Reset Control Systems: Stabilization by Nearly-Periodic Reset

Abstract In this paper the class of linear impulsive systems is considered. These systems are those with a continuous linear dynamics for all time, except at a sequence of instants. When such a discrete time occurs, the state undergoes a jump, or more precisely follows a discrete linear dynamics. The sequence of time instants, when a discrete dynamics occurs, is nearly-periodic only, i.e. it is distant from a periodic sequence to an uncertain distance. This paper succeeds to state tractable conditions to analyze the stability, and to design reset matrices such that the hybrid system is globally asymptotically stable to the origin. The approach is based on a polytopic embedding of the uncertain dynamics. An example illustrates the main stability result.

[1]  J. C. Clegg.  A nonlinear integrator for servomechanisms , 1958, Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry.

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

[3]  Luca Zaccarian,et al.  Stability properties of reset systems , 2008, Autom..

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

[5]  Rafal Goebel,et al.  Smooth patchy control Lyapunov functions , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[6]  L. Zaccarian,et al.  First order reset elements and the Clegg integrator revisited , 2005, Proceedings of the 2005, American Control Conference, 2005..

[7]  Ji-Woong Lee,et al.  Uniform stabilization of discrete-time switched and Markovian jump linear systems , 2006, Autom..

[8]  Sophie Tarbouriech,et al.  Guaranteed stability for nonlinear systems by means of a hybrid loop , 2010 .

[9]  Wassim M. Haddad,et al.  Non-linear impulsive dynamical systems. Part II: Stability of feedback interconnections and optimality , 2001 .

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

[11]  Alfonso Baños,et al.  Delay-dependent stability of reset systems , 2010, Autom..

[12]  Wassim M. Haddad,et al.  Non-linear impulsive dynamical systems. Part I: Stability and dissipativity , 2001 .

[13]  Jamal Daafouz,et al.  Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties , 2001, Syst. Control. Lett..

[14]  Wassim M. Haddad,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control , 2006 .

[15]  Laurentiu Hetel,et al.  LMI control design for a class of exponential uncertain systems with application to network controlled switched systems , 2007, 2007 American Control Conference.

[16]  I. Horowitz,et al.  Non-linear design for cost of feedback reduction in systems with large parameter uncertainty † , 1975 .

[17]  Orhan Beker,et al.  Plant with integrator: an example of reset control overcoming limitations of linear feedback , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[18]  Rafal Goebel,et al.  Hybrid Feedback Control and Robust Stabilization of Nonlinear Systems , 2007, IEEE Transactions on Automatic Control.

[19]  M. C. D. Oliveiraa,et al.  A new discrete-time robust stability condition ( , 1999 .