⚡ electrical engineering

Model-Free Disturbance Observer with Online Modification: Listening to MFDOOM

This paper proposes MFDOOM, a model-free disturbance observer with online modification that augments Data-Enabled Predictive Control (DeePC) by correcting output predictions using a continuously updated Hankel matrix to reduce tracking errors and online-update burdens for systems subject to disturbances generated by autonomous linear time-invariant dynamics.

Original authors: Nadav Barak, Christian Grussler

Published 2026-07-09

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CC0 1.0

Original authors: Nadav Barak, Christian Grussler

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 trying to drive a car to a specific destination (the "reference") while navigating a road that is constantly being shaken by invisible, rhythmic bumps (the "disturbance").

In the world of robotics and automation, this is a classic problem. You need a controller that can steer the car perfectly despite the bumps.

The Old Way: Learning by Heart (DeePC)

Traditionally, engineers used a method called DeePC (Data-Enabled Predictive Control). Think of this like a student who has memorized a massive textbook of "what happens when I press the gas pedal."

How it works: The student looks at the last few seconds of driving data and predicts the next few seconds based on that history.
The Problem: If the road suddenly starts shaking (a new disturbance), the student's textbook is wrong. The student keeps trying to drive based on old rules, gets confused, and the car swerves.
The "Fixes" that existed:
- ODeePC (Online DeePC): This is like the student frantically rewriting their textbook while driving. They keep adding new pages to account for the bumps. But to do this, they have to constantly wiggle the steering wheel just to see how the car reacts (injecting noise), which makes the ride bumpy and computationally exhausting.
- MDeePC (Reduced-Order DeePC): This student tries to be smarter by only adding new pages if the bumps are totally new. But the textbook still keeps getting thicker and heavier, making it harder to read quickly as time goes on.

The New Way: The "Disturbance Detective" (MFDOOM)

The authors of this paper propose a new method called MFDOOM (Model-Free Disturbance Observer with Online Modification).

Instead of rewriting the entire textbook (the main controller), MFDOOM hires a specialized detective whose only job is to watch the "mistakes."

Here is how it works using a simple analogy:

The Fixed Teacher: The main controller (DeePC) stays exactly the same. It keeps using its original, trusted textbook. It doesn't need to be rewritten or updated constantly.
The Detective's Notebook: The system starts a separate, small notebook dedicated only to recording the difference between what the teacher predicted and what actually happened.
- Teacher says: "We should be at position X."
- Reality says: "We are actually at position X + 5 because of a bump."
- Detective writes down: "Error = +5."
Learning the Pattern: The detective doesn't just write down random numbers. They look at the history of errors in their notebook. If the errors are rhythmic (like a sine wave bump), the detective realizes, "Ah! The road shakes every 2 seconds."
The Correction: Before the teacher makes a prediction, the detective whispers, "Hey, based on the pattern in my notebook, the next bump will push us 5 units to the right. Let's subtract that from your prediction."

Why is this better?

The paper claims three main advantages, which we can visualize as:

Less Noise: Unlike the "ODeePC" student who has to wiggle the steering wheel to learn, the MFDOOM detective learns purely by watching the errors. The main driver doesn't need to be disturbed to learn.
Lighter Load: The main textbook never changes size. The detective's notebook is small and only tracks the errors, not the whole car's physics. This makes the computer calculations much faster and lighter.
Smarter Adaptation: If the detective's notebook is full of garbage data (noise), the system has a "volume knob" (called an adaptive gain). It can turn down the detective's voice so the main driver ignores bad advice. If the pattern is clear, it turns the volume up.

The Results

The authors tested this on a simulated electric motor (like the one in a drone or a fan) that was being shaken by rhythmic vibrations.

The Result: The MFDOOM method kept the motor's speed much closer to the target than the other methods. It had fewer "wobbles" and recovered from errors much faster.
The Catch: This detective is specifically good at tracking disturbances that follow a predictable, rhythmic pattern (like a sine wave or a ramp). If the disturbance is completely chaotic and random, the detective might struggle, but for the specific "autonomous" disturbances the paper tested, it worked beautifully.

In short: Instead of trying to rewrite the entire rulebook of the car every time the road changes, MFDOOM keeps the rulebook fixed and adds a smart, lightweight "error-correction layer" that learns the rhythm of the bumps and cancels them out.

Technical Summary: Model-Free Disturbance Observer with Online Modification (MFDOOM)

1. Problem Statement

Data-Enabled Predictive Control (DeePC) offers a model-free alternative to Model Predictive Control (MPC) by predicting system behavior using Hankel matrices constructed from input-output data. However, DeePC performance is heavily dependent on the relevance of the collected data. When systems are subject to disturbances—particularly those generated by autonomous linear time-invariant (LTI) systems (e.g., sinusoids, ramps, or steps)—the measured plant behavior deviates from the nominal model. This leads to inevitable prediction errors.

Existing strategies to handle these issues include:

Regularized DeePC: Improves robustness but often degrades nominal tracking performance.
Online DeePC (ODeePC): Continuously updates Hankel matrices with new measurements. However, this requires persistently exciting (PE) inputs (often achieved by injecting noise), which drives the system away from the reference, and risks "erasing" the system model if old data is shifted out before new data enriches the model.
Online Reduced-Order DeePC (MDeePC): Uses mosaic Hankel matrices to avoid shifting out data. While it avoids some discontinuity issues, the matrix size grows over time, increasing computational burden, and it still struggles with the trade-off between model richness and stability.

The core problem addressed is how to correct output predictions for time-varying disturbances without corrupting the baseline DeePC predictor or incurring excessive computational and excitation costs.

2. Methodology: MFDOOM

The paper proposes Model-Free Disturbance Observer with Online Modification (MFDOOM). This method augments the standard DeePC framework with a dedicated, continuously updated disturbance observer that operates independently of the nominal behavior matrices.

Core Mechanism

MFDOOM treats the prediction error ( $d = \bar{y} - \hat{y}$ ) as a trajectory generated by an autonomous LTI system ( $B_d$ ). Instead of modifying the primary behavior matrices ( $U, Y$ ) used for nominal control, MFDOOM constructs a separate Hankel matrix ( $D$ ) from historical prediction errors.

The optimization problem in MFDOOM (Equation 11) minimizes a cost function that includes a correction term:
$\min_{g, u, y, \sigma_y, d, g_d} J_{QR}(y - \Gamma d, u, r) + \dots$
Subject to:

Nominal Constraints: Standard DeePC constraints linking inputs/outputs to the fixed behavior matrices ( $U, Y$ ).
Error Dynamics Constraints: A new constraint linking the current prediction error history ( $d_{ini}$ ) and future error prediction ( $d$ ) to the error behavior matrix ( $D$ ):
$\begin{bmatrix} D_p \\ D_f \end{bmatrix} g_d = \begin{bmatrix} d_{ini} \\ d \end{bmatrix}$

Key Algorithmic Features

Fixed Baseline: The nominal behavior matrices ( $U, Y$ ) remain fixed, allowing for pre-computation of DeePC terms.
Adaptive Correction: The error matrix $D$ is updated online using a sliding window of prediction errors.
Activation Threshold: Data collection for the error matrix begins only when the prediction error exceeds a numerical noise threshold ( $\epsilon$ ).
Adaptive Gain ( $\Gamma$ ): An optional mechanism (Algorithm 4) scales the correction term based on the estimation error ( $e_d = |d_a - d^*|$ ). If the error estimation is unreliable, $\Gamma$ approaches zero, effectively reverting the controller to a regularized DeePC to ensure transient robustness.

3. Key Contributions

Separation of Concerns: MFDOOM decouples the nominal predictor from the disturbance adaptation. Unlike ODeePC or MDeePC, it does not modify the matrices representing the system's nominal dynamics, preventing the injection of noise into the baseline model.
Reduced Computational Burden: The online update is restricted to the smaller prediction-error matrix $D$ , which does not expand in size over time, unlike the growing matrices in MDeePC.
No Persistent Excitation Requirement: Unlike ODeePC, MFDOOM does not require the nominal control input to be persistently exciting. The necessary richness is shifted to the observation of the prediction error itself.
Theoretical Foundation: The method relies on Proposition 1, which establishes that if the error behavior is generated by an autonomous LTI system and the Hankel matrix of errors has full rank, future error trajectories can be linearly predicted from past errors.

4. Simulation Results

The authors evaluated MFDOOM on a DC motor velocity control task under sinusoidal and time-varying amplitude harmonic disturbances, comparing it against MPC (no disturbance), ODeePC, and MDeePC.

Tracking Performance: MFDOOM demonstrated significantly lower Root Mean Square Error (RMSE) compared to ODeePC and MDeePC. In the sinusoidal disturbance test, MFDOOM achieved an RMSE of 0.06, compared to 0.34 for ODeePC and 0.19 for MDeePC.
Convergence: MFDOOM exhibited faster convergence to near-zero tracking error after the prediction error dynamics were learned.
Robustness to Noise: When sensor noise was added, MFDOOM maintained stability and low error, whereas MDeePC required significantly higher regularization ( $K_g = 10,000$ ) to remain stable, which degraded performance.
Time-Varying Disturbances: In a scenario with time-varying amplitude disturbances, MFDOOM required no gain adjustments and maintained accurate compensation, while ODeePC suffered from discontinuities when the disturbance was toggled.

5. Significance and Claims

The paper claims that MFDOOM offers a superior approach for controlling systems subject to autonomous LTI disturbances within the DeePC framework. By isolating the disturbance compensation to a dedicated error observer, the method achieves:

Improved Tracking: Reduced tracking error compared to existing online DeePC variants.
Lower Online Burden: Reduced computational cost by avoiding the continuous expansion or shifting of large nominal behavior matrices.
Stability: Avoids the need for continuous noise injection (PE inputs) to maintain model validity, thereby preventing unnecessary deviation from reference trajectories.

The authors conclude that while the current formulation is optimized for disturbances generated by autonomous LTI systems, the separation of the baseline predictor and the adaptive correction mechanism provides a promising foundation for future extensions to broader classes of time-varying systems.