Real-Time Numerical Differentiation of Sampled Data Using Adaptive Input and State Estimation

TL;DR

This work tackles real-time numerical differentiation under unknown signal and noise characteristics by introducing adaptive input estimation with an adaptive Kalman filter (AIE/ASE). The method jointly estimates the input derivative $d_k$ and system state $x_k$ while online-adjusting covariances via residuals, and it encompasses NSE, SSE, and ASE variants. Across two-tone harmonic signals and CarSim vehicle data, AIE/ASE consistently outperforms conventional baselines such as Backward-Difference, Savitzky–Golay, and High-Gain-Observer methods, especially when noise characteristics vary. The proposed framework enables robust, model-free derivative estimation in digital control, with practical impact on real-time estimation tasks and vehicle dynamics applications.

Abstract

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.

Figures (18)

• Figure 1: Timing diagram for causal numerical differentiation. The causal numerical differentiator uses data obtained at step $k$ to estimate the derivative of the signal $y$. Because of the computation time $T_{\rm c},$ the estimate $\hat{y}_k^{(q)}$ of $y_k^{(q)}$ is not available until step $k+1$. In this case, the delay is $\delta = 1$ step. For a noncausal differentiator, $\delta \ge 2.$
• Figure 2: Relative RMSE $\rho_{k_{\rm f}}^{(1)}$ of the estimate of the first derivative versus SNR, where $k_{{\rm f}} = 2000$ steps, for BD, SG, HGO/1, and HGO/2. For the first derivative, the red dashed line denotes the delay floor for $\delta=1$, and the black dashed line denotes the delay floor for $\delta=3$.
• Figure 3: Relative RMSE $\rho_{k_{\rm f}}^{(2)}$ of the estimate of the second derivative versus SNR, where $k_{{\rm f}} = 2000$ steps, for BD, SG, HGO/1, and HGO/2. For the second derivative, the red dashed line denotes the delay floor for $\delta=1$, and the black dashed line denotes the delay floor for $\delta=3$.
• Figure 4: Block diagram of AIE/NSE. The unknown input $d$ is the signal whose estimates are desired, $v$ is sensor noise, and $y$ is the noisy measurement. In this version of AIE, $V_1$ is fixed at a user-chosen value and $V_2$ is fixed at its true value. The state estimator is thus not adaptive.
• Figure 5: Block diagram of AIE/SSE. In this version of AIE, $V_1$ is adapted and $V_2$ is fixed at its true value. The state estimator is thus semi-adaptive.

Theorems & Definitions (2)

• Example 6.1
• Example 7.1