Equivalent Stability Notions, Lyapunov Inequality, and Its Application in Discrete-Time Linear Systems with Stochastic Dynamics Determined by an i.i.d. Process
Provides a theoretical foundation for stability analysis and control synthesis in stochastic linear systems, but the contribution is incremental as it extends existing Lyapunov-based methods to a specific i.i.d. setting.
The paper proves equivalence of three stability notions for discrete-time linear systems with i.i.d. stochastic dynamics and derives a necessary and sufficient Lyapunov inequality condition, which is then solved as a standard linear matrix inequality for state feedback synthesis.
This paper is concerned with stability analysis and synthesis for discrete-time linear systems with stochastic dynamics. Equivalence is first proved for three stability notions under some key assumptions on the randomness behind the systems. In particular, we use the assumption that the stochastic process determining the system dynamics is independent and identically distributed (i.i.d.) with respect to the discrete time. Then, a Lyapunov inequality condition is derived for stability in a necessary and sufficient sense. Although our Lyapunov inequality will involve decision variables contained in the expectation operation, an idea is provided to solve it as a standard linear matrix inequality; the idea also plays an important role in state feedback synthesis based on the Lyapunov inequality. Motivating numerical examples are further discussed as an application of our approach.