, Tenreiro Machado JA (2017) Stabilization of fractional-order systems subject to saturation element using fractional dynamic output feedback sliding mode control. J Comput Nonlinear Dyn 12(3):031014-1
Shahri ES, Alaviyan AA, 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
Yang S, Hu C, Yu J, Jiang H (2020) Exponential stability of fractional-order impulsive control systems with applications in synchronization. IEEE Trans Cybern 50(7):3157–3168
Li Yu, Cao Y, Fan Y (2020) Generalized Mittag–Leffler quadrature methods for fractional differential equations. Comput Appl Math 39(3):1–16
Shahri ESA, Alfi A, Tenreiro Machado JA (2020) Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation. Appl Math Model 81:663–672
Liu H, Pan Y, Li S, Chen Y (2017) Adaptive fuzzy backstepping control of fractional-order nonlinear systems. IEEE Trans Syst Man Cybern Syst 47(8):2209–2217
Gong P, Lan W (2018) Adaptive robust tracking control for uncertain nonlinear fractional-order multi-agent systems with directed topologies. Automatica 92:92–99
Sharafian A, Sharifi A, Zhang W (2020) Fractional sliding mode based on RBF neural network observer: application to HIV infection mathematical model. Comput Math Appl 79(11):3179–3188
Razzaghian A, Moghaddam RK, Pariz N (2020) Adaptive neural network conformable fractional-order nonsingular terminal sliding mode control for a class of second-order nonlinear systems. IETE J Res. https://doi.org/10.1080/03772063.2020.1791743
Yang J, Chen W, Li S (2011) Nonlinear disturbance observer-based robust control for systems with mismatched disturbances/uncertainties. IET Control Theory Appl 5(18):2053–2062
Xiang W, Chen F (2011) An adaptive sliding mode control scheme for a class of chaotic systems with mismatched perturbations and input nonlinearities. Commun Nonlinear Sci Numer Simul 16:1–9
Yang J, Zolotas A, Chen W-H, Michail K, Li S (2011) Robust control of nonlinear MAGLEV suspension system with mismatched uncertainties via DOBC approach. ISA Trans 50(3):389–396
Li F, Wu L, Shi P, Lim C-C (2015) State estimation and sliding mode control for semi-Markovian jump systems with mismatched uncertainties. Automatica 51:385–393
Wang J, Li S, Yang J, Wu B, Li Q (2015) Extended state observer-based sliding mode control for PWM-based DC–DC buck power converter systems with mismatched disturbances. IET Control Theory Appl 9(4):579–586
Edwards C, Spurgeon S (1998) Sliding mode control: theory and applications. CRC Press, Boca Raton
Wu YQ, Yu XH, Man ZH (1998) Terminal sliding mode control design for uncertain dynamic systems. Syst Control Lett 34(5):281–287
Dadras S, Momeni HR (2014) Fractional-order dynamic output feedback sliding mode control design for robust stabilization of uncertain fractional-order nonlinear systems. Asian J Control 16(2):489–497
Yin C, Chen Y, Zhong S-M (2014) Fractional-order sliding mode based extremum seeking control of a class of nonlinear systems. Automatica 50(12):3173–3181
Wang Y, Gu L, Xu Y, Cao X (2016) Practical tracking control of robot manipulators with continuous fractional-order nonsingular terminal sliding mode. IEEE Trans Ind Electron 63(10):6194–6204
Song S, Zhang B, Xia J, Zhang Z (2020) Adaptive backstepping hybrid fuzzy sliding mode control for uncertain fractional-order nonlinear systems based on finite-time scheme. IEEE Trans Syst Man Cybern Syst 50(4):1559–1569
Yang J, Li S, Yu X (2013) Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans Ind Electron 60(1):160–169
Ginoya D, Shendge PD, Phadke SB (2014) Sliding mode control for mismatched uncertain systems using an extended disturbance observer. IEEE Trans Ind Electron 61(4):1983–1992
Yang J, Li S, Su J, Yu X (2013) Continuous nonsingular terminal sliding mode control for systems with mismatched disturbances. Automatica 49(7):2287–2291
Muthukumar P, Balasubramaniam P, Ratnavelu K (2018) Sliding mode control for generalized robust synchronization of mismatched fractional order dynamical systems and its application to secure transmission of voice messages. ISA Trans 82:51–61
Wang J, Shao C, Chen Y-Q (2018) Fractional order sliding mode control via disturbance observer for a class of fractional order systems with mismatched disturbance. Mechatronics 53:8–19
Shi SL, Li JX, Fang YM (2019) Fractional-disturbance-observer-based sliding mode control for fractional order system with matched and mismatched disturbances. Int J Control Autom Syst 17(5):1184–1190
Wang J et al (2019) Fractional-order DOB-sliding mode control for a class of noncommensurate fractional-order systems with mismatched disturbances. Math Methods Appl Sci. https://doi.org/10.1002/mma.5850
Razzaghian A, Moghaddam RK, Pariz N (2020) Fractional-order nonsingular terminal sliding mode control via a disturbance observer for a class of nonlinear systems with mismatched disturbances. J Vib Control. https://doi.org/10.1177/1077546320925263
Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Elsevier, Amsterdam
Li Y, Chen Y, Podlubny I (2009) Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8):1965–1969
Li Y, Chen Y, Podlubny I (2010) Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag– Leffler stability. Comput Math Appl 59(5):1810–1821
Aghababa MP (2013) A novel terminal sliding mode controller for a class of non-autonomous fractional-order systems. Nonlinear Dyn 73(1):679–688
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Razzaghian, A., Kardehi Moghaddam, R. & Pariz, N. Disturbance observer-based fractional-order nonlinear sliding mode control for a class of fractional-order systems with matched and mismatched disturbances. Int. J. Dynam. Control 9, 671–678 (2021). https://doi.org/10.1007/s40435-020-00691-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-020-00691-2