Equivalent Characterizations of Input-to-State Stability for Stochastic Discrete-Time Systems

Input-to-state stability (ISS) for stochastic difference inclusions is studied. First, ISS in probability relative to a compact set is defined. Subsequently, several equivalent characterizations are given. For example, ISS in probability is shown to be equivalent to global asymptotic stability in probability when the disturbance takes values in a ball whose radius is determined by a sufficiently small, but unbounded, function of the distance of the state to the compact set. In turn, a recent converse Lyapunov theorem for global asymptotic stability in probability provides an equivalent Lyapunov characterization. Finally, robust ISS in probability is defined and is shown to give another equivalent characterization.

[1]  Miroslav Krstic,et al.  Stabilization of stochastic nonlinear systems driven by noise of unknown covariance , 2001, IEEE Trans. Autom. Control..

[2]  John Tsinias,et al.  The concept of 'Exponential input to state stability' for stochastic systems and applications to feedback stabilization , 1999 .

[3]  Andrew R. Teel,et al.  Lyapunov conditions certifying stability and recurrence for a class of stochastic hybrid systems , 2013, Annu. Rev. Control..

[4]  J. Tsinias.  Stochastic input-to-state stability and applications to global feedback stabilization , 1998 .

[5]  Eduardo Sontag,et al.  On characterizations of the input-to-state stability property , 1995 .

[6]  A. Subbaraman,et al.  Ac onverse Lyapunov theorem for asymptotic stability in probability , 2012 .

[7]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[8]  Zhong-Ping Jiang,et al.  A converse Lyapunov theorem for discrete-time systems with disturbances , 2002, Syst. Control. Lett..

[9]  Andrew R. Teel,et al.  A Matrosov Theorem for Adversarial Markov Decision Processes , 2013, IEEE Transactions on Automatic Control.

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

[11]  Andrew R. Teel,et al.  A converse Lyapunov theorem for strong global recurrence , 2013, Autom..

[12]  Eduardo Sontag,et al.  New characterizations of input-to-state stability , 1996, IEEE Trans. Autom. Control..

[13]  Debasish Chatterjee,et al.  Towards ISS disturbance attenuation for randomly switched systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[14]  Xue-Jun Xie,et al.  Adaptive backstepping controller design using stochastic small-gain theorem , 2007, Autom..

[15]  Jifeng Zhang,et al.  A notion of stochastic input-to-state stability and its application to stability of cascaded stochastic nonlinear systems , 2008 .

[16]  Dragan Nesic,et al.  Changing supply functions in input to state stable systems: the discrete-time case , 2001, IEEE Trans. Autom. Control..

[17]  Bert Fristedt,et al.  A modern approach to probability theory , 1996 .

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

[19]  J. Tsinias,et al.  Notions of exponential robust stochastic stability, ISS and their Lyapunov characterizations , 2003 .

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