Robust H static output feedback control of discrete-time switched polytopic linear systems with average dwell-time

  • JianBin Qiu
  • Gang Feng
  • Jie Yang


This paper investigates the problem of robust exponential H static output feedback controller design for a class of discrete-time switched linear systems with polytopic-type time-varying parametric uncertainties. The objective is to design a switched static output feedback controller guaranteeing the exponential stability of the resulting closed-loop system with a minimized exponential H performance under average dwell-time switching scheme. Based on a parameter-dependent discontinuous switched Lyapunov function combined with Finsler’s lemma and Dualization lemma, some novel conditions for exponential H performance analysis are first proposed and in turn the static output feedback controller designs are developed. It is shown that the controller gains can be obtained by solving a set of linear matrix inequalities (LMIs), which are numerically efficient with commercially available software. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approaches.


static output feedback switched systems average dwell-time time-varying uncertainty H control linear matrix inequality 


  1. 1.
    Branicky M S, Borkar V, Mitter S. A unified framework for hybrid control: model and optimal control theory. IEEE Trans Automat Control, 1998, 43(1): 31–45zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Liberzon D, Morse A S. Basic problems in stability and design of switched systems. IEEE Control Syst Mag, 1999, 19(5): 59–70CrossRefGoogle Scholar
  3. 3.
    Decarlo R A, Branicky M S, Pettersson S, et al. Perspectives and results on the stability and stabilizability of hybrid systems. Proc IEEE, Special Issue Hybrid Syst, 2000, 88(7): 1069–1082Google Scholar
  4. 4.
    Sun Z, Ge S S. Analysis and synthesis of switched linear control systems. Automatica, 2005, 41(2): 181–195zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lin H, Antsaklis P J. Stability and stabilizability of switched linear systems: a short survey of recent results. In: Proc 20th IEEE Int Symp Intell Control, Limassol, Cyprus, 2005. 24–29Google Scholar
  6. 6.
    Narendra K S, Balakrishnan J. Improving transient response of adaptive control systems using multiple models and switching. IEEE Trans Autom Control, 1994, 39(9): 1861–1866zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Xi Z, Feng G, Jiang Z P, et al. A switching algorithm for global exponential stabilization of uncertain chained systems. IEEE Trans Autom Control, 2003, 48(10): 1793–1798CrossRefMathSciNetGoogle Scholar
  8. 8.
    Feng G. An approach to adaptive control of fuzzy dynamic systems. IEEE Trans Fuzzy Syst, 2002, 10(2): 268–275CrossRefGoogle Scholar
  9. 9.
    Sun Z, Ge S S, Lee T H. Controllability and reachability criteria for switched linear systems. Automatica, 2002, 38(5): 775–786zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Xie G, Wang L. Controllability and stabilizability of switched linear-systems. Syst Control Lett, 2003, 48(2): 135–155zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Peleties P, Decarlo R A. Asymptotic stability of m-switched systems using Lyapunov-like functions. In: Proc Amer Control Conf, Boston, MA, USA, 1991. 1679–1684Google Scholar
  12. 12.
    Branicky M S. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Control, 1998, 43(4): 475–482zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ye H, Michel A N, Hou L. Stability theory for hybrid dynamical systems. IEEE Trans Autom Control, 1998, 43(4): 461–474zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Johansson M, Rantzer A. Computation of piecewise Lyapunov quadratic functions for hybrid systems. IEEE Trans Autom Control, 1998, 43(4): 555–559zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Feng G. Stability analysis of piecewise discrete-time linear systems. IEEE Trans Autom Control, 2002, 47(7): 1108–1112CrossRefGoogle Scholar
  16. 16.
    Hespanha J P, Morse A S. Stability of switched systems with average dwell-time. In: Proc 38th IEEE Conf Decision Control, Phoenix, Arizona, USA, 1999. 2655–2660Google Scholar
  17. 17.
    Zhai G, Hu B, Yasuda K, et al. Disturbance attenuation properties of time-controlled switched systems. J Franklin Inst, 2001, 338(7): 765–779zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pettersson S. Synthesis of switched linear systems. In: Proc 42rd IEEE Conf Decision Control, Hawaii, USA, 2003. 5283–5288Google Scholar
  19. 19.
    Sun X M, Zhao J, Hill D J. Stability and L 2-gain analysis for switched delay systems: a delay dependent method. Automatica, 2006, 42(10): 1769–1774zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lin H, Antsaklis P J. Switching stabilizability for continuous-time uncertain switched linear systems. IEEE Trans Autom Control, 2007, 52(4): 633–646CrossRefMathSciNetGoogle Scholar
  21. 21.
    Lin H, Antsakis P J. Hybrid state feedback stabilization with L 2 performance for discrete-time switched linear systems. Int J Control, 2008, 81(7): 1114–1124zbMATHCrossRefGoogle Scholar
  22. 22.
    Daafouz J, Bernussou J. Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties. Syst Control Lett, 2001, 43(5): 355–359zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Daafouz J, Riedinger P, Iung C. Stability and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans Autom Control, 2002, 47(1): 1883–1887CrossRefMathSciNetGoogle Scholar
  24. 24.
    Daafouz J, Bernussou J. Robust dynamic output feedback control for switched systems. In: Proc 41st IEEE Conf Decision Control, Las Vegas, Nevada, USA, 2002. 4389–4394Google Scholar
  25. 25.
    Xie D, Wang L, Hao F, et al. LMI approach to L 2-gain analysis and control synthesis of uncertain switched systems. IEE Proc Control Theory Appl, 2004, 151(1): 21–28CrossRefGoogle Scholar
  26. 26.
    Fang L, Lin H, Antsaklis P J. Stabilization and performance analysis for a class of switched systems. In: Proc 43rd IEEE Conf Decision Control, Atlantis, Paradise Island, Bahamas, 2004. 3265–3270Google Scholar
  27. 27.
    De Oliveira M C, Bernussou J, Geromel J C. A new discrete-time robust stability condition. Syst Control Lett, 1999, 37(4): 261–265zbMATHCrossRefGoogle Scholar
  28. 28.
    Peaucelle D, Arzelier D, Bachelier O, et al. A new robust D-stability condition for real convex polytopic uncertainties. Syst Control Lett, 2000, 40(1): 21–30zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Gao H, Lam J, Xie L, et al. New approach to mixed H 2/H filtering for polytopic discrete-time systems. IEEE Trans Signal Process, 2005, 53(8): 3183–3192CrossRefMathSciNetGoogle Scholar
  30. 30.
    Xie L, Lu L, Zhang D, et al. Improved robust H 2 and H filtering for uncertain discrete-time systems. Automatica, 2004, 40(5): 873–880zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Duan Z, Zhang J, Zhang C, et al. Robust H 2 and H filtering for uncertain linear systems. Automatica, 2006, 42(11): 1919–1926zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Gao H, Meng X, Chen T. A new design of robust H 2 filters for uncertain systems. Syst Control Lett, 2008, 57(7): 585–593zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Gao H, Zhang L, Shi P, et al. Stability and stabilization of switched linear discrete-time systems with polytopic uncertainties. In: Proc Amer Control Conf, Minnesota, USA, 2006. 5953–5958Google Scholar
  34. 34.
    Zhang L, Shi P, Boukas E K, et al. H control of switched linear discrete-time systems with polytopic uncertainties. Optimal Control Appl Methods, 2006, 27(5): 273–291CrossRefMathSciNetGoogle Scholar
  35. 35.
    Zhang L, Shi P, Wang C, et al. Robust H filtering for switched linear discrete-time systems with polytopic uncertainties. Int J Adapt Control Signal Process, 2006, 20(6): 291–304zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Zhang L, Boukas E K, Shi P. Exponential H filtering for uncertain discrete-time switched linear systems with average dwell time: a μ-dependent approach. Int J Robust Nonlinear Control, 2008, 18(11): 1188–1207CrossRefMathSciNetGoogle Scholar
  37. 37.
    Qiu J, Feng G, Yang J. Robust mixed H 2/H filtering design for discrete-time switched polytopic linear systems. IET Control Theory Appl, 2008, 2(5): 420–430CrossRefMathSciNetGoogle Scholar
  38. 38.
    Zhang L, Shi P, Boukas E K, et al. Robust L 2-L filtering for switched linear discrete time-delay systems with polytopic uncertainties. IET Control Theory Appl, 2007, 1(3): 722–730CrossRefMathSciNetGoogle Scholar
  39. 39.
    Qiu J, Feng G, Yang J. New results on robust energy-to-peak filtering for discrete-time switched polytopic linear systems with time-varying delay. IET Control Theory Appl, 2008, 2(9): 795–806CrossRefMathSciNetGoogle Scholar
  40. 40.
    Bara G I, Boutayeb M. Switched output feedback stabilization of discrete-time switched systems. In: Proc 45th IEEE Conf Decision Control, San Diego, CA, USA, 2006. 2667–2672Google Scholar
  41. 41.
    Bara G I. Robust switched static output feedback control for discrete-time switched linear systems. In: Proc 46th IEEE Conf Decision Control, New Orleans, LA, USA, 2007. 4986–4992Google Scholar
  42. 42.
    Lee K H, Lee J H, Kwon W H. Sufficient LMI conditions for H output feedback stabilization of linear discrete-time systems. IEEE Trans Autom Control, 2006, 51(4): 675–680CrossRefMathSciNetGoogle Scholar
  43. 43.
    Dong J, Yang G H. Robust static output feedback control for linear discrete-time systems with time-varying uncertainties. Syst Control Lett, 2008, 57(2): 123–131zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Boyd S, El Ghaoui L, Feron E, et al. Linear Matrix Inequality in Systems and Control Theory. Philadelphia, PA: SIAM, 1994Google Scholar
  45. 45.
    Gahinet P, Nemirovski A, Laud A, et al. LMI Control Toolbox User’s Guide. Natick, MA: Mathworks, 1995Google Scholar
  46. 46.
    El Ghaoui L, Niculescu S I, eds. Advances in Linear Matrix Inequalities Methods in Control. Philadelphia, PA: SIAM, 2000Google Scholar
  47. 47.
    De Oliveira M C, Skelton R E. Stability tests for constrained linear systems. In: Reza Moheimani S O, eds. Perspectives in Robust Control. Ser Lect Notes Control Inf Sci, Vol 268. New York: Springer-Verlag, 2001. 241–257CrossRefGoogle Scholar

Copyright information

© Science in China Press and Springer Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Space Control and Inertial Technology Research CenterHarbin Institute of TechnologyHarbinChina
  2. 2.Department of Manufacturing Engineering and Engineering ManagementCity University of Hong KongKowloon, Hong KongChina
  3. 3.Department of Precision Machinery and InstrumentationUniversity of Science and Technology of ChinaHefeiChina

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