Stabilisation of second-order LTI switched positive systems

The stabilisation problem of second-order switched positive systems consisting of two unstable subsystems is considered in this article. By considering the vector fields and geometric characteristics, a necessary and sufficient condition for the stabilisability of second-order switched positive systems with two unstable subsystems is provided. Furthermore, it is shown via this condition that neither second-order switched positive systems consisting of two subsystems with unstable nodes nor second-order switched positive systems consisting of one subsystem with unstable nodes and the other with a saddle point can be stabilised via any switching law.

[1]  Guisheng Zhai,et al.  Practical stability and stabilization of hybrid and switched systems , 2004, IEEE Trans. Autom. Control..

[2]  Moussa Balde,et al.  Stability of planar switched systems: the nondiagonalizable case , 2006, math/0610401.

[3]  Bruce A. Francis,et al.  Stabilizing a linear system by switching control with dwell time , 2002, IEEE Trans. Autom. Control..

[4]  Robert Shorten,et al.  On the Stability of Switched Positive Linear Systems , 2007, IEEE Transactions on Automatic Control.

[5]  Pingyuan Cui,et al.  Stabilization for a class of second-order switched systems , 2005 .

[6]  Jun Zhao,et al.  On stability, L 2 -gain and H 8 control for switched systems , 2008 .

[7]  Robin J. Evans,et al.  Stability results for switched controller systems , 1999, Autom..

[8]  Hai Lin.  Hybrid Output Feedback Stabilization for LTI Systems With Single Output , 2008, IEEE Transactions on Automatic Control.

[9]  Huijun Gao,et al.  Asynchronously switched control of switched linear systems with average dwell time , 2010, Autom..

[10]  Jun Zhao,et al.  On stability, L2-gain and Hinfinity control for switched systems , 2008, Autom..

[11]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[12]  S. Pettersson,et al.  Synthesis of Switched Linear Systems handling Sliding Motions , 2005, Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, 2005..

[13]  Hai Lin,et al.  Necessary and sufficient conditions for regional stabilisability of generic switched linear systems with a pair of planar subsystems , 2010, Int. J. Control.

[14]  Jamal Daafouz,et al.  Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach , 2002, IEEE Trans. Autom. Control..

[15]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[16]  Robert Shorten,et al.  On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems , 2007, IEEE Transactions on Automatic Control.

[17]  S. Shankar Sastry,et al.  Stabilization of planar switched linear systems using polar coordinates , 2010, HSCC '10.

[18]  Shigemasa Takai,et al.  Improving closed-loop stability of second-order LTI systems by hybrid static output feedback , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[19]  R. Decarlo,et al.  Perspectives and results on the stability and stabilizability of hybrid systems , 2000, Proceedings of the IEEE.

[20]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[21]  Robert Shorten,et al.  On linear co-positive Lyapunov functions for sets of linear positive systems , 2009, Autom..

[22]  Franco Blanchini,et al.  Discrete‐time control for switched positive systems with application to mitigating viral escape , 2011 .

[23]  Michael Margaliot,et al.  On the Stability of Positive Linear Switched Systems Under Arbitrary Switching Laws , 2009, IEEE Transactions on Automatic Control.

[24]  Xuping Xu,et al.  Stabilization of second-order LTI switched systems , 2000 .

[25]  Ugo V. Boscain,et al.  Stability of Planar Switched Systems: The Linear Single Input Case , 2002, SIAM J. Control. Optim..

[26]  A. Morse,et al.  Stability of switched systems with average dwell-time , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[27]  G. Zhai,et al.  Quadratic stabilizability of switched linear systems with polytopic uncertainties , 2003 .

[28]  Michael Margaliot,et al.  Necessary and sufficient conditions for absolute stability: the case of second-order systems , 2003 .

[29]  S. Rinaldi,et al.  Positive Linear Systems: Theory and Applications , 2000 .

[30]  Patrizio Colaneri,et al.  Stabilization of continuous-time switched linear positive systems , 2010, Proceedings of the 2010 American Control Conference.

[31]  Xingwen Liu,et al.  Stability Analysis of Switched Positive Systems: A Switched Linear Copositive Lyapunov Function Method , 2009, IEEE Transactions on Circuits and Systems II: Express Briefs.

[32]  S. Pettersson,et al.  Stabilization of hybrid systems using a min-projection strategy , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).