A non-integer sliding mode controller to stabilize fractional-order nonlinear systems. (English) Zbl 1486.93010
Summary: In this study, we examine the stabilization of fractional-order chaotic nonlinear dynamical systems with model uncertainties and external disturbances. We used the sliding mode controller by a new approach for controlling and stabilization of these systems. In this research, we replaced a continuous function with the sign function in the controller design and the sliding surface to suppress chattering and undesirable vibration effects. The advantages of the proposed control method are rapid convergence to the equilibrium point, the absence of chattering and unwanted oscillations, high resistance to uncertainties, and the possibility of applying this method to most fractional order chaotic systems. We applied the direct method of Lyapunov stability theory and the frequency distributed model to prove the stability of the slip surface and closed loop system. Finally, we simulated this method on two commonly used and practical chaotic systems and presented the results.
MSC:
| 93B12 | Variable structure systems |
| 34A08 | Fractional ordinary differential equations |
| 93C40 | Adaptive control/observation systems |
| 34H10 | Chaos control for problems involving ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional-order system; uncertainty; chattering; Lyapunov theory; sliding mode control; frequency distributed modelReferences:
| [1] | Roohi, M.; Aghababa, M. P.; Haghighi, A. R., Switching adaptive controllers to control fractional-order complex systems with unknown structure and input nonlinearities, Complexity, 21, 2, 211-223 (2015) |
| [2] | Pham, V.-T.; Kingni, S. T.; Volos, C.; Jafari, S.; Kapitaniak, T., A simple three-dimensional fractional-order chaotic system without equilibrium: dynamics, circuitry implementation, chaos control and synchronization, AEÜ, Int. J. Electron. Commun., 78, 220-227 (2017) |
| [3] | Asl, M. S.; Javidi, M., Numerical evaluation of order six for fractional differential equations: stability and convergency, Bull. Belg. Math. Soc. Simon Stevin, 26, 2, 203-221 (2019) · Zbl 07094825 |
| [4] | Biagini, F.; Øksendal, B.; Sulem, A.; Wallner, N., An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 460, 2041, 347-372 (2004) · Zbl 1043.60044 |
| [5] | Kam, S. I.; Nguyen, Q. P.; Li, Q.; Rossen, W. R., Dynamic simulations with an improved model for foam generation, SPE J., 12, 1, 35-48 (2007) |
| [6] | Shi, L.; Yu, Z.; Mao, Z.; Xiao, A., A directed continuous time random walk model with jump length depending on waiting time, Sci. World J., 2014 (2014) |
| [7] | Gabano, J.-D.; Poinot, T.; Kanoun, H., Identification of a thermal system using continuous linear parameter-varying fractional modelling, IET Control Theory Appl., 5, 7, 889-899 (2011) |
| [8] | Ivanov, D. V.; Sandler, I. L.; Kozlov, E. V., Identification of fractional linear dynamical systems with autocorrelated errors in variables by generalized instrumental variables, IFAC-PapersOnLine, 51, 32, 580-584 (2018) |
| [9] | Hu, X.; Zou, H.; Tao, J.; Gao, F., Multimodel fractional predictive functional control design with application on an industrial heating furnace, Ind. Eng. Chem. Res., 57, 42, 14182-14190 (2018) |
| [10] | Wang, Y.; Luo, G.; Gu, L.; Li, X., Fractional-order nonsingular terminal sliding mode control of hydraulic manipulators using time delay estimation, J. Vib. Control, 22, 19, 3998-4011 (2016) |
| [11] | Weitzner, H.; Zaslavsky, G. M., Some applications of fractional equations, Commun. Nonlinear Sci. Numer. Simul., 8, 3-4, 273-281 (2003) · Zbl 1041.35073 |
| [12] | Laskin, N., Fractional Schrödinger equation, Phys. Rev. E, 66, 5 (2002) |
| [13] | Zubair, M.; Mughal, M. J.; Naqvi, Q. A., Electromagnetic wave propagation in fractional space, Electromagnetic Fields and Waves in Fractional Dimensional Space, 27-60 (2012), Berlin: Springer, Berlin · Zbl 1244.78001 |
| [14] | Tarasov, V. E.; Trujillo, J. J., Fractional power-law spatial dispersion in electrodynamics, Ann. Phys., 334, 1-23 (2013) |
| [15] | Luo, Y.; Chen, Y. Q.; Pi, Y. G., Experimental study of fractional order proportional derivative controller synthesis for fractional order systems, Mechatronics, 21, 1, 204-214 (2011) |
| [16] | Gheisarnejad, M.; Khooban, M. H., Design an optimal fuzzy fractional proportional integral derivative controller with derivative filter for load frequency control in power systems, Trans. Inst. Meas. Control, 41, 9, 2563-2581 (2019) |
| [17] | Yan, Y.; Kou, C., Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay, Math. Comput. Simul., 82, 9, 1572-1585 (2012) · Zbl 1253.92037 |
| [18] | Aghababa, M. P.; Borjkhani, M., Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme, Complexity, 20, 2, 37-46 (2014) |
| [19] | Cohen, I.; Golding, I.; Ron, I. G.; Ben-Jacob, E., Biofluiddynamics of lubricating bacteria, Math. Methods Appl. Sci., 24, 17-18, 1429-1468 (2001) · Zbl 1097.76618 |
| [20] | Ahmad, W. M.; El-Khazali, R., Fractional-order dynamical models of love, Chaos Solitons Fractals, 33, 4, 1367-1375 (2007) · Zbl 1133.91539 |
| [21] | Song, L.; Xu, S.; Yang, J., Dynamical models of happiness with fractional order, Commun. Nonlinear Sci. Numer. Simul., 15, 3, 616-628 (2010) · Zbl 1221.93234 |
| [22] | Teng, L.; Iu, H. H.; Wang, X.; Wang, X., Chaotic behavior in fractional-order memristor-based simplest chaotic circuit using fourth degree polynomial, Nonlinear Dyn., 77, 1-2, 231-241 (2014) |
| [23] | Hasani-Marzooni, M.; Hosseini, S. H., Trading strategies for wind capacity investment in a dynamic model of combined tradable green certificate and electricity markets, IET Gener. Transm. Distrib., 6, 4, 320-330 (2012) |
| [24] | Khooban, M. H.; Gheisarnejad, M.; Farsizadeh, H.; Masoudian, A.; Boudjadar, J., A new intelligent hybrid control approach for DC-DC converters in zero-emission ferry ships, IEEE Trans. Power Electron., 35, 6, 5832-5841 (2019) |
| [25] | Khooban, M.-H.; Gheisarnejad, M.; Vafamand, N.; Boudjadar, J., Electric vehicle power propulsion system control based on time-varying fractional calculus: implementation and experimental results, IEEE Trans. Intell. Veh., 4, 2, 255-264 (2019) |
| [26] | Azami, A.; Naghavi, S. V.; Tehrani, R. D.; Khooban, M. H.; Shabaninia, F., State estimation strategy for fractional order systems with noises and multiple time delayed measurements, IET Sci. Meas. Technol., 11, 1, 9-17 (2017) |
| [27] | Khooban, M.-H.; Gheisarnejad, M.; Vafamand, N.; Jafari, M.; Mobayen, S.; Dragicevic, T.; Boudjadar, J., Robust frequency regulation in mobile microgrids: HIL implementation, IEEE Syst. J., 13, 4, 4281-4291 (2019) |
| [28] | Dehghani, M.; Khooban, M. H.; Niknam, T.; Rafiei, S. M.R., Time-varying sliding mode control strategy for multibus low-voltage microgrids with parallel connected renewable power sources in islanding mode, J. Energy Eng., 142, 4 (2016) |
| [29] | Veysi, M.; Soltanpour, M. R.; Khooban, M. H., A novel self-adaptive modified bat fuzzy sliding mode control of robot manipulator in presence of uncertainties in task space, Robotica, 33, 10, 2045-2064 (2015) |
| [30] | Khooban, M. H.; Niknam, T.; Blaabjerg, F.; Dehghani, M., Free chattering hybrid sliding mode control for a class of non-linear systems: electric vehicles as a case study, IET Sci. Meas. Technol., 10, 7, 776-785 (2016) |
| [31] | Khooban, M., Secondary load frequency control of time-delay stand-alone microgrids with electric vehicles, IEEE Trans. Ind. Electron., 65, 9, 7416-7422 (2018) · doi:10.1109/TIE.2017.2784385 |
| [32] | Dabiri, A.; Butcher, E. A.; Poursina, M.; Nazari, M., Optimal periodic-gain fractional delayed state feedback control for linear fractional periodic time-delayed systems, IEEE Trans. Autom. Control, 63, 4, 989-1002 (2017) · Zbl 1390.93324 |
| [33] | Haghighi, A. R.; Pourmahmood Aghababa, M.; Roohi, M., Robust stabilization of a class of three-dimensional uncertain fractional-order non-autonomous systems, Int. J. Ind. Math., 6, 2, 133-139 (2014) |
| [34] | Lin, D.; Wang, X.; Yao, Y., Fuzzy neural adaptive tracking control of unknown chaotic systems with input saturation, Nonlinear Dyn., 67, 4, 2889-2897 (2012) · Zbl 1243.93063 |
| [35] | Shi, K.; Wang, J.; Zhong, S.; Tang, Y.; Cheng, J., Non-fragile memory filtering of T-S fuzzy delayed neural networks based on switched fuzzy sampled-data control, Fuzzy Sets Syst., 394, 40-64 (2020) · Zbl 1452.93021 · doi:10.1016/j.fss.2019.09.001 |
| [36] | Shi, K.; Wang, J.; Tang, Y.; Zhong, S., Reliable asynchronous sampled-data filtering of T-S fuzzy uncertain delayed neural networks with stochastic switched topologies, Fuzzy Sets Syst., 381, 1-25 (2020) · Zbl 1464.93081 · doi:10.1016/j.fss.2018.11.017 |
| [37] | Shi, K.; Wang, J.; Zhong, S.; Tang, Y.; Cheng, J., Hybrid-driven finite-time \(H_{\infty}\) sampling synchronization control for coupling memory complex networks with stochastic cyber attacks, Neurocomputing, 387, 241-254 (2020) · doi:10.1016/j.neucom.2020.01.022 |
| [38] | Esfahani, Z.; Roohi, M.; Gheisarnejad, M.; Dragičević, T.; Khooban, M.-H., Optimal non-integer sliding mode control for frequency regulation in stand-alone modern power grids, Appl. Sci., 9, 16 (2019) |
| [39] | Cai, N.; Jing, Y.; Zhang, S., Modified projective synchronization of chaotic systems with disturbances via active sliding mode control, Commun. Nonlinear Sci. Numer. Simul., 15, 6, 1613-1620 (2010) · Zbl 1221.37211 |
| [40] | Zare, K.; Mardani, M. M.; Vafamand, N.; Khooban, M. H.; Sadr, S. S.; Dragičević, T., Fuzzy-logic-based adaptive proportional-integral sliding mode control for active suspension vehicle systems: Kalman filtering approach, Inf. Technol. Control, 48, 4, 648-659 (2019) |
| [41] | Khooban, M. H.; Niknam, T.; Sha-Sadeghi, M., A time-varying general type-II fuzzy sliding mode controller for a class of nonlinear power systems, J. Intell. Fuzzy Syst., 30, 5, 2927-2937 (2016) · Zbl 1361.93033 |
| [42] | Xi, H.; Yu, S.; Zhang, R.; Xu, L., Adaptive impulsive synchronization for a class of fractional-order chaotic and hyperchaotic systems, Optik, Int. J. Light Electron Opt., 125, 9, 2036-2040 (2014) |
| [43] | Roohi, M.; Zhang, C.; Chen, Y., Adaptive model-free synchronization of different fractional-order neural networks with an application in cryptography, Nonlinear Dyn., 100, 4, 3979-4001 (2020) · Zbl 1516.93137 · doi:10.1007/s11071-020-05719-y |
| [44] | Aghababa, M. P.; Haghighi, A. R.; Roohi, M., Stabilisation of unknown fractional-order chaotic systems: an adaptive switching control strategy with application to power systems, IET Gener. Transm. Distrib., 9, 14, 1883-1893 (2015) |
| [45] | Yin, C.; Dadras, S.; Zhong, S.-M.; Chen, Y., Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach, Appl. Math. Model., 37, 4, 2469-2483 (2013) · Zbl 1349.93237 · doi:10.1016/j.apm.2012.06.002 |
| [46] | Yin, C.; Dadras, S.; Zhong, S.-M., Design an adaptive sliding mode controller for drive-response synchronization of two different uncertain fractional-order chaotic systems with fully unknown parameters, J. Franklin Inst., 349, 10, 3078-3101 (2012) · Zbl 1255.93038 · doi:10.1016/j.jfranklin.2012.09.009 |
| [47] | Mofid, O.; Mobayen, S.; Khooban, M. H., Sliding mode disturbance observer control based on adaptive synchronization in a class of fractional-order chaotic systems, Int. J. Adapt. Control Signal Process., 33, 3, 462-474 (2019) · Zbl 1417.93102 |
| [48] | Lin, T.-C.; Lee, T.-Y., Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control, IEEE Trans. Fuzzy Syst., 19, 4, 623-635 (2011) |
| [49] | Wang, Y.; Gu, L.; Xu, Y.; Cao, X., Practical tracking control of robot manipulators with continuous fractional-order nonsingular terminal sliding mode, IEEE Trans. Ind. Electron., 63, 10, 6194-6204 (2016) |
| [50] | Yin, C.; Huang, X.; Chen, Y.; Dadras, S.; Zhong, S.-M.; Cheng, Y., Fractional-order exponential switching technique to enhance sliding mode control, Appl. Math. Model., 44, 705-726 (2017) · Zbl 1443.93023 |
| [51] | Yu, X.; Kaynak, O., Sliding-mode control with soft computing: a survey, IEEE Trans. Ind. Electron., 56, 9, 3275-3285 (2009) |
| [52] | Boiko, I.; Fridman, L.; Pisano, A.; Usai, E., Analysis of chattering in systems with second-order sliding modes, IEEE Trans. Autom. Control, 52, 11, 2085-2102 (2007) · Zbl 1366.93378 |
| [53] | Ziaratban, R.; Haghighi, A. R.; Reihani, P., Design of a no-chatter fractional sliding mode control approach for stabilization of non-integer chaotic systems, Int. J. Ind. Math., 12, 3, 215-223 (2020) |
| [54] | Soltanpour, M. R.; Khooban, M. H., A particle swarm optimization approach for fuzzy sliding mode control for tracking the robot manipulator, Nonlinear Dyn., 74, 1-2, 467-478 (2013) · Zbl 1281.93067 |
| [55] | Bartolini, G.; Pisano, A.; Usai, E., Second-order sliding-mode control of container cranes, Automatica, 38, 10, 1783-1790 (2002) · Zbl 1011.93523 |
| [56] | Fridman, L.; Levant, A., Higher order sliding modes, Sliding Mode Control in Engineering, 53-102 (2002) |
| [57] | Asl, M. S.; Javidi, M., An improved PC scheme for nonlinear fractional differential equations: error and stability analysis, J. Comput. Appl. Math., 324, 101-117 (2017) · Zbl 1369.65087 · doi:10.1016/j.cam.2017.04.026 |
| [58] | Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (1998), Amsterdam: Elsevier, Amsterdam · Zbl 0922.45001 |
| [59] | Tian, X.; Fei, S., Robust control of a class of uncertain fractional-order chaotic systems with input nonlinearity via an adaptive sliding mode technique, Entropy, 16, 2, 729-746 (2014) |
| [60] | Roohi, M.; Khooban, M.-H.; Esfahani, Z.; Aghababa, M. P.; Dragicevic, T., A switching sliding mode control technique for chaos suppression of fractional-order complex systems, Trans. Inst. Meas. Control, 41, 10, 2932-2946 (2019) · doi:10.1177/0142331219834606 |
| [61] | Aghababa, M. P., A novel terminal sliding mode controller for a class of non-autonomous fractional-order systems, Nonlinear Dyn., 73, 1-2, 679-688 (2013) · Zbl 1281.93045 |
| [62] | Wang, B.; Ding, J.; Wu, F.; Zhu, D., Robust finite-time control of fractional-order nonlinear systems via frequency distributed model, Nonlinear Dyn., 85, 4, 2133-2142 (2016) · Zbl 1349.93345 |
| [63] | Zhang, S.; Yu, Y.; Wang, H., Mittag-Leffler stability of fractional-order Hopfield neural networks, Nonlinear Anal. Hybrid Syst., 16, 104-121 (2015) · Zbl 1325.34016 · doi:10.1016/j.nahs.2014.10.001 |
| [64] | Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29, 1-4, 3-22 (2002) · Zbl 1009.65049 |
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