⚡ electrical engineering

Safe Data-Driven Control and Dynamical Learning via Constrained Neural Architectures and Koopman Operators

This paper presents an integrated framework combining Constrained Neural ODEs, measurement-induced geometric structures, and Koopman operator theory to create data-driven control models with computable stability certificates, achieving significant improvements in tracking accuracy and robustness for safety-critical systems.

Original authors: Lin Feng, Xin He

Published 2026-05-26

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Lin Feng, Xin He

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 teaching a robot to drive a car or balance a heavy weight. You want the robot to learn from experience (data) rather than being programmed with every single rule of physics. However, there's a big problem: standard "learning" robots are like wild horses. They might run perfectly on the track you trained them on, but if the wind blows or the road gets slippery, they might panic, spin out, or crash because they have no mathematical guarantee that they will stay safe.

This paper presents a new way to build these learning robots so they come with a built-in "seatbelt" and a "safety manual" that proves they will behave, even when things get messy.

Here is how the authors did it, broken down into simple concepts:

1. The Problem: Learning Without Guarantees

Usually, when we use AI to learn how a system moves, we just say, "Here is the data, guess the pattern." The AI guesses well, but it doesn't know why it's safe. If you push it too hard, it might break. The authors wanted to build a system where safety isn't just a lucky guess; it's a mathematically proven fact.

2. The Solution: Three Safety Layers

The authors combined three different tools to create a "certified" learning system. Think of this as building a car with three specific safety features:

Feature A: The "Self-Correcting Engine" (ControlSynth Neural ODEs)
Imagine a car engine that, no matter how you twist the steering wheel, automatically steers itself back toward the center of the lane. The authors created a special type of AI (called a Neural ODE) that is forced by math rules to always pull itself back to stability. They proved this using a mathematical checklist (called an LMI) that ensures the system never spirals out of control, no matter how complex the driving gets.
Feature B: The "Weather-Adaptive Suspension" (ICODE)
Real-world driving isn't just about the car; it's about the road, wind, and rain. Standard AI often treats these as hidden secrets. This paper teaches the AI to treat the weather (like wind or slippery roads) as an explicit guest in the conversation. They built a system where the AI can instantly adjust its "suspension" based on the current weather conditions, ensuring it stays on the road even when the environment changes suddenly.
Feature C: The "Safety Net" (Koopman Operators & ISS)
Sometimes, the AI learns a model that is too complicated to understand. To fix this, the authors used a trick called the "Koopman Operator." Imagine taking a messy, tangled ball of yarn (nonlinear chaos) and stretching it out into a straight, neat line (linear order). Once the data is straightened out, it's much easier to check if it's safe. They then ran a "safety test" (Input-to-State Stability) on this straightened line. If the test passes, the system gets a "Safety Certificate" proving it can handle bumps and disturbances without crashing.

3. Putting It All Together: The "Smart Driver"

Once they built this super-safe, mathematically certified model, they put it inside a controller called ICODE-MPPI.

Think of this controller as a driver who doesn't just drive, but constantly runs mental simulations. Before making a turn, the driver imagines 1,500 different versions of the future (some with wind, some with slippery roads). Because their "engine" (the model) is certified to be safe, these mental simulations don't go crazy. The driver picks the best path from these safe simulations.

Because the model is certified, the driver doesn't have to worry about the simulations exploding into chaos. This leads to much smoother, more accurate driving.

4. The Results: Does It Work?

The authors tested this on two things:

A self-driving car trying to stay in its lane while the road friction changes (like going from dry asphalt to ice).
A heavy spring-mass system (like a bouncy ball) being pushed by random forces.

The findings were impressive:

Accuracy: The new system reduced tracking errors by 61% for the car and 54% for the spring system compared to standard, uncertified AI.
Reliability: The "safety checks" (LMIs) passed 100% of the time, even when the disturbances were strong.
Speed: The system was fast enough to run in real-time (about 8 milliseconds per step), meaning it could actually be used in a real car or robot right now.

The Bottom Line

This paper doesn't just say, "Our AI learns well." It says, "Our AI learns well, and here is the mathematical proof that it will never lose control, even when the world gets messy." By combining self-correcting math, weather-aware adjustments, and a safety-net verification system, they created a learning robot that is both smart and safe.

Technical Summary: Safe Data-Driven Control and Dynamical Learning via Constrained Neural Architectures and Koopman Operators

1. Problem Formulation

The deployment of learning-based models in safety-critical control systems faces a fundamental challenge: standard regression architectures lack mathematical guarantees regarding stability and safety. In domains such as autonomous vehicles and robotic manipulators, systems exhibit nonlinearities, time delays, and parameter variations that resist analytical description. While Neural Ordinary Differential Equations (Neural ODEs) and Koopman operator methods (e.g., Extended Dynamic Mode Decomposition, EDMD) have advanced the ability to learn system trajectories from data, they typically fail to provide formal certificates of stability or safety. Consequently, models that perform well on training distributions may diverge under distributional shifts or adversarial disturbances, rendering them unsuitable for reliable deployment.

The paper addresses the need for a data-driven framework that produces models with computable stability certificates rather than conjectured ones. The specific objectives are:

Convergence: The learned model must be globally contracting, ensuring trajectories from different initial conditions converge exponentially.
Manifold Confinement: Predictions must remain within physically admissible manifolds (e.g., respecting road friction or load constraints).
Input-to-State Stability (ISS): The mapping from external disturbances to state errors must be ISS with a computable $L_2$ -gain, quantifying robustness to bounded disturbances.

2. Methodology

The authors propose a unified pipeline integrating four distinct components: ControlSynth Neural ODEs, the ICODE architecture, measurement-induced bundle structures, and Koopman ISS verification.

A. ControlSynth Neural ODEs with Bundle Constraints

The unknown system drift is parameterized as a ControlSynth Neural ODE (CSODE). This architecture enforces global convergence through tractable Linear Matrix Inequalities (LMIs).

Structure: The latent state dynamics are defined as $\dot{z} = A_\theta z + \Phi_\theta(z, \xi) + B_\theta u$ , where $\Phi_\theta$ is an auxiliary sub-network capturing nonlinear mode coupling.
Stability Constraint: A quadratic sector condition is imposed on $\Phi_\theta$ (Persidskii-type inequality). This ensures the Jacobian is uniformly negative definite in a specific metric, certifying global convergence via LMIs.
Bundle Constraints: To enforce physical constraints, a fiber bundle framework is incorporated. An encoder $\Phi(x, \xi)$ is trained with a bundle-aware loss that penalizes latent predictions drifting off the environment-conditioned fiber. This induces measurement-aware Control Barrier Functions (CBFs) that adapt to local sensing conditions.

B. ICODE: Extrinsic Input Modeling

The ICODE architecture specializes the CSODE by explicitly modeling extrinsic environmental inputs ( $\xi$ , e.g., wind, friction) as arguments rather than hidden parameters.

It introduces a bilinear coupling term between the state and $\xi$ , allowing the effective system matrix to be modulated physically interpretable ways (e.g., shifting damping matrices based on tire-road friction).
Sufficient conditions for uniform contraction over varying environments are derived via coupled LMIs.

C. Koopman ISS Verification

As an alternative to Neural ODEs, the paper utilizes a data-driven Koopman representation via EDMD.

Verification: Instead of modifying the identification procedure, the paper applies a post-hoc verification step. It checks the feasibility of a convex LMI to certify that the $L_2$ -gain from disturbances to state errors is bounded.
Sensitivity Analysis: A new proposition quantifies how the optimal ISS gain ( $\gamma^\star$ ) degrades with dictionary truncation error, providing a guide for dictionary design.

D. ICODE-MPPI Controller

The certified models are embedded into a Model Predictive Path Integral (MPPI) controller.

Mechanism: A continuous-time residual ICODE network is appended to the nominal predictor to compensate for unmodeled drift during stochastic rollouts.
Variance Reduction: Theoretical analysis (Proposition 2) demonstrates that using certified predictors with bounded ISS gains limits the variance of cost samples in the MPPI rollout. This prevents the "variance explosion" seen in uncertified recurrent networks, leading to smoother and more reliable control updates.

3. Key Contributions

The paper synthesizes existing tools into a unified pipeline and provides specific theoretical and practical additions:

Combined Stability Analysis: A proof (Theorem 1) showing that the convergence certificate of ControlSynth and the bundle constraint jointly imply ISS of the composite Neural ODE model, with an explicit gain formula.
Koopman Sensitivity Result: A formalization (Proposition 1) showing that enlarging the dictionary monotonically improves (or preserves) the ISS certificate, offering a practical guide for dictionary construction.
Variance Reduction Theory: A proof (Proposition 2) linking tighter ISS certificates to reduced MPPI rollout variance, theoretically justifying the use of certified predictors in stochastic control loops.
Comprehensive Validation: Numerical experiments on two benchmarks (vehicle path-tracking and a nonlinear mechanical oscillator) with ablation studies isolating the contribution of each structural constraint.

4. Experimental Results

The framework was evaluated on two benchmarks: a kinematic bicycle model subject to time-varying friction and wind, and a Duffing oscillator. Four methods were compared: Vanilla NODE, Uncertified Koopman, Koopman + ISS, and the full CSODE-ICODE-MPPI pipeline.

Performance: The full pipeline achieved a 61% reduction in tracking RMSE on the vehicle benchmark and a 54% reduction on the Duffing oscillator benchmark compared to the Vanilla NODE baseline.
Robustness: Under persistent disturbances, the full pipeline maintained the lowest Root Mean Square Error (RMSE) across all disturbance levels. The uncertified methods exhibited super-linear error degradation or intermittent divergence.
Stability Guarantees: The LMI feasibility rate was 100% across all test episodes, confirming that the models retained their stability guarantees under distribution shifts.
Ablation Study: Removing individual components (bundle constraints, ICODE coupling, or ISS certification) resulted in significant performance degradation (18–22% RMSE increase) or increased cost variance (factor of 3.1), confirming that each component contributes independently to the system's robustness.
Real-Time Feasibility: The full pipeline operated with a step time of 8.3 ms (dominated by CSODE integration), fitting comfortably within a 20 Hz real-time control budget.

5. Significance and Claims

The paper claims to bridge the gap between expressive data-driven modeling and rigorous safety guarantees. Its primary significance lies in:

Computable Certificates: Moving beyond conjectured stability to providing explicit, computable ISS certificates via convex optimization (LMIs).
Integrated Framework: Successfully combining Neural ODEs, geometric bundle structures, and Koopman theory into a single, end-to-end control architecture.
Practical Deployment: Demonstrating that these theoretical guarantees translate to tangible improvements in tracking accuracy and robustness in the presence of parametric uncertainty and persistent disturbances, without sacrificing real-time performance.

The authors note that while the current framework handles bounded deterministic signals, extending the verification pipeline to stochastic disturbances and exploring input-convex neural networks for automatic Lyapunov satisfaction remain open directions for future work.