⚡ electrical engineering

Learning a Contracting KKL-observer with Local Optimal Guarantees

This paper proposes a deep learning-based methodology to design a Kazantzis-Kravaris-Luenberger (KKL) observer that structurally enforces contraction for global stability while locally mimicking the optimality of a Minimum Energy Estimator, validated through numerical simulations on nonlinear benchmarks.

Original authors: Clara Lucía Galimberti, Johan Peralez, Daniele Astolfi, Vincent Andrieu, Madiha Nadri

Published 2026-05-14

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Clara Lucía Galimberti, Johan Peralez, Daniele Astolfi, Vincent Andrieu, Madiha Nadri

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 trying to navigate a ship through a thick, foggy storm. You can see the water splashing against the hull (your measurements), but you can't see the ship's actual position, speed, or direction (the hidden state). You need a "navigator" to guess where the ship is based on the splashes and your knowledge of how ships move.

In the world of engineering, this is called state estimation. For simple ships, this is easy. But for complex, non-linear systems (like a drone fighting wind gusts or a chemical plant reacting unpredictably), the math gets incredibly messy.

This paper introduces a new, smarter way to build that navigator. Here is the breakdown in simple terms:

1. The Old Way: The "Best Guess" Navigator

Engineers have a tool called the KKL Observer. Think of it as a translator. It takes the messy, real-world data and translates it into a "hidden language" (a latent space) where the rules are simple and predictable.

The Problem: To make this translator work, you have to choose the rules of that hidden language. Usually, engineers just pick rules that seem okay (heuristic choices). It's like guessing the recipe for a cake without knowing the chemistry. Sometimes it works great; other times, if the wind blows hard (noise), the guess goes wildly off track.

2. The New Idea: Learning the "Perfect" Translator

The authors say, "Why guess? Let's teach a computer to learn the perfect rules."
They want to build a KKL observer that does two things simultaneously:

Global Stability: It must always bring the ship back to the right path, no matter how far off course it starts (like a self-correcting rudder).
Local Optimality: Once the ship is close to the right path, the navigator should be incredibly precise, ignoring the fog as much as possible.

3. The Secret Sauce: The "Mortensen" Blueprint

To teach the computer what "perfect" looks like, they use a theoretical concept called the Mortensen Observer.

The Analogy: Imagine the Mortensen Observer is the "God Mode" navigator. It knows the absolute best path by calculating the energy required to explain every single splash of water. It is mathematically perfect but too heavy and slow to run on a real ship (it's like trying to solve a super-complex puzzle in real-time).
The Innovation: The authors realized that while we can't run the "God Mode" navigator, we can look at its local behavior. When the ship is already close to the right path, the "God Mode" navigator acts in a very specific, efficient way.
The Goal: They want to train their new AI navigator to mimic this specific, efficient behavior when things are going well, while keeping the self-correcting rudder for when things go wrong.

4. How They Did It: The Neural Network Gym

They used Deep Learning (AI) to build this navigator.

The Architecture: They didn't just throw a standard AI at the problem. They built a special type of AI network (called NodeRENs) that is structurally forced to be stable. Think of it as building a robot with springs that physically prevent it from falling over, no matter how you push it. This guarantees the "Global Stability" part.
The Training: They taught the AI two lessons at once:
1. The Math Lesson: "You must follow the laws of physics (the KKL equations)."
2. The Behavior Lesson: "When you are close to the truth, your corrections should look exactly like the 'God Mode' navigator's corrections."

5. The Results: Testing in the Storm

They tested this new AI navigator on two famous, tricky mathematical systems (the Van der Pol oscillator and the Duffing oscillator). You can think of these as "stormy seas" for math.

The Test: They threw noise (fog) and disturbances (wind) at the systems.
The Outcome: The new observer worked beautifully.
- It recovered quickly from bad starting points (thanks to the structural stability).
- It tracked the true path very accurately even with noisy data (thanks to mimicking the optimal behavior).
- They could even tune it: if they told the AI to trust the model more, it smoothed out the noise; if they told it to trust the measurements more, it reacted faster to changes.

Summary

The paper presents a method to teach an AI to be a perfect navigator. It combines the safety of a rigid, self-correcting system with the precision of a theoretical "perfect" estimator. Instead of guessing how to tune the system, they use Deep Learning to learn the exact rules that make the system both safe and super-accurate, specifically for nonlinear systems where traditional math fails.

Technical Summary: Learning a Contracting KKL-observer with Local Optimal Guarantees

Problem Statement
The Kazantzis-Kravaris-Luenberger (KKL) framework offers a general approach to nonlinear state estimation by immersing system dynamics into a stable latent space. While recent advances have utilized machine learning to approximate the KKL transformation (solving the associated nonlinear partial differential equation), a fundamental challenge remains: the selection of the latent dynamics $\phi$ . In practice, these dynamics are often chosen heuristically (e.g., linear diagonal systems), which may ensure global stability but frequently results in sub-optimal performance regarding steady-state sensitivity to measurement noise and unmodeled disturbances. The core problem addressed is how to systematically design the latent dynamics to achieve local optimality without compromising the global contraction guarantees inherent to the KKL theory.

Methodology
The authors propose a learning-based strategy that synthesizes the global robustness of KKL observers with the local optimality of the Minimum Energy Estimator (MEE), also known as the Mortensen observer. The methodology proceeds in three main stages:

Theoretical Characterization of Local Behavior:
The paper derives a first-order approximation of the reconstructed state dynamics $\dot{\hat{x}}$ for a general KKL observer. This analysis reveals that the local behavior is governed by an effective gain matrix $\Psi(\hat{x})$ . The authors demonstrate that for the observer to mimic the optimal local behavior of a Mortensen filter, this gain must match a specific optimal term $\Psi_{opt}(\hat{x})$ . This optimal term is derived from the solution to a Partial Differential Riccati Equation (PDRE) associated with the Information Matrix $P(x)$ .
Learning Strategy with Structural Constraints:
To realize this theoretically optimal behavior, the authors employ Deep Learning to approximate three key components: the immersion map $\hat{T}$ , the reconstruction map $\hat{\tau}$ , and the latent dynamics $\hat{\phi}$ .
- Contraction Guarantee: To ensure global stability (Requirement R2), the latent dynamics $\hat{\phi}$ are parameterized using NodeRENs (Neural Ordinary Differential Equations with specific structural constraints), which mathematically enforce the contraction property by design.
- Optimality Approximation: Since solving the PDRE for $P(x)$ is intractable for general nonlinear systems, a Multi-Layer Perceptron (MLP) is first trained to approximate the inverse information matrix $\hat{P}^{-1}$ by minimizing the residual of the transformed PDRE.
- Joint Optimization: The immersion map, reconstruction map, and latent dynamics are jointly trained to minimize a composite loss function. This loss includes:
  - $\ell_{pde}$ : Penalizing deviations from the KKL PDE.
  - $\ell_{inv}$ : Ensuring the reconstruction map is a left-inverse of the immersion map.
  - $\ell_{opt}$ : Forcing the effective gain of the learned observer to match the optimal gain derived from the approximated $\hat{P}^{-1}$ .

Key Contributions

Local Optimality Condition: The paper derives a specific condition on the latent dynamics such that a KKL observer locally mimics the behavior of a Minimum Energy Estimator (Mortensen observer) when the estimation error is small.
Contraction-Based Learning Framework: It introduces a methodology to learn KKL observers using neural networks that structurally enforce contraction (via NodeRENs), thereby guaranteeing global robustness while optimizing local performance.
Approximation of Optimal Filtering: The work bridges the gap between optimal filtering theory (which typically requires solving infinite-dimensional HJB equations) and practical observer design by using data-driven approximation of the Information Matrix and the associated optimal gain structure.

Results
The proposed methodology was validated through numerical simulations on two nonlinear benchmarks: the Van der Pol oscillator and the reverse Duffing oscillator.

Global Stability: The learned observers demonstrated robust convergence from arbitrary initial conditions, attributed to the enforced contraction property of the latent dynamics.
Noise Handling: The estimators accurately tracked system states in the presence of significant measurement noise.
Tunability: Simulations showed that by adjusting the weighting matrices ( $R$ ) in the training loss, the observer's behavior could be tuned. A smaller $R$ led the observer to track noisy dynamics more closely, while a larger $R$ caused it to rely more heavily on the system model, demonstrating the ability to trade off between model fidelity and noise rejection.

Significance and Claims
The authors claim that this work provides a "principled characterization" of latent dynamics for KKL observers, moving beyond heuristic selection. By combining global contraction guarantees with a learning objective that targets local optimality, the proposed strategy offers a systematic way to design observers that are both robust to initialization and sensitive to noise in a controlled manner. The paper modestly positions these results as a demonstration on academic benchmarks, noting that future work will focus on establishing approximation bounds for the learned models and extending the theory to discrete-time dynamics. The approach is presented as a solution for systems where analytical solutions to the KKL PDE are out of reach and standard observers may struggle with noise performance.