Abstract
In this paper, an adaptive nonlinear delayed feedback control is proposed for stabilizing an unstable periodic orbit (UPO) of a class of uncertain chaotic systems using the sliding mode approach. The chaotic system is subjected to external disturbances with unknown bounds. Using the Lyapunov stability theorem and the concept of sliding mode control, an adaptive delayed feedback controller has been designed. The controller and adaptation law guarantee the asymptotic stability of the closed loop system on a periodic trajectory which can be sufficiently close to the UPO of the chaotic system. The capability of the proposed control method in stabilizing the UPO of chaotic systems is established analytically and the effectiveness of the proposed scheme is examined numerically by applying it to the chaotic Duffing and Gyro systems.
Similar content being viewed by others
References
Banerjee S, Rondoni L (2015) Applications of chaos and nonlinear dynamics in science and engineering. Springer, Berlin
Bruno R, Iu HC, Feki M (2004) Adaptive time-delayed feedback for chaos control in a PWM single phase inverter. J Circuits Syst Comput 13(03):519–534
Chekan JA, Nojoumian MA, Merat K, Salarieh H (2017) Chaos control in lateral oscillations of spinning disk via linear optimal control of discrete systems. J Vibr Control 23(1):103–110
Chen B, Liu XP, Ge SS, Lin C (2012) Adaptive fuzzy control of a class of nonlinear systems by fuzzy approximation approach. IEEE Trans Fuzzy Syst 20(6):1012–1021
El-Dessoky MM, Yassen MT, Aly ES (2014) Bifurcation analysis and chaos control in Shimizu–Morioka chaotic system with delayed feedback. Appl Math Comput 243(1):283–297
Hajiloo R, Salarieh H, Alasti A (2017) Chaos control in delayed phase space constructed by the Takens embedding theory. Commun Nonlinear Sci Numer Simul 54:453–465
Kawai Y, Tsubone T (2012) Chaos control based on stability transformation method for unstable periodic orbits. Nonlinear Theory Appl IEICE 3(2):246–256
Khanesar MA, Oniz Y, Kaynak O, Gao H (2016) Direct model reference adaptive fuzzy control of networked SISO nonlinear systems. IEEE/ASME Trans Mechatron 21(1):205–213
Lehnert J, Hovel P, Flukert V, Guzenko YP, Fradkov A, Scholl E (2011) Adaptive tuning of feedback gain in time-delayed feedback control. Chaos Interdiscip J Nonlinear Sci 21(4):043111
Li SY, Yang CH, Lin CT, Ko LW, Chiu TT (2012) Adaptive synchronization of chaotic systems with unknown parameters via new backstepping strategy. Nonlinear Dyn 70(3):2129–2143
Li H, Shi P, Yao D, Wu L (2016) Observer-based adaptive sliding mode control for nonlinear Markovian jump systems. Automatica 64:133–142
Lin CJ, Yang SK, Yau HT (2012) Chaos suppression control of a coronary artery system with uncertainties by using variable structure control. Comput Math Appl 64(5):988–995
Liu D, Yan G (2013) Stabilization of discrete-time chaotic systems via improved periodic delayed feedback control based on polynomial matrix right coprime factorization. Nonlinear Dyn 74(4):1243–1252
Mehta N, Henderson R (1991) Controlling chaos to generate aperiodic orbits. Phys Rev A 44(8):4861–4865
Miladi Y, Feki M, Derbel N (2015) Stabilizing the unstable periodic orbits of a hybrid chaotic system using optimal control. Commun Nonlinear Sci Numer Simul 20(3):1043–1056
Mobayen S (2015a) An LMI-based robust controller design using global nonlinear sliding surfaces and application to chaotic systems. Nonlinear Dyn 79(2):1075–1084
Mobayen S (2015b) Design of LMI-based global sliding mode controller for uncertain nonlinear systems with application to Genesio’s chaotic system. Complexity 21(1):94–98
Moon FC (2004) Chaotic and fractal dynamics: an introduction for applied scientists and engineers. Wiley-VCH Verlag GmbH, New York
Ott E, Grebogi C, Yorke J (1990) Controlling chaos. Phys Rev Lett 64(11):1196–1199
Pyragas K (1846) Delayed feedback control of chaos. Philos Trans R Soc Lond A Math Phys Eng Sci 206(364):2309–2334
Pyragas V, Pyragas K (2011) Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay. Phys Lett A 375(44):3866–3871
Rega G, Lenci S, Thompson J (2010) Controlling chaos: the OGY method, its use in mechanics, and an alternative unified framework for control of non-regular dynamics, in nonlinear dynamics and chaos: advances and perspectives. Springer, Berlin, pp 211–269
Roopaei M, Jahromi MZ (2008) Synchronization of a class of chaotic systems with fully unknown parameters using adaptive sliding mode approach. Chaos 18(4):043112
Sadeghian H, Merat K, Salarieh H, Alasty A (2011) On the fuzzy minimum entropy control to stabilize the unstable fixed points of chaotic maps. Appl Math Model 35(3):1016–1023
Salarieh H, Alasty A (2008) Delayed feedback control of chaotic spinning disk via minimum entropy approach. Nonlinear Anal Theory Methods Appl 69(10):3273–3280
Salarieh H, Alasty A (2009) Chaos control in uncertain dynamical systems using nonlinear delayed feedback. Chaos Soliton Fract 41(1):67–71
Slotine JJE, Li W (1991) Applied nonlinear control. Prentice-Hall, Englewood Cliffs
Tong X, Mrad N (2011) Chaotic motion of a symmetric gyro subjected to a harmonic base excitation. J Appl Mech 68(4):681–684
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hajiloo, R., Shahrokhi, M. & Salarieh, H. Adaptive Delayed Feedback Chaos Control of a Class of Uncertain Chaotic Systems Subject to External Disturbance. Iran J Sci Technol Trans Mech Eng 43 (Suppl 1), 517–525 (2019). https://doi.org/10.1007/s40997-018-0174-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40997-018-0174-9