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.
This paper is concerned with the technique called discrete-time noncausal linear periodically time-varying (LPTV) scaling for robust stability analysis. Noncausal LPTV scaling has already been shown to be effective for reducing the conservativeness of robustness analysis in theoretical and numerical ways. However, there still remain some issues to be resolved for further understanding and exploiting noncausal LPTV scaling, e.g., its relationship with the conventional analysis approach of causal linear time-invariant scaling. In this paper, by introducing the key idea of shift-invariance in lifting-timing, we discuss the difference and corresponding relationship between the conventional approach and noncausal LPTV scaling.
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