  Real-time numerical differentiation plays a crucial role in many digital
control algorithms, such as PID control, which requires numerical
differentiation to implement derivative action. This paper addresses the
problem of numerical differentiation for real-time implementation with minimal
prior information about the signal and noise using adaptive input and state
estimation. Adaptive input estimation with adaptive state estimation (AIE/ASE)
is based on retrospective cost input estimation, while adaptive state
estimation is based on an adaptive Kalman filter in which the input-estimation
error covariance and the measurement-noise covariance are updated online. The
accuracy of AIE/ASE is compared numerically to several conventional numerical
differentiation methods. Finally, AIE/ASE is applied to simulated vehicle
position data generated from CarSim.
