Open Access Open Access  Restricted Access Subscription or Fee Access

Modeling and Stabilization Fuzzy Control Strategies of a Nonlinear Cart-Pendulum System: a Comparative Study

Larbi Kharroubi(1*), Wahid Nouibat(2), Mohamed Ouslim(3)

(1) Laboratory of Power Electronics, Solar Energy and Automatic (LEPESA). University of Science and Technology of Oran (USTO), Algeria
(2) Laboratory of Power Electronics, Solar Energy and Automatic (LEPESA). University of Science and Technology of Oran (USTO), Algeria
(3) Laboratory of Microsystems and Embedded Systems (LMSE). University of Science and Technology of Oran (USTO), Algeria
(*) Corresponding author



In this paper, we propose and validate an unsimplified nonlinear state-space model of the cart-pendulum system. Then, we present several fuzzy control strategies in order to stabilize and control both the pendulum angle in upright position and the cart position into a desired position. These strategies are built upon two separate fuzzy logic controllers (FLC), having the same characteristics. To achieve this goal, we used various structures of two-input fuzzy controllers such as fuzzy proportional-derivative (FPD), fuzzy proportional-integral (FPI), FPI plus FPD and FPI derivative (FPID), designed by two basic fuzzy inference systems, i.e., Mamdani or zero-order Takagi-Sugeno. Then, the effectiveness of different strategies was tested and compared using several performance integration indices. Many tests were carried out and the obtained simulation results proved that the strategy, based on the zero-order Takagi-Sugeno FPID structures, could be applied to control the coupled variables of a nonlinear system. Furthermore, the proposed unsimplified nonlinear state-space model of the cart-pendulum system, might be used for validating new nonlinear control strategies.
Copyright © 2015 Praise Worthy Prize - All rights reserved.


Nonlinear Cart-Pendulum System; Nonlinear State-Space Model; Stabilization; Two-Input Fuzzy Controllers; Performance Integration Indices

Full Text:



M. Nie, W. W. Tan, Stable adaptive fuzzy PD plus PI controller for nonlinear uncertain systems, Fuzzy Sets and Systems 179(2011) 1-19.

H. Boubertakh, M. Tadjine, P. Y. Glorennec, S. Labiod, Tuning fuzzy PD and PI controllers using reinforcement learning, ISA Transactions 49 (2010) 543-551.

T. Bouarar, K. Guelton, N. Manamanni, Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi-Sugeno descriptors, ISA Transactions 49 (2010) 447-461.

E. Sivaraman, S. Arulselvi, Modeling of an inverted pendulum based on fuzzy clustering techniques, Expert Systems with Applications 38 (2011) 13942–13949.

Y. Farzaneh, A.A. Tootoonchi, A novel data reduction method for Takagi–Sugeno fuzzy system design based on statistical design of experiment, Applied Soft Computing 9 (2009) 1367–1376.

M. J. Mahmoodabadi, S. A. Mostaghim, A. Bagheri, N. Nariman-zadeha, Pareto optimal design of the decoupled sliding mode controller for an inverted pendulum system and its stability simulation via Java programming, Mathematical and Computer Modelling 57 (2013) 1070-1082.

J. J. Wang, Simulation studies of inverted pendulum based on PID controllers, Simulation Modelling Practice and Theory 19 (2011) 440–449.

J. Yi, N. Yubazaki, Stabilization fuzzy control of inverted pendulum systems, Artificial Intelligence in Engineering 14 (2000) 153–163.

O. Boubaker, The Inverted Pendulum Benchmark in Nonlinear Control Theory: A Survey, International Journal of Advanced Robotic Systems, Vol. 10, 233: 2013.

O. Boubaker, The inverted pendulum: A fundamental benchmark in control theory and robotics, Journal of Electronic Systems, Vol. 2, N. 4, 2012.

Feedback Instruments Ltd, Digital Pendulum Control Experiments 33-936S.

H.T. Dorrah, W.I. Gabr, Development of new Consolidity Theory for systems’ analysis and design in fully fuzzy environment, Expert Systems with Applications 39 (2012) 1191–1199.

B.J. Choi, S.W. Kwak, B.K. Kim, Design and Stability Analysis of Single-Input Fuzzy Logic Controller, IEEE Trans. Syst., Man, Cybern.-Part B: Cybern., Vol. 30, No. 2, pp. 303-309, April 2000.

G. Li, X. Liu, Dynamic characteristic prediction of inverted pendulum under the reduced-gravity space environments, Acta Astronautica 67 (2010) 596-604.

M. Zekri, S. Sadri, F. Sheikholeslam, Adaptive fuzzy wavelet network control design for nonlinear systems, Fuzzy Sets and Systems 159 (2008) 2668-2695.

A. K. Yadav, P. Gaur, A.P. Mittal and M. Anzar, Comparative Analysis of Various Control Techniques for Inverted Pendulum, Power Electronics (IICPE), 2010 India International Conference on. PP. 1-6, Jan. 2011.

B.M. Al-Hadithi, A. Jiménez, F. Matia , A new approach to fuzzy estimation of Takagi–Sugeno model and its applications to optimal control for nonlinear systems, Applied Soft Computing 12 (2012) 280–290.

L. A. Zadeh, Fuzzy Sets, Information and Control 8, 338-353 (1965).

C.C. Lee, Fuzzy Logic in Control Systems: Fuzzy Logic Controller, Part I & Part II, IEEE Trans. Syst. Man, Cybern., vol. 20, no. 2, pp. 404-435, 1990.

H. Ying, W. Siler and J. J. Buckley, Fuzzy Control Theory: A Nonlinear Case, Automatica, Vol. 26, No. 3, pp.513-520, 1990.

H. Ying, An Analytical Study on Structure, Stability and Design of General Nonlinear Takagi-Sugeno Fuzzy Control Systems, Automatica, Vol. 34, No. 12, pp.1617-1623, 1998.

H. Ying, The Simplest Fuzzy Controllers using Different Inference Methods are Different Nonlinear Proportional-integral Controllers with Variable Gains, Automatica, Vol. 29, No. 6, pp.1579-1589, 1993.

Bouchiba, F., Nouibat, W., Neuro-fuzzy navigation of a mobile robot in an unknown environment, (2015) International Review of Automatic Control (IREACO), 8 (3), pp. 220-227.

J. J. Buckley, H. Ying, Expert fuzzy controller, Fuzzy Sets and Systems 44 (1991) 373-390.

W. Siler and H. Ying, Fuzzy Control Theory: The Linear Case, Fuzzy Sets and Systems 33 (1989) 275-290.

E. H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. IEE Contr. Sci., Vol. 121, pp.1585-1588, 1974.

T. Takagi and M. Sugeno, Fuzzy Identification of Systems and its Application to Modelling and Control, IEEE Trans. Syst., Man, Cybern., Vol. SMC-15, No. 1, pp. 116–132, 1985.

G.K.I. Mann, B.G. Hu, R.G. Gosine, Analysis of Direct Action Fuzzy PID Controller Structures, IEEE Transactions on Systems, Man, and Cybern.-Part B: Cybern., vol. 29, No. 3, pp. 371-388, 1999.

B.G. Hu, G.K.I. Mann, R.G. Gosine, A Systematic Study of Fuzzy PID Controllers-Function-Based Evaluation Approach, IEEE Trans. Fuzzy Syst., vol. 9, No. 5, pp. 699-712, 2001.

G.K.I. Mann, B.G. Hu, R.G. Gosine, Two-Level Tuning of Fuzzy PID Controllers, IEEE Trans. Systems Man Cybern.-Part B, vol. 31, No. 2, pp. 263-269, 2001.

H. X. Li, H. B. Gatland, and A. W. Green, Fuzzy Variable Structure Control, IEEE Trans. Syst., Man, Cybern.-Part B: Cybern., Vol. 27, No. 2, pp. 306-312, 1997.

H.X. Li and H.B. Gatland, Conventional fuzzy control and its enhancement, IEEE Trans. Syst. Man, Cybern.-Part B, vol.26, No. 5, pp. 791-797, 1996.

H. X. Li, A Comparative Design and Tuning for Conventional Fuzzy Control, IEEE Trans. Syst., Man, Cybern.-Part B: Cybern., Vol. 27, No. 5, pp. 884-889, 1997.

H.X. Li, H.B. Gatland, Enhanced methods of fuzzy logic control, in Proc. 4th IEEE Int. Conf. Fuzzy Systems. Japan, pp. 331-336, 1995.

Manuel A. Duarte-Mermoud, Rodrigo A. Prieto, Performance index for quality response of dynamical systems, ISA Transactions 43 (2004) 133-151.

K. Astrom, T. Haggluno, PID Controllers: Theory, Design, and Tuning (Instrument Society of America, 1995).

J.H. Lee, H.J. Shin, S.J. Lee, S. Jung, Balancing control of a single-wheel inverted pendulum system using air blowers: Evolution of Mechatronics capstone design, Mechatronics 23 (2013) 926–932.

P.J. MacVicar-Whelan, Fuzzy sets for man-machine interaction, Int. J. Man-Machine Studies, Vol. 8, pp. 687-697, 1976.


  • There are currently no refbacks.

Please send any question about this web site to
Copyright © 2005-2020 Praise Worthy Prize