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Adaptive differentiating filter: case study of PID feedback control

Alexey Pavlov, Michael Ruderman

Abstract

This paper presents an adaptive causal discrete-time filter for derivative estimation, exemplified by its use in estimating relative velocity in a mechatronic application. The filter is based on a constrained least squares estimator with window adaptation. It demonstrates low sensitivity to low-amplitude measurement noise, while preserving a wide bandwidth for large-amplitude changes in the process signal. Favorable performance properties of the filter are discussed and demonstrated in a practical case study of PID feedback controller and compared experimentally to a standard linear low-pass filter-based differentiator and a robust sliding-mode based homogeneous differentiator.

Adaptive differentiating filter: case study of PID feedback control

Abstract

This paper presents an adaptive causal discrete-time filter for derivative estimation, exemplified by its use in estimating relative velocity in a mechatronic application. The filter is based on a constrained least squares estimator with window adaptation. It demonstrates low sensitivity to low-amplitude measurement noise, while preserving a wide bandwidth for large-amplitude changes in the process signal. Favorable performance properties of the filter are discussed and demonstrated in a practical case study of PID feedback controller and compared experimentally to a standard linear low-pass filter-based differentiator and a robust sliding-mode based homogeneous differentiator.

Paper Structure

This paper contains 20 sections, 1 theorem, 24 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Adaptive Differentiating Filter (ADF)
  3. ADF concept
  4. ADF algorithm
  5. Tools for efficient ADF implementation
  6. Adding a point to the right: $W\rightarrow W^+$
  7. Removing a point from the left: $W\rightarrow W^-$
  8. Calculation of least squares fit
  9. Properties of ADF filter
  10. Tuning and performance
  11. Memory requirements
  12. PID control case study
  13. PID feedback with symmetrical optimum
  14. Linear filtered differentiation
  15. Robust finite-time differentiator
  16. ...and 5 more sections

Key Result

Theorem 1

Inequality (linineq) is feasible if and only if

Figures (6)

  • Figure 2: Shaped loop transfer function $C(j\omega)G(j\omega)$ with identified system plant $G(j\omega)$ and PID control $C(j\omega)$ designed by symmetrical optimum.
  • Figure 3: Actuator system in laboratory setting with translational motion $x$.
  • Figure 4: Measured and identified FRF of the system.
  • Figure 5: Measured noisy output and its distribution at the idle state ($u=0$) in (a), and exemplary $\dot{x}(t)$ estimate by means of LDF and ADF in (b).
  • Figure 6: Numerically determined FRF of three differentiators.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1