Open Access Open Access  Restricted Access Subscription or Fee Access

Chaos Control of Third Order Phase Locked Loop Using Sliding Mode Nonlinear Controller

(*) Corresponding author

Authors' affiliations



The nonlinear dynamics of a third-order phase locked loop (PLL) reveals a chaotic behavior for range values of loop parameter. Using bifurcation theory, it has been verified through mathematical analysis and simulation that such behavior pulls the PLL to an undesirable and unstable out-of-lock state. This paper aims to control such chaotic behavior and drive the PLL to the in-lock state, by applying a nonlinear controller based on sliding mode control theory to the PLL under consideration. The design process involves two phases. In the first phase, a sliding surface is designed such that the sliding motion meets the design specifications. In the second phase, a high-speed switching control law is selected such that the system state trajectory reaches the sliding surface. The validity of the proposed controller is tested through simulations. Results show the effectiveness of the designed nonlinear controller in stabilizing the system and driving the PLL to the phase locked state.
Copyright © 2019 Praise Worthy Prize - All rights reserved.


Bifurcation; Chaos; Nonlinear Controller; Phase Locked Loop; Sliding Mode

Full Text:



K. Watada,, T. Endo, and H. Seishi, Shilnikov Orbits in an Autonomous Third-Order Chaotic Phase-Locked Loop, IEEE Transactions on Circuits and Systems–I: Fundamental Theory and Applications, Vol. 45 :979-983, 1998.

B. Harb, and A. Harb, Chaos and Bifurcation in Third-Order Phase Locked Loop, Chaos, Solitons & Fractals Vol. 19: 667-672., 2004.

Kumar, A., Ramana, N., Design of a New Discrete Sliding Mode Optimal Controller for Load Frequency Control in Multi Area Deregulated Power System, (2013) International Review on Modelling and Simulations (IREMOS), 6 (4), pp. 1183-1189.

P. Singh, K. Singh, and B. Roy, Chaos control in biological system using recursive backstepping sliding mode control. The European Physical Journal Special Topics, Volume 227, Issue 7–9: 731–746, October 2018.

Moghaddam, M., Mojallali, H., Neural Network Based Modeling and Predictive Position Control of Traveling Wave Ultrasonic Motor Using Chaotic Genetic Algorithm, (2013) International Review on Modelling and Simulations (IREMOS), 6 (2), pp. 370-379.

B. I. Shakhtarin, Y.A. Sidorkina,and A.V. Khodunin,. Analysis of chaotic synchronization of cascade coupled systems for phase-locked loop using wavelet analysis. J. Commun. Technol. Electron, Volume 62, Issue 11: 1262–1270, November 2017.

S.Tidjani, R. Lozi, and L. Chua. Hidden bifurcation in the Multispiral Chua Attractor. International Journal of Bifurcation and Chaos, Vol. 26, Issue 14 : 1630039-1 1630023-26., 2017.

G. Zhang, J. Ma, A. Alsaedi, B. Ahmad and F. Alzahrani Dynamical behavior and application in Josephson Junction coupled by memristor, Applied Mathematics and Computation archive Volume 321, Issue C ; 290-299, March 2018.

G, Chen, N.V. Kuznetsov, G. Leonov, and T.V. Mokaev, Hidden attractors on one path: Glukhovskya Dolzhansky, Lorenz, and Rabinovich systems. International Journal of Bifurcation and Chaos. 27(8): 1750115: 1-9, 2017.

K. Pusuluri and A. Shilnikov, Homoclinic chaos and its organization in a nonlinear optics model, Phys. Rev. E, Vol. 98, Issue 4, 1-7, 2018.

Sbiaa, F., Zeghid, M., Kotel, S., Tourki, R., Machhout, M., Baganne, A., A Chaos-Based Approach for Correcting the Security Level of a Block Cipher Algorithm, (2016) International Review on Computers and Software (IRECOS), 11 (2), pp. 92-101.

Hreshee, S., Abdullah, H., Jawad, A., A High Security Communication System Based on Chaotic Scrambling and Chaotic Masking, (2018) International Journal on Communications Antenna and Propagation (IRECAP), 8 (3), pp. 257-264.

E. Ott, Chaos in Dynamical Systems (Cambridge University Press-1993).

E. Ott, C. Grebogi, and J. A. Yorke, Controling chaotic dynamical systems, in CHAOS: Soviet-American Prespectives on Nonlinear Science, D. K. Campbell, ed.,: 153-172, 1990.

P. So, and E. Ott, Controlling Chaos Using Time Delay Coordinates via Stabilization of Unstable Periodic Orbits, Phys. Rev. E 51 : 2955-2962.

Pyragas, K., Continuous Control of Chaos by Self-Controlling Feedback, Physical Letters A, Vol. 170: 421-428,1992.

Y. Braiman and I. Goldhirsch, Taming chaotic dynamics with weak periodic perturbations, Phys. Rev. Lett. 66, 2545-2548, 1991.

A. Hubler, Adaptive control of chaotic systems, Helv Phys. Acta 62 : 343-346, 1989.

E. H. Abed and J.-H Fu, Local Feedback Stabilization and Bifurcation Control, Hopf Bifurcation. Systems and Control Letters: 11-17, 1986.

A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics (John Wiley-1994).

S.Luo and Y. Song, Chaos analysis-based adaptive backstepping control of the microelectromechanical resonators with constrained output and uncertain time delay, IEEE Transactions on Industrial Electronics, Vol. 63, Issue 10 :6217–6225, 2016.

L. Shaohua, L. Shaobo and F. Tajaddodianfar. Chaos and Nonlinear Feedback Control of the Arch Micro-Electro-Mechanical System. Journal of Systems Science and Complexity, Vol. 31, Issue 6: 1510–1524, December 2018.

L. Shaohua and S. Yongduan , Chaos Analysis-Based Adaptive Backstepping Control of the Microelectromechanical Resonators With Constrained Output and Uncertain Time Delay, IEEE Transactions on Industrial Electronics, Vol. 63 , Issue 10 : 6217 – 6225, Oct. 2016.

A. Mina, Y.Wang, G. Peng, E. Wang, and J. Auth, Bifurcations, chaos and adaptive backstepping sliding mode control of a power system with excitation limitation, AIP Advances 6, 085214 : 1-11 (2016).

H. Liu, Y. Pan, Li, and Y. Chen, Adaptive fuzzy backstepping control of fractional-order nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst. Vol. 47, Issue (8) : 2209–2217, 2017.

O. Calvo and J. Cartwright, Fuzzy Control of Chaos, International Journal of Bifurcation and Chaos, Vol. 8, (Issue 8) : 1743-1747, 1998.

M.George, Hu, Bao-Gang, and G. Raymond ,Analysis of Direct Action Fuzzy PID Controller Structures, IEEE Trans. System and Cybernetics, Vol. 29, (Issue 3) : 371-388, June 1999.

J. Zhu and K. Khayati, On a new adaptive sliding mode control for mimo nonlinear systems with uncertainties of unknown bounds. International Journal of Robust and Nonlinear Control, Vol. 27, Issue 6 : 942-962, 2017.

S. Mobayen, D. Baleanu, and F. Tchier, Second-order fast terminal sliding mode control design based on LMI for a class of non-linear uncertain systems and its application to chaotic systems. Journal of Vibration and Control, Vol. 23, Issue 18: 2912–2925, 2017.

Slotine, J. and W. Li, Applied nonlinear control (Prentice Hall, 1991).

V.I. Utkin, Variable structure systems with sliding modes, IEEE Trans. on Automatic Control, Vol.22, (Issue 2) : 212-222, 1977.

J. Xu, C-C Lim and P. Shi. Sliding mode control of singularly perturbed systems and its application in quad-rotors. International Journal of Control, Vol. 92, Issue 6: 1325-1334., 2019.

Reddak, M., Berdai, A., Gourma, A., Boukherouaa, J., Belfiqih, A., Enhanced Sliding Mode MPPT and Power Control for Wind Turbine Systems Driven DFIG (Doubly-Fed Induction Generator), (2016) International Review of Automatic Control (IREACO), 9 (4), pp. 207-215.

Lahssan, B., Bakhti, M., Bououlid Idrissi, B., Flexible Manipulator State Reconstruction Using Luenberger and First Order Sliding Mode Observers Under Parametric Uncertainties, (2016) International Review on Modelling and Simulations (IREMOS), 9 (6), pp. 427-434.

Tesfahunegn, S., Hajizadeh, A., Undeland, T., Ulleberg, O., Vie, P., Modelling and Control of Grid-Connected PV/Fuel Cell/Battery Hybrid Power System, (2018) International Journal on Energy Conversion (IRECON), 6 (5), pp. 168-177.

Taleb, M., Cherkaoui, M., Optimal Control of Active and Reactive Powers in Wind Energy Conversion Systems Using Particle Swarm Optimization and Adaptive Sliding Mode Control, (2018) International Review of Automatic Control (IREACO), 11 (5), pp. 248-254.

Mazari, M., Boudinar, A., Mazari, B., Effect of Open Circuit Fault on PMSM Drive Controlled by Sliding Mode Control and Feedback Linearization Using Time and Frequency Analysis, (2018) International Review on Modelling and Simulations (IREMOS), 11 (4), pp. 235-244.


  • There are currently no refbacks.

Please send any question about this web site to
Copyright © 2005-2024 Praise Worthy Prize