⚡ electrical engineering

EDMD-Based Robust Observer Synthesis for Nonlinear Systems

This paper proposes a data-driven robust observer synthesis method for continuous-time nonlinear systems that utilizes Extended Dynamic Mode Decomposition (EDMD) to construct an approximate linear lifted model, where the resulting modeling errors are bounded via reproducing kernel Hilbert space theory and mitigated through Linear Matrix Inequality (LMI) optimization to ensure desired observation performance.

Original authors: Xiuzhen Ye, Wentao Tang

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiuzhen Ye, Wentao Tang

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 stormy, unpredictable ocean. The waves (the system's behavior) are chaotic and non-linear, meaning they don't follow simple, straight lines. You have a map, but it's a bit blurry, and your compass (the sensors) only tells you a few things about the water's surface, not the depth or the currents underneath.

Your goal is to build a "ghost ship" (an observer) inside your control room that perfectly mimics the real ship's movement, so you can steer it safely even when you can't see everything outside.

This paper presents a clever new way to build that ghost ship using a method called Koopman Operator Theory and EDMD (Extended Dynamic Mode Decomposition). Here is the breakdown in simple terms:

1. The Problem: Trying to Fit a Square Peg in a Round Hole

Non-linear systems (like weather, chemical reactors, or the ship in a storm) are messy. They twist and turn in ways that are hard to predict with simple math.

The Old Way: Scientists try to approximate these messy systems by "lifting" them into a higher dimension. Imagine taking a tangled ball of yarn and stretching it out until it looks like a straight line.
The Catch: When you stretch that yarn, it never becomes perfectly straight. There's always a little bit of "wiggle" or error left over. In the past, engineers often ignored this wiggle or assumed it was random noise. But this paper says, "No, that wiggle is predictable!"

2. The Solution: The "Sector" Safety Net

The authors realized that this "wiggle" (the error between the real messy system and their simplified linear model) isn't random chaos. It behaves like a rubber band.

The Analogy: Imagine the error is a rubber band attached to the center of your map. The further you get from the center, the tighter the rubber band pulls, but it always pulls within a specific cone shape (a "sector").
Why this matters: Because the error is predictable (it stays within that cone), you can design a safety net. You don't need to know the exact error; you just need to know the maximum it could ever be.

3. The Two Types of "Wiggle"

The paper identifies two reasons why the map isn't perfect:

The "Dictionary" Error (Structural): You tried to describe a complex painting using only a limited set of Lego bricks. No matter how many bricks you use, you can't perfectly recreate the Mona Lisa. This is the error from using a finite list of functions.
The "Data" Error (Stochastic): You only looked at 100 snapshots of the ocean instead of a million. Your sample might have missed a giant wave. This is the error from having limited data.

The authors proved that both of these errors behave like that predictable rubber band (the sector bound).

4. Building the Ghost Ship (The Observer)

Now that they know the error is a "rubber band," they can use a standard, reliable tool from linear algebra (called LMIs or Linear Matrix Inequalities) to build the observer.

Think of this as setting up a guardrail. The guardrail is designed to be wide enough to catch the rubber band if it snaps, but tight enough to keep the ghost ship on the right path.
The Tuning Knob: The engineers can choose how aggressive the ghost ship should be.
- Slow and Steady: The ghost ship follows the real ship gently. It's safe but might be slow to react.
- Fast and Furious: The ghost ship tries to catch up instantly. It reacts quickly but might "jitter" or overshoot a bit before settling down.

5. The Results: Testing the Ship

The authors tested this on three different scenarios:

A Perfectly Stable System: Like a ship in calm water. The method worked perfectly, even with very little data.
A Messy Chemical Plant: Like a ship in choppy waves where the "Lego bricks" (dictionary) didn't perfectly fit the shape. The rubber band safety net caught the errors, and the ghost ship still tracked the real ship accurately.
A Limit Cycle (Van der Pol Oscillator): This is like a ship that naturally wants to spin in a circle (a limit cycle). This is the hardest case because the system is unstable. The method adjusted the "guardrails" to minimize the worst-case error, successfully tracking the spinning ship.

The Big Takeaway

This paper is like giving engineers a universal adapter.
Previously, if you wanted to use simple linear tools to control a complex, non-linear system, you had to hope the approximation was good enough. This paper says: "Don't hope. Calculate the worst-case error, treat it like a rubber band, and build a robust controller that works even if the map is slightly wrong."

It turns a messy, scary non-linear problem into a clean, solvable linear problem with a built-in safety margin.

1. Problem Statement

The paper addresses the challenge of designing state observers for continuous-time nonlinear systems in a data-driven context.

Context: Traditional nonlinear control often assumes full state availability, but in practice, only a subset of outputs is measurable. State observers are required to infer unmeasured states.
Challenge: While the Koopman operator framework allows for the linearization of nonlinear dynamics via "lifting" (mapping states to a higher-dimensional space of observables), practical implementation relies on Extended Dynamic Mode Decomposition (EDMD) with a finite dictionary of basis functions.
Core Issue: Using a finite dictionary introduces an inevitable approximation error (structural mismatch) between the true infinite-dimensional Koopman dynamics and the identified finite-dimensional linear model. Existing literature often treats this error conservatively or ignores the stochastic nature of finite data samples. The authors aim to rigorously characterize this error and synthesize a robust observer that accounts for both deterministic (structural) and stochastic (sampling) errors.

2. Methodology

The proposed methodology integrates Reproducing Kernel Hilbert Space (RKHS) theory with Linear Matrix Inequalities (LMIs) to achieve robust observer synthesis.

A. Theoretical Foundation: Sectorial Error Bound

The authors establish a rigorous bound on the prediction error of the Koopman model identified via generator EDMD (gEDMD).

RKHS Framework: They utilize a specific linear-radial kernel within an RKHS formulation that preserves the equilibrium point at the origin. This allows the state space to be "lifted" into a feature space where the dynamics are approximately linear.
Error Characterization: The total error in the lifted linear model ( $\dot{z} = Az + r$ $\overset{z}{˙} = A z + r$ ) is decomposed into:
1. Deterministic Error: Arising from the projection of the infinite-dimensional operator onto a finite dictionary.
2. Stochastic Error: Arising from the finite number of data samples used for identification.
Key Theoretical Result: The authors prove that the total remainder term $r(x)$ is sectorially bounded by the lifted state $z$ . Specifically, $\|r(x)\| \leq c_r \|z\|$ , where $c_r$ is a constant dependent on the dictionary size, sample size, and confidence level. This "sectorial" property is crucial for robust control design.

B. Robust Observer Synthesis

Based on the sectorial error bound, the observer design is formulated as a convex optimization problem.

Observer Structure: A Luenberger-type observer is designed in the lifted space:
$\dot{\hat{z}} = A\hat{z} + L(y - C\hat{z})$
where $A$ is the identified gEDMD matrix, and $L$ is the gain to be determined.
Stability Analysis: The error dynamics are analyzed using Lyapunov functions. The design ensures that the observation error decays exponentially or that the $L_2$ -gain from the modeling error to the observation error is minimized.
LMI Formulation:
- Case 1 (Asymptotically Stable Systems): The synthesis is formulated as an LMI to guarantee a prescribed exponential decay rate ( $\alpha$ ) for the observation error.
- Case 2 (Unstable/Limit Cycle Systems): The synthesis minimizes the $L_2$ -gain ( $\gamma$ ) from the modeling error to the observation error, ensuring bounded error even if the system does not converge to the origin.

3. Key Contributions

Rigorous Error Quantification: The paper provides a novel theoretical proof that the approximation error of gEDMD, when using a linear-radial kernel, is sectorially bounded by the lifted state. This unifies deterministic structural errors and stochastic sampling errors into a single robust framework.
Robust Observer Synthesis: It proposes a systematic method to design observers using Linear Matrix Inequalities (LMIs) that explicitly account for the identified error bounds. This moves beyond simple "plug-and-play" EDMD to a robust control-theoretic approach.
Handling Unstable Systems: The framework extends to non-asymptotically stable systems (e.g., limit cycles) by focusing on $L_2$ -gain minimization rather than just asymptotic stability.
Elementary Linear Systems Approach: The work demonstrates how to apply elementary linear systems theory (via Koopman lifting) to nonlinear observation without requiring complex operator calculus, provided the error bounds are respected.

4. Numerical Results

The authors validated the method on three distinct numerical examples:

Example 1: Stable System with Invariant Lifting
- A system where the chosen dictionary yields an exact linear representation.
- Result: The observer successfully achieved desired exponential convergence rates ( $\alpha = 0.1$ and $0.99$). Higher $\alpha$ led to faster convergence but larger initial "peaking" errors, confirming the trade-off.
Example 2: Stable System without Invariant Lifting
- A chemical process (two CSTRs in series) where the dictionary is not invariant (approximation error exists).
- Result: Despite the non-invariant dictionary and a significant sectorial error bound ( $c_r \approx 0.67$ ), the synthesized observer guaranteed that estimation errors decayed to negligible levels ( $<10^{-4}$ ).
Example 3: Limit Cycle System (Van der Pol Oscillator)
- An unstable system exhibiting a limit cycle.
- Result: Using the $L_2$ -gain formulation, the observer successfully tracked the limit cycle. The study showed that increasing the polynomial degree of the dictionary (from 4 to 8) significantly reduced the error bound ( $c_r$ ) and improved tracking accuracy (RMSE dropped from 0.123 to 0.022).

5. Significance and Conclusion

This paper bridges the gap between data-driven modeling (Koopman/EDMD) and robust control theory.

Theoretical Impact: It resolves the issue of "mismatch" in finite-dimensional Koopman approximations by characterizing it as a sector-bounded uncertainty, making it amenable to standard robust control tools (LMIs).
Practical Impact: It offers a reliable, algorithmic pathway (Algorithm 1) for designing state observers for complex nonlinear systems using only input-output data, without requiring full knowledge of the system dynamics.
Future Directions: The authors note that while the current method relies on a pre-assigned dictionary, future work could explore kernel-based methods to optimize the dictionary itself, potentially eliminating the structural mismatch entirely. Additionally, addressing scenarios where state data is completely unavailable for model identification remains an open challenge.