Distributed Design of Robust Kalman Filters over Corrupted Channels

The robust state estimation problem is on how to design robust filters for estimating an unknown state of uncertain systems. This paper considers this problem for multi-agent systems with multiplicative noise and degraded measurements over corrupted channels. Employing a covariance intersection fusion method, we propose a distributed robust Kalman filter with stochastic gains, which enables a sequence of upper bounds of conditional mean square error given channel noise to be calculated online. Considering the limitation of step-wise optimization, for better performance, we propose a switching fusion scheme based on a sliding window method, which provides an online design of covariance intersection weights by solving a semi-definite programming problem. Compared to the filter fusing latest estimates, the one based on the switching fusion method has a smaller upper bound of the conditional mean square error. We present a robust collective observability condition, which degenerates to the traditional collective observability condition for time-varying stochastic systems if there is no measurement degradation or multiplicative noise. Under this condition and strong connectivity, we prove that the mean square errors of two filters are both uniformly upper bounded by a constant matrix over a finite transient time, which depends on the system observability and the network size. Different to existing results, some requirements including stability for the systems and observability of the sub-systems are not needed for our results. Finally, a numerical simulation is provided to validate the theoretical results.