Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation

Sierociuk, D. ; Dzieliński, A.

International Journal of Applied Mathematics and Computer Science, Tome 16 (2006) no. 1, pp. 129-140.

Résumé

This paper presents a generalization of the Kalman filter for linear and nonlinear fractional order discrete state-space systems. Linear and nonlinear discrete fractional order state-space systems are also introduced. The simplified kalman filter for the linear case is called the fractional Kalman filter and its nonlinear extension is named the extended fractional Kalman filter. The background and motivations for using such techniques are given, and some algorithms are discussed. The paper also shows a simple numerical example of linear state estimation. Finally, as an example of nonlinear estimation, the paper discusses the possibility of using these algorithms for parameters and fractional order estimation for fractional order systems. Numerical examples of the use of these algorithms in a general nonlinear case are presented.

Détail

Keywords: discrete fractional state-space systems, fractional Kalman filter, parameter estimation, order estimation, extended fractional Kalman filter
Mots-clés : przestrzeń fazowa, układ ułamkowy, filtr Kalmana, estymacja parametryczna

Bibliographie

[1] Bologna M. and Grigolini P. (2003): Physics of Fractal Operators.-New York: Springer-Verlag.

[2] Brown R.G. and Hwang P.Y.C. (1997): Introduction to RandomSignals and Applied Kalman Filtering with Matlab Exercises and Solutions. - New York: Wiley.

[3] Cois O., Oustaloup A., Battaglia E. and Battaglia J.-L. (2000): Non-integer model from modal decomposition for time domain system identification. - Proc. Symp. System Identification, SYSID 2000, Santa Barbara, CA, Vol. 3, p. 989-994.

[4] Cois O., Oustaloup A., Poinot T. and Battaglia J.-L. (2001): Fractional state variable filter for system identification by fractional model. - Proc. European Contr. Conf., ECC'2001, Porto, Portugal, pp. 2481-2486.

[5] Engheta N. (1997): On the role of fractional calculus in electromagnetic theory. - IEEE Trans. Antenn. Prop., Vol. 39, No. 4, pp. 35-46.

[6] Ferreira N.M.F. and Machado J.A.T. (2003): Fractional-order hybrid control of robotic manipulators. - Proc. 11-th Int. Conf. Advanced Robotics, ICAR'2003, Coimbra, Portugal, pp. 393-398.

[7] Gałkowski K. (2005): Fractional polynomials and nD systems. - Proc. IEEE Int. Symp. Circuits and Systems, ISCAS' 2005, Kobe, Japan, CD-ROM.

[8] Haykin S. (2001): Kalman Filtering and Neural Networks. - New York: Wiley.

[9] Hilfer R., Ed. (2000): Application of Fractional Calculus in Physics. -Singapore: World Scientific.

[10] Jun S.C. (2001): A note on fractional differences based on a linear combination between forward and backward differences. -Comput. Math. Applic., Vol. 41, No. 3, pp. 373-378.

[11] Kalman R.E. (1960): A new approach to linear filtering and prediction problems. - Trans. ASME J. Basic Eng., Vol. 82, Series D, pp. 35-45.

[12] Miller K.S. and Ross B. (1993): An Introduction to the Fractional Calculus and Fractional Differenctial Equations. - New York: Wiley.

[13] Moshrefi-Torbati M. and Hammond K. (1998): Physical and geometrical interpretation of fractional operators. - J. Franklin Inst., Vol. 335B, No. 6, pp. 1077-1086.

[14] Nishimoto K. (1984): Fractional Calculus. - Koriyama: Decartes Press.

[15] Oldham K.B. and Spanier J. (1974): The Fractional Calculus. -New York: Academic Press.

[16] Ostalczyk P. (2000): The non-integer difference of the discretetime function and its application to the control system synthesis.- Int. J. Syst. Sci., Vol. 31, No. 12, pp. 1551-1561.

[17] Ostalczyk P. (2004a): Fractional-Order Backward Difference Equivalent Forms Part I - Horner's Form. - Proc. 1-st IFAC Workshop Fractional Differentation and its Applications, FDA'04, Enseirb, Bordeaux, France, pp. 342-347.

[18] Ostalczyk P. (2004b): Fractional-Order Backward Difference Equivalent Forms Part II - Polynomial Form.-Proc. 1-st IFAC Workshop Fractional Differentation and its Applications, FDA'04, Enseirb, Bordeaux, France, pp. 348-353.

[19] Oustaloup A. (1993): Commande CRONE.-Paris: Hermés.

[20] Oustaloup A. (1995): La dérivation non entiére. - Paris: Hermés.

[21] Podlubny I. (1999): Fractional Differential Equations. - San Diego: Academic Press.

[22] Podlubny I. (2002): Geometric and physical interpretation of fractional integration and fractional differentiation. - Fract. Calc. Appl. Anal., Vol. 5, No. 4, pp. 367-386.

[23] Podlubny I., Dorcak L. and Kostial I. (1997): On fractiona derivatives, fractional-order systems and PIλDμ- controllers. - Proc. 36-th IEEE Conf. Decision and Control, San Diego, CA, pp. 4985-4990.

[24] Poinot T. and Trigeassou J.C. (2003): Modelling and simulation of fractional systems using a non integer integrator. - Proc. Design Engineering Technical Conferences, DETC'03, and Computers and Information in Engineering Conference, ASME 2003, Chicago, IL, pp. VIB-48390.

[25] Reyes-Melo M.E., Martinez-Vega J.J., Guerrero-Salazar C.A. and Ortiz-Mendez U. (2004a): Application of fractional calculus to modelling of relaxation phenomena of organic dielectric materials. - Proc. Int. Conf. Solid Dielectrics, Toulouse, France, Vol. 2, pp. 530-533.

[26] Reyes-Melo M.E., Martinez-Vega J.J., Guerrero-Salazar C.A. and Ortiz-Mendez U. (2004b): Modelling of relaxation phenomena in organic dielectric materials. application of differential and integral operators of fractional order. - J. Optoel. Adv. Mat., Vol. 6, No. 3, pp. 1037-1043.

[27] Riewe F. (1997): Mechanics with fractional derivatives. - Phys. Rev. E, Vol. 55, No. 3, pp. 3581-3592.

[28] Riu D., Retiére N. and Ivanés M. (2001): Turbine generator modeling by non-integer order systems.- Proc. IEEE Int. Electric Machines and Drives Conference, IEMDC 2001, Cambridge , MA, pp. 185-187.

[29] Samko S.G., Kilbas A.A. and Maritchev O.I. (1993): Fractional Integrals and Derivative. Theory and Applications. - London: Gordon Breach.

[30] Schutter J., Geeter J., Lefebvre T. and Bruynickx H. (1999): Kalman filters: A tutorial. - Available at http://citeseer.ist.psu.edu/443226.html

[31] Sierociuk D. (2005a): Fractional order discrete state-space system Simulink toolkit user guide. - Available at http://www.ee.pw.edu.pl/˜dsieroci/fsst/fsst.htm

[32] Sierociuk D. (2005b): Application of a fractional Kalman filter to parameter estimation of a fractional-order system. - Proc. 15-th Nat. Conf. Automatic Control, Warsaw, Poland, Vol. 1, pp. 89-94, (in Polish).

[33] Sjöberg M. and Kari L. (2002): Non-linear behavior of a rubber isolator system using fractional derivatives.-Vehicle Syst. Dynam., Vol. 37, No. 3, pp. 217-236.

[34] Suarez J.I., Vinagre B.M. and Chen Y.Q. (2003): Spatial path tracking of an autonomous industrial vehicle using fractional order controllers. - Proc. 11-th Int. Conf. Advanced Robotics, ICAR 2003, Coimbra, Portugal, pp. 405-410.

[35] Sum J.P.F., Leung C.S. and Chan L.W. (1996): Extended Kalman filter in recurrent neural network training and pruning. - Tech. Rep., Department of Computer Science and Engineering, Chinese University of Hong Kong, CS-TR-96-05.

[36] Vinagre B.M., Monje C.A. and Calderón A.J. (2002): Fractional order systems and fractional order control actions.-Lecture 3 IEEE CDC02 TW#2: Fractional calculus Applications in Automatic Control and Robotics'.

[37] Vinagre B.M. and Feliu V. (2002): Modeling and control of dynamic system using fractional calculus: Application to electrochemical processes and flexible structures. - Proc. 41-st IEEE Conf. Decision and Control, Las Vegas, NV, pp. 214-239.

[38] Zaborovsky V. and Meylanov R. (2001): Informational network traffic model based on fractional calculus. - Proc. Int. Conf. Info-tech and Info-net, ICII 2001, Beijing, China, Vol. 1, pp. 58-63.