Skip to main content
SpringerLink
Log in

Some Open Problems on Simultaneous Stabilization of Linear Systems

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

Simultaneous stabilization of linear systems is a fundamental issue in the system and control theory, and is of theoretical and practical significance. In this paper, the authors review the recent research progress and the state-of-art results on simultaneous stabilization of single-input single-output linear time-invariant systems. Especially, the authors list the ever best results on the parameters involved in the well known “French Champagne Problem” and “Belgian Chocolate Problem” from the point of view of mathematical theoretical analysis and numerical calculation. And the authors observed that Boston claimed the lower bound of δ can be enlarged to 0.976461 in 2012 is not accurate. The authors hope it will inspire further study on simultaneous stabilization of several linear systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blondel V, Simultaneous Stabilization of Linear Systems, Lect. Notes Contr. Inf. vol. 191, London: Springer-Verlag, 1994.

    Book  Google Scholar 

  2. Vidyasagar M, Control System Synthesis: A Factorization Approach, The MIT Press, Massachusetts, 1985.

    MATH  Google Scholar 

  3. Saeks R and Murray J, Fractional representation, algebraic geometry and the simultaneous stabilization problem, IEEE T. Automat. Contr., 1982, 27(4): 895–903.

    Article  MathSciNet  MATH  Google Scholar 

  4. Youla D, Bongiorno J, and Jabr H, Modern Wiener-Hopf design of optimal controllers-part I: The single-input-output case, IEEE T. Automat. Contr., 1976, 21(1): 3–13.

    Article  MathSciNet  MATH  Google Scholar 

  5. Kucera V, Discrete Linear Control: The Polynomial Equation Approach, New York: Wiley, 1979.

    MATH  Google Scholar 

  6. Vidyasagar M and Viswanadham N, Algebraic design techniques for reliable stabilization, IEEE T. Automat. Contr., 1982, 27(5): 1085–1095.

    Article  MathSciNet  MATH  Google Scholar 

  7. Youla D, Bongiorno J, and Lu C, Single-loop feedbak stabilization of linear multivariable plants, Automatica, 1974, 10(2): 159–173.

    Article  MathSciNet  MATH  Google Scholar 

  8. Blondel V and Gevers M, Simultaneous stabilization of three linear systems is rationally undecidable, Math Control Signal, 1993, 6(2): 135–145.

    Article  MathSciNet  MATH  Google Scholar 

  9. Blondel V, Gevers M, Mortini R, et al., Simultaneous stabilization of three or more systems: Conditions on the real axis do not suffice, SIAM J. Control Optim., 1994, 32(2): 572–590.

    Article  MathSciNet  MATH  Google Scholar 

  10. Blondel V, Sontag E D, Vidyasagar M, et al., Open Problems in Mathematical Systems and Control Theory, Springer-Verlag, London, 1999.

    Book  MATH  Google Scholar 

  11. Patel V V, Solution to the “Champagne Problem” on the simultaneous stabilization of three plants, Syst. Control Lett., 1999, 37(3): 173–175.

    Article  MATH  Google Scholar 

  12. Patel V V. Deodhare G, and Viswanath T, Some applications of randomized algorithms for control system design, Automatica, 2002, 38(12): 2085–2092.

    Article  MathSciNet  MATH  Google Scholar 

  13. Ahlfors L, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill Book Company, New York, 1973.

    MATH  Google Scholar 

  14. Conway J, Functional of One Complex Variable, Springer-Verlag, New York, 1978.

    Book  Google Scholar 

  15. Nehari Z, Conformal Mapping, McGraw-Hill Book Co., Inc, 1952.

    MATH  Google Scholar 

  16. Leizarowitz A, Kogan J, and Zeheb E, On simultaneous stabilization of linear plants, Lat. Am. Appl. Res., 1999, 29(3/4): 167–174.

    Google Scholar 

  17. Guan Q, Wang L, Xia B, et al., Solution to the generalized champagne problem on simultaneous stabilization of linear systems, Science in China (F), 2007, 50(5): 719–731.

    Article  MathSciNet  MATH  Google Scholar 

  18. Glouzin G, Geometric theory of functions of a complex variable, Translation of Math. Monographs vol 26 American Math. Society, 1969.

    Google Scholar 

  19. Caratheodory C, Sur quelques applications du théorème de Landau-Picard, C. R. Acad. Sci., 1907, 144: 1203–1206.

    MATH  Google Scholar 

  20. Hurwitz A, Über die Anwendung der elliptischen Modulfunktionen auf einen Satz der allgemenien Funktionentheorie, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 1904, 49: 242–253.

    Google Scholar 

  21. Burke J V, Henrion D, Lewis A S, et al., Stabilization via nonsmooth, nonconvex optimization, IEEE Transactions on Automat Control, 2006, 51(11): 1760–1769.

    Article  MathSciNet  Google Scholar 

  22. Rupp R, A covering theorem for a composite class of analytic functions, Complex Variables: Theory and Applications, 1994, 25(1): 35–41.

    Article  MathSciNet  MATH  Google Scholar 

  23. Blondel V, Rupp R, and Shapiro H, On zero and one points of analytic functions, Complex Variables: Theory and Applications, 1995, 28(2): 182–192.

    MathSciNet  MATH  Google Scholar 

  24. Chang Y and Sahinidis N, Global optimization in stabilizing controller design, Journal of Global Optimization, 2007, 38(4): 509–526.

    Article  MathSciNet  MATH  Google Scholar 

  25. He G, Wang L, Xia B, and Yu W, Stabilization of the Belgian chocolate system via low-order controllers, Chinese Control Conference, 2007, 26(3): 88–92.

    Google Scholar 

  26. He G, Wang L, and Yu W, French champagne and Belgian chocolate problems in simultaneous stabilization of linear systems, Proceedings of IFAC World Congress, 2008, 16: 1105–1110.

    Google Scholar 

  27. Guan Q, He G, Wang L, and Yu W, Simultaneous stabilization of linear systems, Control Theory and Applications, 2011, 28(1): 58–73.

    MATH  Google Scholar 

  28. Bergweiler W and Eremenko A, Goldberg’s constants, J. d’Analyse Math, 2013, 119(1): 365–402.

    Article  MathSciNet  MATH  Google Scholar 

  29. Goldberg A, On a theorem of Landau’s type, Teor. Funktsi?i Funkcional. Anal. i Prilo?zen, 1973, 17: 200–206 (in Russian).

    MathSciNet  Google Scholar 

  30. Guan Q, He G, Li W, and Yu W, A remark on “Simultaneous Stabilization of Linear Systems”, 2014 Chinese Control and Decision Conference, 2014, 26: 463–466.

    Google Scholar 

  31. Li W and Yu W, A sharper parameter bound for Belgian Chocolate stabilization problem, Chinese Control and Decision Conference, 2014, 26: 459–462.

    Google Scholar 

  32. Jenkins J, On a problem of A. A. Goldberg, Ann. Univ. Mariae Curie-Skłodowska Sect, 1982/83, 36/37: 83–86.

    MathSciNet  Google Scholar 

  33. Batra P, On small circles containing zeros and ones of analytic functions, Complex Variables Theory Appl, 2004, 49(11): 787–791.

    Article  MathSciNet  MATH  Google Scholar 

  34. Hempel J A and Smith S J, Hyperbolic lengths of geodesics surrounding two punctures, Proc. Amer. Math. Soc., 1988, 103(2): 513–516.

    Article  MathSciNet  MATH  Google Scholar 

  35. Smith S, On the uniformization of the n-punctured disc, PhD. Thesis, University of New England, 1986.

    Google Scholar 

  36. Yang L and Xia B, Inequality Machinery Proving and Theorem Automation Finding, Science Press, Beijing, 2008.

    Google Scholar 

  37. Boston N, On the Belgian chocolate problem and output feedback stabilization: Efficacy of algebraic methods, Fiftieth Annual Allerton Conference Allerton House, 2012, 5: 869–873.

    Google Scholar 

  38. Boston N, Applications of algebra to communications, control, and signal processing, SpringerBriefs in Computer Science, Springer, New York, 2012.

    Google Scholar 

  39. Bertilsson D and Blondel V, Transcendence in simultaneous stabilization, Journal of Mathematical Systems, Estimation, and Control, 1996, 6(3): 1–22.

    MathSciNet  MATH  Google Scholar 

  40. Blondel V and Tsitsiklis J N, A survey of computational complexity results in systems and control, Automatica, 2000, 36(9): 1249–1274.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China

    Wang Li & Wensheng Yu

  2. Intelligent Control Laboratory, College of Engineering, Peking University, Beijing, 100871, China

    Long Wang

  3. College of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Wensheng Yu

Authors
  1. Wang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Long Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wensheng Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wang Li.

Additional information

This paper was supported by the National Natural Science Foundation under Grant Nos. 61370176 and 61571064.

This paper was recommended for publication by Editor ZHOU Kemin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, W., Wang, L. & Yu, W. Some Open Problems on Simultaneous Stabilization of Linear Systems. J Syst Sci Complex 29, 289–299 (2016). https://doi.org/10.1007/s11424-015-4182-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-015-4182-1

Key words

Search

Navigation