Suboptimal linear estimation for continuous-discrete bilinear systems

In this paper we derive a suboptimal estimation for continuous-discrete bilinear systems. One of the motivations of this work is that the bilinear system has the simplest structure in the nonlinear class in some sense. Similar to the Kalman filter, our algorithm includes prediction and updating step. We show rigorously that our algorithm gives an unbiased estimate, the a-priori estimate approaches to the conditional expectation exponentially fast, and the posterior estimate minimizes the conditional variance error in the linear space spanned by the a-priori estimate and the innovation. Our algorithm is also applicable to solve the nonlinear filtering problems. The efficiency of our method is illustrated by the cubic sensor problem and Lorenz system with discrete observation. The results have been compared with the extended Kalman filter and the unscented Kalman filter.