Quadratic Filtering of non-Gaussian Linear Systems with Random Observation Matrices

In this paper we consider the problem of state estimation for linear discrete-time non-Gaussian systems with random observation matrices. This is the model for systems with observation losses due to propagation through unreliable communication channels. Losses may result from intermittent failures that cause packet dropouts, as in the case of networks, or fading phenomena in propagation channel, as in the case of wireless networks. These are common problems in wireless sensor network, or networked control systems. In this paper, we do not make any assumption about the distribution of the observation matrix, thus encompassing a great variety of possible scenarios. We derive the quadratic estimate of the state by means of a recursive algorithm. The solution is obtained by applying the Kalman filter to a suitably augmented system, which is fully observable. The augmented system is constructed as the aggregate of the actual system and the observable part of a system having as state the second Kronecker power of the original state, namely the quadratic system. To extract the observable part of the quadratic system we exploit the knowledge of the rank of the corresponding observability matrix. This approach guarantees the internal stability of the estimation filter.