Optimizing Nonlinear Dynamical Systems through Adaptive Gray-Box Feedback Control

Core Concepts

The core message of this article is to develop a gray-box feedback optimization controller that combines the complementary benefits of model-based and model-free approaches to efficiently optimize the steady-state operation of nonlinear dynamical systems. The proposed controller adaptively fuses approximate sensitivities and model-free gradient estimates to achieve a balanced closed-loop behavior, retaining provable sample efficiency and optimality guarantees for nonconvex problems.

Abstract

The article presents the design and analysis of a gray-box feedback optimization controller for efficiently optimizing the steady-state operation of nonlinear dynamical systems. The key insights are: The gray-box controller merges model-based inexact gradients using approximate sensitivities and model-free gradient estimates through an adaptive convex combination. This allows it to exploit the benefits of both approaches. The authors characterize the conditions on the accuracy of the approximate sensitivities that render the gray-box controller preferable over purely model-based or model-free controllers. They show that the gray-box controller can handle inaccurate sensitivities and relax the accuracy requirement of sensitivity learning. The performance of the gray-box controller is analyzed in terms of the squared gradient norm of the nonconvex objective. It is shown to overcome the sub-optimality issues of model-based controllers and improve the sample efficiency compared to model-free approaches. The gray-box controller is extended to handle time-varying problems with changing objectives, variable disturbances, and input constraints. The authors provide performance certificates in terms of dynamic regret and tracking error, demonstrating the controller's ability to strike a balance between sample efficiency and accuracy despite approximate sensitivities.

Stats

The article does not contain any explicit numerical data or statistics. It focuses on the theoretical design and analysis of the gray-box feedback optimization controller.

Quotes

"The gray-box controller contributes to a balanced closed-loop behavior, which retains provable sample efficiency and optimality guarantees for nonconvex problems." "The gray-box controller overcomes the issue of sub-optimality, which troubles model-based controllers using sensitivities with slowly vanishing errors." "The gray-box controller is shown to strike an excellent balance between sample efficiency and accuracy despite approximate sensitivities."

Key Insights Distilled From

Gray-Box Nonlinear Feedback Optimization

by Zhiy... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04355.pdf

Deeper Inquiries

How can the gray-box controller be extended to handle more general classes of nonlinear dynamical systems, such as those with non-differentiable steady-state maps or non-smooth objective functions?

The extension of the gray-box controller to handle more general classes of nonlinear dynamical systems involves adapting the controller's design to accommodate the specific characteristics of the system. For systems with non-differentiable steady-state maps, the controller can be modified to work with subgradients or subdifferentials instead of gradients. This adjustment allows the controller to handle the non-smoothness of the steady-state map and still make informed decisions based on the available information. Similarly, for systems with non-smooth objective functions, the gray-box controller can be enhanced to incorporate techniques from nonsmooth optimization. This may involve using subgradients or subdifferentials of the objective function to guide the optimization process. By incorporating these non-smooth elements into the controller's update rules, it can effectively navigate the optimization landscape and converge to optimal solutions even in the presence of non-smoothness. Overall, the extension of the gray-box controller to handle more general classes of nonlinear dynamical systems involves adapting its algorithms and methodologies to accommodate the specific challenges posed by non-differentiable steady-state maps and non-smooth objective functions.

What are the potential applications of the gray-box feedback optimization approach beyond the steady-state optimization considered in this article, such as in the context of model predictive control or reinforcement learning?

The gray-box feedback optimization approach has a wide range of potential applications beyond the steady-state optimization discussed in the article. Some of these applications include: Model Predictive Control (MPC): In MPC, the gray-box controller can be used to optimize control inputs over a finite time horizon based on a predictive model of the system. By incorporating real-time measurements and adaptive combination of model-based and model-free components, the gray-box controller can enhance the performance of MPC systems, especially in cases where accurate models are unavailable or uncertain. Reinforcement Learning (RL): In RL, the gray-box controller can be utilized to optimize the policy of an agent interacting with an environment to maximize a cumulative reward. By integrating approximate sensitivities and gradient estimates, the gray-box approach can improve the sample efficiency and stability of RL algorithms, making them more robust to uncertainties and inaccuracies in the system dynamics. Stochastic Optimization: The gray-box controller can also be applied to stochastic optimization problems where the objective function or constraints involve random variables. By adapting the controller's update rules to handle stochastic gradients or noisy measurements, the gray-box approach can effectively optimize under uncertainty and variability. Adaptive Control: In adaptive control systems, the gray-box controller can be used to adjust control parameters based on real-time feedback and a combination of model-based and model-free updates. This adaptive approach allows the controller to adapt to changing system dynamics and disturbances, improving the robustness and performance of the control system. Overall, the gray-box feedback optimization approach has versatile applications in various fields beyond steady-state optimization, offering benefits in terms of adaptability, robustness, and efficiency in optimizing complex systems.

How can the adaptive combination of model-based and model-free components in the gray-box controller be further improved or generalized to handle a wider range of sensitivity estimation quality and problem characteristics?

The adaptive combination of model-based and model-free components in the gray-box controller can be further improved and generalized by considering the following strategies: Dynamic Weighting: Instead of using fixed combination coefficients, dynamic weighting schemes can be employed to adjust the influence of model-based and model-free components based on the quality of sensitivity estimation and problem characteristics. Adaptive algorithms can dynamically tune the coefficients during the optimization process to optimize performance. Ensemble Methods: Utilizing ensemble methods that combine multiple models or estimates can enhance the robustness and accuracy of the sensitivity estimation. By integrating diverse sources of information and estimates, the gray-box controller can make more informed decisions and adapt to a wider range of problem characteristics. Meta-Learning: Incorporating meta-learning techniques can enable the gray-box controller to learn the optimal combination strategy for different problem settings and sensitivity estimation qualities. By leveraging meta-learning algorithms, the controller can adapt its behavior based on past experiences and performance feedback. Transfer Learning: Leveraging transfer learning approaches can allow the gray-box controller to generalize its adaptive combination strategy across different systems and problem domains. By transferring knowledge and insights from one problem to another, the controller can improve its performance and efficiency in handling diverse sensitivity estimation qualities. By implementing these advanced strategies and techniques, the adaptive combination of model-based and model-free components in the gray-box controller can be further enhanced to handle a wider range of sensitivity estimation qualities and problem characteristics, leading to improved optimization performance and robustness.