Mean Square Stability Of System Research Articles

Mean square stability for systems on stochastically generated discrete time scales

We consider dynamic systems which evolve on discrete time domains where the time steps form a sequence of independent, identically distributed random variables. In particular, we classify the mean-square stability of linear systems on these time domains using quadratic Lyapunov functionals. In the case where the system matrix is a function of the time step, our results agree with and generalize stability results found in the Markov jump linear systems literature. In the case where the system matrix is constant, our results generalize, illuminate, and extend to the stochastic realm results in the field of dynamic equations on time scales. In order to help see the factors that contribute to stability, we prove a sufficient condition for the solvability of the Lyapunov equation by appealing to a fixed point theorem of Ran and Reurings. Finally, an example using observer-based feedback control is presented to demonstrate the utility of the results to control engineers who cannot guarantee uniform timing of the system.

Robust stabilisation of uncertain delayed Markovian jump systems and its applications

ABSTRACTThis paper is concerned with the robust stabilsation of uncertain delayed Markovian jump systems. Given a Markovian jump system with time delay and Brownian motion simultaneously, we allow the uncertainty added in the form of additive perturbations and existing in the drift and diffusion sections at the same time. A sufficient condition on the mean square stability of system in the face of such disturbances is obtained, which is similar to small-gain theorem. A kind of partially delay-dependent controller stabilising the resulting closed-loop system is firstly designed to relate to the probability distribution of delay, whose key idea is applied to construct a delayed controller with disordering phenomenon. It is seen that the existence conditions established here could be solved easily. Based on the proposed results, some applications on robust synchronisation of uncertain delayed multi-agent systems with Markovian switching are considered. It is shown that the robust synchronisation of such an uncertain multi-agent network could be achieved by a protocol that each controller being partially delay-dependent or disordering could robustly stabilise a given single Markovian jump system. As for these cases, the proposed protocols could be obtained by solving certain algebraic Riccati equations and inequalities, which also involve weighting factors and depend on the eigenvalues of the Laplacian graph.

Stability analysis and stabilization of networked linear systems with random packet losses

This paper is concerned with the stability analysis and stabilization of networked discrete-time and sampled-data linear systems with random packet losses. Asymptotic stability, mean-square stability, and stochastic stability are considered. For networked discrete-time linear systems, the packet loss period is assumed to be a finite-state Markov chain. We establish that the mean-square stability of a related discrete-time system which evolves in random time implies the mean-square stability of the system in deterministic time by using the equivalence of stability properties of Markovian jump linear systems in random time. We also establish the equivalence of asymptotic stability for the systems in deterministic discrete time and in random time. For networked sampled-data systems, a binary Markov chain is used to characterize the packet loss phenomenon of the network. In this case, the packet loss period between two transmission instants is driven by an identically independently distributed sequence assuming any positive values. Two approaches, namely the Markov jump linear system approach and randomly sampled system approach, are introduced. Based on the stability results derived, we present methods for stabilization of networked sampled-data systems in terms of matrix inequalities. Numerical examples are given to illustrate the design methods of stabilizing controllers.

Relationship between spectral and coefficient criteria of mean-square stability for systems of linear stochastic differential and difference equations

We establish the relationship (equivalence) between the spectral and algebraic (coefficient) criteria (the latter is represented in terms of the Sylvester matrix algebraic equation) of mean-square asymptotic stability for three classes of systems of linear equations with varying random perturbations of coefficients, namely, the lto differential stochastic equations, difference stochastic equations with discrete time, and difference stochastic equations with continuous time.

A property of stable systems of linear stochastic equations

The relationship between exponential mean square stability of systems of linear ordinary differential equations with Gaussian coefficients and the same stability of the corresponding linear stochastic Ito differential equations is obtained.