Bounds for Kalman filtering with intermittent observations

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This paper considers the Kalman filtering problem where sensor signals are transmitted over an unreliable network and thus observation packets could be lost randomly. It has been proven by Sinopoli et al. that there is a critical observation arrival probability such that the expectation of the estimation error covariance is always finite and the equilibrium of the Modified Algebraic Riccati Equation (MARE) gives an upper bound on the mean covariance matrix. In this paper, we analyze the necessary condition on the existence of equilibrium of the MARE and provide an analytical lower bound on the critical value, below which there is no positive definite equilibrium. In contrast to the lower bound which only depends on the unstable eigenvalues on the system matrix, our analytical lower bound is tighter and depends on both the unstable eigenvalues of the system matrix and the rank of the observation matrix. Moreover, any sub-system can give an analytical lower bound, which also depends on the rank of the sub-system's observation matrix. The simulation results show that our lower bound is very tight.