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Bulletin of the Polish Academy of Sciences Technical Sciences

The Journal of Polish Academy of Sciences

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Volume 61, Issue 3 (Sep 2013)

Issues

Fixed final time and free final state optimal control problem for fractional dynamic systems – linear quadratic discrete-time case

A. Dzieliński
  • Electrical Engineering Institute of Control and Industrial Electronics, Warsaw University of Technology, 75 Koszykowa St., 00-662 Warszawa, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ P.M. Czyronis
Published Online: 2013-10-15 | DOI: https://doi.org/10.2478/bpasts-2013-0072

Abstract

The optimization problem for fractional discrete-time systems with a quadratic performance index has been formulated and solved. The case of fixed final time and a free final state has been considered. A method for numerical computation of optimization problems has been presented. The presented method is a generalization of the well-known method for discrete-time systems of integer order. The efficiency of the method has been demonstrated on numerical examples and illustrated by graphs. Graphs also show the differences between the fractional and classical (standard) systems theory. Results for other cases of the fractional system order (coefficient ) and not illustrated with numerical examples have been obtained through a computer algorithm written for this purpose.

Keywords : fractional order systems; discrete-time systems; optimal control; linear quadratic performance index

  • [1] A. Dzieliński, D. Sierociuk, and G. Sarwas, “Some applications of fractional order calculus”, Bull. Pol. Ac.: Tech. 58 (4), 583-592 (2010).Google Scholar

  • [2] D. Sierociuk, A. Dzieliński, G. Sarwas, I. Petras, I. Podlubny, and T. Skovranek, “Modelling heat transfer in heterogeneous media using fractional calculus”, Phil. Trans. R. Soc. A (371), 20130146 (2013).Google Scholar

  • [3] D. Sierociuk and B.M. Vinagre, “Infinite horizon statefeedback LQR controller for fractional systems”, Decision andControl (CDC), 2010 49th IEEE Conf. 1, 6674-6679 (2010).Google Scholar

  • [4] C. Tricaud and Y.Q. Chen, “An approximate method for numerically solving fractional order optimal control problems of general form”, Comput. Math. Appl. 59, 1644-1655 (2010).CrossrefGoogle Scholar

  • [5] C. Tricaud and Y.Q. Chen, “Time-optimal control of systems with fractional dynamics”, Int. J. Differ. Equ. 1, 1-16 (2010).Google Scholar

  • [6] M. Busłowicz, “Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders”, Bull. Pol. Ac.: Tech. 60 (2), 279-284 (2012).Web of ScienceGoogle Scholar

  • [7] R. Bellman, Dynamic Programming, University Press, Princeton, 1957.Google Scholar

  • [8] T. Kaczorek, Control Theory, vol. II, PWN, Warsaw, 1981, (in Polish).Google Scholar

  • [9] F.L. Lewis and V.L. Syrmos, Optimal Control, 2nd ed, Wiley- IEEE, London, 1995.Google Scholar

  • [10] D.S. Naidu, Optimal Control Systems, Electrical Engineering, CRC Press, Inc., Boca Raton, 2002.Google Scholar

  • [11] P. Ostalczyk, Epitome of the Fractional calculus: Theory andIts Applications in Automatics, Lodz University of Technology Publishing House, Łodź, 2008.Google Scholar

  • [12] I. Podlubny, Fractional Differential Equations. An Introductionto Fractional Derivatives, Fractional Differential Equations,Some Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, 1999.Google Scholar

  • [13] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integralsand Derivatives: Theory and Applications, Gordon and Breach Science, New York, 1993.Google Scholar

  • [14] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory andApplications of Fractional Differential Equations, vol. 204, Elsevier Science Inc., New York, 2006.Google Scholar

  • [15] F. Liu, M.M. Meerschaert, S. Momani, N.N. Leonenko, C. Wen, and O.P. Agrawal, “Fractional differential equations”, Int. J. Differ. Equations 464321, CD-ROM (2010).Google Scholar

  • [16] X. Cai and F. Liu, “Numerical simulation of the fractionalorder control system”, J. Appl. Math. Comput. 23, 229-241 (2007).CrossrefGoogle Scholar

  • [17] M.M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations”, J. Computational and Applied Mathematics 172 (1), 65-77 (2004).Google Scholar

  • [18] O.P. Agrawal, “Formulation of Euler - Lagrange equations for fractional variational problems”, J. Mathematical Analysis andApplications 272 (1), 368-379 (2002).Google Scholar

  • [19] O.P. Agrawal, “A general formulation and solution scheme for fractional optimal control problems”, Nonlinear Dynamics 38, 323-337 (2004).CrossrefGoogle Scholar

  • [20] O.P. Agrawal, “Fractional variational calculus and the transversality conditions”, J. Physics Math. Theor. 39, 10375-10384 (2006).CrossrefGoogle Scholar

  • [21] O.P. Agrawal, “Fractional variational calculus in terms of Riesz fractional derivatives”, J. Physics. Math. Theor. 40 (24), 6287-6303 (2007).CrossrefGoogle Scholar

  • [22] O.P. Agrawal, “A general finite element formulation for fractional variational problems”, J. Mathemat. Analysis and Appl. 337 (1), 1-12 (2008).Google Scholar

  • [23] G.S.F. Frederico and D.F.M. Torres, “Fractional conservation laws in optimal control theory”, Nonlinear Dynamics 53 (3), 215-222 (2008).CrossrefWeb of ScienceGoogle Scholar

  • [24] Z. D. Jelicic and N. Petrovacki, “Optimality conditions and a solution scheme for fractional optimal control problems”, JStruct Multidisciplinary Optimization 38 (6), 571-581 (2008).Google Scholar

  • [25] R.K. Biswas and S. Sen, “Fractional optimal control problems with specified final time”, ASME J. Comput. Nonlinear Dyn. 6, 021009.1-021009.6 (2011).Google Scholar

  • [26] R.K. Biswas and S. Sen, “Fractional optimal control problems: a pseudo-state-space approach”, J. Vib. Control 17, 1034-1041 (2010).Web of ScienceGoogle Scholar

  • [27] D. Sierociuk and A. Dzieliński, “Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation”, Int. J. Appl. Math. Comput. Sci. 16 (1), 129-140 (2006).Google Scholar

  • [28] D. Sierociuk, I. Tejado, and B.M. Vinagre, “Improved fractional Kalman filter and its application to estimation over lossy networks”, Signal Processing 91 (3), 542-552 (2011).CrossrefGoogle Scholar

  • [29] T. Kaczorek, “New stability tests of positive standard and fractional linear systems”, Circuits and Systems 2, 261-268 (2011).Google Scholar

  • [30] T. Kaczorek, “Positive linear systems consisting of n subsystems with different fractional orders”, IEEE Trans. on Circuitsand Systems 58, 1203-1210 (2011).Web of ScienceGoogle Scholar

  • [31] T. Kaczorek, Selected Problems of Fractional Systems Theory, Lecture Notes in Control and Information Sciences 411, Springer, Berlin, 2011.Web of ScienceGoogle Scholar

  • [32] T. Kaczorek, “Positive fractional continuous-time linear systems with singular pencils”, Bull. Pol. Ac.: Tech. 60 (1), 9-12 (2012).Web of ScienceGoogle Scholar

  • [33] A. Dzieliński and P. M. Czyronis, “Fixed final time optimal control problem for fractional dynamic systems - linear quadratic discrete-time case”, Advances in Control Theory andAutomation 1, 71-80 (2012).Google Scholar

  • [34] A. Dzieliński and P. M. Czyronis, “Computer algorithms for solving optimization problems for discrete-time fractional systems”, Eur. Control Conf. ECC 1, CD-ROM (2013). Google Scholar

About the article

Published Online: 2013-10-15

Published in Print: 2013-09-01


Citation Information: Bulletin of the Polish Academy of Sciences: Technical Sciences, ISSN (Print) 0239-7528, DOI: https://doi.org/10.2478/bpasts-2013-0072.

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