⚡ electrical engineering

Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator

This paper establishes a global linearization framework for parameterized nonlinear systems with stable equilibrium points using the Koopman operator, demonstrating that the transformation depends continuously on parameters and identifying conditions under which control-affine systems admit a parameter-independent global bilinearization.

Original authors: Natsuki Katayama, Alexandre Mauroy, Yoshihiko Susuki

Published 2026-04-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Natsuki Katayama, Alexandre Mauroy, Yoshihiko Susuki

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 predict the weather. The atmosphere is a chaotic, swirling mess of nonlinear forces. It's incredibly hard to predict because the rules change depending on how hot, windy, or humid it gets. Now, imagine if you could magically translate that chaotic weather into a simple, straight line on a graph where the rules never change. That is the dream of Global Linearization, and this paper is about how to achieve that dream using a mathematical tool called the Koopman Operator.

Here is a simple breakdown of what the authors did, using everyday analogies.

1. The Problem: The "Curved" World

Most real-world systems (like a car, a robot, or a chemical reaction) are nonlinear.

The Analogy: Imagine driving a car. If you turn the steering wheel a little, the car turns a little. If you turn it a lot, the car might spin out. The relationship between your input (steering) and the output (turning) is curved and complicated. It's hard to predict exactly where you'll end up because the "rules of the road" change depending on how fast you're going.

2. The Magic Tool: The Koopman Operator

The Koopman Operator is like a special pair of glasses or a translator.

The Analogy: Instead of looking at the car's position directly (which is messy), the Koopman Operator looks at the car through a "magic lens." Through this lens, the car's chaotic, curved movement looks like a perfectly straight line.
Why it's cool: In the world of straight lines (linear systems), we have superpowers. We can predict the future easily, and we can design controllers (like autopilot) that work perfectly. The Koopman Operator allows us to take a messy, curved problem and turn it into a clean, straight-line problem.

3. The New Discovery: Handling "Knobs" (Parameters)

In the past, scientists mostly used this magic lens on systems that ran on their own (autonomous). But real systems have inputs or knobs that we can turn (like a gas pedal, a temperature setting, or a control signal).

The Challenge: If you turn the gas pedal, the "rules" of the car change. Does the magic lens still work? Does the straight-line view stay straight?
The Paper's Solution: The authors treated these inputs as parameters (like a dial you can turn). They proved that even when you turn the dial, you can still find a "magic lens" that turns the system into a straight line.
The Key Finding: They showed that as you slowly turn the dial (change the parameter), the magic lens changes smoothly. It doesn't snap or break. This means if you have a system that reacts to different settings, you can still predict its behavior globally (everywhere, not just locally) using a simple linear model.

4. The Big Breakthrough: The "Universal" Lens

The authors went a step further. They asked: "Is there a single, fixed magic lens that works no matter how we turn the dial?"

The Analogy: Imagine you have a car with a gas pedal. Usually, the lens you need to see the car's path changes depending on how hard you press the pedal. But, under very specific conditions (related to the mathematical structure of the car's engine), the authors found a way to build one single lens that works for any pedal position.
The Result: This creates a Global Bilinearization.
- Linearization = Turning a curve into a straight line.
- Bilinearization = Turning a complex system into a specific type of simple system where the state and the input interact in a predictable, simple way.
Why it matters: If you can find this "Universal Lens," you don't need to recalculate your controller every time the input changes. You can design one simple controller that works for the whole range of operations.

5. How They Did It (The "Lie Algebra" Secret Sauce)

How did they know when this "Universal Lens" exists? They used a branch of math called Lie Algebra.

The Analogy: Think of the system's components (the engine, the wheels, the steering) as ingredients in a recipe. Lie Algebra is like a set of rules that tells you how these ingredients interact when you mix them.
The authors proved that if the "mixing rules" of the real, messy car match the "mixing rules" of the simple, straight-line model, then the Universal Lens exists. It's like saying, "If the way the ingredients interact in the real cake is mathematically identical to how they interact in the simplified cake recipe, we can use the simplified recipe to bake the real cake perfectly."

Summary: What Does This Mean for You?

Simplification: Complex, messy systems can be turned into simple, predictable linear systems.
Robustness: This works even when you change the settings (inputs) of the system, and the transition is smooth.
Control: For certain types of systems, we can find a single, fixed transformation that makes control design much easier, regardless of how the system is being driven.

In a nutshell: This paper provides a new map and a new compass for navigating the chaotic world of nonlinear control systems. It tells us exactly when we can turn a tangled knot into a straight string, and how to do it smoothly even when we pull on the string. This is a huge step forward for designing better robots, autonomous vehicles, and smart industrial systems.

1. Problem Statement

The paper addresses the challenge of analyzing and controlling nonlinear dynamical systems with inputs using the Koopman operator framework. While the Koopman operator is well-established for autonomous systems (providing global linearizations via eigenfunctions), its application to systems with control inputs remains theoretically underdeveloped.

Specifically, the authors aim to:

Clarify the spectral properties of Koopman operators for systems where inputs are treated as parameters.
Determine conditions under which a nonlinear system can be transformed into a finite-dimensional linear system (global linearization) that depends continuously on the input parameter.
Identify conditions under which this transformation becomes independent of the input, enabling a global bilinearization for control-affine systems. This is crucial because standard parameter-dependent linearizations are often unsuitable for control design (as the transformation matrix changes with the control input).

2. Methodology

The authors employ a combination of spectral theory, perturbation theory of linear operators, and Lie algebraic arguments.

Parameterization of Inputs: The control input $u$ is treated as a piecewise constant parameter. The system is modeled as $\dot{x} = F_u(x)$ , where $F_u$ is a vector field dependent on $u$ .
Koopman Operator Analysis:
- The Koopman generator $L_F$ is defined as the Lie derivative along the vector field.
- The authors analyze the resolvent set and eigenfunctions of the parameterized generator $L_{F_u}$ .
- They utilize perturbation theory (specifically regarding the strong convergence of resolvents and eigenprojections) to prove that Koopman eigenfunctions depend continuously on the parameter $u$ , provided the system has a Globally Exponentially Stable (GES) Equilibrium Point (EP) at the origin.
Spectral Conditions: The analysis relies on two key conditions for the eigenvalues $\lambda_i$ $λ_{i}$ of the Jacobian $D_x F_u(0)$ $D_{x} F_{u} (0)$ :
1. $k$ -nonresonant condition: No linear combination of eigenvalues equals another eigenvalue (up to order $k$ ).
2. $k$ -spectral spread condition: The real parts of eigenvalues are sufficiently separated.
Lie Algebraic Approach for Bilinearization: To achieve input-independent transformations, the authors extend local results to a global setting. They compare the Lie algebra generated by the nonlinear vector fields $\{F, G_1, \dots, G_d\}_L$ with the Lie algebra generated by their linearizations at the origin $\{A, B_1, \dots, B_d\}_L$ .

3. Key Contributions and Results

A. Global Linearization with Continuous Parameter Dependence

Theorem 4 establishes that for a parameterized system $\dot{x} = F_u(x)$ with a GES EP:

If the eigenvalues of the Jacobian satisfy the nonresonant and spectral spread conditions, there exists a diffeomorphism $\psi_u: X \to \mathbb{R}^n$ that linearizes the system:
$\frac{d}{dt}\psi_u(x) = A_u \psi_u(x)$
where $A_u = D_x F_u(0)$ .
Continuity: Crucially, the transformation $\psi_u$ depends continuously on the parameter $u$ in the $C^k$ norm.
Significance: This justifies the use of data-driven methods (like Extended Dynamic Mode Decomposition - EDMD) for control systems, as it guarantees the existence of a continuous map $(x, u) \mapsto \psi_u(x)$ that can be approximated by universal approximators (e.g., neural networks).

B. Global Bilinearization for Control-Affine Systems

Theorem 5 addresses control-affine systems of the form $\dot{x} = F(x) + \sum u_i G_i(x)$ .

Condition: If the Lie algebra generated by the vector fields $\{F, G_1, \dots, G_d\}$ is isomorphic to the Lie algebra generated by their linearizations at the origin $\{A, B_1, \dots, B_d\}$ , then a global bilinearization exists.
Result: There exists a diffeomorphism $\psi$ (independent of $u$ ) such that the transformed state $z = \psi(x)$ satisfies:
$\dot{z} = Az + \sum_{i=1}^d u_i B_i z$
Global vs. Local: Unlike previous geometric control results (e.g., Proposition 4 in the paper) which are often local (valid only near the equilibrium due to Taylor series convergence), this result is global because it relies on the spectral properties of the Koopman operator rather than local analytic expansions.

C. Resolvent and Eigenprojection Convergence

The paper provides rigorous proofs (Theorems 2 and 3) showing that the resolvent $R(\zeta, L_{F_u})$ and the associated eigenprojections $P_{\lambda_u}$ strongly converge as the parameter $u$ approaches a reference value $u_0$ . This mathematical foundation ensures the stability of the Koopman eigenfunctions under parameter variations.

4. Significance and Implications

Theoretical Foundation for Data-Driven Control: The continuity result (Theorem 4) provides the theoretical justification for using EDMD and its variants in control applications. It confirms that finite-dimensional linear representations exist and vary smoothly with inputs, validating the "black-box" learning approach.
Global Bilinearization: The paper bridges the gap between geometric nonlinear control (Lie algebraic methods) and Koopman operator theory. It offers a constructive condition (Lie algebra isomorphism) for achieving a global bilinear model, which is highly desirable for control design (e.g., Model Predictive Control) as bilinear systems are easier to handle than general nonlinear systems.
Handling Inputs: By treating inputs as parameters, the authors clarify how Koopman eigenfunctions behave in the presence of control, moving beyond the limitations of purely autonomous system analysis.
Extension to Stabilizable Systems: Remark 5 suggests that the results can be extended to systems that are not inherently stable but can be stabilized via feedback, broadening the applicability to optimal control problems for bilinear systems.

5. Example Verification

The paper validates the theory with a 2D example:

A system $\dot{x} = F(x) + uG(x)$ is analyzed.
It demonstrates that when the nonresonance condition holds, the eigenfunctions depend continuously on $u$ .
It shows that when $a=1$ (satisfying the Lie algebra isomorphism condition $[F, G] = 0$ ), a global bilinearization is achieved with a transformation $\psi$ that is independent of $u$ , successfully converting the nonlinear system into a bilinear one.

In summary, this paper rigorously extends the Koopman operator framework to parameterized and control-affine systems, providing necessary and sufficient conditions for global linearization and bilinearization, thereby strengthening the theoretical basis for modern data-driven control strategies.