Optimal state filtering and parameter identification for linear systems

This paper presents the optimal filtering and parameter identification problem for linear stochastic systems with unknown multiplicative and additive parameters over linear observations, where unknown parameters are considered Wiener processes. The original problem is reduced to the filtering problem for an extended state vector that incorporates parameters as additional states. The resulting filtering system is bilinear in state, with unmeasured linear part, and linear in observations. The obtained solution is based on the recently derived optimal filter for bilinear-linear states with partially measured linear part over linear observations. The optimal filter for the extended state vector also serves as the optimal identifier for the unknown parameters. In the example, performance of the designed optimal state filter and parameter identifier is verified for linear systems with unknown multiplicative parameter over linear observations. Both, stable and unstable, linear systems are examined.