Minimum energy control of descriptor fractional discrete-time linear systems with two different fractional orders

• Autorzy: Łukasz Sajewski
• Języki publikacji: EN

Abstrakty

EN

Reachability and minimum energy control of descriptor fractional discrete-time linear systems with different fractional orders are addressed. Using the Weierstrass-Kronecker decomposition theorem of the regular pencil, a solution to the state equation of descriptor fractional discrete-time linear systems with different fractional orders is given. The reachability condition of this class of systems is presented and used for solving the minimum energy control problem. The discussion is illustrated with numerical examples.

Słowa kluczowe

EN

minimum energy control; descriptor system; fractional system; discrete-time linear system

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2017
• Tom: 27
• Numer: 1
• Strony: 33-41

Daty

wydano

2017

otrzymano

2016-04-19

poprawiono

2016-09-11

zaakceptowano

2016-12-10

Twórcy

autor

Łukasz Sajewski

• Faculty of Electrical Engineering, Białystok University of Technology, ul. Wiejska 45D, 15-351 Białystok, Poland

Bibliografia

• Campbell, S.L., Meyer, C.D., Rose, N.J. (1976). Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM Journal on Applied Mathematics 31(3): 411-425, DOI: 10.1137/0131035.
• Caputo, M., Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications 1(2): 1-13, DOI: 10.12785/pfda/010201.
• Dai, L. (1989). Singular Control Systems, Lectures Notes in Control and Information Sciences, Vol. 118, Springer-Verlag, Berlin, DOI: 10.1007/BFb0002475.
• Dodig, M. and Stosic, M. (2009). Singular systems state feedbacks problems, Linear Algebra and Its Applications 431(9): 1267-1292, DOI: 10.1016/j.laa.2009.04.024.
• Dzieliński, A., Sierociuk, D. and Sarwas, G. (2009). Ultracapacitor parameters identification based on fractional order model, European Control Conference ECC, Budapest, Hungary, pp. 196-200.
• Ferreira, N.M.F., and Machado, J.A.T. (2003). Fractional-order hybrid control of robotic manipulators, 11th International Conference on Advanced Robotics, ICAR, Coimbra, Portugal, pp. 393-398, DOI: 10.1109/ICSMC.1998.725510.
• Guang-Ren, D. (2010). Analysis and Design of Descriptor Linear Systems, Springer, New York, NY, DOI: 10.1007/978-1-4419-6397-0.
• Kaczorek, T, and Klamka, J. (1986). Minimum energy control of 2D linear systems with variable coefficients, International Journal of Control 44(3): 645-650.
• Kaczorek, T. (1998). Vectors and Matrices in Automation and Electrotechnics, WNT, Warsaw.
• Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer-Verlag, London, DOI: 10.1007/978-1-44710221-2.
• Kaczorek, T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(3): 453-458, DOI: 10.2478/v10175-010-0043-1.
• Kaczorek, T. (2011a). Selected Problems in Fractional Systems Theory, Springer-Verlag, Berlin, DOI: 10.1007/978-3-642-20502-6.
• Kaczorek, T. (2011b). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions Circuits and Systems 58(6): 1203-1210, DOI: 10.1109/TCSI.2010.2096111.
• Kaczorek, T. (2011c). Reduction and decomposition of singular fractional discrete-time linear systems, Acta Mechanica et Automatica 5(5): 1-5.
• Kaczorek, T. (2011d). Singular fractional discrete-time linear systems, Control and Cybernetics 40(3): 1-8.
• Kaczorek, T. (2013a). Descriptor fractional linear systems with regular pencils, International Journal Applied Mathematics and Computer Science 23(2): 309-315, DOI: 10.2478/amcs-2013-0023.
• Kaczorek, T. (2013b). Solution of the state equations of descriptor fractional discrete-time linear systems with regular pencils, Technika Transportu Szynowego 10: 415-422.
• Kaczorek, T. (2013c). Singular fractional continuous-time and discrete-time linear systems, Acta Mechanica et Automatica 7(1): 26-33, DOI: 10.2478/ama-2013-0005.
• Kaczorek, T. (2014). Minimum energy control of fractional descriptor positive discrete-time linear systems, International Journal of Applied Mathematics and Computer Science 24(4): 735-743, DOI: 10.2478/amcs-2014-0054.
• Klamka, J. (1991).Controllability of Dynamical Systems, Kluwer Academic Press, Dordrecht.
• Klamka, J. (2009). Controllability and minimum energy control problem of infinite dimensional fractional discrete-time systems with delays, 1st Asian Conference on Intelligent Information and Database Systems ACIIDS, Dong Hoi, Vietnam, DOI: 10.1109/ACIIDS.2009.53.
• Klamka, J. (2010). Controllability and minimum energy control problem of fractional discrete-time systems, in D. Baleanu et al. (Eds.), New Trends in Nanotechology and Fractional Calculus, Springer-Verlag, New York, NY, pp. 503-509, DOI: 10.1007/978-90-481-3293-5_45.
• Klamka, J. (2014). Controllability and minimum energy control of linear fractional discrete-time infinite-dimensional systems, 11th IEEE International Conference on Control & Automation ICCA, Taichung, Taiwan, pp. 1210-1214, DOI: 10.1109/ICCA.2014.6871094.
• Miller, K.S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY.
• Nishimoto, K. (1984). Fractional Calculus, Decartess Press, Koriama.
• Oldham, K.B. and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY.
• Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.
• Popović, J.K., Pilipović S. and Atanacković, T.M. (2013). Two compartmental fractional derivative model with fractional derivatives of different order, Communications in Nonlinear Science and Numerical Simulation 18(10): 2507-2514, DOI: 10.1016/j.cnsns.2013.01.004.
• Sajewski, Ł. (2015). Solution of the state equation of descriptor fractional continuous-time linear systems with two different fractional, in R. Szewczyk et al. (Eds.), Progress in Automation, Robotics and Measuring Techniques, Advances in Intelligent Systems and Computing, Vol. 350, Springer, Cham, pp. 233-242, DOI: 10.1007/978-3-319-15796-2_24.
• Sajewski, Ł. (2016a). Reachability, observability and minimum energy control of fractional positive continuous-time linear systems with two different fractional orders, Multidimensional Systems and Signal Processing 27(1): 27-41, DOI: 10.1007/s11045-014-0287-2.
• Sajewski, Ł. (2016b). Descriptor fractional discrete-time linear system and its solution-comparison of three different methods, in R. Szewczyk et al. (Eds.), Challenges in Automation, Robotics and Measurement Techniques, Advances in Intelligent Systems and Computing, Vol. 440, Springer, Cham, pp. 37-50, DOI: 10.1007/978-3-319-29357-8_4.
• Sajewski, Ł. (2016c). Descriptor fractional discrete-time linear system with two different fractional orders and its solution, Bulletin of the Polish Academy of Sciences: Technical Sciences 64(1): 15-20, DOI: 10.1515/bpasts-2016-0003.
• Van Doore, P. (1979). The computation of Kronecker's canonical form of a singular pencil, Linear Algebra and Its Applications 27: 103-140, DOI: 10.1016/0024-3795(79)90035-1.
• Van Dooren, P. and Beelen, T. (1988). An improved algorithm for the computation of Kronecker’s canonical form of a singular pencil, Linear Algebra and Its Applications 105: 9-65, DOI: 10.1016/0024-3795(88)90003-1.

Identyfikatory

DOI

10.1515/amcs-2017-0003