⚡ electrical engineering

Gaussian behaviors and stochastic data-driven control

This paper introduces "Gaussian behaviors," a tractable stochastic modeling framework that extends deterministic LTI systems with Gaussian noise to enable convex, data-driven predictive control methods that explicitly account for both aleatoric and epistemic uncertainty through finite-sample confidence bounds.

Original authors: András Sasfi, Alberto Padoan, Ivan Markovsky, Florian Dörfler

Published 2026-07-20

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: András Sasfi, Alberto Padoan, Ivan Markovsky, Florian Dörfler

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

The Art of Guessing Right: When Machines Learn from Mistakes

Imagine you are trying to teach a robot how to drive a car. In the old days, engineers would spend years writing a massive manual describing every possible bump, curve, and wind gust the car might encounter. This is called "modeling." But what if the world is too messy for a manual? What if the car is driving on a planet where the physics change slightly every Tuesday?

This is where a field called data-driven control comes in. Instead of writing a manual, you just show the robot a thousand videos of the car driving. The robot looks at the patterns in the videos and figures out, "Okay, when I turn the wheel left, the car goes left." This works great if the videos are perfect. But in the real world, data is messy. Sometimes the car slips on ice (random noise), and sometimes the robot hasn't seen enough videos to know what to do in a new situation (lack of knowledge).

The big question scientists are asking is: How do we build a controller that doesn't just guess the average path, but also understands how uncertain it is? If the robot is unsure, it should be careful. If it's sure, it can be bold. This paper dives into that exact problem, trying to give machines a better sense of "gut feeling" based on the data they've seen.

The Paper: Giving Robots a "Gut Feeling"

The authors of this paper, András Sasfi and his team, have invented a new way to describe how machines learn from messy data. They call their invention "Gaussian Behaviors."

To understand what this is, imagine you are trying to predict the path of a bouncing ball.

The Deterministic Part: You know the basic rules of physics. If you throw the ball at a certain angle, it should follow a specific, smooth curve. In the paper's language, this is the "nominal behavior." It's the perfect, clean path the ball wants to take.
The Noise Part: But in reality, the wind blows, the floor is bumpy, and the ball might wobble. This is the "Gaussian noise." It's the random wiggle that pushes the ball off its perfect path.

The paper proposes a framework that combines these two. It treats the system not as a rigid machine, but as a "cloud" of possibilities. The center of the cloud is the perfect path, and the size of the cloud represents how much the ball might wiggle. The clever part? They show how to calculate the size and shape of this cloud directly from the data, without needing to know the exact physics of the wind or the floor.

The Magic Trick: Predicting the Future

The authors show that if you have a bunch of past data (like a video of the ball bouncing), you can use a simple math trick to predict where the ball will be next.

The Mean: This is the "best guess." It's the center of the cloud. The paper proves that this best guess is exactly the same as the methods scientists have been using for years (called "Subspace Predictive Control").
The Uncertainty: This is the new magic. The method also calculates how wide the cloud is. If the data you have is old or sparse, the cloud gets huge, telling the robot, "I'm not sure about this!" If the data is fresh and plentiful, the cloud shrinks, and the robot can be more confident.

Two Ways to Drive: The Cautious vs. The Optimist

The paper explores two different ways a robot can use this "cloud" of uncertainty to make decisions.

1. The Cautious Driver (Robust Control)
Imagine you are driving in a foggy forest. You know the road is somewhere in front of you, but you can't see the edges. A cautious driver stays in the middle of the lane, avoiding the trees just in case they are closer than they look.
The paper's "Robust" method does exactly this. It looks at the worst-case scenario inside the uncertainty cloud. It asks, "What is the worst possible path the ball could take, and how do I make sure I don't crash even if that happens?"

The Result: In their tests on a spring-mass system (a fancy way of saying a bouncy toy), this cautious method was much better than the others. While other methods tried to be too bold and ended up overshooting the target, the cautious method stayed close to the plan, even when the data was messy. It essentially said, "I don't trust the data enough to be risky, so I'll take the safe route."

2. The Optimistic Driver (The "DeePC" Connection)
Now, imagine a driver who thinks, "The fog is just an illusion! The road is probably right here, so I'll drive fast."
The paper shows that a popular existing method called DeePC (Data-enabled Predictive Control) acts like this optimistic driver. It assumes the best-case scenario within the uncertainty cloud. The authors found that this explains why DeePC works the way it does: it's essentially betting on luck.

The Catch: The paper suggests that while this optimism can work, it's risky. If the uncertainty is high (the fog is thick), betting on the best case can lead to big mistakes. The authors show that by adjusting a "tuning knob" (a parameter called $\lambda$ ), you can shift from being super optimistic to being more cautious, effectively turning DeePC into a safer driver.

The Secret Weapon: Listening to the Mistakes

The paper also introduces a superpower called "Disturbance Affine Feedback."
Imagine you are driving, and you feel the car drift slightly to the left. A normal driver might just keep steering straight, hoping it corrects itself. But a smart driver immediately turns the wheel slightly to the right to fix the drift as it happens.
The authors show that by letting the robot adjust its steering based on the errors it sees in real-time (the difference between where it thought the ball would be and where it actually was), it can shrink the uncertainty cloud.

The Test: They tested this on a building heating system (HVAC). They simulated a cold day where the outside temperature was unpredictable.
- The "Old Way" (Open-loop) had to keep the heating on high just to be safe, because it couldn't react to mistakes.
- The "New Way" (Feedback) reacted instantly to the temperature changes. Because it could fix its mistakes on the fly, it didn't need to be so cautious.
- The Win: This new method saved 77% of the energy cost compared to the old way, all while keeping the room temperature safe.

What They Didn't Do (And Why It Matters)

It's important to know what this paper doesn't claim.

It doesn't say the robot is now perfect. The simulations show that the methods work well, but they are still based on estimates.
It doesn't claim to solve every type of uncertainty. The paper focuses on "Gaussian" (bell-curve) noise. If the noise is weird and unpredictable in a non-bell-curve way, this specific method might need adjustments.
It explicitly argues against methods that ignore uncertainty. The paper shows that if you pretend the data is perfect (ignoring the "cloud"), you end up with controllers that are either too risky (DeePC without tuning) or too conservative (standard methods).

The Bottom Line

This paper is like giving a robot a pair of glasses that let it see not just the road, but also the fog. It provides a mathematical toolkit to turn raw, messy data into a "confidence map."

If the map says "Foggy," the robot drives slowly and carefully (Robust Control).
If the map says "Clear," the robot drives efficiently.
And if the robot has a "steering wheel" that reacts to mistakes in real-time (Feedback), it can drive even more efficiently, saving energy and money.

The authors demonstrate through simulations that this approach is not just a theoretical idea, but a practical tool that can make machines safer, smarter, and more efficient in the real world. They haven't solved the problem of uncertainty forever, but they've handed engineers a much better map for navigating it.

Technical Summary: Gaussian Behaviors and Stochastic Data-Driven Control

Problem Statement
Data-driven predictive control (DDPC) has emerged as a powerful alternative to model-based Model Predictive Control (MPC) by synthesizing policies directly from measured data, leveraging behavioral systems theory. However, existing DDPC methods are fundamentally based on deterministic Linear Time-Invariant (LTI) behaviors. While these methods often perform well with regularization, they lack an intrinsic stochastic framework. Consequently, uncertainty is typically handled a posteriori through robustification or by ignoring variance, leading to risk-neutral designs that may be overly conservative or fail to account for aleatoric (inherent) and epistemic (data-limited) uncertainties. Furthermore, existing stochastic extensions of behavioral theory are either limited to static systems, computationally intractable (e.g., polynomial chaos expansions), or rely on parametric models that do not fully exploit the non-parametric nature of behavioral theory. There is a need for a stochastic behavioral framework that is both mathematically rigorous and computationally tractable for data-driven control design.

Methodology
The authors propose a framework termed Gaussian Behaviors, which augments a deterministic LTI behavior with a Gaussian noise component.

Gaussian Behaviors Definition: A Gaussian behavior is defined as a pair $(B_L, \Sigma_e)$ , where $B_L$ is a deterministic finite-horizon LTI behavior (a subspace of valid trajectories) and $\Sigma_e$ is a positive semi-definite covariance matrix representing exogenous stochasticity. Any trajectory $w$ is decomposed into a nominal component $\bar{w} \in B_L$ and a noise term $e \sim \mathcal{N}(0, \Sigma_e)$ . This framework is shown to be a tractable subclass of abstract stochastic behaviors and encompasses classical parametric stochastic state-space models.
Identification and Prediction:
- The framework derives a prediction method by conditioning the Gaussian behavior on the known part of a trajectory (initial conditions and future inputs).
- The predictive distribution is identified directly from the sample covariance of trajectory data. The mean of this distribution corresponds to the established subspace predictor, while the covariance captures the prediction uncertainty.
- Crucially, the authors derive a finite-sample confidence bound that accounts for both aleatoric uncertainty (process/measurement noise) and epistemic uncertainty (arising from finite data). This bound defines a confidence ellipsoid for the predicted outputs.
- The method is extended to incorporate uncertain forecasts of external variables (e.g., weather) by treating forecast errors as part of the exogenous noise, allowing for the identification of unstructured uncertainty directly from data.
Control Formulations:
- Chance-Constrained Control (Aleatoric Uncertainty): The authors formulate a stochastic predictive control problem that minimizes the expected cost subject to individual chance constraints on outputs and inputs.
  - Feedforward: Restricting inputs to be deterministic recovers standard Subspace Predictive Control (SPC) when risk is ignored, or a risk-aware variant when constraints are enforced.
  - Disturbance Affine Feedback: The framework optimizes over disturbance affine feedback policies ( $u_f = K e_{pred} + \bar{u}_f$ ). This allows the controller to actively shape the output covariance, reducing conservatism compared to open-loop feedforward strategies. The resulting problem is reformulated as a tractable convex optimization problem.
- Robust Control (Aleatoric + Epistemic Uncertainty): To address finite-data uncertainty, the authors propose a robust formulation that minimizes the worst-case cost over the confidence ellipsoid derived from the finite-sample bound.
  - Using Lagrangian duality, the non-convex min-max problem is upper-bounded by a tractable convex problem. This provides a conservative controller that safeguards against the worst-case realization within the data-driven confidence region.
  - Conversely, an "optimistic" formulation (min-min) is shown to recover the regularized Data-enabled Predictive Control (DeePC) algorithm, providing a stochastic interpretation for DeePC's regularization as an optimistic stance toward prediction uncertainty.

Key Contributions

Gaussian Behaviors Framework: Introduction of a pragmatic, parsimonious stochastic modeling framework within behavioral systems theory that unifies deterministic behaviors with Gaussian noise, encompassing parametric stochastic models.
Data-Driven Prediction with Uncertainty Bounds: Derivation of a linear prediction method identified directly from sample covariance, accompanied by a finite-sample confidence bound that explicitly quantifies both aleatoric and epistemic uncertainty.
Stochastic Control with Feedback: Development of chance-constrained control formulations that optimize over disturbance affine feedback policies, demonstrating that such policies reduce conservatism by actively managing prediction uncertainty.
Robust and Optimistic Interpretations: Derivation of a robust control formulation with a tractable convex upper bound for the worst-case cost, and the demonstration that the optimistic counterpart recovers regularized DeePC, linking existing data-driven methods to stochastic interpretations.

Results
The paper validates the proposed methods through two numerical case studies:

Double Spring-Mass-Damper System: In a scenario with unevenly excited offline data (leading to high epistemic uncertainty in certain input channels), the proposed robust method (GB-Robust) outperformed SPC, DeePC, and a risk-neutral parametric method (MSM-SMPC). GB-Robust acted cautiously, avoiding high-uncertainty input channels and achieving lower realized control costs with smaller variance compared to competitors, which deviated significantly from the predicted mean.
HVAC Building Control: The study demonstrated the integration of uncertain weather forecasts into the predictive distribution. Comparing feedforward vs. disturbance affine feedback policies, the feedback policy allowed the controller to plan trajectories closer to comfort constraints by reacting to realized errors. This resulted in a 77% reduction in closed-loop control cost (heating/cooling effort) compared to the feedforward formulation, while maintaining safety constraints.

Significance and Claims
The authors claim that this work provides a necessary extension of behavioral systems theory to the stochastic setting, enabling the design of simple and pragmatic stochastic data-driven control formulations. The framework bridges the gap between the theoretical elegance of behavioral theory and the practical necessity of handling uncertainty in control design. By recovering SPC and DeePC as special cases (mean-only and optimistic, respectively), the framework unifies existing approaches under a single stochastic umbrella. The proposed robust and feedback-based methods offer a systematic way to handle both types of uncertainty (aleatoric and epistemic) without relying on explicit parametric model identification, thereby reducing the conservatism inherent in traditional data-driven approaches while maintaining computational tractability.