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Dynamics and Control of Initialized Fractional-Order Systems

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Abstract

Due to the importance of historical effects in fractional-order systems,this paper presents a general fractional-order system and control theorythat includes the time-varying initialization response. Previous studieshave not properly accounted for these historical effects. Theinitialization response, along with the forced response, forfractional-order systems is determined. The scalar fractional-orderimpulse response is determined, and is a generalization of theexponential function. Stability properties of fractional-order systemsare presented in the complex w-plane, which is a transformation of thes-plane. Time responses are discussed with respect to pole positions inthe complex w-plane and frequency response behavior is included. Afractional-order vector space representation, which is a generalizationof the state space concept, is presented including the initializationresponse. Control methods for vector representations of initializedfractional-order systems are shown. Finally, the fractional-orderdifferintegral is generalized to continuous order-distributions whichhave the possibility of including all fractional orders in a transferfunction.

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References

  1. Heaviside, O., Electromagnetic Theory, Vol. II, Chelsea Edition (1971), New York, 1922.

  2. Bush, V., Operational Circuit Analysis, Wiley, New York, 1929.

    Google Scholar 

  3. Goldman, S., Transformation Calculus and Electrical Transients, Prentice-Hall, Englewood Cliffs, NJ, 1949.

    Google Scholar 

  4. Holbrook, J. G., Laplace Transforms for Electronic Engineers, 2nd edition, Pergamon Press, New York, 1966.

    Google Scholar 

  5. Starkey, B. J., Laplace Transforms for Electrical Engineers, Iliffe, London, 1965.

    Google Scholar 

  6. Carslaw, H. S. and Jeager, J. C., Operational Methods in Applied Mathematics, 2nd edition, Oxford University Press, Oxford, 1948.

    Google Scholar 

  7. Scott, E. J., Transform Calculus with an Introduction to Complex Variables, Harper, New York, 1955.

    Google Scholar 

  8. Mikusinski, J., Operational Calculus, Pergamon Press, New York, 1959.

    Google Scholar 

  9. Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, San Diego, CA, 1974.

    Google Scholar 

  10. Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

    Google Scholar 

  11. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, CA, 1999.

    Google Scholar 

  12. Bagley, R. L. and Calico, R. A., 'Fractional order state equations for the control of viscoelastic structures', Journal of Guidance, Control, and Dynamics 14(2), 1991, 304–311.

    Google Scholar 

  13. Koeller, R. C., 'Application of fractional calculus to the theory of viscoelasticity', Journal of Applied Mechanics 51, 1984, 299–307.

    Google Scholar 

  14. Koeller, R. C., 'Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics', Acta Mechanica 58, 1986, 251–264.

    Google Scholar 

  15. Skaar, S. B., Michel, A. N., and Miller, R. K., 'Stability of viscoelastic control systems', IEEE Transactions on Automatic Control 33(4), 1988, 348–357.

    Google Scholar 

  16. Ichise, M., Nagayanagi, Y., and Kojima, T., 'An analog simulation of non-integer order transfer functions for analysis of electrode processes', Journal of Electroanalytical Chemistry Interfacial Electrochemistry 33, 1971, 253–265.

    Google Scholar 

  17. Sun, H. H., Onaral, B., and Tsao, Y., 'Application of positive reality principle to metal electrode linear polarization phenomena', IEEE Transactions on Biomedical Engineering 31(10), 1984, 664–674.

    Google Scholar 

  18. Sun, H. H., Abdelwahab, A. A., and Onaral, B., 'Linear approximation of transfer function with a pole of fractional order', IEEE Transactions on Automatic Control 29(5), 1984, 441–444.

    Google Scholar 

  19. Mandelbrot, B., 'Some noises with 1/f spectrum, a bridge between direct current and white noise', IEEE Transactions on Information Theory 13(2), 1967, 289–298.

    Google Scholar 

  20. Hartley, T. T., Lorenzo, C. F., and Qammar, H. K., 'Chaos in a fractional order Chua system', IEEE Transactions on Circuits & Systems: Part I 42(8), 1995, 485–490.

    Google Scholar 

  21. Padovan, J. and Sawicki, J. T., 'Diophantine type fractional derivative representation of structural dynamics, Part I: Formulation', Computational Mechanics 19, 1997, 335–340.

    Google Scholar 

  22. Lorenzo, C. F. and Hartley, T. T., 'Initialized fractional calculus', International Journal of Applied Mathematics 3(3), 2000, 249–266.

    Google Scholar 

  23. Podlubny, I., 'Fractional order systems and PIλDμ controllers', IEEE Transactions on Automatic Control 44(1), 1999, 208–214.

    Google Scholar 

  24. Oustaloup, A., La dérivation non entière: Théorie, synthèse et applications, Hermès, Paris, 1995.

    Google Scholar 

  25. Hartley, T. T. and Lorenzo, C. F., 'Fractional system identification: An approach using continuous orderdistributions', NASA/TM-1999–209640, 1999.

  26. Lorenzo, C. F. and Hartley, T. T., 'Variable order and distributed order fractional operators', Nonlinear Dynamics 29, 2002, 57–98 (this volume).

    Google Scholar 

  27. Hartley, T. T. and Lorenzo. C. F., 'Control of initialized fractional-order systems', NASA/TM-2002–211377, 2002.

  28. Lorenzo, C. F. and Hartley, T. T., 'Initialization, conceptualization, and application in the generalized (fractional) calculus', NASA TP-1998–208415, 1998.

  29. Hartley, T. T. and Lorenzo, C. F., 'A solution to the fundamental linear fractional order differential equation', NASA/TP-1998–208693, 1998.

  30. Glockle, W. G. and Nonnenmacher, T. F., 'Fractional integral operators and Fox functions in the theory of viscoelasticity', Macromolecules 24, 1991, 6426–6434.

    Google Scholar 

  31. Robotnov, Y. N., Elements of Hereditary Solid Mechanics, MIR Publishers, Moscow, 1980.

    Google Scholar 

  32. Robotnov, Y. N., Tables of a Fractional Exponential Function of Negative Parameters and Its Integral, Nauka, Moscow, 1969 [in Russian].

    Google Scholar 

  33. Matignon, D., 'Stability properties for generalized fractional differential systems', ESIAM Proceedings on Fractional Differential Systems 5, 1998, 145–158.

    Google Scholar 

  34. Lorenzo, C. F. and Hartley, T. T., 'Generalized functions for the fractional calculus', NASA TP-1999–209424, 1999.

  35. Lorenzo, C. F. and Hartley, T. T., 'R-function relationships for application in the fractional calculus', NASA TM-2000–210361, 2000.

  36. Raynaud, H. F. and Zergainoh, A., 'State space representation for fractional order controllers', Automatica 36, 2000, 1017–1021.

    Google Scholar 

  37. Kailath, T., Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1980.

    Google Scholar 

  38. Matignon, D. and d'Andrea-Novel, B., 'Observer based controllers for fractional differential equations', in Proceedings of the 36th IEEE Conference on Decision and Control, IEEE, New York, 1996, pp. 4967–4972.

    Google Scholar 

  39. Hotzel, R., 'Contribution à la théorie structurelle et à la commande des systèmes linéaires fractionnaires', Ph.D. Thesis, Université Paris Sud, 1998.

  40. D'Azzo, J. J. and Houpis, C. H., Linear Control Systems, Analysis and Design, McGraw-Hill, New York, 1981.

    Google Scholar 

  41. Maciejowski, J. M., Multivariable Feedback Design, Addison-Wesley, Wickingham, 1989.

    Google Scholar 

  42. Oustaloup, A., La Commande CRONE, Hermès, Paris, 1991.

    Google Scholar 

  43. Oustaloup, A., La commande CRONE: Du scalaire au multivariable, Hermès, Paris, 1999.

    Google Scholar 

  44. Maia, N. M. M., Silva, J. M. M., and Ribeiro, A. M. R., 'On a general model for damping', Journal of Sound and Vibration 218(5), 1998, 749–767.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, The University of Akron, Akron, OH, 44325-3904, U.S.A

    Tom T. Hartley

  2. John H. Glenn Research Center, National Aeronautics and Space Administration, Cleveland, OH, 44135, U.S.A

    Carl F. Lorenzo

Authors
  1. Tom T. Hartley
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  2. Carl F. Lorenzo
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Hartley, T.T., Lorenzo, C.F. Dynamics and Control of Initialized Fractional-Order Systems. Nonlinear Dynamics 29, 201–233 (2002). https://doi.org/10.1023/A:1016534921583

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