Learning Stable Koopman Operators with Convex Constraints for Improved Nonlinear System Modeling

Core Concepts

A novel sufficient condition for the stability of discrete-time linear systems is presented, which can be expressed using piecewise linear constraints. This condition is leveraged to impose stability on a learnable Koopman matrix during the training process using a control barrier function-based projected gradient descent optimization.

Abstract

The paper introduces a new sufficient condition for the asymptotic stability of discrete-time linear systems, which can be expressed using piecewise linear constraints. This condition is then utilized to impose stability on a learnable Koopman matrix during the training process.
Key highlights:

The stability condition can be decoupled by rows of the system matrix, reducing the optimization problem dimensionality.
A control barrier function-based projected gradient descent is proposed to enforce the stability constraints during the iterative learning of the Koopman matrix and observables.
The method is evaluated on the LASA handwriting dataset, showing comparable prediction performance to other recent stable Koopman learning approaches while providing more flexibility in the optimization problem.
The stability constraints are sufficient but not necessary, allowing the model to learn a wider range of dynamics compared to other parameterizations.
Future work includes extending the method to controlled systems, further reducing computation time, and integrating additional physical constraints into the optimization problem.

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Key Insights Distilled From

Learning deep Koopman operators with convex stability constraints

by Marc Mitjans... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15978.pdf

Deeper Inquiries

How can the proposed stability constraints be extended to handle more complex nonlinear dynamics beyond the linear case

To extend the proposed stability constraints to handle more complex nonlinear dynamics beyond the linear case, one can consider incorporating higher-order terms in the constraints. By introducing terms that capture the nonlinear interactions between the system states, the constraints can be formulated to ensure stability in a broader range of dynamical systems. Additionally, utilizing advanced mathematical techniques such as Taylor series expansions or neural network parameterizations can help capture the nonlinear behavior of the system within the stability constraints. This approach would enable the constraints to address the complexities of nonlinear dynamics while still providing a framework for ensuring stability during the learning process.

What are the potential drawbacks or limitations of using a sufficient but not necessary stability condition compared to necessary and sufficient conditions

The use of a sufficient but not necessary stability condition, as proposed in the context, comes with certain drawbacks and limitations. One limitation is that the sufficient condition may lead to a more conservative approach, potentially resulting in a loss of predictive accuracy or model performance. Since the condition is not necessary, there is a possibility of overconstraining the system, leading to suboptimal solutions. Moreover, the lack of necessity in the stability condition may result in the model not capturing all aspects of the system dynamics that are crucial for stability. This could limit the applicability of the model to real-world scenarios where precise stability guarantees are required.

How can the flexibility of the optimization-in-the-loop approach be leveraged to incorporate additional application-specific requirements beyond stability, such as energy efficiency or robustness

The flexibility of the optimization-in-the-loop approach can be leveraged to incorporate additional application-specific requirements beyond stability, such as energy efficiency or robustness, by introducing custom constraints into the optimization problem. For instance, constraints related to energy conservation or system robustness can be formulated and integrated into the optimization process alongside the stability constraints. By including these additional constraints, the optimization algorithm can be guided to learn models that not only ensure stability but also adhere to specific performance criteria relevant to the application domain. This customization allows for the development of tailored models that meet the unique requirements of the system being studied.