Advertisement

Coprime Representations and Feedback Stabilization of Discrete Timevarying Linear Systems

  • Liu LiuEmail author
  • Yufeng Lu
Regular Papers Control Theory and Applications
  • 4 Downloads

Abstract

Within the framework of nest algebra, this paper deals with the stabilization problems of causal, discrete time, time-varying linear systems based on the coprime representations. It is shown that a linear system is stabilizable if and only if it admits a left coprime representation or a right coprime representation. The parametrization of stabilizing controllers, strong stabilizability and simultaneous stabilizability criteria are characterized in terms of left coprime representations alone, or analogously right coprime representations alone.

Keywords

Coprime representation nest algebra stabilization time-varying system 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Feintuch, Robust Control Theory in Hilbert Space, Springer–Verlag, 1998.CrossRefzbMATHGoogle Scholar
  2. [2]
    M. Vidyasagar, Control System Synthesis: A Factorization Approach, MIT Press Ser. Signal Process. Optim. Control, MIT Press, Cambridge, 1985.zbMATHGoogle Scholar
  3. [3]
    Y. Q. Wu and R. Q. Lu, “Event–based control for network systems via integral quadratic constraints,” IEEE Trans. on Circuits and Systems I: Regular Papers, vol. 65, no. 4, pp. 1386–1394, April 2018.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Y. Q. Wu and R. Q. Lu, “Output synchronization and L2–gain analysis for network systems,” IEEE Trans. on Systems Man & Cybernetics: Systems, vol. PP, no. 99, pp. 1–10, October 2017.Google Scholar
  5. [5]
    T. Yang, W. Qiu, Y. Ma, M. Chadli, and L. X. Zhang, “Fuzzy model–based predictive control of dissolved oxygen in activated sludge processes,” Neurocomputing, vol. 136, no. 20, pp. 88–95, July 2014.CrossRefGoogle Scholar
  6. [6]
    H. Liu, P. Shi, H. R. Karimi, and M. Cliuhadli, “Finite–time stability and stabilisation for a class of nonlinear systems with time–varying delay,” International Journal of Systems Science, vol. 47, no. 6, pp. 1433–1444, April 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    K. Mohamed, M. Chadli, and M. Chaabane, “Unknown inputs observer for a class of nonlinear uncertain systems: an LMI approach,” International Journal of Automation and Computing, vol. 9, no. 3, pp. 331–336, June 2012.CrossRefGoogle Scholar
  8. [8]
    C. A. Desoer, R. W. Liu, J. Murray, and R. Saeks, “Feedback system design: the fractional representation approach to analysis and synthesis,” IEEE Trans. on Automatic Control, vol. 25, no. 3, pp. 399–412, 1980.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M. C. Smith, “On stabilization and the existence of coprime factorizations,” IEEE Trans. on Automatic Control, vol. 34, no. 9, pp. 1005–1007, September 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A. Quadrat, “The fractional representation approach to synthesis problems: an algebraic analysis viewpoint. Part II: Internal stabilization,” SIAM J. Control. Optim., vol. 42, no. 1, pp. 300–320, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    W. N. Dale and M. C. Smith, “Stabilizability and existence of system representation for discrete–time timevarying system,” SIAM J. Control. Optim., vol. 31, no. 6, pp. 1538–1557, November 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    R. F. Curtain and M. R. Opmeer, “Normalized doubly coprime factorizations for infinite–dimensional linear systems,” Math. Control Signals Systems, vol. 18, no. 1, pp. 1–31, February 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L. Liu and Y. F. Lu, “Stabilizability, representations and factorizations for the time–varying linear system,” Syst. Control Lett., vol. 66, no. 4, pp. 58–64, April 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    R. Saeks and J. Murray, “Fractional representation, algebraic geometry and the simultaneous stabilization problem,” IEEE Trans. on Automatic Control, vol. 27, no. 4, pp. 895–903, 1982.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    N. Murata and H. Inaba, “A Note on simultaneous stabilization of multivariable control systems,” Preprint of the SICE Symp. on Dynamical System Theory (in Japanese), pp. 225–228, 1989.Google Scholar
  16. [16]
    H. Inaba, R. Abdurusul, and S. Takahashi, “Doubly coprime representation of linear systems and its application to simultaneous stabilization,” IMA Journal of Mathematical Control and Information, vol. 20, no. 1, pp. 21–35, March 2003.MathSciNetCrossRefGoogle Scholar
  17. [17]
    A. Feintuch, “On the strong stabilization of slowly timevarying linear systems,” Syst. Control Lett., vol. 61, no. 1, pp. 112–116, January 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    L. Liu and Y. F. Lu, “Analysis on the time–varying gap of discrete time–varying linear systems,” Operators and Matrices, vol. 11, no. 2, pp. 533–555, June 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Q. Yu, “Simultaneously stabilizing controllers subject to QI subspace constraints,” European Journal of Control, vol. 37, pp. 63–69, September 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    L. Liu and Y. F. Lu, “Transitivity in simultaneous stabilization for a family of time–varying systems,” Syst. Control Lett., vol. 78, no. 3, pp. 55–62, April 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    K. R. Davidson and Y. Q. Ji, “Topological stable rank of nest algebras,” Proceedings of the London Mathematical Society, vol. 98, no. 3, pp. 652–678, May 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Y. F. Lu and T. Gong, “On stabilization for discrete linear time–varying systems,” Syst. Control Lett., vol. 60, no. 12, pp. 1024–1031, December 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    K. R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Series 191, Longman Scientific and Technical Pub. Co., London, NewYork, 1988.Google Scholar
  24. [24]
    J. B. Conway, A Course in Functional Analysis, Springer–Verlag, New York, 1990.zbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianP. R. China

Personalised recommendations