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Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems

Year 2022, Volume: 5 Issue: 1, 10 - 14, 15.03.2022

Ahmet Duman

https://doi.org/10.32323/ujma.1055172

Abstract

In this paper, the problem of dwell time for the Hurwitz stability of switched linear systems is considered. Dwell time is determined based on the solution of Lyapunov matrix equation for the Hurwitz stability of switched linear differential systems. A numerical example illustrating the efficiency of theorem has been given.

Keywords

Differential equation systems Hurwitz stability Switched system Dwell time

References

  • [1] S. Solmaz, R. Shorten, K. Wulff, F. O’Cairbre, A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control, Automatica, 44(9) (2008), 2358-2363.
  • [2] A. Balluchi, M. D. Benedetto, C. Pinello, C. Rossi, A. Sangiovanni-Vincentelli, Cut-off in Engine Control, A Hybrid System Approach, Proceedings of the 36th IEEE Conference on Decision and Control, (1997), 4720–4725.
  • [3] B. E. Bishop, M. W. Spong, Control of Redundant Manipulators Using Logic-Based Switching, Proceedings of the 36th IEEE Conference on Decision and Control, (1998), 16–18.
  • [4] W. Zhang, M. S. Branicky, S. M. Phillips, Stability of Networked Control Systems, IEEE Control Systems Magazine, 21(1) (1998), 84–99.
  • [5] D. Z. Chen, Y. J. Guo, Advances on Switched Systems, Control Theory and Applications, 22(6) (2005), 954–960.
  • [6] H. Lin and P. J. Antsaklis, Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results, IEEE Trans. Automat. Contr., 54(2) (2009), 308-322.
  • [7] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, Proc. of the 38th Conf. on Decision and Control, (1999), 2655–2660.
  • [8] O¨ . Karabacak, Dwell time and average dwell time methods based on the cycle ratio of the switching graph, Systems Control Lett., 62(2013), 1032–1037.
  • [9] A. Ya. Bulgakov, An effectively calculable parameter for the stability quality of systems of linear differential equations with constant coefficients, Siberian Math. J., 21(1980), 339–347.
  • [10] A. Ya. Bulgakov, Matrix Computations with Guaranteed Accuracy in Stability Theory, Selc¸uk University, The Research Center of Applied Mathematics, Konya, 1995.
  • [11] S. K. Godunov, Ordinary Differential Equations with Constant Coefficients, Translations of Mathematical Monographs, 169, American Mathematical Society, Providence, RI, 1997.
  • [12] H. Bulgak, Pseudoeigenvalues, spectral portrait of a matrix and their connections with different criteria of stability, in Error Control and Adaptivity in Scientific Computing, eds. H. Bulgak and C. Zenger, NATO Science Series, Series C: Mathematical and Physical Sciences, 536 (Kluwer Academic Publishers, Dordrecht, 1999), 95–124.
  • [13] A. Duman, K. Aydın, Sensitivity of Hurwitz stability of linear differential equation systems with constant coefficients, Int. J. Geom. Methods Mod. Phys., 14 (2017), 1750084.
  • [14] S. K. Godunov, Modern Aspects of Linear Algebra, Translations of Mathematical Monographs, 175, American Mathematical Society, Providence, RI, 1998.
  • [15] Y. M. Nechepurenko, Bounds for the matrix exponential based on the Lyapunov equation and limits of yhe ,Hausdorff set, Computational Mathematics and Mathematical Physics, 42(2) (2002), 125-134.

Year 2022, Volume: 5 Issue: 1, 10 - 14, 15.03.2022

Ahmet Duman

https://doi.org/10.32323/ujma.1055172

Abstract

References

  • [1] S. Solmaz, R. Shorten, K. Wulff, F. O’Cairbre, A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control, Automatica, 44(9) (2008), 2358-2363.
  • [2] A. Balluchi, M. D. Benedetto, C. Pinello, C. Rossi, A. Sangiovanni-Vincentelli, Cut-off in Engine Control, A Hybrid System Approach, Proceedings of the 36th IEEE Conference on Decision and Control, (1997), 4720–4725.
  • [3] B. E. Bishop, M. W. Spong, Control of Redundant Manipulators Using Logic-Based Switching, Proceedings of the 36th IEEE Conference on Decision and Control, (1998), 16–18.
  • [4] W. Zhang, M. S. Branicky, S. M. Phillips, Stability of Networked Control Systems, IEEE Control Systems Magazine, 21(1) (1998), 84–99.
  • [5] D. Z. Chen, Y. J. Guo, Advances on Switched Systems, Control Theory and Applications, 22(6) (2005), 954–960.
  • [6] H. Lin and P. J. Antsaklis, Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results, IEEE Trans. Automat. Contr., 54(2) (2009), 308-322.
  • [7] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, Proc. of the 38th Conf. on Decision and Control, (1999), 2655–2660.
  • [8] O¨ . Karabacak, Dwell time and average dwell time methods based on the cycle ratio of the switching graph, Systems Control Lett., 62(2013), 1032–1037.
  • [9] A. Ya. Bulgakov, An effectively calculable parameter for the stability quality of systems of linear differential equations with constant coefficients, Siberian Math. J., 21(1980), 339–347.
  • [10] A. Ya. Bulgakov, Matrix Computations with Guaranteed Accuracy in Stability Theory, Selc¸uk University, The Research Center of Applied Mathematics, Konya, 1995.
  • [11] S. K. Godunov, Ordinary Differential Equations with Constant Coefficients, Translations of Mathematical Monographs, 169, American Mathematical Society, Providence, RI, 1997.
  • [12] H. Bulgak, Pseudoeigenvalues, spectral portrait of a matrix and their connections with different criteria of stability, in Error Control and Adaptivity in Scientific Computing, eds. H. Bulgak and C. Zenger, NATO Science Series, Series C: Mathematical and Physical Sciences, 536 (Kluwer Academic Publishers, Dordrecht, 1999), 95–124.
  • [13] A. Duman, K. Aydın, Sensitivity of Hurwitz stability of linear differential equation systems with constant coefficients, Int. J. Geom. Methods Mod. Phys., 14 (2017), 1750084.
  • [14] S. K. Godunov, Modern Aspects of Linear Algebra, Translations of Mathematical Monographs, 175, American Mathematical Society, Providence, RI, 1998.
  • [15] Y. M. Nechepurenko, Bounds for the matrix exponential based on the Lyapunov equation and limits of yhe ,Hausdorff set, Computational Mathematics and Mathematical Physics, 42(2) (2002), 125-134.
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ahmet Duman

Publication Date March 15, 2022
Submission Date January 8, 2022
Acceptance Date February 21, 2022
Published in Issue Year 2022 Volume: 5 Issue: 1

Cite

APA Duman, A. (2022). Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems. Universal Journal of Mathematics and Applications, 5(1), 10-14. https://doi.org/10.32323/ujma.1055172
AMA Duman A. Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems. Univ. J. Math. Appl. March 2022;5(1):10-14. doi:10.32323/ujma.1055172
Chicago Duman, Ahmet. “Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems”. Universal Journal of Mathematics and Applications 5, no. 1 (March 2022): 10-14. https://doi.org/10.32323/ujma.1055172.
EndNote Duman A (March 1, 2022) Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems. Universal Journal of Mathematics and Applications 5 1 10–14.
IEEE A. Duman, “Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems”, Univ. J. Math. Appl., vol. 5, no. 1, pp. 10–14, 2022, doi: 10.32323/ujma.1055172.
ISNAD Duman, Ahmet. “Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems”. Universal Journal of Mathematics and Applications 5/1 (March 2022), 10-14. https://doi.org/10.32323/ujma.1055172.
JAMA Duman A. Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems. Univ. J. Math. Appl. 2022;5:10–14.
MLA Duman, Ahmet. “Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems”. Universal Journal of Mathematics and Applications, vol. 5, no. 1, 2022, pp. 10-14, doi:10.32323/ujma.1055172.
Vancouver Duman A. Dwell Time for the Hurwitz Stability of Switched Linear Differential Equation Systems. Univ. J. Math. Appl. 2022;5(1):10-4.
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