Advertisement

Descriptor Case and System Augmentation

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

Abstract

Chapter  3 is an extension of the SV-LMIs of the preceding chapter both in terms of generalization of the results and for further conservatism reduction. First the SV results are generalized to descriptor systems. Not only this result is valuable in itself, but, combined to a model manipulation technique, an infinite sequence of SV-LMIs can be build. This sequence of SV-LMIs is shown to be easy to construct and proved to be of decreasing conservatism. Tests are provided for checking if the conservatism gap vanishes. On examples it is shown that the conservatism gap indeed vanishes and this is obtained early elements of the sequence (i.e., the convergence is rather fast).

Keywords

Robust Stability System Augmentation Impulsive Mode Quadratic Stability Linear Descriptor System 
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.
    Hou M, Müller PC (1999) Causal observability of descriptor systems. IEEE Trans Autom Control 44(1):158–163CrossRefzbMATHGoogle Scholar
  2. 2.
    Ishihara JY, Terra MH (2001) Impulse controllability and observability of rectangular descriptor systems. IEEE Trans Autom Control 46(6):991–994MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Hou M (2004) Controllability and elimination of impulsive modes in descriptor systems. IEEE Trans Autom Control 49(10):1723–1727CrossRefGoogle Scholar
  4. 4.
    Verghese G, Levy B, Kailath T (1981) A generalized state-space for singular systems. IEEE Trans Autom Control 26(4):811–831MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Takaba K, Morihira N, Katayama T (1995) A generalized Lyapunov theorem for descriptor system. Syst Control Lett 24(1):49–51MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Masubuchi I, Shimemura E (1997) An LMI condition for stability of implicit systems. In: IEEE conference on decision and control, pp 779–780Google Scholar
  7. 7.
    Uezato E, Ikeda M (1999) Strict LMI conditions for stability, robust stabilization, and \(H_\infty \) control of descriptor systems. In: IEEE conference on decision and control, pp 4092–4097Google Scholar
  8. 8.
    Ishihara JY, Terra MH (2002) On the lyapunov theorem for singular systems. IEEE Trans Autom Control 47(11):1926–1930MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chaabane M, Bachelier O, Souissi M, Mehdi D (2006) Stability and stabilization of continuous descriptor systems: an LMI approach. Math Probl Eng. doi: 10.1155/MPE/2006/39367
  10. 10.
    Peaucelle D (2007) Quadratic separation for uncertain descriptor system analysis, strict LMI conditions. In: IEEE conference on decision and control. Section 5 of the manuscript is erroneous. New Orleans. A corrected version is available at http://www.laas.fr/peaucell/papers/cdc07a.pdf
  11. 11.
    Trofino A (2000) Robust stability and domain of attraction of uncertain nonlinear systems. In: American control conference, Chicago, pp 3707–3711Google Scholar
  12. 12.
    Coutinho D, Trofino A, Fu M (2002) Guaranteed cost control of uncertain nonlinear systems via polynomial Lyapunov functions. IEEE Trans Autom Control 47(9):1575–1580MathSciNetCrossRefGoogle Scholar
  13. 13.
    Masubuchi I, Akiyama T, Saeki M (2003) Synthesis of output-feedback gain-scheduling controllers based on descriptor LPV system representation. In: IEEE conference on decision and control, pp 6115–6120Google Scholar
  14. 14.
    Doyle J, Packard A, Zhou K (1991) Review of LFTs, LMIs and \(\mu \). In: IEEE conference on decision and control, Brignton, England, pp 1227–1232Google Scholar
  15. 15.
    Hecker S, Varga A (2004) Generalized LFT-based representation of parametric uncertain models. Eur J Control 10(4):326–337MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Peaucelle D, Ebihara Y (2014) Robust stability analysis of discrete-time systems with parametric and switching uncertainties. In: Proceedings of the 19th IFAC world congress, pp 724–729Google Scholar
  17. 17.
    Skelton RE, Iwasaki T, Grigoriadis K (1998) A unified approach to linear control design., Systems and Control. Taylor and Francis, LondonGoogle Scholar
  18. 18.
    Scherer CW (2006) LMI relaxations in robust control. Eur J Control 12:3–29MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Parillo PA (2003) Semidefinite programming relaxations for semialgebraic problems. Math Program 96(2):293–320MathSciNetCrossRefGoogle Scholar
  20. 20.
    Henrion D, Garulli A (eds) (2005) Positive polynomials in control. Lecture notes in control and information sciences, vol 312. Springer, BerlinGoogle Scholar
  21. 21.
    Scherer CW, Hol CWJ (2006) Matrix sum-of-squares relaxations for robust semi-definite programs. Math Program 107(1–2):189–211MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chesi G, Garulli A, Tesi A, Vicino A (2003) Robust stability of polytopic systems via polynomially parameter-dependent Lyapunov functions. In: IEEE conference on decision and control, Maui, Hawaii, USAGoogle Scholar
  23. 23.
    Chesi G, Garulli A, Tesi A, Vicino A (2005) Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach. IEEE Trans Autom Control 50(3):365–370MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chesi G (2010) LMI techniques for optimization over polynomials in control: a survey. IEEE Trans Autom Control 55(11):2500–2510MathSciNetCrossRefGoogle Scholar
  25. 25.
    Oliveira RCLF, de Oliveira MC, Peres PLD (2008) Convergent LMI relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter-dependent Lyapunov functions. Syst Control Lett 57:680–689MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Bliman PA (2001) A convex approach to robust stability for linear systems with uncertain scalar parameters. Technical Report RR-4316, INRIA, Rocquencourt, France. http://www.inria.fr/rrrt/rr-4316.html
  27. 27.
    Bliman P-A (2004) A convex approach to robust stability for linear systems with uncertain scalar parameters. SIAM J Control Optim 42:2016–2042MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tempo R, Calafiore G, Dabbene F (2005) Randomized algorithms for analysis and control of uncertain systems. Springer, LondonCrossRefzbMATHGoogle Scholar
  29. 29.
    Scherer CW (2005) Relaxations for robust linear matrix inequality problems with verifications for exactness. SIAM J Matrix Anal Appl 27(2):365–395MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ebihara Y, Onishi Y, Hagiwara T (2009) Robust performance analysis of uncertain LTI systems: dual LMI approach and verifications for exactness. IEEE Trans Autom Control 54(5):938–951MathSciNetCrossRefGoogle Scholar
  31. 31.
    Balakrishnan V, Vandenberghe L (2003) Semidefinite programming duality and linear time-invariant systems. IEEE Trans Autom Control 48(1):30–41MathSciNetCrossRefGoogle Scholar
  32. 32.
    Boyd S, Balakrishnan V (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  33. 33.
    Löfberg J (2004) YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the IEEE computer aided control system design, pp 284–289Google Scholar
  34. 34.
    Pittet C, Arzelier D (2006) DEMETER: a benchmark for robust analysis and control of the attitude of flexible microsatellites. In: IFAC symposium on robust control design, Toulouse, FranceGoogle Scholar
  35. 35.
    Peaucelle D, Bortott A, Gouaisbaut F, Arzelier D, Pittet C (2010) Robust analysis of DEMETER benchmark via quadratic separation. In: IFAC symposium on automatic control in aerospace, NaraGoogle Scholar
  36. 36.
    Biannic J-M, Pittet C, Roos C (2010) Lpv analysis of switched controllers in satellite attitude control systems. In: AIAA guidance, navigation, and control conference, TorontoGoogle Scholar
  37. 37.
    Biannic J-M, Roos C, Pittet C (2011) LPV analysis of switched controllers for attitude control systems. J Guid Control Dyn 34(5):1561–1566CrossRefGoogle Scholar
  38. 38.
    Tregouët J-F, Arzelier D, Peaucelle D, Ebihara Y, Pittet C, Falcoz A (2011) Periodic h2 synthesis for spacecraft attitude control with magnetorquers and reaction wheels. In: IEEE conference on decision and control, OrlandoGoogle Scholar
  39. 39.
    Tregouët J-F, Arzelier D, Peaucelle D, Pittet C, Zaccarian L (2014) Reaction wheels desaturation using magnetorquers and static input allocation. IEEE Trans Control Syst Technol. Submitted (LAAS Tech. Report N\(\,^{\circ }\)13472)Google Scholar
  40. 40.
    Peaucelle D, Drouot A, Pittet C, Mignot J (2011) Simple adaptive control without passivity assumptions and experiments on satellite attitude control DEMETER benchmark. In: IFAC world congressGoogle Scholar
  41. 41.
    Luzi AR, Peaucelle D, Biannic J-M, Pittet C, Mignot J (2014) Structured adaptive attitude control of a satellite. Int J Adapt Control Signal ProcessGoogle Scholar
  42. 42.
    Pittet C, Mignot J, Fallet C (1999) LMI based multi-objective \(H_\infty \) control of flexible microsatellites. In: IEEE conference on decision and control, Sydney, AustraliaGoogle 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