⚡ electrical engineering

Adaptive differentiating filter: case study of PID feedback control

This paper introduces and experimentally validates an adaptive causal discrete-time filter based on constrained least squares estimation that effectively balances noise rejection and bandwidth preservation for derivative estimation in PID feedback control, outperforming standard low-pass and sliding-mode differentiators in a mechatronic case study.

Original authors: Alexey Pavlov, Michael Ruderman

Published 2026-03-31

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CC BY 4.0

Original authors: Alexey Pavlov, Michael Ruderman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are driving a car, but your speedometer is broken and replaced by a shaky, jittery camera that only shows you your position every fraction of a second. To drive smoothly, you need to know your speed (how fast your position is changing).

If you just look at the difference between two camera frames, the tiny shakes in the camera (noise) make your calculated speed look like it's jumping wildly between 0 and 100 mph. If you try to smooth out those jumps by averaging them over a long time, your speed reading becomes too slow to react to sudden stops or accelerations.

This is the classic problem engineers face: How do you calculate a rate of change (like speed) from a noisy signal without losing the ability to react quickly?

This paper introduces a clever new tool called the Adaptive Differentiating Filter (ADF) to solve this problem. Here is how it works, explained simply:

1. The Problem with "Fixed" Filters

Most standard filters are like a rigid pair of sunglasses.

If the lenses are too dark (strong filtering), they block out the noise, but they also make the world look blurry and slow. You can't see a sudden stop coming.
If the lenses are too light (weak filtering), you see the world clearly, but the glare (noise) is blinding.

Engineers usually have to pick one setting and stick with it, even if the road conditions change.

2. The New Solution: "Smart, Shape-Shifting Glasses"

The authors created a filter that acts like smart, shape-shifting glasses. It doesn't have a fixed setting; it changes its "lens" in real-time based on what it sees.

When the road is smooth (low noise): The filter looks at a long stretch of the road (a large window). By averaging over a long distance, it ignores the tiny bumps and gives you a very smooth, accurate speed reading.
When the road gets bumpy or you hit a pothole (high dynamics): The filter instantly realizes, "Hey, things are changing fast!" It shrinks its view to a short, sharp look at just the immediate past. This allows it to react instantly to the pothole without being confused by the noise.

The Magic Trick:
The filter uses a mathematical "rule of thumb." It asks: "Can I draw a straight line through these recent data points without the line being too far off from any single point?"

If yes, it keeps adding more points to its view (making the window bigger) to smooth out the noise.
If no (because the signal is changing too fast for a straight line), it stops adding points and calculates the speed based on the current window.

3. The Real-World Test: The PID Controller

To prove this works, the authors tested it on a PID controller. Think of a PID controller as the "autopilot" for a machine (in this case, a motor moving a heavy object).

The autopilot needs to know the object's velocity to stop it precisely.
They tested three different ways to guess the velocity:
1. Standard Filter (LDF): The rigid sunglasses. Good at smoothing, but slow to react.
2. Robust Sliding-Mode (RED): A very tough, aggressive method. Good at ignoring noise, but can be a bit jittery.
3. The New Adaptive Filter (ADF): The smart glasses.

The Results:
When the motor had to stop suddenly (a "step" command):

The Standard Filter was a bit sluggish.
The Robust Filter worked well but made the motor vibrate slightly (wasting energy).
The Adaptive Filter (ADF) was the winner. It stopped the motor smoothly, with less overshoot (it didn't swing past the target) and less vibration. It used less energy because it didn't waste power fighting the noise.

The Bottom Line

This paper presents a "smart" way to calculate speed from shaky data. Instead of choosing between "smooth but slow" or "fast but noisy," this new filter adapts on the fly. It acts like a skilled driver who knows when to look far ahead for a smooth ride and when to focus intently on the immediate road for a sharp turn.

This makes it perfect for robots, drones, and industrial machines that need to move precisely without getting confused by sensor noise.

1. Problem Statement

The core challenge addressed is the estimation of signal derivatives (e.g., velocity) from noisy measurements in real-time control systems.

The Trade-off: Fixed-structure filters (like standard low-pass filters) face a fundamental conflict: reducing measurement noise typically introduces phase lag (delay), while preserving fast signal transients (bandwidth) makes the estimate highly sensitive to noise.
Control Context: In PID feedback loops, the derivative term ( $D$ ) requires an accurate estimate of the first derivative ( $\dot{x}$ ). However, sensors rarely provide direct velocity measurements; they must be derived from position data, which is often contaminated by noise.
Limitations of Existing Methods:
- Linear Low-Pass Differentiators (LDF): Require tuning a cutoff frequency; often too slow for large transients or too noisy for small signals.
- Robust Sliding-Mode Differentiators (RED): While robust, they can be computationally intensive and may introduce high-frequency oscillations (chattering) in the control signal.
- Fixed Window Methods: Cannot adapt to sudden changes in signal dynamics, leading to either excessive smoothing (underestimating peaks) or noise amplification.

2. Methodology: The Adaptive Differentiating Filter (ADF)

The authors propose a causal, discrete-time adaptive filter based on constrained least-squares estimation with dynamic window adaptation.

Core Concept

Instead of using a fixed time window, the ADF dynamically adjusts the window size ( $R$ ) based on the local behavior of the signal:

Constraint: The filter assumes the signal can be approximated by a linear function $x(t) = k(t-t_l) + b$ within a window $W$ with an absolute error tolerance $\delta$ .
Adaptation Logic:
- If the signal is smooth (low dynamics), the window size $R$ expands to the maximum allowed ( $R_{max}$ ) to maximize noise averaging.
- If the signal exhibits rapid changes (high dynamics), the window size $R$ shrinks to ensure the linear approximation remains valid within the tolerance $\delta$ .
Optimization: At each time step, the filter solves two sub-problems:
- Window Optimization: Find the largest $R$ such that the linear fit error $\le \delta$ .
- Estimation: Solve the constrained least-squares problem to find the slope ( $k^*$ , the derivative estimate) and intercept ( $b^*$ , the signal estimate).

Efficient Implementation

To ensure real-time feasibility on embedded hardware, the authors developed specific algorithms:

Feasibility Check (Theorem 1): A mathematical theorem allows checking if a window is valid (i.e., if a linear fit exists within $\delta$ ) without explicitly calculating the least-squares coefficients. This check uses pre-calculated bounds ( $m$ and $M$ ) derived from pairwise differences of data points.
Incremental Updates: The algorithm updates the bounds $m$ and $M$ incrementally when adding a new data point or removing an old one, avoiding $O(N^2)$ recalculations.
Analytic Solution: The constrained least-squares problem is solved using an analytic formula, allowing for pre-computation of vectors for fixed sampling rates.

3. Key Contributions

Novel Adaptive Architecture: A causal filter that decouples window adaptation from derivative calculation, optimizing the trade-off between noise rejection and transient response automatically.
Computational Efficiency: The derivation of efficient recursive algorithms for window feasibility checking and least-squares solving makes the filter suitable for real-time implementation on standard control boards (e.g., Speedgoat).
Practical Validation: The method is not just theoretical; it is rigorously tested in a physical mechatronic system (voice-coil motor) and compared against two state-of-the-art alternatives.

4. Experimental Results

The study utilized a voice-coil motor actuator with a sampling frequency of 2 kHz. The system was controlled using a PID regulator tuned via the "symmetrical optimum" method. Three differentiators were compared:

LDF: Linear Low-Pass Filter based differentiator.
RED: Robust Exact Differentiator (Sliding-mode based).
ADF: The proposed Adaptive Differentiating Filter.

Key Findings:

Noise Sensitivity: The ADF demonstrated significantly lower sensitivity to measurement noise compared to the LDF and RED.
Transient Response: The ADF preserved large-amplitude changes in the signal (high bandwidth) without the phase lag typical of fixed low-pass filters.
Control Performance:
- Step Response: All three methods achieved similar tracking, though ADF showed slightly lower overshoot.
- Control Effort ( $u(t)$ ): This was the most critical differentiator. The LDF and RED produced control signals with higher oscillatory patterns (richer frequency components), leading to higher energy consumption and system stress. The ADF produced a much smoother control signal, reducing mechanical loading and energy usage while maintaining stability.
Frequency Response: The ADF maintained a flat gain up to higher frequencies compared to the LDF and avoided the bandwidth limitations (breakdown around 100 rad/sec) observed in the RED.

5. Significance and Conclusion

The paper demonstrates that adaptive windowing is a superior strategy for derivative estimation in noisy, real-time control environments compared to fixed-parameter or sliding-mode approaches.

Practical Impact: The ADF allows for the use of standard, noisy position sensors in high-performance motion control without the need for expensive velocity sensors or complex tuning procedures.
System Efficiency: By providing a cleaner derivative estimate, the ADF reduces unnecessary control actuation (chattering), thereby extending the lifespan of mechanical components and reducing energy consumption.
Future Work: The authors note that further analysis of convergence properties and optimal tuning strategies for varying noise levels is planned.

In summary, the proposed ADF offers a robust, computationally efficient, and self-tuning solution for the critical problem of derivative estimation in PID control, outperforming traditional linear and robust non-linear differentiators in experimental settings.