Application of Fractional Calculus in the Control of Heat Systems

Isabel S. Jesus; J. A. Tenreiro Machado

doi:10.20965/jaciii.2007.p1086

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JACIII Vol.11 No.9 pp. 1086-1091

doi: 10.20965/jaciii.2007.p1086

(2007)

Paper:

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Application of Fractional Calculus in the Control of Heat Systems

Isabel S. Jesus and J. A. Tenreiro Machado

Dept. of Electrotechnical Engineering, Institute of Engineering of Porto / GECAD, Rua Dr. António Bernardino de Almeida, 431, 4200-072 Porto, Portugal

Received:

March 7, 2007

Accepted:

June 14, 2007

Published:

November 20, 2007

Keywords:

fractional calculus, control, diffusion systems, ISE, ITSE, IAE, ITAE

Abstract

The PID controller is by far the most dominating form of feedback in use in the process industries, due to its functional simplicity and performance. In this work, we apply a generalization of the PID, namely the fractional controller PID^β, to the heat diffusion system. For the PID^β tuning are used four performance indices, to find the optimum controller settings by taking advantage of the fractional order β. The effect of actuator saturation and the required control energy are also analyzed.

Cite this article as:

I. Jesus and J. Machado, “Application of Fractional Calculus in the Control of Heat Systems,” J. Adv. Comput. Intell. Intell. Inform., Vol.11 No.9, pp. 1086-1091, 2007.

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References

[1] K. B. Oldham and J. Spanier, “The fractional calculus,” Academic press, London, 1974.
[2] I. Podlubny, “Fractional differential equations,” Academic Press, San Diego, 1999.
[3] R. Courant and D. Hilbert, “Methods of Mathematical Physics, Partial Differential Equations,” Wiley Interscience II, New York, 1962.
[4] I. Podlubny, “Fractional-order systems and PI^λD^μ controllers,” IEEE Trans. Automatic Control, Vol.44(1), pp. 208-214, 1999.
[5] Y.-Q. Chen, C.-H. Hu, and K. L. Moore, “Relay Feedback Tuning of Robust PID controllers with Iso-Damping Property,” IEEE Transactions on systems, man, and cybernetics – part B, 35(1), pp. 23-31, 2003.
[6] R. C. Dorf and R. H. Bishop, “Modern Control Systems,” Addison-Wesley, New York, 1990.
[7] R. S. Barbosa, J. A. T. Machado, and I. M. Ferreira, “Tuning of PID controllers based on Bode’s ideal transfer function,” Nonlinear Dynamics, 38(1/4), pp. 305-321, 2004.
[8] I. Petráš and B. M. Vinagre, “Practical application of digital fractional order controller to temperature control,” Acta Montanistica Slovaca, 7(2), pp. 131-137, 2002.
[9] J. T. Machado, I. Jesus, J. B. Cunha, and J. K. Tar, “Fractional Dynamics and Control of Distributed Parameter Systems,” Intelligent Systems at the Service of Mankind, Vol.2, pp. 295-305, 2006.
[10] J. Crank, “The Mathematics of Diffusion,” Oxford Univ. Press, London, 1956.
[11] C. F. Gerald and P. O. Wheatley, “Applied Numerical Analysis,” Addison-Wesley, USA, 1999.
[12] I. S. Jesus, R. S. Barbosa, J. A. T. Machado, and J. B. Cunha, “Strategies for the Control of Heat Diffusion Systems Based on Fractional Calculus,” Proc. IEEE Int. Conf. on Computational Cybernetics, Estonia, 2006.
[13] I. S. Jesus, J. A. T. Machado, and J. B. Cunha, “Fractional Dynamics and Control of Heat Diffusion Systems,” Proc. The 26th IASTED Int. Conf. on Modelling, Identification and Control – MIC 2007, Innsbruck, Austria, 2007.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] K. B. Oldham and J. Spanier, “The fractional calculus,” Academic press, London, 1974.

[2] [2] I. Podlubny, “Fractional differential equations,” Academic Press, San Diego, 1999.

[3] [3] R. Courant and D. Hilbert, “Methods of Mathematical Physics, Partial Differential Equations,” Wiley Interscience II, New York, 1962.

[4] [4] I. Podlubny, “Fractional-order systems and PI^λD^μ controllers,” IEEE Trans. Automatic Control, Vol.44(1), pp. 208-214, 1999.

[5] [5] Y.-Q. Chen, C.-H. Hu, and K. L. Moore, “Relay Feedback Tuning of Robust PID controllers with Iso-Damping Property,” IEEE Transactions on systems, man, and cybernetics – part B, 35(1), pp. 23-31, 2003.

[6] [6] R. C. Dorf and R. H. Bishop, “Modern Control Systems,” Addison-Wesley, New York, 1990.

[7] [7] R. S. Barbosa, J. A. T. Machado, and I. M. Ferreira, “Tuning of PID controllers based on Bode’s ideal transfer function,” Nonlinear Dynamics, 38(1/4), pp. 305-321, 2004.

[8] [8] I. Petráš and B. M. Vinagre, “Practical application of digital fractional order controller to temperature control,” Acta Montanistica Slovaca, 7(2), pp. 131-137, 2002.

[9] [9] J. T. Machado, I. Jesus, J. B. Cunha, and J. K. Tar, “Fractional Dynamics and Control of Distributed Parameter Systems,” Intelligent Systems at the Service of Mankind, Vol.2, pp. 295-305, 2006.

[10] [10] J. Crank, “The Mathematics of Diffusion,” Oxford Univ. Press, London, 1956.

[11] [11] C. F. Gerald and P. O. Wheatley, “Applied Numerical Analysis,” Addison-Wesley, USA, 1999.

[12] [12] I. S. Jesus, R. S. Barbosa, J. A. T. Machado, and J. B. Cunha, “Strategies for the Control of Heat Diffusion Systems Based on Fractional Calculus,” Proc. IEEE Int. Conf. on Computational Cybernetics, Estonia, 2006.

[13] [13] I. S. Jesus, J. A. T. Machado, and J. B. Cunha, “Fractional Dynamics and Control of Heat Diffusion Systems,” Proc. The 26th IASTED Int. Conf. on Modelling, Identification and Control – MIC 2007, Innsbruck, Austria, 2007.