On eigenvalue sets and convergence rate of Itô stochastic systems with Markovian switching

This paper is concerned with stability analysis and stabilization of Itô stochastic systems with Markovian switching. A couple of eigenvalue sets for some positive operator associated with the stochastic system under study are defined to characterize its stability in the mean square sense. Properties for these eigenvalue sets are established based on which we show that the spectral abscissa of these eigenvalues sets are the same and thus these eigenvalue sets are equivalent in the sense of characterizing the stability of the system. Also, it is shown that the guaranteed convergence rate of the Markovian jump Itô stochastic systems can be determined by some eigenvalue set. Finally, a linear matrix inequality based approach is proposed to design controllers such that the closed-loop system has guaranteed convergence rate. Some numerical examples are carried out to illustrate the effectiveness of the proposed approach. The research in this paper opens several perspectives for future work stated as some open problems.

[1]  Dong Yue,et al.  Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching , 2005, IEEE Transactions on Automatic Control.

[2]  J. Lam,et al.  Spectral properties of sums of certain Kronecker products , 2009 .

[3]  James Lam,et al.  Analysis and Synthesis of Markov Jump Linear Systems With Time-Varying Delays and Partially Known Transition Probabilities , 2008, IEEE Transactions on Automatic Control.

[4]  Xingyu Wang,et al.  Sliding mode control for Itô stochastic systems with Markovian switching , 2007, Autom..

[5]  Yuguang Fang,et al.  Stochastic stability of jump linear systems , 2002, IEEE Trans. Autom. Control..

[6]  Xuerong Mao,et al.  Robust stability and controllability of stochastic differential delay equations with Markovian switching , 2004, Autom..

[7]  Keith J. Burnham,et al.  On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters , 2002, IEEE Trans. Autom. Control..

[8]  Xuerong Mao,et al.  On Input-to-State Stability of Stochastic Retarded Systems With Markovian Switching , 2009, IEEE Transactions on Automatic Control.

[9]  T. Damm,et al.  Detectability, Observability, and Asymptotic Reconstructability of Positive Systems , 2009 .

[10]  James Lam,et al.  Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers , 2006, Autom..

[11]  Yuanqing Xia,et al.  On designing of sliding-mode control for stochastic jump systems , 2006, IEEE Transactions on Automatic Control.

[12]  James Lam,et al.  Necessary and Sufficient Conditions for Analysis and Synthesis of Markov Jump Linear Systems With Incomplete Transition Descriptions , 2010, IEEE Transactions on Automatic Control.

[13]  El-Kébir Boukas,et al.  Control of Singular Systems with Random Abrupt Changes , 2008 .

[14]  H. Chizeck,et al.  Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control , 1990 .

[15]  El-Kébir Boukas,et al.  Stochastic Switching Systems: Analysis and Design , 2005 .

[16]  V. Dragan,et al.  Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise , 2002 .

[17]  Kiev,et al.  Stability and Stabilization of Nonlinear Systems with Random Structure , 2002 .

[18]  John Lygeros,et al.  Stabilization of a class of stochastic differential equations with Markovian switching , 2005, Syst. Control. Lett..

[19]  H. Schneider.  Positive operators and an inertia theorem , 1965 .

[20]  Lixian Zhang,et al.  H∞ estimation for discrete-time piecewise homogeneous Markov jump linear systems , 2009, Autom..

[21]  Peng Shi,et al.  Robust filtering for jumping systems with mode-dependent delays , 2006, Signal Process..

[22]  D. Elworthy.  ASYMPTOTIC METHODS IN THE THEORY OF STOCHASTIC DIFFERENTIAL EQUATIONS , 1992 .

[23]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[24]  Xuerong Mao,et al.  Exponential stability of stochastic delay interval systems with Markovian switching , 2002, IEEE Trans. Autom. Control..

[25]  A. Berman,et al.  Nonnegative matrices in dynamic systems , 1979 .

[26]  Lihua Xie,et al.  Interval Stability and Stabilization of Linear Stochastic Systems , 2009, IEEE Transactions on Automatic Control.

[27]  James Lam,et al.  Filtering for Nonlinear Genetic Regulatory Networks With Stochastic Disturbances , 2008, IEEE Transactions on Automatic Control.

[28]  James Lam,et al.  Non-Fragile Exponential Stability Assignment of Discrete-Time Linear Systems With Missing Data in Actuators , 2009, IEEE Transactions on Automatic Control.