A Stochastic Lyapunov Theorem with Application to Stability Analysis of Networked Control Systems

The source of randomness in stochastic systems is an input with stochastic behavior as treated in the existing literature. Special types of stochastic processes such as the Wiener process or the Brownian motion have served as an adequate model of such an input for years. The body of stochastic systems theory is elegantly shaped around such input models. An example is the Itô’s formula. With development of new applications, we are faced with various phenomena that are more demanding from a stochastic modeling approach. To cope with this problem we restate the stochastic Lyapunov theorem such that it can be applied to a wider class of stochastic systems. In this paper stochastic systems are considered without imposing assumptions on the nature of the stochastic input and the way it affects the sample trajectories. Lyapunov stability theorem is represented for this type of systems in terms of a stability notion that generalizes the notion of stability in moments. As a result, the new theorem finds a larger domain of applications while it can be reduced to some known versions of the stochastic Lyapunov theorem. As an application, an existing deterministic result for nonlinear networked control systems is extended to a more practical probabilistic setting which extends the available analysis tools for checking the stability of continuous-time nonlinear networked control systems in the stochastic setting. The results are applied to a two-channel magnetic levitation system which is controlled over a local communication network to obtain a bound on the rate of transmission failures due to the presence of noise in the industrial environment.