Asymptotic Behavior of the Pseudo-Covariance Matrix of a Robust State Estimator with Intermittent Measurements

Ergodic properties and asymptotic stationarity are investigated in this paper for the pseudo-covariance matrix (PCM) of a recursive state estimator which is robust against parametric uncertainties and is based on plant output measurements that may be randomly dropped. When the measurement dropping process is described by a Markov chain and the modified plant is both controllable and observable, it is proved that if the dropping probability is less than 1, this PCM converges to a stationary distribution that is independent of its initial values. A convergence rate is also provided. In addition, it has also been made clear that when the initial value of the PCM is set to the stabilizing solution of the algebraic Riccati equation related to the robust state estimator without measurement dropping, this PCM converges to an ergodic process. Based on these results, two approximations are derived for the probability distribution function of the stationary PCM, as well as a bound of approximation errors. A numerical example is provided to illustrate the obtained theoretical results. Key Words—ergodicity, networked system, random measurement dropping, recursive state estimation, robustness, sensitivity penalization, stationary distribution.