This paper presents a series of new results in finite and infinite-memory modeling of discrete-time fractional differences. The introduced normalized finite fractional difference is shown to properly approximate its fractional difference original, in particular in terms of the steady-state properties. A stability analysis is also presented and a recursive computation algorithm is offered for finite fractional differences. A thorough analysis of computational and accuracy aspects is culminated with the introduction of a perfect finite fractional difference and, in particular, a powerful adaptive finite fractional difference, whose excellent performance is illustrated in simulation examples.
Bandrowski, B., Karczewska, A. and Rozmej, P. (2010). Numerical solutions to integral equations equivalent to differential equations with fractional time, International Journal of Applied Mathematics and Computer Science 20(2): 261-269, DOI: 10.2478/v10006-010-0019-1.
Barbosa, R. and Machado, J. (2006). Implementation of discrete-time fractional-order controllers based on LS approximations, Acta Polytechnica Hungarica 3(4): 5-22.
Busłowicz, M. and Kaczorek, T. (2009). Simple conditions for practical stability of positive fractional discrete-time linear systems, International Journal of Applied Mathematics and Computer Science 19(2): 263-269, DOI: 10.2478/v10006-009-0022-6.
Chen, Y., Vinagre, B. and Podlubny, I. (2003). A new discretization method for fractional order differentiators via continued fraction expansion, Proceedings of DETC’2003, ASME Design Engineering Technical Conferences, Chicago, IL, USA, Vol. 340, pp. 349-362.
Debeljković, D.L., Aleksendric´, M., Yi-Yong, N. and Zhang, Q. (2002). Lyapunov and nonlyapunov stability of linear discrete time delay systems, Facta Universitatis Mechanical Engineering 14(9-10): 1147-1160.
Delavari, H., Ranjbar, A., Ghaderi, R. and Momani, S. (2010). Fractional order control of a coupled tank, Nonlinear Dynamics 61(3): 383-397.
Dzieliński, A. and Sierociuk, D. (2008). Stability of discrete fractional order state-space systems, Journal of Vibration and Control 14(9-10): 1543-1556.
Guermah, S., Djennoune, S. and Bettayeb, M. (2010). A new approach for stability analysis of linear discrete-time fractional-order systems, in D. Baleanu, Z.B. Güvenç and J.A.T. Machado (Eds.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer, Dordrecht, pp. 151-162.
Hunek, W.P. and Latawiec, K.J. (2011). A study on new right/left inverses of nonsquare polynomial matrices, International Journal of Applied Mathematics and Computer Science 21(2): 331-348, DOI: 10.2478/v10006-011-0025-y.
Kaczorek, T. (2008). Practical stability of positive fractional discrete-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 313-317.
Latawiec, K.J. (2004). The Power of Inverse Systems in Linear and Nonlinear Modeling and Control, Opole University of Technology Press, Opole.
Liavas, A.P. and Regalia, P. (1999). On the numerical stability and accuracy of the conventional recursive least squares algorithm, IEEE Transactions on Signal Processing 47(1): 88-96.
Maione, G. (2006). A digital, noninteger order, differentiator using laguerre orthogonal sequences, International Journal of Intelligent Control and Systems 11(2): 77-81.
Miller, K. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Willey, New York, NY.
Momani, S. and Odibat, Z. (2007). Numerical approach to differential equations of fractional order, Journal of Computational and Applied Mathematics 207(1): 96 - 110.
Monje, C., Chen, Y., Vinagre, B., Xue, D. and Feliu, V. (2010). Fractional-order Systems and Controls, Springer-Verlag, London.
Oldham, K. and Spanier, J. (1974). The Fractional Calculus, Academic Press, Orlando, FL.
Ortigueira, M.D. (2000). Introduction to fractional linear systems, II: Discrete-time case, IEE Proceedings on Vision, Image and Signal Processing 147(1): 71-78.
Ostalczyk, P. (2000). The non-integer difference of the discrete-time function and its application to the control system synthesis, International Journal of Systems Science 31(12): 1551-1561.
Ostalczyk, P. (2010). Stability analysis of a discrete-time system with a variable-fractional-order controller, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(4): 613-619.
Petráš, I., Dorčák, L. and Koštial, I. (2000). The modelling and analysis of fractional-order control systems in discrete domain, Proceedings of the International Carpa-tian Control Conference, High Tatras, Slovak Republic, pp. 257-260.
Petráš, I. and Vinagre, B. (2002). Practical application of digital fractional-order controller to temperature control, Acta Montanistica Slovaca 7(2): 131-137.
Podlubny, I. (1999). Fractional Differential Equations, Academic Press, Orlando, FL.
Riu, D., Retiére, N. and Ivanes, M. (2001). Turbine generator modeling by non-integer order systems, IEEE International Conference on Electric Machines and Drives, Cambridge, MA, USA, pp. 185-187.
Saeedi, H., Mollahasani, N., Moghadam, M.M. and Chuev, G.N. (2011). An operational Haar wavelet method for solving fractional Volterra integral equations, International Journal of Applied Mathematics and Computer Science 21(3): 535-547, DOI: 10.2478/v10006-011-0042-x.
Sierociuk, D. and Dzielin´ski, A. (2006). Fractional Kalman filter algorithm for states, parameters and order of fractional system estimation, International Journal of Applied Mathematics and Computer Science 16(1): 129-140.
Stanisławski, R. (2009). Identification of open-loop stable linear systems using fractional orthonormal basis functions, Proceedings of the 14th International Conference on Methods and Models in Automation and Robotics, Mie˛dzyzdroje, Poland, pp. 935-985.
Stanisławski, R. and Latawiec, K.J. (2010). Modeling of open-loop stable linear systems using a combination of a finite fractional derivative and orthonormal basis functions, Proceedings of the 15th International Conference on Methods and Models in Automation and Robotics, Mie˛dzyz-droje, Poland, pp. 411-414.
Stanisławski, R. and Latawiec, K.J. (2011). Finite approximations of a discrete-time fractional derivative, 16th International Conference on Methods and Models in Automation and Robotics, Mie˛dzyzdroje, Poland, pp. 142-145.
Stojanovic, S.B. and Debeljkovic, D.L. (2010). Simple stability conditions of linear discrete time systems with multiple delay, Serbian Journal of Electrical Engineering 7(1): 69-79.
Sun, H., Chen, W. and Chen, Y. (2009). Variable-order fractional differential operators in anomalous diffusion modeling, Physica A: Statistical Mechanics and Its Applications 388(21): 4586-4592.
Tseng, C., Pei, S. and Hsia, S. (2000). Computation of fractional derivatives using Fourier transform and digital FIR diferentiator, Signal Processing 80(1): 151-159.
Valério, D. and Sá da Costa, J. (2011). Variable-order fractional derivatives and their numerical approximations, Signal Processing 91(3): 470-483.
Verhaegen, M. H. (1989). Round-off error propagation in four generally-applicable, recursive, least-squares estimation schemes, Automatica 25(3): 437-444.
Vinagre, B., Podlubny, I., Hernandez, A. and Feliu, V. (2000). Some approximations of fractional order operators used in control theory and applications, Fractional Calculus & Applied Analysis 3(3): 945-950.
Zaborowsky, V. and Meylaov, R. (2001). Informational network traffic model based on fractional calculus, Proceedings of the International Conference on Info-tech and Info-net, ICII 2001, Beijing, China, Vol. 1, pp. 58-63.