Advertisement

Introduction

  • Yoshio EbiharaEmail author
  • Dimitri Peaucelle
  • Denis Arzelier
Chapter
Part of the Communications and Control Engineering book series (CCE)

Abstract

Chapter 1 is dedicated to the origins of the “\(S\)-variable approach.” We trace the contributions of independent authors that participated to establish the fundamentals and justify our choice for the denomination “\(S\)-variable approach.” This attentive detailed study is the occasion to show several interpretations of the \(S\)-variables with respect to technical results such as Finsler’s lemma and elimination lemma . The chapter concludes with the brief exposure of all the problems to be tackled in the remaining part of the book, justifying the importance of these selected problems.

Keywords

Variable Approach Controller Synthesis Lyapunov Inequality Linear System Analysis Hurwitz Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Geromel JC, de Oliveira MC, Hsu L (1998) LMI characterization of structural and robust stability. Linear Algebra Appl 285:68–80CrossRefGoogle Scholar
  2. 2.
    de Oliveira MC, Geromel JC, Hsu L (1999) LMI characterization of structural and robust stability: The discrete-time case. Linear Algebra Appl 296(1–3):27–38MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    de Oliveira MC, Bernussou J, Geromel JC (1999) A new discrete-time stability condition. Syst Control Lett 37(4):261–265CrossRefzbMATHGoogle Scholar
  4. 4.
    Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM Studies in Applied Mathematics, PhiladelphiaCrossRefzbMATHGoogle Scholar
  5. 5.
    Peaucelle D, Arzelier D, Bachelier O, Bernussou J (2000) A new robust D-stability condition for real convex polytopic uncertainty. Syst Control Lett 40(1):21–30MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Oliveira MC, Skelton RE (2001) Stability tests for constrained linear systems. In: Reza Moheimani S.O. (ed) Lecture notes in control and information sciences, pp 241–257. Springer-Verlag, New YorkGoogle Scholar
  7. 7.
    Henrion D, Arzelier D, Peaucelle D (2003) Positive polynomial matrices and improved LMI robustness conditions. Automatica 39(8):1479–1485MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Leite V, Peres P (2003) An improved LMI condition for robust D-stability of uncertain polytopic systems. IEEE Trans Autom Control 48(3):500–504MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ebihara Y, Hagiwara T (2005) A dilated LMI approach to robust performance analysis of linear time-invariant uncertain systems. Automatica 41(11):1933–1941MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fridman E, Shaked U (2002) An improved stabilization method for linear time-delay systems. IEEE Trans Autom Control 47(11):1931–1937MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fridman E, Shaked U (2002) A descriptor system approach to \(H_\infty \) control of time-delay systems. IEEE Trans Autom Control 47:253–270MathSciNetCrossRefGoogle Scholar
  12. 12.
    Geromel JC, de Oliveira MC, Bernussou J (2002) Robust filtering of discrete-time linear systems with parameter dependent lyapunov functions. SIAM J Control Optim 41:700–711MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    de Oliveira MC, Geromel JC, Bernussou J (2002) Extended \(H_2\) and \(H_\infty \) norm characterizations and controller parametrizations for discrete-time systems. Int J Control 75:666–679CrossRefzbMATHGoogle Scholar
  14. 14.
    Apkarian P, Tuan HD, Bernussou J (2001) Continuous-time analysis and \(H_2\) multi-channel synthesis with enhanced LMI characterizations. IEEE Trans Autom Control 46(12):1941–1946MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ebihara Y, Hagiwara T (2004) New dilated LMI characterizations for continuous-time multiobjective controller synthesis. Automatica 40(11):2003–2009MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Farges C, Peaucelle D, Arzelier D, Daafouz J (2007) Robust \(H_2\) performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs. Syst Control Lett 56:159–166Google Scholar
  17. 17.
    Ebihara Y, Peaucelle D, Arzelier D (2011) Periodically time-varying memory state-feedback controller synthesis for discrete-time linear systems. Automatica 47(1):14–25MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ebihara Y (2013) Periodically time-varying memory state-feedback for robust \(H_2\) control of uncertain discrete-time linear systems. Asian J Control 15(2):409–419MathSciNetCrossRefGoogle Scholar
  19. 19.
    Trégouët JF, Peaucelle D, Arzelier D, Ebihara Y (2013) Periodic memory state-feedback controller: New formulation, analysis and design results. IEEE Trans Autom Control 58(8):1986–2000Google Scholar
  20. 20.
    Rantzer A (1996) On the Kalman-Yakubovitch-Popov lemma. Syst Control Lett 28:7–10MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Iwasaki T, Hara S (2005) Generalized KYP lemma: Unified frequency domain inequalities with design applications. IEEE Trans Autom Control 50(1):41–59MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ebihara Y, Maeda K, Hagiwara T (2008) Generalized S-procedure for inequality conditions on one-vector-lossless sets and linear system analysis. SIAM J Control Optim 47(3):1547–1555MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shimomura T, Takahashi M, Fujii T (2001) Extended-space control design with parameter-dependent lyapunov functions. In: Proceedings of the conference on decision and control, pp 2157–2162Google Scholar
  24. 24.
    Pipeleers Goele, Demeulenaere Bram, Swevers Jan, Vandenberghe Lieven (2009) Extended LMI characterizations for stability and performance of linear systems. Syst Control Lett 58(7):510–518MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sebe N (2007) A new dilated LMI characterization and iterative control system synthesis. In: Proceedings of the 11th IFAC symposium on large scale complex systems theory and applications, pp 250–255, 2007Google Scholar
  26. 26.
    Fujisaki Y, Befekadu GK (2009) Reliable decentralised stabilisation of multi-channel systems: A design method via dilated LMIs and unknown disturbance observers. Int J Control 82(11):2040–2050MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Feng Y, Yagoubiab M, Chevrel P (2010) Dilated LMI characterisations for linear time-invariant singular systems. Int J Control 83(11):2276–2284CrossRefzbMATHGoogle Scholar
  28. 28.
    Bara GL (2011) Dilated LMI conditions for time-varying polytopic descriptor systems: The discrete-time case. Int J Control 84(6):1010–1023MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sajjadi-Kia S, Jabbari F (2012) On bounded real matrix inequality dilation. Int J Control 85(10):1593–1601MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sato M, Peaucelle D (2007) Comparison between SOS approach and slack variable approach for non-negativity check of polynomial functions: Single variable case. In: Proceedings of the american control conference, New-York, July 2007Google Scholar
  31. 31.
    Sato M, Peaucelle D (2007) Comparison between SOS approach and slack variable approach for non-negativity check of polynomial functions: Multiple variable case. In: Proceedings of the european control conference, KosGoogle Scholar
  32. 32.
    Sato M, Peaucelle D (2007) Robust stability/performance analysis for uncertain linear systems via multiple slack variable approach: Polynomial LTIPD systems. In: Proceedings of the IEEE conference on decision and control, New-OrleansGoogle Scholar
  33. 33.
    Peaucelle D, Sato M (2009) LMI tests for positive definite polynomials: Slack variable approach. IEEE Trans Autom Control 54(4):886–891MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2015

Authors and Affiliations

  • Yoshio Ebihara
    • 1
    Email author
  • Dimitri Peaucelle
    • 2
  • Denis Arzelier
    • 2
  1. 1.Department of Electrical EngineeringKyoto UniversityKyotoJapan
  2. 2.Laboratory for Analysis and Architecture of Systems ScienceNational Centre for Scientific ResearchToulouseFrance

Personalised recommendations