аналитика - Algorithms and Data Structures - # Data-Driven Modeling of Nonlinear Systems

Data-Driven Representation of Nonlinear Systems with Koopman Linear Embedding

Q: How can the data-driven representation be extended to nonlinear systems with only an approximate Koopman linear embedding?

The extension of the data-driven representation to nonlinear systems with only an approximate Koopman linear embedding can be approached by leveraging the insights from the existing framework of Willems' fundamental lemma and the properties of the Koopman operator. In cases where an exact Koopman linear embedding is not available, one can still utilize the concept of lifting functions to create an approximate representation of the system dynamics. The key is to identify a set of lifting functions that can capture the essential dynamics of the nonlinear system, even if they do not provide a perfect linear representation. This can be achieved through techniques such as dynamic mode decomposition (DMD) or extended dynamic mode decomposition (EDMD), which allow for the approximation of the Koopman operator based on observed data. By collecting a sufficiently rich trajectory library that includes various input-output pairs, one can construct a data-driven model that approximates the behavior of the nonlinear system. Moreover, the proposed approach emphasizes the importance of persistent excitation, which ensures that the trajectory library is rich enough to capture the dynamics of the system. By ensuring that the trajectory library includes a diverse set of trajectories, one can mitigate the effects of approximation errors and improve the robustness of the data-driven representation. This approach allows for the development of predictive control strategies that can effectively manage nonlinear systems, even when the Koopman linear embedding is only approximate.

Q: What are the potential limitations of the proposed approach when dealing with highly nonlinear or high-dimensional systems?

The proposed approach, while powerful, does have potential limitations when applied to highly nonlinear or high-dimensional systems. One significant challenge is the curse of dimensionality, which refers to the exponential increase in the volume of the input space as the number of dimensions increases. In high-dimensional systems, collecting a sufficiently rich trajectory library becomes increasingly difficult, as the number of required samples grows rapidly to ensure adequate coverage of the state space. Additionally, highly nonlinear systems may exhibit complex dynamics that are not easily captured by linear approximations. The reliance on lifting functions to create a Koopman linear embedding may lead to significant modeling errors if the chosen functions do not adequately represent the underlying dynamics. This can result in poor predictive performance and control outcomes, particularly in regions of the state space that are underrepresented in the trajectory library. Furthermore, the requirement for persistent excitation can be challenging to satisfy in practice, especially in systems where the input-output relationships are highly sensitive to initial conditions or external disturbances. If the trajectory library does not meet the persistent excitation condition, the resulting data-driven model may lack robustness and fail to generalize well to unseen scenarios.

Основные понятия

A data-driven representation of nonlinear systems that admit a Koopman linear embedding can be constructed directly from input-output data, bypassing the need to identify the lifting functions.

Аннотация

The paper presents an extended Willems' fundamental lemma for nonlinear systems that admit a Koopman linear embedding. The key contributions are:

Characterization of the relationship between the trajectory space of the nonlinear system and its Koopman linear embedding. This shows that the trajectory space of the Koopman linear embedding can be represented by a linear combination of rich-enough trajectories from the original nonlinear system.
Introduction of a new notion of persistent excitation for nonlinear systems that accounts for the lifted state in the Koopman linear embedding. This enables the data-driven representation to be constructed directly from the input-output data of the nonlinear system.
Establishment of a data-driven representation adapted from Willems' fundamental lemma for nonlinear systems with a Koopman linear embedding. This representation bypasses the need to identify the lifting functions, which is a challenging task in Koopman-based modeling.

The data-driven representation can be directly utilized in predictive control for nonlinear systems. The results also illustrate the importance of both the width (more trajectories) and depth (longer trajectories) of the trajectory library in ensuring the accuracy of the data-driven model.

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arxiv.org

Статистика

The nonlinear system considered has the following dynamics:
x1,k+1 = 0.99x1,k
x2,k+1 = 0.9x2,k + x2^1,k + x3^1,k + x4^1,k + uk
The output is yk = xk.

Цитаты

"Koopman operator theory and Willems' fundamental lemma both can provide (approximated) data-driven linear representation for nonlinear systems."
"Our data-driven representation can be directly utilized in predictive control, without the need of identifying the linear model."

Ключевые выводы из

Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding

by Xu S... в arxiv.org 09-26-2024

https://arxiv.org/pdf/2409.16389.pdf

Willems' Fundamental Lemma for Nonlinear Systems with Koopman Linear Embedding

Дополнительные вопросы

How can the data-driven representation be extended to nonlinear systems with only an approximate Koopman linear embedding?

The extension of the data-driven representation to nonlinear systems with only an approximate Koopman linear embedding can be approached by leveraging the insights from the existing framework of Willems' fundamental lemma and the properties of the Koopman operator. In cases where an exact Koopman linear embedding is not available, one can still utilize the concept of lifting functions to create an approximate representation of the system dynamics.
The key is to identify a set of lifting functions that can capture the essential dynamics of the nonlinear system, even if they do not provide a perfect linear representation. This can be achieved through techniques such as dynamic mode decomposition (DMD) or extended dynamic mode decomposition (EDMD), which allow for the approximation of the Koopman operator based on observed data. By collecting a sufficiently rich trajectory library that includes various input-output pairs, one can construct a data-driven model that approximates the behavior of the nonlinear system.
Moreover, the proposed approach emphasizes the importance of persistent excitation, which ensures that the trajectory library is rich enough to capture the dynamics of the system. By ensuring that the trajectory library includes a diverse set of trajectories, one can mitigate the effects of approximation errors and improve the robustness of the data-driven representation. This approach allows for the development of predictive control strategies that can effectively manage nonlinear systems, even when the Koopman linear embedding is only approximate.

What are the potential limitations of the proposed approach when dealing with highly nonlinear or high-dimensional systems?

The proposed approach, while powerful, does have potential limitations when applied to highly nonlinear or high-dimensional systems. One significant challenge is the curse of dimensionality, which refers to the exponential increase in the volume of the input space as the number of dimensions increases. In high-dimensional systems, collecting a sufficiently rich trajectory library becomes increasingly difficult, as the number of required samples grows rapidly to ensure adequate coverage of the state space.
Additionally, highly nonlinear systems may exhibit complex dynamics that are not easily captured by linear approximations. The reliance on lifting functions to create a Koopman linear embedding may lead to significant modeling errors if the chosen functions do not adequately represent the underlying dynamics. This can result in poor predictive performance and control outcomes, particularly in regions of the state space that are underrepresented in the trajectory library.
Furthermore, the requirement for persistent excitation can be challenging to satisfy in practice, especially in systems where the input-output relationships are highly sensitive to initial conditions or external disturbances. If the trajectory library does not meet the persistent excitation condition, the resulting data-driven model may lack robustness and fail to generalize well to unseen scenarios.

How can the insights from this work on the importance of trajectory library size be leveraged to develop more efficient data collection strategies for data-driven modeling of complex systems?

The insights from this work highlight the critical role of trajectory library size in ensuring the accuracy and reliability of data-driven models for complex systems. To develop more efficient data collection strategies, one can focus on several key aspects:

Targeted Data Collection: Instead of random sampling, data collection efforts can be strategically focused on regions of the state space that are known to exhibit significant dynamics or where the system behavior is less understood. This targeted approach can help ensure that the trajectory library captures the most relevant dynamics, thereby improving the quality of the data-driven model.

Adaptive Sampling: Implementing adaptive sampling techniques can enhance data collection efficiency. By continuously evaluating the performance of the data-driven model, one can identify areas where additional data is needed and prioritize data collection in those regions. This iterative process allows for the dynamic adjustment of the trajectory library based on the evolving understanding of the system.

Utilization of Simulations: In scenarios where real-world data collection is costly or time-consuming, simulations can be employed to generate synthetic data that complements the real data. By simulating a variety of operating conditions and disturbances, one can enrich the trajectory library and ensure that it encompasses a wide range of system behaviors.

Incorporating Domain Knowledge: Leveraging domain-specific knowledge can guide the selection of lifting functions and the design of experiments. By understanding the underlying physics or mechanics of the system, one can make informed decisions about which trajectories to prioritize, thus enhancing the efficiency of data collection.

Balancing Width and Depth: The findings emphasize the importance of both the width (number of trajectories) and depth (length of trajectories) of the trajectory library. Data collection strategies should aim to achieve a balance between these two aspects, ensuring that the library is sufficiently rich to meet the persistent excitation condition while also being manageable in terms of data acquisition and processing.

By implementing these strategies, researchers and practitioners can optimize their data collection efforts, leading to more accurate and robust data-driven models for complex nonlinear systems.