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Adaptive Real-Time Numerical Differentiation with Variable-Rate Forgetting and Exponential Resetting

Shashank Verma, Brian Lai, Dennis S. Bernstein

TL;DR

The paper tackles real-time numerical differentiation for digital PID control in the presence of nonstationary sensor noise. It extends adaptive input and state estimation (AISE) with recursive least squares using variable-rate forgetting and exponential resetting (VRF-ER) to enable rapid adaptation while keeping the covariance well-conditioned. The VRF-ER mechanism bounds the covariance via a resetting matrix and uses a data-driven forgetting strategy, improving derivative estimates in PID control and velocity estimation in collision-avoidance scenarios, as demonstrated on digital PID and CarSim simulations. The results show AISE-VRF-ER outperforms conventional filtering and standard AISE by achieving lower RMSE and maintaining stable covariance, underscoring its practical value for robust, real-time control under changing noise characteristics.

Abstract

Digital PID control requires a differencing operation to implement the D gain. In order to suppress the effects of noisy data, the traditional approach is to filter the data, where the frequency response of the filter is adjusted manually based on the characteristics of the sensor noise. The present paper considers the case where the characteristics of the sensor noise change over time in an unknown way. This problem is addressed by applying adaptive real-time numerical differentiation based on adaptive input and state estimation (AISE). The contribution of this paper is to extend AISE to include variable-rate forgetting with exponential resetting, which allows AISE to more rapidly respond to changing noise characteristics while enforcing the boundedness of the covariance matrix used in recursive least squares.

Adaptive Real-Time Numerical Differentiation with Variable-Rate Forgetting and Exponential Resetting

TL;DR

The paper tackles real-time numerical differentiation for digital PID control in the presence of nonstationary sensor noise. It extends adaptive input and state estimation (AISE) with recursive least squares using variable-rate forgetting and exponential resetting (VRF-ER) to enable rapid adaptation while keeping the covariance well-conditioned. The VRF-ER mechanism bounds the covariance via a resetting matrix and uses a data-driven forgetting strategy, improving derivative estimates in PID control and velocity estimation in collision-avoidance scenarios, as demonstrated on digital PID and CarSim simulations. The results show AISE-VRF-ER outperforms conventional filtering and standard AISE by achieving lower RMSE and maintaining stable covariance, underscoring its practical value for robust, real-time control under changing noise characteristics.

Abstract

Digital PID control requires a differencing operation to implement the D gain. In order to suppress the effects of noisy data, the traditional approach is to filter the data, where the frequency response of the filter is adjusted manually based on the characteristics of the sensor noise. The present paper considers the case where the characteristics of the sensor noise change over time in an unknown way. This problem is addressed by applying adaptive real-time numerical differentiation based on adaptive input and state estimation (AISE). The contribution of this paper is to extend AISE to include variable-rate forgetting with exponential resetting, which allows AISE to more rapidly respond to changing noise characteristics while enforcing the boundedness of the covariance matrix used in recursive least squares.
Paper Structure (10 sections, 1 theorem, 43 equations, 15 figures, 2 tables)

This paper contains 10 sections, 1 theorem, 43 equations, 15 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. ADAPTIVE INPUT AND STATE ESTIMATION
  3. Input Estimation
  4. Recursive Least Squares with Variable-Rate Forgetting and Exponential Resetting
  5. State Estimation
  6. Adaptive Input and State Estimation
  7. NUMERICAL EXAMPLES
  8. Digital PID Control
  9. Target tracking for collision avoidance
  10. CONCLUSIONS

Key Result

Proposition II.1

Let $k \ge 0$ and let $\eta_k \in [\eta_L,\eta_U]$ and $V_{2,k} \ge 0$ be given by covmin. If ${\mathcal{J}}_{{\rm f},k}$, defined in J_f_positive, is nonempty, then, for any $\beta \in [0,1]$, $\eta_k$ and $V_{2,k}$ are given by where If ${\mathcal{J}}_{{\rm f},k}$ is empty, then $\eta_k$ and $V_{2,k}$ are given by

Figures (15)

  • Figure 1: Servo loop consisting of discrete-time first-order-lag-plus-dead-time dynamics with a discrete-time PID controller, where $\eta_{k}$ is the sensor noise.
  • Figure 2: Servo loop consisting of discrete-time first-order-lag-plus-dead-time dynamics with a discrete-time PID controller combined with adaptive differentiation (PID/AD).
  • Figure 3: Example \ref{['pid']}: (a) shows the step response $y_k$ of the PI controller in the absence of sensor noise, where, without the derivative action, the settling time is 20 sec. (b) shows the control $u_k$.
  • Figure 4: Example \ref{['pid']}: (a) shows the step response $\Bar{y}_k$ of the PID controller in the absence of sensor noise using BD for numerical differentiation. (b) shows the control $u_k$.
  • Figure 5: Example \ref{['pid']}: (a) shows the step response $y_{k}$ of the PID controller in the presence of sensor noise using BD for numerical differentiation. The steady-state response is noisy. The ${\rm RMSE}$ is $0.1904$. (b) shows the control $u_k$.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Proposition II.1
  • proof
  • Example III.1
  • Example III.2