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Global stability of nonlinear feedback systems with fractional positive linear parts. (English) Zbl 1471.93219

In this paper, the author establishes a sufficient condition for the global stability of certain nonlinear continuous-time feedback systems with positive, not necessarily asymptotically stable, fractional linear parts. To introduce the fractional parts, the Caputo definition of a fractional derivative is employed.
The characteristics \(u=f(e)\) of the nonlinear elements are assumed to satisfy the estimation \(k_1e\leq f(e)\leq k_2e\) for positive \(k_1,k_2\), where the gain \(k_1\) is chosen so that the transfer function is asymptotically stable. By applying the Lyapunov method to the fractional nonlinear system and using Hurwitz estimations, it is shown that this system is globally asymptotically stable provided that the Nyquist plot of the linear part is located on the right-hand side of the circle \((-\frac{1}{k_1},-\frac{1}{k_2})\), the formal definition of the latter being given {\em ad hoc}. This result can be regarded as an extension of the Kudrewicz theorem to the case of fractional nonlinear systems with positive linear parts. The practical utility of the above-mentioned global stability criterion is illustrated by means of a suitable numerical example.

MSC:

93D20 Asymptotic stability in control theory
93B52 Feedback control
93C28 Positive control/observation systems
93C10 Nonlinear systems in control theory
26A33 Fractional derivatives and integrals
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[1] Berman A. and Plemmons R.J. (1994). Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, MA. · Zbl 0815.15016
[2] Borawski K. (2017). Modification of the stability and positivity of standard and descriptor linear electrical circuits by state feedbacks, Electrical Review93(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 Science19(2): 263-269, DOI: 10.2478/v10006-009-0022-6. · Zbl 1167.93019
[4] Farina L. and Rinaldi S. (2000). Positive Linear Systems: Theory and Applications, Wiley, New York, NY. · Zbl 0988.93002
[5] Kaczorek T. (2019a). 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. · Zbl 1416.93149
[6] Kaczorek T. (2019b). Global stability of nonlinear feedback systems with positive linear parts, International Journal of Nonlinear Sciences and Numerical Simulation20(5): 575-579, DOI: 10.1515/ijnsns-2018-0189. · Zbl 07168305
[7] Kaczorek T. (2017). Superstabilization of positive linear electrical circuit by state-feedbacks, Bulletin of the Polish Academy of Sciences: Technical Sciences65(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 Sciences64(3): 491-494.
[9] Kaczorek T. (2015a). Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems, Computational Problems of Electrical Engineering5(1): 11-16.
[10] Kaczorek T. (2015b). Stability of fractional positive nonlinear systems, Archives of Control Sciences25(4): 491-496. · Zbl 1446.93066
[11] Kaczorek T. (2012). Positive fractional continuous-time linear systems with singular pencils, Bulletin of the Polish Academy of Sciences: Technical Sciences60(1): 9-12.
[12] Kaczorek T. (2011a). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuits and Systems58(7): 1203-1210. · Zbl 1468.94663
[13] Kaczorek T. (2011b). Selected Problems of Fractional Systems Theory, Springer, Berlin. · Zbl 1221.93002
[14] Kaczorek T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences58(3): 453-458. · Zbl 1220.78074
[15] Kaczorek T. (2002). Positive 1D and 2D Systems, Springer, London. · Zbl 1005.68175
[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. · Zbl 1451.93149
[17] Kaczorek T. and Rogowski K. (2015). Fractional Linear Systems and Electrical Circuits, Springer, Cham. · Zbl 1354.93001
[18] Kudrewicz J. (1964). Stability of nonlinear systems with feedback, Avtomatika i Telemechanika25(8): 821-837, (in Russian).
[19] Lyapunov A.M. (1963). The General Problem of Motion Stability, Gostechizdat, Moscow, (in Russian). · Zbl 0112.31801
[20] Leipholz H. (1970). Stability Theory, Academic Press, New York, NY. · Zbl 0291.73030
[21] Mitkowski W. (2008). Dynamical properties of Metzler systems, Bulletin of the Polish Academy of Sciences: Technical Sciences56(4): 309-312.
[22] Ostalczyk P. (2016). Discrete Fractional Calculus, World Scientific, River Edge, NJ. · Zbl 1354.93003
[23] Podlubny I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA. · Zbl 0924.34008
[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. · Zbl 1440.93210
[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 Sciences65(5): 709-714.
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