Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays

This paper is concerned with analyzing input-to-state stability (ISS) and integral-ISS (iISS) for nonlinear impulsive systems with delays. Razumikhin-type theorems are established which guarantee ISS/iISS for delayed impulsive systems with external input affecting both the continuous dynamics and the discrete dynamics. It is shown that when the delayed continuous dynamics are ISS/iISS but the discrete dynamics governing the impulses are not, the ISS/iISS property of the impulsive system can be retained if the length of the impulsive interval is large enough. Conversely, when the delayed continuous dynamics are not ISS/iISS but the discrete dynamics governing the impulses are, the impulsive system can achieve ISS/iISS if the sum of the length of the impulsive interval and the time delay is small enough. In particular, when one of the delayed continuous dynamics and the discrete dynamics are ISS/iISS and the others are stable for the zero input, the impulsive system can keep ISS/iISS no matter how often the impulses occur. Our proposed results are evaluated using two illustrative examples to show their effectiveness.

[1]  Wu-Hua Chen,et al.  Robust H∞ control of uncertain linear impulsive stochastic systems , 2008 .

[2]  David Angeli,et al.  A characterization of integral input-to-state stability , 2000, IEEE Trans. Autom. Control..

[3]  Zhengguo Li,et al.  Analysis and design of impulsive control systems , 2001, IEEE Trans. Autom. Control..

[4]  Yuan Wang,et al.  Stabilization in spite of matched unmodeled dynamics and an equivalent definition of input-to-state stability , 1996, Math. Control. Signals Syst..

[5]  Zhengguo Li,et al.  Input-to-state stabilization of switched nonlinear systems , 2001, IEEE Trans. Autom. Control..

[6]  Zhong-Ping Jiang,et al.  Input-to-state stability for discrete-time nonlinear systems , 1999 .

[7]  L. Praly,et al.  Stabilization by output feedback for systems with ISS inverse dynamics , 1993 .

[8]  Eduardo Sontag.  Comments on integral variants of input-to-state stability , 1998 .

[9]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[10]  A. Teel.  Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem , 1998, IEEE Trans. Autom. Control..

[11]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[12]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[13]  Zhong-Ping Jiang,et al.  Small-gain theorem for ISS systems and applications , 1994, Math. Control. Signals Syst..

[14]  David Angeli,et al.  Integral Input to State Stability for cascaded systems , 2006 .

[15]  Tao Yang,et al.  Impulsive Systems and Control: Theory and Applications , 2001 .

[16]  Linda Bushnell,et al.  Asymptotic behavior of nonlinear networked control systems , 2001, IEEE Trans. Autom. Control..

[17]  Eduardo Sontag,et al.  Further Equivalences and Semiglobal Versions of Integral Input to State Stability , 1999, math/9908066.

[18]  Dragan Nesic,et al.  Input-output stability properties of networked control systems , 2004, IEEE Transactions on Automatic Control.

[19]  Debasish Chatterjee,et al.  Input-to-state stability of switched systems and switching adaptive control , 2007, Autom..

[20]  Eduardo Sontag.  Comments on integral variants of ISS , 1998 .

[21]  A. Teel,et al.  Results on input-to-state stability for hybrid systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[22]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..

[23]  Xinhe Xu,et al.  Stability of Impulsive Differential Systems , 1997 .

[24]  Dragan Nesic,et al.  Input-to-state stability of networked control systems , 2004, Autom..

[25]  P. Pepe,et al.  A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems , 2006, Syst. Control. Lett..