Stability of fractional positive continuous-time linear systems with state matrices in integer and rational powers

• Autorzy: Kaczorek T.

Identyfikatory

DOI

10.1515/bpasts-2017-0034

• Języki publikacji: EN

Abstrakty

EN

The stability of fractional standard and positive continuous-time linear systems with state matrices in integer and rational powers is addressed. It is shown that the fractional systems are asymptotically stable if and only if the eigenvalues of the state matrices satisfy some conditions imposed on the phases of the eigenvalues. The fractional standard systems are unstable if the state matrices have at least one positive eigenvalue.

Słowa kluczowe

EN

stability; fractional; positive continuous-time; linear system; integers; rational powers; order

PL

stabilność; pozytywny; liniowy; czas ciągły; system

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2017
• Tom: Vol. 65, nr 3
• Strony: 305--311
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Kaczorek T.

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

Bibliografia

• [1] L. Farina and S. Rinaldi: “Positive Linear Systems; Theory and Applications”, J. Wiley, New York, 2000.
• [2] T. Kaczorek: “Positive 1D and 2D Systems”, Springer-Verlag, London, 2001.
• [3] K. B. Oldham and J. Spanier: “The Fractional Calculus”, Academic Press, New York, 1974.
• [4] P. Ostalczyk: “Epitome of the fractional calculus: Theory and its Applications in Automatics”, Wydawnictwo Politechniki Łódzkiej, Łódź, 2008 (in Polish).
• [5] I. Podlubny: “Fractional Differential Equations”, Academic Press, San Diego, 1999.
• [6] T. Kaczorek: “Asymptotic stability of positive fractional 2D linear systems”, Bull. Pol. Ac.: Tech. 57(3), 289‒292 (2009).
• [7] T. Kaczorek: “Minimum energy control of fractional positive continuous-time linear systems”, Bull. Pol. Ac.: Tech. 61(4), 803‒807 (2013).
• [8] T. Kaczorek: “Selected Problems of Fractional Systems Theory”, Springer-Verlag, Berlin, 2012.
• [9] P. J. Antsaklis, A. N. Michel: “Linear Systems”, Birkhauser, Boston, 2006.
• [10] A. G. Radwan, A. M. Soliman, A. S. Elwakil and A. Sedeek: “On the stability of linear systems with fractional-order elements”, Chaos, Solitones and Fractals 40(5), 2317‒2328 (2009).
• [11] S. H. Żak: “Systems and Control”, Oxford University Press, New York, 2003.
• [12] M. Busłowicz: “Stability of linear continuous time fractional order systems with delays of the retarded type”, Bull. Pol. Ac.: Tech. 56(4), 319‒324 (2008).
• [13] E. J. Solteiro Pires, J. A. Tenreiro Machado and P. B. Moura Oliveira: “Fractional dynamics in genetic algorithms”, Workshop on Fractional Differenation and its Application 2, 414‒419 (2006).
• [14] B. M. Vinagre, C. A. Monje and A. J. Calderon: “Fractional order systems and fractional order control actions”, Lecture 3 IEEE CDC’02 TW#2: Fractional calculus Applications in Automatic Control and Robotics, 2002.
• [15] T. Kaczorek: “Practical stability of positive fractional discrete-time linear systems”, Bull. Pol. Ac.: Tech. 56(4), 313‒317 (2008).
• [16] A. Dzieliński, D. Sierociuk, G. Sarwas: “Ultracapacitor parameters identification based on fractional order model”, Proc ECC’09, Budapest, 2009.
• [17] T. Kaczorek: “Positive linear systems consisting of n subsystems with different fractional orders”, IEEE Trans. Circuits and Systems 58(6), 1203‒1210 (2011).
• [18] T. Kaczorek: “Fractional positive continuous-time systems and their reachability”, Int. J. Appl. Math. Comput. Sci. 18(2), 223‒228 (2008).
• [19] J. Klamka: “Controllability and minimum energy control problem of fractional discrete-time systems”, Chapter in “New Trends in Nanotechology and Fractional Calculus”, Eds. D. Baleanu, Z. B. Guvenc, J. A. Tenreiro Machado, Springer-Verlag, New York, 503‒509 (2010).
• [20] Ł. Sajewski: “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 (2016).
• [21] F. R. Gantmacher: “The Theory of Matrices”. Chelsea Pub. Comp., London, 1959.
• [22] T. Kaczorek: “Vectors and Matrices in Automation and Electrotechnics”, WNT, Warszawa, 1998 (in Polish).
• [23] T. Kaczorek: “Determination of the set of Metzler matrices for given stable polynomials”, Measurement Automation Monitoring 58(5), 407‒412 (2012).

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).