Positivity and stability of descriptor linear systems with interval state matrices

2018;
: pp. 7-17
Authors:
1
Białystok University of Technology

The positivity and asymptotic stability of descriptor linear continuous-time and discrete-time systems with interval state matrices and interval polynomials are investigated. Necessary and sufficient conditions for the positivity of descriptor continuous-time and discrete-time linear systems are established. It is shown that the convex linear combination of polynomials of positive linear systems is also the Hurwitz polynomial. The Kharitonov theorem is extended to the positive descriptor linear systems with interval state matrices. Necessary and sufficient conditions for the asy mptotic stability of descriptor positive linear systems have been also established. The considerations have been illustrated by numerical examples.

  1. A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994.
  2. M. Busłowicz, “Stability of linear continuous-time fractional order systems with delays of the retarded type“, Bull. Pol. Acad. Sci. Tech., vol. 56, no. 4, pp. 319-324, 2008.
  3. M. Busłowicz, “Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders“, Bull. Pol. Acad. Sci. Tech., vol. 60, no. 2, pp. 279-284, 2012.
  4. M. Busłowicz and T. Kaczorek, “Simple conditions for practical stability of positive fractional discrete-time linear systems“, Int. J. Appl. Math. Comput. Sci., vol. 19, no. 2, pp. 263-169, 2009.
  5. L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications. New York, USA: John Wiley, 2000.
  6. T. Kaczorek, “Analysis of positivity and stability of fractional discrete-time nonlinear systems“, Bull. Pol. Acad. Sci. Tech., vol. 64, no. 3, pp. 491-494, 2016.
  7. T. Kaczorek, “Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems“, Computational Problems of Electrical Engineering, vol. 5, no. 1, pp. 11-16, 2015.
  8. T. Kaczorek, “Application of Drazin inverse to analysis of descriptor fractional discrete-time linear systems with regular pencils“, Int. J. Appl. Math. Comput. Sci., vol. 23, no. 1, 29-34, 2013.
  9. T. Kaczorek, “Descriptor positive discrete-time and continuous-time nonlinear systems“, Proc. of SPIE, vol. 9290, 2014.
  10. T. Kaczorek, “Fractional positive continuous-time linear systems and their reachability“, Int. J. Appl. Math. Comput. Sci., vol. 18, no. 2, pp. 223-228, 2008,.
  11. T. Kaczorek, Positive 1D and 2D Systems. London, UK: Springer-Verlag, 2002.
  12. T. Kaczorek, “Positive linear systems with different fractional orders“, Bull. Pol. Acad. Sci. Techn., vol. 58, no. 3, pp. 453-458, 2010.
  13. T. Kaczorek, “Positivity and stability of discrete-time nonlinear systems“, in Proc. IEEE 2nd International Conference on Cybernetics, pp. 156-159, 2015.
  14. T. Kaczorek, “Positive linear systems consisting of n subsystems with different fractional orders“, IEEE Trans. on Circuits and Systems, vol. 58, no. 7, pp. 1203-1210, 2011.
  15. T. Kaczorek, “Positive fractional continuous-time linear systems with singular pencils“, Bull. Pol. Acad. Sci. Techn., vol. 60, no. 1, pp. 9-12, 2012.
  16. T. Kaczorek, “Positive singular discrete-time linear systems“, Bull. Pol. Acad. Sci. Tech., vol. 45, no. 4, pp. 619-631, 1997.
  17. T. Kaczorek, “Positivity and stability of discrete-time nonlinear systems“, in Proc. IEEE 2nd International Conference on Cybernetics, pp. 156-159, 2015.
  18. T. Kaczorek, “Stability of fractional positive nonlinear systems“, Archives of Control Sciences, vol. 25, no. 4, pp. 491-496, 2015.
  19. T. Kaczorek, “Stability of interval positive continuous-time linear systems“, Bull. Pol. Acad. Sci. Techn., vol. 66, no. 1, 2018.
  20. T. Kaczorek, Theory of Control and Systems. Warsaw, Poland: PWN, 1993. (Polish)
  21. V.L. Kharitonov, “Asymptotic stability of an equilibrium position of a family of systems of differential equations, Differentsialnye uravneniya, vol. 14, pp. 2086-2088, 1978.
  22. L. Sajewski, “Descriptor fractional discrete-time linear system and its solution – comparison of three different methods“, Challenges in Automation, Robotics and Measurement Techniques, Advances in Intelligent Systems and Computing, vol. 440, pp. 37-50, 2016.
  23. L. Sajewski, “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.
  24. H. Zhang, D. Xie, H. Zhang and G. Wang, “Stability analysis for discrete-time switched systems with unstable subsystems by a mode-dependent average dwell time approach“, ISA Transactions, vol. 53, pp. 1081-1086, 2014.
  25. J. Zhang, Z. Han, H. Wu, and J. Hung, “Robust stabilization of discrete-time positive switched systems with uncertainties and average dwell time switching“, Circuits Syst. Signal Process., vol. 33, pp. 71-95, 2014.
  26. W. Xiang-Jun, W. Zheng-Mao, and L. Jun-Guo, “Stability analysis of a class of nonlinear fractional-order systems“, IEEE Trans. Circuits and Systems-II, Express Briefs, vol. 55, no. 11, pp. 1178-1182, 2008.