Conditionally-linear filtering using conditionally-orthogonal projection

In this work a general discrete-time dynamical not necessarily linear system with additive not necessarily Gaussian noises is considered and the problem of estimating the unknown states using the past history of the known states is addressed. The proposed approach is based on the concept of conditionally-orthogonal projection, where the conditional expectation operator given the known states' history, rather than the unconditional expectation, is used to define orthogonality. This approach, called conditionally-linear approach, is a direct extension of the standard linear filtering approach, and is more adequate than the latter for on-line implementation since it allows for the filter gains to be function of the incoming observations. In the case of an underlying discretization of a continuous-time system with Gaussian additive measurement noises, the proposed discrete-time algorithm is formally shown to be asymptotically equivalent to the continuous-time optimal non-linear filter, when the discretization step is very small. For conditionally-Gaussian systems, this filter coincides with the standard conditionally-Gaussian filter. The advantage of the conditionally-linear filter over the standard linear filter is analyzed and illustrated when applied to identification of a constant discrete parameter.