This article presents an adaptive nonlinear delayed feedback control scheme for stabilizing the unstable periodic orbit of unknown fractional-order chaotic systems. The proposed control framework uses the Lyapunov approach and sliding mode control technique to guarantee that the closed-loop system is asymptotically stable on a periodic trajectory sufficiently close to the unstable periodic orbit of the system. The proposed method has two significant advantages. First, it employs a direct adaptive control method, making it easy to implement this method on systems with unknown parameters. Second, the framework requires only the period of the unstable periodic orbit. The robustness of the closed-loop system against system uncertainties and external disturbances with unknown bounds is guaranteed. Simulations on fractional-order duffing and gyro systems are used to illustrate the effectiveness of the theoretical results. The simulation results demonstrate that our approach outperforms the previously developed linear feedback control method for stabilizing unstable periodic orbits in fractional-order chaotic systems, particularly in reducing steady-state error and achieving faster convergence of tracking error.
MousaMASolimanMSalehMA, et al. Biohybrid soft robots, e-skin, and bioimpedance potential to build up their applications: a review. IEEE Access2020; 8: 184524–184539.
2.
AzimiradVRamezanlouMTSotubadiSV, et al. A consecutive hybrid spiking-convolutional (CHSC) neural controller for sequential decision making in robots. Neurocomputing2022; 490: 319–336.
3.
SahooSRayM. Active control of nonlinear transient vibration of laminated composite beams using triangular SCLD treatment with fractional order derivative viscoelastic model. J Dyn Syst Meas Control2019; 141(11).
4.
EnghetaN. On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag1996; 44(4): 554–566.
5.
ShalalfehLBogdanPJonckheereE. Modeling of PMU data using ARFIMA models. In: 2018 Clemson University power systems conference (PSC), Charleston, SC, 4–7 September 2018, pp.1–6. New York: IEEE.
6.
YuSFengYYangX. Extended state observer–based fractional order sliding-mode control of piezoelectric actuators. Proc IMechE Part I: J Systems and Control Engineering2021; 235(1): 39–51.
7.
YaghootiBHosseinzadehMSinopoliB. Constrained control of semilinear fractional-order systems: application in drug delivery systems. In: 2020 IEEE conference on control technology and applications (CCTA), Montreal, QC, 24–26 August 2020, pp.833–838. New York: IEEE.
8.
YaghootiBSinopoliB. Inferring dynamics of discrete-time, fractional-order control-affine nonlinear systems. In: 2023 American control conference (ACC), San Diego, CA, 31 May 2023–2 June 2023, pp.935–940. New York: IEEE.
9.
AzarATVaidyanathanSOuannasA. Fractional order control and synchronization of chaotic systems, volume 688. Cham: Springer, 2017.
10.
ShahvaliMNaghibi-SistaniMBAskariJ. Distributed adaptive dynamic event-based consensus control for nonlinear uncertain multi-agent systems. Proc IMechE Part I: J Systems and Control Engineering2022; 236(9): 1630–1648.
11.
ZamaniHJohari MajdVKhandaniK. Formation tracking control of fractional-order multi-agent systems with fixed-time convergence. Proc IMechE Part I: J Systems and Control Engineering2022; 236: 1618–1629.
12.
LoriAARDaneshMAmiriP, et al. Transportation of an unknown cable-suspended payload by a quadrotor in windy environment under aerodynamics effects. In: 2021 7th international conference on control, instrumentation and automation (ICCIA), Tabriz, Iran, 23–24 February 2021, pp.1–6. New York: IEEE.
13.
AmiriPDayyaniMLoriAAR, et al. Fuzzy–sliding mode versus integral sliding mode controller for a quadrotor with mass uncertainty under effect of wind. In: 2020 28th Iranian conference on electrical engineering (ICEE), Tabriz, Iran, 4–6 August 2020, pp.1–7. New York: IEEE.
14.
AytenKKCiplakMHDumluA. Implementation a fractional-order adaptive model-based PID-type sliding mode speed control for wheeled mobile robot. Proc IMechE Part I: J Systems and Control Engineering2019; 233(8): 1067–1084.
15.
MachadoJTLopesAM. On the mathematical modeling of soccer dynamics. Commun Nonlinear Sci Numer Simul2017; 53: 142–153.
16.
ShahvaliMNaghibi-SistaniMBAskariJ. Adaptive fault compensation control for nonlinear uncertain fractional-order systems: static and dynamic event generator approaches. J Franklin Inst2021; 358(12): 6074–6100.
17.
HuongDCThuanMV. Mixed and passive control for fractional-order nonlinear systems via LMI approach. Acta Appl Math2020; 170(1): 37–52.
18.
EssaMAboelelaMAHassanMAM. Application of fractional order controllers on experimental and simulation model of hydraulic servo system. In: AzarAVaidyanathanSOuannasA (eds) Fractional order control and synchronization of chaotic systems. Cham: Springer, 2017; pp.277–324.
19.
WangYLiBYanF, et al. Practical adaptive fractional-order nonsingular terminal sliding mode control for a cable-driven manipulator. Int J Robust Nonlinear Control2019; 29(5): 1396–1417.
20.
YaghootiBSalariehH. Robust adaptive fractional order proportional integral derivative controller design for uncertain fractional order nonlinear systems using sliding mode control. Proc IMechE Part I: J Systems and Control Engineering2018; 232(5): 550–557.
21.
AskariIBadnavaBWoodruffT, et al. Sampling-based nonlinear MPC of neural network dynamics with application to autonomous vehicle motion planning. In: 2022 American control conference (ACC), Atlanta, GA, 8–10 June 2022, pp.2084–2090. New York: IEEE.
22.
AskariIZengSFangH. Nonlinear model predictive control based on constraint-aware particle filtering/smoothing. In: 2021 American control conference (ACC), New Orleans, LA, 25–28 May 2021, pp.3532–3537. New York: IEEE.
23.
AzimiradVSotubadiSVSharifiFJ. Optimizing the parameters of spiking neural networks for mobile robot implementation. In: 2020 10th international conference on computer and knowledge engineering (ICCKE), Mashhad, Iran, 29–30 October 2020, pp.30–34. New York: IEEE.
24.
NicotraMMGaroneE. The explicit reference governor: a general framework for the closed-form control of constrained nonlinear systems. IEEE Control Syst Mag2018; 38(4): 89–107.
25.
AghababaMPAghababaHP. The rich dynamics of fractional-order gyros applying a fractional controller. Proc IMechE Part I: J Systems and Control Engineering2013; 227(7): 588–601.
26.
YaghootiBSiahi ShadbadASafaviK, et al. Adaptive synchronization of uncertain fractional-order chaotic systems using sliding mode control techniques. Proc IMechE Part I: J Systems and Control Engineering2020; 234(1): 3–9.
27.
FaridYRuggieroF. Finite-time disturbance reconstruction and robust fractional-order controller design for hybrid port-Hamiltonian dynamics of biped robots. Robot Auton Syst2021; 144: 103836.
28.
VuMZengS. Iterative optimal control synthesis for nonlinear switching systems. In: 2021 American control conference (ACC), New Orleans, LA, 25–28 May 2021, pp.998–1003. New York: IEEE.
29.
OshimaKYanaoT. Spatial unstable periodic quasi-satellite orbits and their applications to spacecraft trajectories. Celest Mech Dyn Astron2019; 131: 1–32.
30.
HeWHuangYWangJ, et al. Homotopy method for optimal motion planning with homotopy class constraints. IEEE Control Syst Lett2022; 7: 1045–1050.
31.
RahimiMASalariehHAlastyA. Stabilizing periodic orbits of fractional order chaotic systems via linear feedback theory. Appl Math Model2012; 36(3): 863–877.
32.
SadeghianHSalariehHAlastyA, et al. On the control of chaos via fractional delayed feedback method. Comput Math Appl2011; 62(3): 1482–1491.
33.
NaceriAMansouriNCharefA. Prediction-based feedback control of fractional order system. In: 2008 IEEE international symposium on industrial electronics, Cambridge, 30 June 2008–2 July 2008, pp.908–912. New York: IEEE.
34.
ZhengY. Fuzzy prediction-based feedback control of fractional order chaotic systems. Optik2015; 126(24): 5645–5649.
35.
ShahvaliMNaghibi-SistaniMBAskariJ. Dynamic event-triggered control for a class of nonlinear fractional-order systems. IEEE Trans Circuits Syst II: Express Br2022; 69(4): 2131–2135.
36.
RabahKLadaciSLashabM. State feedback with fractional PIλDµ control structure for genesio-tesi chaos stabilization. In: 2015 16th international conference on sciences and techniques of automatic control and computer engineering (STA), Monastir, Tunisia, 21–23 December 2015, pp.328–333. New York: IEEE.
37.
DancaMFFečkanMChenG. Impulsive stabilization of chaos in fractional-order systems. Nonlinear Dyn2017; 89(3): 1889–1903.
38.
LayeghiHArjmandMTSalariehH, et al. Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control. Chaos Solit Fractals2008; 37(4): 1125–1135.
39.
PodlubnyI. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Amsterdam, Netherland: Elsevier, 1998.
40.
MatignonD. Stability results for fractional differential equations with applications to control processing. Computational Eng in Sys Appl1996; 2: 963–968.
41.
LiYChenYPodlubnyI. Stability of fractional-order nonlinear dynamic systems: lyapunov direct method and generalized Mittag–Leffler stability. Comput Math Appl2010; 59(5): 1810–1821.
42.
Duarte-MermoudMAAguila-CamachoNGallegosJA, et al. Using general quadratic lyapunov functions to prove lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simul2015; 22(1–3): 650–659.
43.
ZhangRLiuY. A new Barbalat’s lemma and lyapunov stability theorem for fractional order systems. In: 2017 29th Chinese control and decision conference (CCDC), Chongqing, China, 28–30 May 2017, pp.3676–3681. New York: IEEE.
44.
LiCDengW. Remarks on fractional derivatives. Appl Math Comput2007; 187(2): 777–784.
DiethelmKFordNJFreedAD, et al. Algorithms for the fractional calculus: a selection of numerical methods. Comput Methods Appl Mech Eng2005; 194(6–8): 743–773.
