# Fractional kalman filter algorithm for the states parameters and order of fractional system estimation

Dominik Sierociuk; Andrzej Dzieliński

International Journal of Applied Mathematics and Computer Science (2006)

- Volume: 16, Issue: 1, page 129-140
- ISSN: 1641-876X

## Access Full Article

top## Abstract

top## How to cite

topSierociuk, Dominik, and Dzieliński, Andrzej. "Fractional kalman filter algorithm for the states parameters and order of fractional system estimation." International Journal of Applied Mathematics and Computer Science 16.1 (2006): 129-140. <http://eudml.org/doc/207770>.

@article{Sierociuk2006,

abstract = {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.},

author = {Sierociuk, Dominik, Dzieliński, Andrzej},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {linebreak extended fractional Kalman filter; discrete fractional state-space systems; parameters estimation; fractional Kalman filter; order estimation; extended fractional Kalman filter},

language = {eng},

number = {1},

pages = {129-140},

title = {Fractional kalman filter algorithm for the states parameters and order of fractional system estimation},

url = {http://eudml.org/doc/207770},

volume = {16},

year = {2006},

}

TY - JOUR

AU - Sierociuk, Dominik

AU - Dzieliński, Andrzej

TI - Fractional kalman filter algorithm for the states parameters and order of fractional system estimation

JO - International Journal of Applied Mathematics and Computer Science

PY - 2006

VL - 16

IS - 1

SP - 129

EP - 140

AB - 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.

LA - eng

KW - linebreak extended fractional Kalman filter; discrete fractional state-space systems; parameters estimation; fractional Kalman filter; order estimation; extended fractional Kalman filter

UR - http://eudml.org/doc/207770

ER -

## References

top- Bologna M. and Grigolini P. (2003): Physics of Fractal Operators. - New York: Springer-Verlag.
- Brown R.G. and Hwang P.Y.C. (1997): Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises and Solutions. -New York: Wiley. Zbl0868.93002
- 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.
- 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.
- Engheta N. (1997): On the role of fractional calculus in electromagnetic theory. - IEEE Trans. Antenn. Prop., Vol. 39, No. 4, pp. 35-46.
- 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.
- Gałkowski K. (2005): Fractional polynomials and nD systems. - Proc. IEEE Int. Symp. Circuits and Systems, ISCAS'2005, Kobe, Japan, CD-ROM.
- Haykin S. (2001): Kalman Filtering and Neural Networks. - New York: Wiley.
- Hilfer R., Ed. (2000): Application of Fractional Calculus in Physics. - Singapore: World Scientific. Zbl0998.26002
- 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. Zbl0984.65018
- 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.
- Miller K.S. and Ross B. (1993): An Introduction to the Fractional Calculus and Fractional Differenctial Equations. - New York: Wiley. Zbl0789.26002
- Moshrefi-Torbati M. and Hammond K. (1998): Physical and geometrical interpretation of fractional operators. - J. Franklin Inst., Vol. 335B, No. 6, pp. 1077-1086. Zbl0989.26004
- Nishimoto K. (1984): Fractional Calculus. - Koriyama: Decartes Press. Zbl0605.26006
- Oldham K.B. and Spanier J. (1974): The Fractional Calculus. - New York: Academic Press. Zbl0292.26011
- Ostalczyk P. (2000): The non-integer difference of the discrete-time function and its application to the control system synthesis. - Int. J. Syst. Sci., Vol. 31, No. 12, pp. 1551-1561. Zbl1080.93592
- Ostalczyk P. (2004a): Fractional-Order Backward Difference Equivalent Forms Part I - Horners Form. - Proc. 1-st IFAC Workshop Fractional Differentation and its Applications, FDA'04, Enseirb, Bordeaux, France, pp. 342-347.
- 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.
- Oustaloup A. (1993): Commande CRONE. -Paris: Hermes.
- Oustaloup A. (1995): La derivation non entiere. - Paris: Hermes. Zbl0864.93004
- Podlubny I. (1999): Fractional Differential Equations. - San Diego: Academic Press. Zbl0924.34008
- Podlubny I. (2002): Geometric and physical interpretation of fractional integration and fractional differentiation. - Fract. Calc. Appl. Anal., Vol. 5, No. 4, pp. 367-386. Zbl1042.26003
- Podlubny I., Dorcak L. and Kostial I. (1997): On fractional derivatives, fractional-order systems and PI^λ D^μ-controllers. - Proc. 36-th IEEE Conf. Decision and Control, San Diego, CA, pp. 4985-4990.
- 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. Zbl1145.94372
- 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.
- 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.
- Riewe F. (1997): Mechanics with fractional derivatives. - Phys. Rev. E, Vol. 55, No. 3, pp. 3581-3592.
- Riu D., Retiere N. and Ivanes 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.
- Samko S.G., Kilbas A.A. and Maritchev O.I. (1993): Fractional Integrals and Derivative. Theory and Applications. - London: Gordon and Breach.
- Schutter J., Geeter J., Lefebvre T. and Bruynickx H. (1999): Kalman filters: A tutorial. -Available at http://citeseer.ist.psu.edu/443226.html
- 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
- Sierociuk D. (2005b): Application of a fractional Kalman filter toparameter estimation of a fractional-order system. -Proc. 15-th Nat. Conf. Automatic Control, Warsaw, Poland,Vol. 1, pp. 89-94, (in Polish).
- Sjoberg 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.
- 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.
- 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.
- Vinagre B.M., Monje C.A. and Calderon 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'.
- 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.
- 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.

## Citations in EuDML Documents

top- Tomasz Barszcz, Piotr Czop, Estimation of feedwater heater parameters based on a grey-box approach
- Tadeusz Kaczorek, Fractional positive continuous-time linear systems and their reachability
- Said Guermah, Said Djennoune, Maamar Bettayeb, Controllability and observability of linear discrete-time fractional-order systems
- Tadeusz Kaczorek, Reachability of cone fractional continuous-time linear systems
- Rafał Stanisławski, Krzysztof J. Latawiec, Normalized finite fractional differences: Computational and accuracy breakthroughs

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.