Controllability and Observability of Linear Discrete-Time Fractional-Order Systems

Guermah, Said; Djennoune, Said; Bettayeb, Maamar

doi:10.2478/v10006-008-0019-6

Abstract

In this paper we extend some basic results on the controllability and observability of linear discrete-time fractional-order systems. For both of these fundamental structural properties we establish some new concepts inherent to fractional-order systems and we develop new analytical methods for checking these properties. Numerical examples are presented to illustrate the theoretical results.

References

Antsaklis P.J. and Michel A.N. (1997). Linear Systems, McGraw-Hill, New York.
Search in Google Scholar
Åström K. J. and Wittenmark B. (1996). Computer- Controlled Systems, Theory and Design, 3rd Ed., Prentice Hall Inc., New Jersey.
Search in Google Scholar
Axtell M. and Bise E. M. (1990). Fractional calculus applications in control systems, Proceedings of the IEEE 1990 National Aerospace and Electronics Conference, New York, USA, pp. 536-566.
Search in Google Scholar
Battaglia J. L., Cois O., Puigsegur L. and Oustaloup A. (2001). Solving an inverse heat conduction problem using a non-integer identified model, International Journal of Heat and Mass Transfer, 44(14): 2671-2680.10.1016/S0017-9310(00)00310-0
Search in Google Scholar
Bettayeb M. and Djennoune S. (2006). A note on the controllability and the observability of fractional dynamical systems, Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Workshop Applications, Porto, Portugal, pp. 506-511.
Search in Google Scholar
Boukas E.K. (2006). Discrete-time systems with time-varying time delay: Stability and stabilizability, Mathematical Problems in Engineering, bf 2006 (ID42489): 1-10.10.1155/MPE/2006/42489
Search in Google Scholar
Cois O., Oustaloup A., Battaglia E. and Battaglia J.L. (2002). Non integer model from modal decomposition for time domain identification, 41st IEEE CDC'2002 Tutorial Workshop 2, Las Vegas, USA.
Search in Google Scholar
Debeljković D. Lj., Aleksendrić M., Yi-Yong N. and Zhang Q. L. (2002). Lyapunov and non-Lyapunov stability of linear discrete time delay systems, Facta Universitatis, Series: Mechanical Engineering 1(9): 1147-1160.
Search in Google Scholar
Dorĉák L., Petras I. and Kostial I. (2000). Modeling and analysis of fractional-order regulated systems in the state-space, Procedings of International Carpathian Control Conference, High Tatras, Slovak Republic, pp. 185-188.
Search in Google Scholar
Dzieliński A. and Sierociuk D. (2005). Adaptive feedback control of fractional order discrete state-space systems, Proceedings of the 2005 International Conference on Computational Intelligence for Modelling, Control and Automation, and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA-IAWTIC'05), Vienna Austria, pp. 804-809.
Search in Google Scholar
Dzieliński A. and Sierociuk D. (2006). Observer for discrete fractional order systems, Proceedings of the 2nd IFAC Workshop on Fractional Differentiation Applications, Porto, Portugal, pp. 524-529.
Search in Google Scholar
Dzieliński A. and Sierociuk D. (2007). Reachability, controllability and observability of the fractional order discrete statespace system, Proceedings of the IEEE/IFAC International Conference on Methods and Models in Automation and Robotics, MMAR'2007, Szczecin, Poland, pp. 129-134.
Search in Google Scholar
Gorenflo R. and Mainardi F. (1997). Fractional calculus: Integral and differential equations of fractional order, in (A. Carpintieri and F. Mainardi, Eds.) Fractals and Fractional Calculus in Continuum Mechanics, Vienna, New York, Springer Verlag.10.1007/978-3-7091-2664-6_6
Search in Google Scholar
Hanyga A. (2003). Internal variable models of viscoelasticity with fractional relaxation laws, Proceddings of Design Engineering Technical Conference, Mechanical Vibration and Noise, 48395, American Society of Mechanical Engineers, Chicago, USA.10.1115/DETC2003/VIB-48395
Search in Google Scholar
Hotzel R. and Fliess M. (1998). On linear system with a fractional derivation: Introductory theory and examples, Mathematics and Computers in Simulation 45(3): 385-395.10.1016/S0378-4754(97)00118-3
Search in Google Scholar
Ichise M., Nagayanagi Y. and Kojima T. (1971). An analog simulation of non integer order transfer functions for analysis of electrode processes, Journal of Electroanalytical Chemistry 33(2): 253-265.10.1016/S0022-0728(71)80115-8
Search in Google Scholar
Kilbas A. A., Srivasta H.M. and Trujillo J. J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
Search in Google Scholar
Lakshmikantham D. T. V. (1998). Theory of Difference Equations: Numerical Methods and Applications, Academic Press, New York.
Search in Google Scholar
Manabe S. (1960). The non-integer integral and its application to control systems, Japanese Institute of Electrical Engineers Journal 80(860): 589-597.
Search in Google Scholar
Matignon D. (1994). Reprèsentation en variables d'ètat de modèles de guides d'ondes avec dèrivation fractionnaire, Ph.D. thesis, Universitè Paris XI, France.
Search in Google Scholar
Matignon D., d'Andrèa Novel B., Depalle P. and Oustaloup A. (1994). Viscothermal Losses in Wind Instruments: A Non-Integer Model, Academic Verlag, Berlin.
Search in Google Scholar
Matignon D. and d'Andrèa-Novel B. (1996). Some results on controllability and observability of finite-dimensional fractional differential systems, Proceedings of the IMACS, IEEE SMC Conference, Lille, France, pp. 952-956.
Search in Google Scholar
Matignon D. (1996). Stability results on fractional differential with application to control processing, Proceedings of the IAMCS, IEEE SMC Conference, Lille, France, pp. 963-968.
Search in Google Scholar
Miller K. S. and Ross B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations,Wiley, New York.
Search in Google Scholar
Mittag-Leffler G. (1904). Sur la reprèsentation analytique d'une branche uniforme d'une fonction monogène, Acta Mathematica 29: 10-181.10.1007/BF02403200
Search in Google Scholar
Oldham K. B. and Spanier J. (1974). The Fractional Calculus, Academic Press, New York.
Search in Google Scholar
Oustaloup A. (1983). Systèmes asservis linèaires d'ordre fractionnaire, Masson, Paris.
Search in Google Scholar
Oustaloup A. (1995). La Dèrivation non entière: Thèorie, synthèse et applications, Hermès, Paris.
Search in Google Scholar
Peng Y., Guangming X. and Long W. (2003). Controllability of linear discrete-time systems with time-delay in state, available at http://dean.pku.edu.cn/bksky/1999tzlwj/4.pdf
Search in Google Scholar
Podlubny I. (1999). Fractional Differential Equations, Academic Press, San Diego.
Search in Google Scholar
Raynaud H. F., Zergainoh A. (2000). State-space representation for fractional-order controllers, Automatica 36(7): 1017-1021.10.1016/S0005-1098(00)00011-X
Search in Google Scholar
Sabatier J., Cois O. and Oustaloup A. (2002). Commande de systèmes non entiers par placement de pôles, Deuxième Confèrence Internationale Francophone d'Automatique, CIFA, Nantes, France.
Search in Google Scholar
Samko S. G., Kilbas A. A. and Marichev O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam.
Search in Google Scholar
Sierociuk D. and Dzieliński A. (2006). Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation, International Journal of Applied Mathematics and Computer Science 16(1): 129-140.
Search in Google Scholar
Valerio D. and Sa da Costa J. (2004). Non-integer order control of a flexible robot, Proceedings of the IFAC Workshop on Fractional Differentiation and its Applications, FDA'04, Bordeaux, France.
Search in Google Scholar
Vinagre B. M., Monje C. A. and Caldero A. J. (2002). Fractional order systems and fractional order actions, Tutorial Workshop 2: Fractional Calculus Applications in Automatic Control and Robotics, 41st IEEE CDC, Las Vegas, USA.
Search in Google Scholar

Controllability and Observability of Linear Discrete-Time Fractional-Order Systems

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