On Koopman Resolvents and Frequency Response of Nonlinear Systems

Here is an explanation of the paper using simple language and creative analogies.

The Big Idea: Turning Chaos into a Symphony

Imagine you are trying to understand how a complex machine works. If the machine is simple and predictable (like a clock), you can easily predict how it will react if you push a button. In engineering, we call this a Linear Time-Invariant (LTI) system. We have a perfect tool for analyzing these: the Frequency Response. It tells us, "If I wiggle the input at this speed, how much will the output wiggle, and will it be in sync or delayed?"

But what if the machine is nonlinear? Think of a rollercoaster, a stock market, or a weather system. These are messy. If you push them, they don't just wiggle; they might spin, jump, or behave in weird, unpredictable ways. For a long time, engineers didn't have a clean, mathematical way to describe how these messy systems react to rhythmic inputs. They had to rely on messy approximations or look at the problem in the "time domain" (watching the movie frame-by-frame), which is hard to visualize.

This paper introduces a new "magic lens" called the Koopman Operator. It allows us to take that messy, nonlinear rollercoaster and view it through a lens that makes it look like a simple, linear orchestra. With this lens, we can finally draw the famous Bode Plot (a graph showing gain and phase) for nonlinear systems, just like we do for simple clocks.

The Core Concept: The "Shadow" of the System

1. The Problem: The Nonlinear Rollercoaster

Imagine a nonlinear system as a rollercoaster car going through a loop. If you push the car (the input), it doesn't just speed up linearly. It might loop faster, slow down, or even flip. Because of this complexity, if you push it with a steady rhythm, the output might not just be a simple wave; it might be a wave with extra "harmonics" (like a musical note that also has a high-pitched squeal on top).

2. The Solution: The Koopman "Shadow"

The authors use a mathematical trick called the Koopman Operator.

The Analogy: Imagine the rollercoaster is a 3D object in a dark room. You can't see the object itself easily. But, if you shine a light from a specific angle, the shadow it casts on the wall is a perfect, simple 2D circle.
The Math: The Koopman operator is that light. It takes the complex, nonlinear movement of the system (the 3D object) and projects it onto a "shadow space" (an infinite-dimensional linear space). In this shadow world, the messy nonlinear rules disappear, and the system behaves like a simple, linear line.

3. The "Skew-Product": Adding the Rhythm

To study how the system reacts to a specific rhythm (like a sine wave input), the authors create a special version of the system called a Skew-Product.

The Analogy: Imagine the rollercoaster is on a stage. Usually, the stage is dark. To study the rhythm, they turn on a giant, rotating spotlight that spins at a constant speed (the input frequency). Now, the system isn't just the car; it's the car plus the spinning spotlight.
By treating the spinning spotlight as part of the system, they turn a "forced" system (one being pushed from outside) into an "autonomous" system (one that moves on its own). This makes the math much easier to handle.

The Main Discovery: The "Ghost Frequencies"

Once they have this "shadow" view of the system with the spinning spotlight, they look for Resonances.

In the shadow world, the system has specific "natural frequencies" where it likes to vibrate. The authors prove that:

If you push the system at a frequency $\omega$ , the output will contain waves at $\omega$ , $2\omega $,$ 3\omega$, etc. (Harmonics).
Sometimes, it might even contain waves at $\omega/2$ or $\omega/3$ (Subharmonics).

The paper provides a formula to calculate exactly how strong these waves are and how much they are delayed (phase).

The "Koopman Mode": Think of this as the "volume knob" for each specific frequency. The paper shows that these knobs are actually just the mathematical "residues" (a specific value you get when you zoom in on a spike in the math) of the Koopman operator.

Why is this cool?
Because now, instead of running a million computer simulations to see how a nonlinear system reacts, you can calculate these "volume knobs" directly. You can then draw a Bode Plot (a graph of Gain vs. Frequency) for a nonlinear system.

Gain: How loud the output is compared to the input.
Phase: How much the output lags behind the input.

Real-World Examples from the Paper

The authors tested this on two examples:

The Simple Linear Case: They checked their math on a simple spring. As expected, their new method gave the exact same answer as the old, classic method. This proved their "new lens" works correctly.
The Nonlinear Case: They looked at a system where one part squared the other (a classic nonlinearity).
- The Result: When they pushed the system at frequency $\omega$ , the first part of the system ( $x_2$ ) wiggled at $\omega$ . But the second part ( $x_1$ ), because of the squaring nonlinearity, ignored the $\omega$ frequency and only wiggled at $2\omega$ (twice the speed).
- The Visualization: They drew a Bode plot for this. It showed that for the second part, the "Gain" was zero at the input frequency and peaked at double the frequency. This is something traditional linear tools couldn't see clearly, but the Koopman method revealed instantly.

When Does This Work? (The "Sufficient Conditions")

The paper admits this magic lens doesn't work for every chaotic mess. It works best in three scenarios:

Linear Systems: (Obviously, it works perfectly here).
Globally Stable Systems: Systems that eventually settle down into a predictable, repeating loop (like a pendulum that stops swinging and settles).
Ergodic Systems: Systems that wander around a specific area (like a chaotic attractor) but visit every part of it over time in a statistical sense.

If the system is too wild (like a system that explodes or behaves chaotically without settling into a pattern), the math gets harder, but the authors lay the groundwork for future research there.

Summary: Why Should You Care?

Before this paper, analyzing the "sound" (frequency response) of a complex, nonlinear machine was like trying to tune a radio in a storm—you could guess, but it was messy.

This paper gives engineers a new tuning fork. By using the Koopman operator, they can:

Turn complex nonlinear problems into linear ones.
Predict exactly how a system will react to rhythmic inputs.
Draw standard "Bode Plots" for nonlinear systems, allowing for better design of controllers for things like drones, power grids, and biological systems.

In short: They found a way to make the messy world of nonlinear dynamics look as clean and predictable as a simple clock, so we can finally tune it properly.

Here is a detailed technical summary of the paper "On Koopman Resolvents and Frequency Response of Nonlinear Systems" by Susuki et al.

1. Problem Statement

The concept of frequency response is a cornerstone of control engineering for Linear Time-Invariant (LTI) systems, allowing for analysis via Bode plots (gain and phase) and synthesis in the frequency domain. However, for nonlinear systems, a rigorous, general frequency response theory analogous to the LTI case has been lacking.

Current Limitations: Existing approaches for nonlinear frequency response (e.g., describing functions, harmonic balance, Volterra series) are primarily phenomenological or based on time-domain/state-space formulations. They often lack a unified theoretical framework that mirrors the spectral properties of LTI systems.
Goal: The authors aim to establish a rigorous frequency response formulation for nonlinear systems that is guided by the Koopman operator framework, specifically utilizing the Koopman resolvent, to enable frequency-domain analysis (including Bode plots) for nonlinear dynamics.

2. Methodology

The paper employs the Koopman operator theory, which lifts nonlinear dynamics into an infinite-dimensional linear space of observables. The methodology proceeds through the following steps:

A. System Modeling via Skew-Product Form

To analyze the frequency response under a periodic input $u(t) = u_0 e^{i\omega t}$ , the authors convert the non-autonomous nonlinear system into an equivalent autonomous system using a skew-product form:
$\begin{cases} \dot{x} = F(x, u) \\ \dot{u} = i\omega u \\ y = g(x, u) \end{cases}$
This augmentation treats the input as a dynamic variable, allowing the application of autonomous Koopman operator theory.

B. Koopman Generator and Resolvent

Koopman Generator ( $L_{forced}$ ): Defined for the skew-product system using gradient and differential operators.
Koopman Resolvent ( $R(s; L_{forced})$ ): Defined as $(sI - L_{forced})^{-1}$ .
Laplace Transform Connection: Building on previous work [7], the Laplace transform of the output $\hat{y}(s)$ is expressed as the action of the Koopman resolvent on the observable $g$ :
$\hat{y}(s; x_0, u_0) = [R(s; L_{forced})g](x_0, u_0)$

C. Spectral Characterization of Frequency Response

The authors define the frequency response based on the spectral properties (specifically the point spectrum/eigenvalues) of the Koopman generator $L_{forced}$ :

Harmonic Response ( $n\omega$ ): Associated with eigenvalues $in\omega$ .
Subharmonic Response ( $\omega/n$ ): Associated with eigenvalues $i\omega/n$ .
Koopman Modes: The frequency response coefficients are identified as Koopman modes (residues of the resolvent at the poles).

3. Key Contributions

Novel Formulation: A rigorous definition of frequency response for nonlinear systems derived directly from the Koopman resolvent. This generalizes the classical LTI approach to the nonlinear domain.
Complex-Valued Response & Bode Plots: The derived frequency response $H_n(\omega; g)$ is a complex-valued function of the driving frequency. This allows for the construction of Bode plots (magnitude and phase) for nonlinear systems, visualizing gain and phase characteristics similar to linear systems.
Theoretical Link to Koopman Modes: The paper proves that the frequency response is mathematically equivalent to the Koopman mode associated with the specific eigenfrequency. This bridges the gap between spectral decomposition and frequency response analysis.
Sufficient Conditions for Existence: The paper establishes sufficient conditions for the existence of these frequency responses across three classes of dynamics:
- LTI Dynamics: Proves the formulation reduces to the classical transfer function $C(sI-A)^{-1}B$ .
- Globally Stable Dynamics: Shows that if a globally stable periodic solution exists and the system is analytic, the resolvent has a simple pole at the target frequency.
- Ergodic Dynamics: Demonstrates that for systems with compact ergodic attractors, the spectral expansion of the resolvent yields the frequency response, provided the eigenvalue is isolated.

4. Results and Examples

Theoretical Proofs:
- Theorem 1 & 2: Provide integral and limit formulas to calculate the frequency response coefficients ( $H_n$ and $H_{1/n}$ ) by evaluating the residue of the Koopman resolvent at the poles $in\omega$ and $i\omega/n$ .
- Lemma 1 & 2: Establish that the mappings $u \to u^n$ and $u \to u^{1/n}$ correspond to eigenfunctions of the forced Koopman generator with eigenvalues $in\omega$ and $i\omega/n$ , respectively.
Case Studies:
- 1D Linear Case: The formulation successfully recovers the standard linear frequency response $b/(i\omega - a)$ .
- 2D Nonlinear Case: A system with a quadratic nonlinearity ( $\dot{x}_1 = a_1 x_1 + x_2^2$ $\overset{x}{˙}_{1} = a_{1} x_{1} + x_{2}^{2}$ ) is analyzed.
  - The output $x_2$ (linear subsystem) shows a fundamental frequency response $H_1$ .
  - The output $x_1$ (nonlinear subsystem) exhibits no fundamental response ( $H_1=0$ ) but a strong second-harmonic response ( $H_2$ ) due to the $x_2^2$ term.
  - Bode Plots: The authors successfully generated Bode plots for these nonlinear responses, showing $x_1$ behaving as a 3rd-order lag element and $x_2$ as a 1st-order lag element.
  - Validation: The nonlinear results were verified by "lifting" the system to a higher-dimensional linear space (using states $x_1, x_2, u, x_2^2, \dots$ ) and solving the resulting linear transfer function, confirming the Koopman-based derivation.

5. Significance and Impact

Unification of Frameworks: This work unifies the analysis of nonlinear systems with the intuitive, frequency-domain tools traditionally reserved for linear systems. It provides a "physically intuitive" method for analyzing nonlinear dynamics.
Numerical Feasibility: By identifying frequency responses as Koopman modes, the paper opens the door to using Dynamic Mode Decomposition (DMD) and other data-driven Koopman methods to estimate frequency responses directly from data, without requiring explicit analytical models.
Design and Synthesis: The ability to draw Bode plots for nonlinear systems suggests new pathways for the synthesis and control of nonlinear plants, potentially enabling robust control design based on gain and phase margins in the nonlinear regime.
Theoretical Rigor: It moves beyond phenomenological approximations (like describing functions) to a mathematically rigorous spectral theory grounded in operator semigroups.

In summary, Susuki et al. successfully extend the concept of frequency response to nonlinear systems by leveraging the Koopman resolvent, proving that nonlinear frequency responses are spectral properties (Koopman modes) of an augmented autonomous system, and demonstrating their utility through analytical examples and Bode plot generation.