Extended Recursive Three-Step Filter for Linear Discrete-Time Systems with Dual-Unknown Inputs

This paper proposes two new extended recursive three-step filters for linear discrete systems with dual-unknown inputs, which can simultaneously estimate unknown input and state. Extended recursive three-step filter 1 (ERTSF1) introduces an innovation for obtaining the estimates of the unknown input in the measurement equation, then derives the estimates of the unknown input in the state equation. After that, it uses the already obtained estimates of the dual-unknown inputs to correct the one-step prediction of the state, and finally, it obtains the minimum-variance unbiased estimate of the system state. Extended recursive three-step filter 2 (ERTSF2) establishes a unified innovation feedback model, then applies linear minimum-variance unbiased estimation to obtain the estimates of the system state and the dual-unknown inputs to refine a more concise recursive filter. Numerical Simulation Ex-ample demonstrates the effectiveness and superiority of the two filters in this paper compared with the traditional method. The battery state of charge estimation results demonstrate the effectiveness of ERTSF2 in practical applications.