Detectability, Observability and Lyapunov-Type Theorems of Linear Discrete Time-Varying Stochastic Systems with Multiplicative Noise

TL;DR

This paper introduces new concepts of detectability and observability for linear discrete-time stochastic systems with multiplicative noise, establishing Lyapunov-type theorems and stability criteria.

Contribution

It proposes novel detectability and observability notions, along with Lyapunov theorems, for analyzing stability of stochastic systems with multiplicative noise.

Findings

New detectability and observability concepts introduced.

Lyapunov-type theorems established for stability analysis.

Conditions for exponential stability in mean square are provided.

Abstract

The objective of this paper is to study detectability, observability and related Lyapunov-type theorems of linear discrete-time time-varying stochastic systems with multiplicative noise. Some new concepts such as uniform detectability, ${\cal K}^{\infty}$-exact detectability (resp. ${\cal K}^{WFT}$-exact detectability, ${\cal K}^{FT}$-exact detectability, ${\cal K}^{N}$-exact detectability) and ${\cal K}^{\infty}$-exact observability (resp. ${\cal K}^{WFT}$-exact observability, ${\cal K}^{FT}$-exact observability, ${\cal K}^{N}$-exact observability) are introduced, respectively, and nice properties associated with uniform detectability, exact detectability and exact observability are also obtained. Moreover, some Lyapunov-type theorems associated with generalized Lyapunov equations and exponential stability in mean square sense are presented under uniform detectability, ${\cal K}^{N}$-exact observability and ${\cal K}^{N}$-exact detectability, respectively.