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
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 PIDβ, to the heat diffusion system. For the PIDβ 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.
-  K. B. Oldham and J. Spanier, “The fractional calculus,” Academic press, London, 1974.
-  I. Podlubny, “Fractional differential equations,” Academic Press, San Diego, 1999.
-  R. Courant and D. Hilbert, “Methods of Mathematical Physics, Partial Differential Equations,” Wiley Interscience II, New York, 1962.
-  I. Podlubny, “Fractional-order systems and PIλDμ controllers,” IEEE Trans. Automatic Control, Vol.44(1), pp. 208-214, 1999.
-  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.
-  R. C. Dorf and R. H. Bishop, “Modern Control Systems,” Addison-Wesley, New York, 1990.
-  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.
-  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.
-  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.
-  J. Crank, “The Mathematics of Diffusion,” Oxford Univ. Press, London, 1956.
-  C. F. Gerald and P. O. Wheatley, “Applied Numerical Analysis,” Addison-Wesley, USA, 1999.
-  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.
-  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.
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