Optimal State Filtering and Parameter Identification for Linear Stochastic Time-Delay Systems

This paper presents the optimal joint state filtering and parameter identification problem for linear stochastic time-delay systems with unknown parameters. The original identification problem is reduced to the optimal filtering problem for incompletely measured polynomial (bilinear) timedelay system states over linear observations with an arbitrary, not necessarily invertible, observation matrix, where the unknown parameters are considered standard Wiener processes and incorporated as additional states in the extended state vector. The obtained solution is based on the designed optimal filter for incompletely measured bilinear time-delay states over linear observations, taking into account that 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 a linear time-delay system with an unknown multiplicative parameter over linear observations. Both, stable and unstable, linear systems are examined.