Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders

Kaczorek, Tadeusz; Sajewski, Łukasz

doi:10.34768/amcs-2020-0047

Abstract

Descriptor and standard linear continuous-time systems with different fractional orders are investigated. Descriptor systems are analyzed making use of the Drazin matrix inverse. Necessary and sufficient conditions for the pointwise completeness and pointwise degeneracy of descriptor continuous-time linear systems with different fractional orders are derived. It is shown that (i) the descriptor linear continuous-time system with different fractional orders is pointwise complete if and only if the initial and final states belong to the same subspace, (ii) the descriptor linear continuous-time system with different fractional orders is not pointwise degenerated in any nonzero direction for all nonzero initial conditions. Results are reported for the case of two different fractional orders and can be extended to any number of orders.

References

Bingi, K., Ibrahim, R., Karsiti, M.N., Hassam, S.M. and Harindran, V.R. (2019). Frequency response based curve fitting approximation of fractional-order PID controllers, International Journal of Applied Mathematics and Computer Science29(2): 311–326, DOI: 10.2478/amcs-2019-0023.10.2478/amcs-2019-0023
Search in Google Scholar Back to article
Borawski, K. (2018). Analysis of the positivity of descriptor continuous-time linear systems by the use of Drazin inverse matrix method, in R. Szewczyk et al. (Eds), Automation 2018, Springer, Cham, pp. 172–182.10.1007/978-3-319-77179-3_16
Search in Google Scholar Back to article
Busłowicz, M. (2012). Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences60(2): 279–284.10.2478/v10175-012-0037-2
Search in Google Scholar Back to article
Campbell, S.L., Meyer, C.D. and Rose, N.J. (1976). Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM Journal on Applied Mathematics31(3): 411–425.10.1137/0131035
Search in Google Scholar Back to article
Dai, L. (1989). Singular Control Systems, Springer, Berlin.10.1007/BFb0002475
Search in Google Scholar Back to article
Djennoune, S., Bettayeb, M. and Al-Saggaf, U.M. (2019). Synchronization of fractional-order discrete-time chaotic systems by an exact delayed state reconstructor: Application to secure communication, International Journal of Applied and Mathematics and Computer Science29(1): 179–194, DOI: 10.2478/amcs-2019-0014.10.2478/amcs-2019-0014
Search in Google Scholar Back to article
Dzieliński, A., Sierociuk, D. and Sarwas, G. (2009). Ultracapacitor parameters identification based on fractional order model, Proceedings of the European Control Conference, Budapest, Hungary, pp. 196–200.
Search in Google Scholar Back to article
Fahmy, M.M. and O’Reilly, J. (1989). Matrix pencil of closed-loop descriptor systems: Infinite-eigenvalues assignment, International Journal of Control49(4): 1421–1431.10.1080/00207178908559713
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
Guang-Ren, D. (2010). Analysis and Design of Descriptor Linear Systems, Springer, New York, NY.10.1007/978-1-4419-6397-0_3
Search in Google Scholar Back to article
Kaczorek, T. (2009). Fractional positive linear systems, Kybernetes: The International Journal of Systems and Cybernetics38(7/8): 1059–1078.10.1108/03684920910976826
Search in Google Scholar Back to article
Kaczorek, T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences58(3): 453–458.10.2478/v10175-010-0043-1
Search in Google Scholar Back to article
Kaczorek, T. (2011). Selected Problems in Fractional Systems Theory, Springer, Berlin.10.1007/978-3-642-20502-6
Search in Google Scholar Back to article
Kaczorek, T. (2014). Drazin inverse matrix method for fractional descriptor continuous-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences62(3): 409–412.10.2478/bpasts-2014-0042
Search in Google Scholar Back to article
Kaczorek, T. (2015). Pointwise completeness and pointwise degeneracy of fractional descriptor continuous-time linear systems with regular pencils, Bulletin of the Polish Academy of Sciences: Technical Sciences63(1): 169–172.10.1515/bpasts-2015-0019
Search in Google Scholar Back to article
Kaczorek, T. (2019). Absolute stability of a class of fractional positive nonlinear systems, International Journal of Applied Mathematics and Computer Science29(1): 93–98, DOI: 10.2478/amcs-2019-0007.10.2478/amcs-2019-0007
Search in Google Scholar Back to article
Kaczorek, T. (2020). Global stability of positive standard and fractional nonlinear feedback systems, Bulletin of the Polish Academy of Sciences: Technical Sciences68(2): 285–288.
Search in Google Scholar Back to article
Kaczorek, T. and Busłowicz, M. (2009). Pointwise completeness and pointwise degeneracy of linear continuous-time fractional order systems, Journal of Automation, Mobile Robotics and Intelligent Systems3(1): 8–11.
Search in Google Scholar Back to article
Kaczorek, T. and Rogowski, K. (2015). Fractional Linear Systems and Electrical Circuits, Springer, Cham.10.1007/978-3-319-11361-6
Search in Google Scholar Back to article
Korobov, A.A. (2017). On pointwise degenerate linear delay-differential systems with nonnilpotent passive matrices, Journal of Applied and Industrial Mathematics11(3): 369–380.10.1134/S1990478917030085
Search in Google Scholar Back to article
Kucera, V. and Zagalak, P. (1988). Fundamental theorem of state feedback for singular systems, Automatica24(5): 653–658.10.1016/0005-1098(88)90112-4
Search in Google Scholar Back to article
Metel’skii, A.V. and Karpuk, V.V. (2009). On properties of pointwise degenerate linear autonomous control systems I, Automation and Remote Control70(10): 1613–1625.10.1134/S0005117909100038
Search in Google Scholar Back to article
Miller, K.S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY.
Search in Google Scholar Back to article
Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.
Search in Google Scholar Back to article
Sajewski, Ł. (2015). Minimum energy control of fractional positive continuous-time linear systems with two different fractional orders and bounded inputs, in K. Latawiec et al. (Eds), Advances in Modelling and Control of Non-integer-Order Systems, Springer, Cham, pp. 171–181.10.1007/978-3-319-09900-2_16
Search in Google Scholar Back to article
Sajewski, Ł. (2016). Reachability, observability and minimum energy control of fractional positive continuous-time linear systems with two different fractional orders, Multidimensional Systems and Signal Processing27(1): 27–41.10.1007/s11045-014-0287-2
Search in Google Scholar Back to article
Trzasko, W. (2014). Pointwise completeness and pointwise degeneracy of linear continuous-time systems with different fractional orders, in R. Szewczyk et al. (Eds), Recent Advances in Automation, Robotics and Measuring Techniques, Springer, Cham, pp. 307–316.10.1007/978-3-319-05353-0_30
Search in Google Scholar Back to article

Pointwise completeness and pointwise degeneracy of fractional standard and descriptor linear continuous-time systems with different fractional orders

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