Based on piecewise quadratic Lyapunov function techniques, this paper first provides a stability condition for the uncertain piecewise discrete-time linear system, which is a feasibility problem of linear matrix inequalities. Then the control methods for uncertain piecewise discrete-time linear systems via state feedback and output feedback controllers are proposed. It is shown that both the state feedback and output feedback controllers can be obtained by solving a set of bilinear matrix inequalities. A numerical example is given to illustrate the effectiveness of the developed methods
BemporadAFerrari-TrecateGMorariM (2000) Observability and controllability of piecewise affine and hybrid systems. IEEE Transactions on Automatic Control45: 1864–1876.
2.
BoydSEl GhaouiLFeronEBalakrishnanV (1994) Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM.
3.
ChenMFengG (2004) Linear-matrix-inequality-based approach to H∞ controller synthesis of uncertain continuous-time piecewise linear systems. IEE Proceedings—Control Theory and Applications151: 295–301.
4.
ConcalvesJ (2000) Global stability analysis of on/off systems. In Proceedings of the 39th IEEE Conference on Decision and Control, pp. 1382–1387.
5.
FengG (2002a) Controller design and analysis of uncertain piecewise-linear systems. IEEE Transactions on Circuits and Systems. I, Fundamental Theory and Applications49: 224–232.
6.
FengG (2002b) Stability analysis of piecewise discrete-time linear systems. IEEE Transaction on Automatic Control47: 1108–1112.
7.
FengG (2003) Observer based output feedback controller design of piecewise discrete-time linear systems. IEEE Transactions on Circuits and Systems. I, Fundamental Theory and Applications50: 448–451.
8.
FengG (2005) Robust filtering design of piecewise discrete time linear systems. IEEE Transactions on Signal Processing53: 599–605.
9.
FukudaMKojimaM (2001) Branch-and-cute algorithms for the bilinear matrix inequality eigenvalue problem. Computational Optimization and Applications19: 79–105.
10.
GahinetPNemirovskiALaubAChilaliM (1995) The LMI Control Toolbox. Natwick, MA: The MathWorks Inc.
11.
GaoYHLiuZYChenH (2010) Observer based controller design of discrete-time piecewise affine systems. Asian Journal of Control12: 558–567.
12.
HassibiABoydS (1998) Quadratic stabilization and control of piecewise-linear systems. In Proceedings of the American Control Conference, pp. 3659–3664.
13.
ImuraJIVan der SchaftA (2000) Characterization of well-posedness of piecewise-linear systems. IEEE Transactions on Automatic Control45: 1600–1619.
14.
JohanssonM (2003) Piecewise Linear Control Systems. Berlin: Springer-Verlag.
15.
JohanssonMRantzerA (1998) Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Transactions on Automatic Control43: 555–559.
16.
KantnerM (1997) Robust stability of piecewise linear discrete time systems. In Proceedings of the American Control Conference, pp. 1241–1245.
17.
RantzerAJohanssonM (2000) Piecewise linear quadratic optimal control. IEEE Transactions on Automatic Control45: 629–637.
18.
RodriguesLHassibiAHowJP (2000) Output feedback controller synthesis for piecewise-affine systems with multiple equilibria. In Proceedings of the American Control Conference, pp. 1784–1789.
19.
RodriguesLHowJP (2001) Observer-based control of piecewise-affine systems. In Proceedings of the 40th IEEE Conference on Decision and Control, pp. 1366–1371.
20.
SongXMZhouSSZhangBY (2008) A cone complementarity linearization approach to robust H∞ controller design for continuous-time piecewise linear systems with linear fractional uncertainties. Nonlinear Analysis: Hybrid Systems2: 1264–1274.
21.
WangZDWeiGLFengG (2009) Reliable H∞ control for discrete-time piecewise linear systems with infinite distributed delays. Automatica45: 2991–2994.
22.
XiangZDWangRH (2010) Robust H∞ control for a class of uncertain switched systems using delta operator. Transactions of the Institute of Measurement and Control32: 331–344.
23.
XieL (1996) Output feedback H∞ control of systems with parameter uncertainty. International Journal of Control63: 741–750.
24.
XieLFuMSouzaC (1992) H∞ control and quadratic stabilization of systems with parameter uncertainty via output feedback. IEEE Transactions on Automatic Control37: 1253–1256.
25.
ZhangBXuS (2007) Robust H∞ filtering for uncertain discrete piecewise time-delay systems. International Journal of Control80: 636–645.
26.
ZhangHFengGDangCStability analysis and H∞ control for uncertain stochastic piecewise-linear systems. IET Control Theory and Applications20093: 1059–1069.
27.
ZhangJXTangWS (2007) Output feedback controller design for uncertain piecewise linear systems. Journal of Control Theory and Applications5: 199–206.
28.
ZhangJXTangWS (2008) Output feedback H∞ control for uncertain piecewise linear systems. Journal of Dynamic and Control Systems14: 121–144.
29.
ZhangJXTangWS (2009a) Output feedback optimal guaranteed cost control of uncertain piecewise linear systems. International Journal of Robust and Nonlinear Control19: 569–590.
30.
ZhangJXTangWS (2009b) Analysis and control for a new chaotic system via piecewise linear feedback. Chaos, Solitons and Fractals42: 2181–2190.
