Abstract
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.
Similar content being viewed by others
References
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): 31–45
Liberzon D, Morse A S. Basic problems in stability and design of switched systems. IEEE Control Syst Mag, 1999, 19(5): 59–70
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): 1069–1082
Sun Z, Ge S S. Analysis and synthesis of switched linear control systems. Automatica, 2005, 41(2): 181–195
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. 24–29
Narendra K S, Balakrishnan J. Improving transient response of adaptive control systems using multiple models and switching. IEEE Trans Autom Control, 1994, 39(9): 1861–1866
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): 1793–1798
Feng G. An approach to adaptive control of fuzzy dynamic systems. IEEE Trans Fuzzy Syst, 2002, 10(2): 268–275
Sun Z, Ge S S, Lee T H. Controllability and reachability criteria for switched linear systems. Automatica, 2002, 38(5): 775–786
Xie G, Wang L. Controllability and stabilizability of switched linear-systems. Syst Control Lett, 2003, 48(2): 135–155
Peleties P, Decarlo R A. Asymptotic stability of m-switched systems using Lyapunov-like functions. In: Proc Amer Control Conf, Boston, MA, USA, 1991. 1679–1684
Branicky M S. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans Autom Control, 1998, 43(4): 475–482
Ye H, Michel A N, Hou L. Stability theory for hybrid dynamical systems. IEEE Trans Autom Control, 1998, 43(4): 461–474
Johansson M, Rantzer A. Computation of piecewise Lyapunov quadratic functions for hybrid systems. IEEE Trans Autom Control, 1998, 43(4): 555–559
Feng G. Stability analysis of piecewise discrete-time linear systems. IEEE Trans Autom Control, 2002, 47(7): 1108–1112
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. 2655–2660
Zhai G, Hu B, Yasuda K, et al. Disturbance attenuation properties of time-controlled switched systems. J Franklin Inst, 2001, 338(7): 765–779
Pettersson S. Synthesis of switched linear systems. In: Proc 42rd IEEE Conf Decision Control, Hawaii, USA, 2003. 5283–5288
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): 1769–1774
Lin H, Antsaklis P J. Switching stabilizability for continuous-time uncertain switched linear systems. IEEE Trans Autom Control, 2007, 52(4): 633–646
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): 1114–1124
Daafouz J, Bernussou J. Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties. Syst Control Lett, 2001, 43(5): 355–359
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): 1883–1887
Daafouz J, Bernussou J. Robust dynamic output feedback control for switched systems. In: Proc 41st IEEE Conf Decision Control, Las Vegas, Nevada, USA, 2002. 4389–4394
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): 21–28
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. 3265–3270
De Oliveira M C, Bernussou J, Geromel J C. A new discrete-time robust stability condition. Syst Control Lett, 1999, 37(4): 261–265
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): 21–30
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): 3183–3192
Xie L, Lu L, Zhang D, et al. Improved robust H 2 and H ∞ filtering for uncertain discrete-time systems. Automatica, 2004, 40(5): 873–880
Duan Z, Zhang J, Zhang C, et al. Robust H 2 and H ∞ filtering for uncertain linear systems. Automatica, 2006, 42(11): 1919–1926
Gao H, Meng X, Chen T. A new design of robust H 2 filters for uncertain systems. Syst Control Lett, 2008, 57(7): 585–593
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. 5953–5958
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): 273–291
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): 291–304
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): 1188–1207
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): 420–430
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): 722–730
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): 795–806
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. 2667–2672
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. 4986–4992
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): 675–680
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): 123–131
Boyd S, El Ghaoui L, Feron E, et al. Linear Matrix Inequality in Systems and Control Theory. Philadelphia, PA: SIAM, 1994
Gahinet P, Nemirovski A, Laud A, et al. LMI Control Toolbox User’s Guide. Natick, MA: Mathworks, 1995
El Ghaoui L, Niculescu S I, eds. Advances in Linear Matrix Inequalities Methods in Control. Philadelphia, PA: SIAM, 2000
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. 241–257
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region of China under Project CityU/112907
Rights and permissions
About this article
Cite this article
Qiu, J., Feng, G. & Yang, J. Robust H ∞ static output feedback control of discrete-time switched polytopic linear systems with average dwell-time. Sci. China Ser. F-Inf. Sci. 52, 2019–2031 (2009). https://doi.org/10.1007/s11432-009-0197-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11432-009-0197-3