On the Accuracy of Numerical Differentiation Using High-Gain Observers and Adaptive Input Estimation

The ability to control a system is often enhanced by feeding back the derivatives of sensor signals, such as estimates of velocity and acceleration when only position is measured. Within this context, signal differentiation must be performed causally, that is, using only current and past data and with minimal computational latency. This paper formulates causal differentiation as a sampled-data input-estimation problem, where the plant is a cascade of integrators. Adaptive input estimation based on retrospective-cost optimization is considered, where the innovations from the Kalman filter is used to drive the online adaptation. Using backward-difference differentiation (BDD) as a baseline comparison, high-gain observers (HGO) with bilinear discretization and retrospective cost input estimation (RCIE) are applied to harmonic signals under various noise levels for single and double differentiation. These methods are then applied to experimental position data of a small rover for estimating its velocity and acceleration. Neither method uses information about the noise statistics, and no analog or digital filtering is used for noise suppression.