Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
-
Tadeusz Kaczorek
Abstract
The global stability of continuous-time fractional orders nonlinear feedback systems with positive linear parts and interval state matrices is investigated. New sufficient conditions for the global stability of this class of positive feedback nonlinear systems are established. The effectiveness of these new stability conditions is demonstrated on simple example.
Acknowledgements
This work was supported by National Science Centre in Poland under work No. 2017/27/B/ST7/02443.
References
[1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. SIAM (1994).10.1137/1.9781611971262Suche in Google Scholar
[2] K. Borawski, Modification of the stability and positivity of standard and descriptor linear electrical circuits by state feedbacks. Electrical Review 93, No 11 (2017), 176–180; doi: 10.15199/48.2017.11.36.Suche in Google Scholar
[3] M. Buslowicz, T. Kaczorek, Simple conditions for practical stability of positive fractional discrete-time linear systems. Int. J. Appl. Math. Comput. Sci. 19, No 2 (2009), 263–269; doi: 10.2478/v10006-009-0022-6.Suche in Google Scholar
[4] L. Farina, S. Rinaldi, Positive Linear Systems; Theory and Applications. J. Wiley, New York (2000).10.1002/9781118033029Suche in Google Scholar
[5] T. Kaczorek, Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems. Computational Problems of Electrical Engineering 5, No 1 (2015), 11–16.Suche in Google Scholar
[6] T. Kaczorek, Global stability of nonlinear feedback systems with positive linear parts. Intern. J. of Nonlin. Sci. and Numer. Simul. 20, No 5 (2019), 575–579; doi: 10.1515/ijnsns-2018-0189.Suche in Google Scholar
[7] T. Kaczorek, Positive 1D and 2D Systems. Springer, London (2002).10.1007/978-1-4471-0221-2Suche in Google Scholar
[8] T. Kaczorek, Positive linear systems with different fractional orders. Bull. Pol. Acad. Sci. Techn. 58, No 3 (2010), 453–458; doi: 10.2478/v10175-010-0043-1.Suche in Google Scholar
[9] T. Kaczorek, Positive linear systems consisting of n subsystems with different fractional orders. IEEE Trans. on Circuits and Systems 58, No 7 (2011), 1203–1210; doi: 10.1109/TCSI.2010.2096111.Suche in Google Scholar
[10] T. Kaczorek, Selected Problems of Fractional Systems Theory. Springer, Berlin (2011).10.1007/978-3-642-20502-6Suche in Google Scholar
[11] T. Kaczorek, Stability of fractional positive nonlinear systems. Archives of Control Sciences 25, No 4 (2015), 491–496; doi: 10.1515/acsc-2015-0031.Suche in Google Scholar
[12] T. Kaczorek, K. Borawski, Stability of positive nonlinear systems. In: Proc. 22nd Intern. Conf. Methods and Models in Automation and Robotics, Miedzyzdroje, Poland (2017), 369–396; doi: 10.1109/MMAR.2017.8046890.Suche in Google Scholar
[13] T. Kaczorek, K. Rogowski, Fractional Linear Systems and Electrical Circuits. Springer, Cham (2015).10.1007/978-3-319-11361-6Suche in Google Scholar
[14] T. Kaczorek, Ł. Sajewski, Relationship between controllability and observability of standard and fractional different orders discrete-time linear system. Fract. Calc. Appl. Anal. 22, No 1 (2019), 158–169; DOI:10.1515/fca-2019-0010; https://www.degruyter.com/journal/key/FCA/22/1/html.Suche in Google Scholar
[15] J. Kudrewicz, Stability of nonlinear systems with feedbacks. Av-tomatika i Telemechanika 25, No 8 (1964) (in Russian).Suche in Google Scholar
[16] J.-G. Lu, Y.Q. Chen, Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties. Fract. Calc. Appl. Anal. 16, No 1 (2013), 142–157; DOI:10.2478/s13540-013-0010-2; https://www.degruyter.com/journal/key/FCA/16/1/html.Suche in Google Scholar
[17] A.M. Lyapunov, General Problem of Stability Movement. Gostechiz-dat, Moscow (1963) (in Russian).Suche in Google Scholar
[18] H. Leipholz, Stability Theory. Academic Press, New York (1970).Suche in Google Scholar
[19] W. Mitkowski, Dynamical properties of Metzler systems. Bull. Pol. Acad. Sci. Techn. 56, No 4 (2008), 309–312.Suche in Google Scholar
[20] P. Ostalczyk, Discrete Fractional Calculus. World Scientific, River Edge, NJ (2016).10.1142/9833Suche in Google Scholar
[21] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Suche in Google Scholar
[22] A. Ruszewski, Stability of discrete-time fractional linear systems with delays. Archives of Control Sciences 29, No 3 (2019), 549–567; doi: 10.24425/acs.2019.130205.Suche in Google Scholar
[23] A. Ruszewski, Practical and asymptotic stabilities for a class of delayed fractional discrete-time linear systems. Bull. Pol. Acad. Sci. Techn. 67, No 3 (2019), 509–515; doi: 10.24425/bpas.2019.128426.Suche in Google Scholar
[24] Ł. Sajewski, Decentralized stabilization of descriptor fractional positive continuous-time linear systems with delays. In: Proc. 22nd Intern. Conf. Methods and Models in Automation and Robotics, Miedzyzdroje, Poland (2017), 482–487; doi: 10.1109/MMAR.2017.8046875.Suche in Google Scholar
[25] Ł. Sajewski, Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller. Bull. Pol. Acad. Sci. Techn. 65, No 5 (2017), 709–714; doi: 10.1515/bpasts-2017-00ZZ.Suche in Google Scholar
© 2021 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial Survey
- In memory of the honorary founding editors behind the FCAA success story
- Research Paper
- Short time coupled fractional fourier transform and the uncertainty principle
- (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response
- Sharp asymptotics in a fractional Sturm-Liouville problem
- Multi-term fractional integro-differential equations in power growth function spaces
- Galerkin method for time fractional semilinear equations
- Müntz sturm-liouville problems: Theory and numerical experiments
- Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u
- Nonlinear convolution integro-differential equation with variable coefficient
- An efficient localized collocation solver for anomalous diffusion on surfaces
- Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach
- Sliding methods for the higher order fractional laplacians
- Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
Artikel in diesem Heft
- Frontmatter
- Editorial Survey
- In memory of the honorary founding editors behind the FCAA success story
- Research Paper
- Short time coupled fractional fourier transform and the uncertainty principle
- (N + α)-Order low-pass and high-pass filter transfer functions for non-cascade implementations approximating butterworth response
- Sharp asymptotics in a fractional Sturm-Liouville problem
- Multi-term fractional integro-differential equations in power growth function spaces
- Galerkin method for time fractional semilinear equations
- Müntz sturm-liouville problems: Theory and numerical experiments
- Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u
- Nonlinear convolution integro-differential equation with variable coefficient
- An efficient localized collocation solver for anomalous diffusion on surfaces
- Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach
- Sliding methods for the higher order fractional laplacians
- Global stability of fractional different orders nonlinear feedback systems with positive linear parts and interval state matrices
