Abstract
A method for decentralized stabilization of fractional positive descriptor discrete-time linear systems is proposed. Necessary and sufficient conditions for the positivity and decentralized stabilization of the fractional positive descriptor discrete-time linear systems are established. The efficiency of proposed method is demonstrated on numerical example.
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References
Ramai, M.A., Tadeo, F.: Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circ. Syst. II Expr. Briefs 54(2), 151–155 (2007)
Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics (SIAM) (1994)
Bru, R., Coll, C., Sanchez, E.: About positively discrete-time singular systems. In: Mastorakis, M.E. (ed.) System and Control: Theory and Applications, pp. 44–48. World Scientific and Engineering Society, Athens (2000)
Bru, R., Coll, C., Romero-Vivo, S., Sanchez, E.: Some problems about structural properties of positive descriptor systems, Positive systems. In: Lecture Notes in Control and Information Sciences, vol. 294, pp. 233–240. Springer, Berlin (2003)
Campbell, S.L., Meyer, C.D., Rose, N.J.: Applications of the Drazin inverse to linear systems of differential equations with singular constructions. SIAMJ Appl. Math. 31(3), 411–425 (1976)
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 1–13 (2015)
Dai, L.: Singular control systems. In: Lectures Notes in Control and Information Sciences. Springer, Berlin (1989)
Dodig, M., Stosic, M.: Singular systems state feedbacks problems. Linear Algebra Appl. 431(8), 1267–1292 (2009)
Duan, G.R.: Analysis and Design of Descriptor Linear Systems. Springer, New York (2010)
Fahmy, M.M., ÓReill, J.: Matrix pencil of closed-loop descriptor systems: infinite-eigenvalues assignment. Int. J. Control 49(4), 1421–1431 (1989)
Farina, L., Rinaldi, S.: Positive Linear Systems. Willey, New York (2000)
Giorgio, G., Zuccotti, C.: Metzlerian and generalized metzlerian matrices: some properties and economic applications. J. Math. Res. 7(2), 42–55 (2015)
Kaczorek, T.: Application of Drazin inverse to analysis of descriptor fractional discrete-time linear systems with regular pencils. Int. J. Appl. Math. Comput. Sci. 23(1), 29–34 (2013)
Kaczorek, T.: Decentralized stabilization of positive descriptor continuous-time linear systems. In: 21st International Conference on System Theory, Control and Computing, 19–21 October 2017, Sinaia, Romania 2017. (submitted)
Kaczorek, T.: Decentralized stabilization of fractional positive descriptor and linear system. Int. J. Appl. Math. Comput. Sci. (2017).(submitted)
Kaczorek, T.: Descriptor positive discrete-time and continuous-time nonlinear systems. In: Proceedings of SPIE, vol. 9290 (2014)
Kaczorek, T.: Minimum energy control of fractional descriptor positive discrete-time linear systems. Int. J. Appl. Math. Comput. Sci. 24(4), 735–743 (2014)
Kaczorek, T.: Positive linear systems with different fractional orders. Bull. Pol. Acad. Sci. Techn. Sci. 58(3), 453–458 (2010)
Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002)
Kaczorek, T.: Positive fractional continuous-time linear systems with singular pencil. Bull. Pol. Acad. Sci. Tech. Sci. 60(1), 9–12 (2012)
Kaczorek, T.: Positive singular discrete time linear systems. Bull. Pol. Acad. Sci. Tech. Sci. 45(4), 619–631 (1997)
Kaczorek, T.: Selected Problems of Fractional Systems Theory. Springer, Berlin (2011)
Kaczorek, T.: Singular fractional discrete-time linear systems. Control Cybern. 40(3), 753–761 (2011)
Losada, J., Nieto, J.: Properties of a new fractional derivative without singular kernel. Prog. Fractional Differ. Appl. 1(2), 87–92 (2015)
Oldham, K.B., Spanier, J.: The Fractional Calculus. Academmic Press, New York (1974)
Ostalczyk, P.: Epitome of the fractional calculus: Theory and its Applications in Automatics, Wydawnictwo Politechniki Łódzkiej, Łódź (2008). (in Polish)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Sajewski, Ł.: Descriptor fractional discrete-time linear system and its solution – comparison of three different methods. In: Challenges in Automation, Robotics and Measurement Techniques, Advances in Intelligent Systems and Computing, vol. 440, pp. 37–50 (2016)
Sajewski, Ł.: Descriptor fractional discrete-time linear system with two different fractional orders and its solution. Bull. Pol. Ac. Tech. 64(1), 15–20 (2016)
Sajewski, Ł.: Decentralized stabilization of fractional positive descriptor continuous-time linear systems with delays. In: Proceedings of 22nd International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, 28–31 August 2017 (2017). (submitted)
Virnik, E.: Stability analysis of positive descriptor systems. Linear Algebra Appl. 429, 2640–2659 (2008)
Acknowledgment
This work was supported by National Science Centre in Poland under work No. 2014/13/B/ST7/03467.
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Kaczorek, T. (2019). Decentralized Stabilization of Fractional Positive Descriptor Discrete-Time Linear Systems. In: Ostalczyk, P., Sankowski, D., Nowakowski, J. (eds) Non-Integer Order Calculus and its Applications. RRNR 2017. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham. https://doi.org/10.1007/978-3-319-78458-8_1
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