Global stability of nonlinear feedback systems with fractional positive linear parts

Kaczorek, Tadeusz

International Journal of Applied Mathematics and Computer Science, Tome 30 (2020) no. 3, pp. 493-499.

Résumé

The global (absolute) stability of nonlinear systems with fractional positive and not necessarily asymptotically stable linear parts and feedbacks is addressed. The characteristics u = f(e) of the nonlinear parts satisfy the condition k₁e ≤ f(e) ≤ k2e for some positive k₁ and k₂. It is shown that the fractional nonlinear systems are globally asymptotically stable if the Nyquist plots of the fractional positive linear parts are located on the right-hand side of the circles (−1/k₁,−1/k₂).

Détail

Keywords: global stability, fractional positive system, nonlinear system, feedback system
Mots-clés : stateczność globalna, układ ułamkowy, układ dodatni, układ nieliniowy

Bibliographie

[1] Berman A. and Plemmons R.J. (1994). Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, MA.

[2] Borawski K. (2017). Modification of the stability and positivity of standard and descriptor linear electrical circuits by state feedbacks, Electrical Review 93(11): 176–180.

[3] Busłowicz M. and Kaczorek T. (2009). Simple conditions for practical stability of positive fractional discrete-time linear systems, International Journal of Applied and Mathematics and Computer Science 19(2): 263–269, DOI: 10.2478/v10006-009-0022-6.

[4] Farina L. and Rinaldi S. (2000). Positive Linear Systems: Theory and Applications, Wiley, New York, NY.

[5] Kaczorek T. (2019a). Absolute stability of a class of fractional positive nonlinear systems, International Journal of Applied Mathematics and Computer Science 29(1): 93–98, DOI: 10.2478/amcs-2019-0007.

[6] Kaczorek T. (2019b). Global stability of nonlinear feedback systems with positive linear parts, International Journal of Nonlinear Sciences and Numerical Simulation 20(5): 575–579, DOI: 10.1515/ijnsns-2018-0189.

[7] Kaczorek T. (2017). Superstabilization of positive linear electrical circuit by state-feedbacks, Bulletin of the Polish Academy of Sciences: Technical Sciences 65(5): 703–708.

[8] Kaczorek T. (2016). Analysis of positivity and stability of fractional discrete-time nonlinear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 64(3): 491–494.

[9] Kaczorek T. (2015a). Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems, Computational Problems of Electrical Engineering 5(1): 11–16.

[10] Kaczorek T. (2015b). Stability of fractional positive nonlinear systems, Archives of Control Sciences 25(4): 491–496.

[11] Kaczorek T. (2012). Positive fractional continuous-time linear systems with singular pencils, Bulletin of the Polish Academy of Sciences: Technical Sciences 60(1): 9–12.

[12] Kaczorek T. (2011a). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuits and Systems 58(7): 1203–1210.

[13] Kaczorek T. (2011b). Selected Problems of Fractional Systems Theory, Springer, Berlin.

[14] Kaczorek T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(3): 453–458.

[15] Kaczorek T. (2002). Positive 1D and 2D Systems, Springer, London.

[16] Kaczorek T. and Borawski K. (2017). Stability of positive nonlinear systems, 22nd International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 564–569, DOI: 10.1109/MMAR.2017.8046890.

[17] Kaczorek T. and Rogowski K. (2015). Fractional Linear Systems and Electrical Circuits, Springer, Cham.

[18] Kudrewicz J. (1964). Stability of nonlinear systems with feedback, Avtomatika i Telemechanika 25(8): 821–837, (in Russian).

[19] Lyapunov A.M. (1963). The General Problem of Motion Stability, Gostechizdat, Moscow, (in Russian).

[20] Leipholz H. (1970). Stability Theory, Academic Press, New York, NY.

[21] Mitkowski W. (2008). Dynamical properties of Metzler systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 56(4): 309–312.

[22] Ostalczyk P. (2016). Discrete Fractional Calculus, World Scientific, River Edge, NJ.

[23] Podlubny I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.

[24] Ruszewski A. (2019). Stability conditions for fractional discrete-time state-space systems with delays, 24th International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 185–190, DOI: 10.1109/MMAR.2019.8864689.

[25] Sajewski L. (2017a). Decentralized stabilization of descriptor fractional positive continuous-time linear systems with delays, 22nd International Conference on Methods and Models in Automation and Robotics, Miedzyzdroje, Poland, pp. 482-487.

[26] Sajewski L. (2017b). Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller, Bulletin of the Polish Academy of Sciences: Technical Sciences 65(5): 709–714.