Research Papers

Model Predictive Control of Fractional Order Systems

[+] Author and Article Information
Aymen Rhouma

Université de Tunis El Manar,
Ecole Nationale d'Ingénieurs de Tunis,
LR11ES20 Laboratoire Analyse,
Conception et Commande des Systèmes,
Tunis 1002, Tunisia
Faculté des Sciences de Tunis,
Tunis 2092, Tunisia
e-mail: aymenrh@yahoo.fr

Faouzi Bouani

Université de Tunis El Manar,
Ecole Nationale d'Ingénieurs de Tunis,
LR11ES20 Laboratoire Analyse,
Conception et Commande des Systèmes,
Tunis 1002, Tunisia
e-mail: faouzi.bouani@enit.rnu.tn

Badreddine Bouzouita

Université de Tunis El Manar,
Ecole Nationale d'Ingénieurs de Tunis,
LR11ES20 Laboratoire Analyse,
Conception et Commande des Systèmes,
Tunis 1002, Tunisia
Université de Sousse,
Ecole Nationale d'Ingénieurs de Sousse,
Sousse 4054, Tunisia
e-mail: badreddine.bouzouita@enit.rnu.tn

Mekki Ksouri

Université de Tunis El Manar,
Ecole Nationale d'Ingénieurs de Tunis,
LR11ES20 Laboratoire Analyse,
Conception et Commande des Systèmes,
Tunis 1002, Tunisia
e-mail: mekki.ksouri@enit.rnu.tn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 25, 2013; final manuscript received January 10, 2014; published online February 13, 2014. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 9(3), 031011 (Feb 13, 2014) (7 pages) Paper No: CND-13-1188; doi: 10.1115/1.4026493 History: Received July 25, 2013; Revised January 10, 2014

This paper provides the model predictive control (MPC) of fractional order systems. The direct method will be used as internal model to predict the future dynamic behavior of the process, which is used to achieve the control law. This method is based on the Grünwald–Letnikov's definition that consists of replacing the noninteger derivation operator of the adopted system representation by a discrete approximation. The performances and the efficiency of this approach are illustrated with practical results on a thermal system and compared to the MPC based on the integer ARX model.

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Bagley, R., and Calico, R., 1991, “Fractional Order State Equations for the Control of Viscoelastically Damped Structures,” Journal of Guidance, Control, and Dynamics, 14, pp. 304–311. [CrossRef]
Zhang, Y., Tian, Q., Chen, L., and Yang, J., 2009, “Simulation of a Viscoelastic Flexible Multibody System Using Absolute Nodal Coordinate and Fractional Derivative Methods,” Multibody System Dynamics, 21, pp. 281–303. [CrossRef]
Yuste, S., Abad, E., and Lindenberg, K., 2010, “Application of Fractional Calculus to Reaction-Subdiffusion Processes and Morphogen Gradient Formation,” e-print arXiv, 1006.2661, Cornell University Library, Ithaca, NY. Available at: http://arxiv.org/abs/1006.2661
Mainardi, F., Raberto, M., Gorenflo, R., and Scalas, E., 2000, “Fractional Calculus and Continuous-Time Finance II: The Waiting-Time Distribution,” Physica A: Statistical Mechanics and its Applications, 287, pp. 468–481. [CrossRef]
Victor, S., Melchior, P., and Oustaloup, A., 2010, “Robust Path Tracking Using Flatness for Fractional Linear MIMO Systems: A Thermal Application,” Computers and Mathematics With Applications, Vol. 59, Elsevier, New York, pp. 1667–1678.
Sun, H. H., Charef, A., Tsao, Y., and Onaral, B., 1992, “Analysis of Polarization Dynamics by Singularity Decomposition Method,” Ann. Biomed. Eng., 20, pp. 321–335. [CrossRef] [PubMed]
Shantanu, D., 2008, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag, Berlin.
Victor, S., Malti, R., Melchior, P., and Oustaloup, A., 2011, “Instrumental Variable Identification of Hybrid Fractional Box-Jenking Models,” 18th IFAC World Congress, Milano, Italy.
Sommacal, L., Melchior, P., Dossat, A., Petit, J., Cabelguen, J. M., Oustaloup, A., and Ijspeert, A. J., 2007, “Improvement of the Muscle Fractional Multimodel for Low-Rate Stimulation,” Biomedical Signal Processing and Control, Vol. 2, Elsevier, New York, pp. 226–233.
Ionescu, C. M., Machado, J. A. T., and De Keyser, R., 2011, “Modeling of the Lung Impedance Using a Fractional-Order Ladder Network With Constant Phase Elements,” IEEE Trans. Biomed. Circuits and Systems, 5(1), pp. 83–89. [CrossRef]
Oustaloup, A., 1988, From fractality to non-integer derivation through recursivity, a property common to these two concepts: a fundamental idea from a new process control strategy, Proceeding of the 12th IMACS World Congress, Paris, July 18–22, Vol. 3, pp. 203–208.
Oustaloup, A., 1991, La Commande CRONE (Commande Robuste d'Ordre Non Entier), Paris, Hermès.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, New York.
Raynaud, H. F., and Zergainoh, A., 2000, “State-Space Representation for Fractional Order Controllers,” Automatica, 36, pp. 1017–1021. [CrossRef]
Monje, C. A., and Feliu, V., 2004, The Fractional-Order Lead Compensator, IEEE International Conference on Computational Cybernetics, Vienna, Austria.
Agrawal, O. P., 2004, “A General Formulation and Solution Scheme for Fractional Optimal Control Problems,” Nonlinear Dynamics, 38, pp. 323–337. [CrossRef]
Vinagre, B. M., Petras, I., Podlubny, I., and Chen, Y. Q., 2002, “Using Fractional-Order Adjustment Rules and Fractional-Order Reference Models in Model Reference Adaptive Control,” Nonlinear Dynamics, 29, pp. 269–279. [CrossRef]
Dadras, S., and Momeni, H. R., 2012, “Fractional Terminal Sliding Mode Control Design for a Class of Dynamical Systems With Uncertainty,” Commun. Nonlinear Sci. Numer. Simul., 17, pp. 367–377. [CrossRef]
Ying, L., Yang, Q. C., Chun, Y. W., and You, G. P., 2010, “Tuning Fractional Order Proportional Integral Controllers for Fractional Order Systems,” Journal of Process Control, 20, pp. 823–831. [CrossRef]
Hitay, O., Catherine, B., and André, R. F., 2012, “PID Controller Design for Fractional-Order Systems With Time Delays,” Syst. Control Lett., 61, pp. 18–23. [CrossRef]
Fukushima, H., Kim, T., and Sugie, T., 2007, “Adaptive Model Predictive Control for a Class of Constrained Linear Systems Based on Comparison Model,” Automatica, 43, pp. 301–308. [CrossRef]
Camacho, E. F., and Bordons, C., 2004, Model Predictive Control, Springer-Verlag, Berlin.
Tavazoei, M. S., 2010, “A Note on Fractional-Order Derivatives of Periodic Functions,” Automatica, 46, pp. 945–948. [CrossRef]
Hajiloo, A., Nariman-zadeh, N., and Moeini, A., 2012, “Pareto Optimal Robust Design of Fractional-Order PID Controllers for Systems With Probabilistic Uncertainties,” Mechatronics, 22, pp. 788–801.
Rodriguez, E., Echeverria, J. C., and Alvarez-Ramirez, J., 2009, “1/f Fractal Noise Generation From Grunwald–Letnikov Formula,” Chaos, Solitons and Fractals, 39, pp. 882–888. [CrossRef]
Miller, K. S., and Ross, B., 1993, An Introduction to the Fractional Calculus and Fractional Differential Equation, John Wiley and Son, New York.
Oustaloup, A., Olivier, C., and Ludovic, L., 2005, Representation et Identification Par Modele Non Entier, Lavoisier, Paris.
Trigeassou, J. C., Poinot, P., Lin, J., Oustaloup, A., and Levron, F., 1999, “Modelling and Identification of a Non-Integer Order System,” in ECC, Karlsruhe, Germany.
Oustaloup, A., 1995, La Dérivation Non-Entiere, Hermès, Paris.
Oustaloup, A., Levron, F., Mathieu, B., and Nanot, F. M., 2000, “Frequency Band Complex Non Integer Differentiator: Characterization and Synthesis,” IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications, 47, pp. 25–40. [CrossRef]
Monje, C. A., Chen, Y. Q., Vinagre, B. M., Dingyu, X., and Vicente, F., 2010, Fractional-Order Ssystems and Control: Fundamentals and Applications: Advances in Industrial Control, Springer, London.
Boucher, P., and Dumur, D., 1996, La Commande Prédictive, technip edition, Paris.
Malti, R., Victor, S., and Oustaloup, A., 2008, “Advances in System Identification Using Fractional Models,” ASME J. Comput. Nonlinear Dyn., 3(2), p. 021401. [CrossRef]
Cois, O., 2002, “Systèmes Linéaires Non Entiers et Identification par Modèle Non Entier: Application en Thermique,” Ph.D. thesis, Université Bordeaux1, Talence, France.
Malti, R., Victor, S., Oustaloup, A., and Garnier, H., 2008, “An Optimal Instrumental Variable Method for Continuous-Time Fractional Model Identification,” 17th IFAC World Congress, Seoul, South Korea, pp. 14379–14384. Available at: http://hal.archives-ouvertes.fr/docs/00/32/75/63/PDF/last_srivc_frac.pdf
Gabano, J. D., and Poinot, T., 2011, “Fractional Modelling and Identification of Thermal Systems,” Signal Processing, 91, pp. 531–541. [CrossRef]


Grahic Jump Location
Fig. 2

Errors for two methods

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Fig. 3

Closed-loop responses

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Fig. 5

Open loop step responses

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Fig. 6

Identification data

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Fig. 8

Closed-loop results with Hp = 8

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Fig. 9

Closed-loop results with Hp = 12

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Fig. 10

Closed-loop results with fractional model

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Fig. 11

Closed-loop results with integer model



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