洞見 - Algorithms and Data Structures - # Data-Driven Robust and Adaptive Model Predictive Control

Robust and Adaptive Data-Driven Model Predictive Control for Unknown Linear Systems

Q: How can the conservatism of the proposed data-driven min-max MPC schemes be further reduced, for example, by considering more general control laws or alternative cost upper bounding functions

To reduce the conservatism of the proposed data-driven min-max MPC schemes, several strategies can be considered: General Control Laws: Instead of restricting the control law to a linear state-feedback form, more general control laws can be explored. Nonlinear or adaptive control laws can provide more flexibility in handling system uncertainties and constraints, potentially leading to less conservative solutions. Alternative Cost Functions: The choice of the cost function plays a crucial role in determining the performance and conservatism of the MPC scheme. By exploring alternative cost functions that better capture the system dynamics and constraints, it may be possible to design more efficient and less conservative controllers. Data Augmentation: Incorporating additional information or data sources can help improve the accuracy of the system model and reduce conservatism. This could involve integrating online measurements, sensor fusion techniques, or advanced data processing algorithms to enhance the system representation. Robust Optimization: Utilizing robust optimization techniques can help account for uncertainties in a more structured manner, leading to controllers that are more resilient to variations in the system dynamics. Robust MPC formulations can provide guarantees under a wider range of operating conditions. By incorporating these strategies and exploring advanced control design methodologies, the conservatism of the data-driven min-max MPC schemes can be further reduced, leading to more effective and adaptive control strategies.

Q: What are the potential challenges and limitations in applying the proposed data-driven min-max MPC schemes to high-dimensional or nonlinear systems

Applying the proposed data-driven min-max MPC schemes to high-dimensional or nonlinear systems poses several challenges and limitations: Curse of Dimensionality: High-dimensional systems require a large number of parameters to be estimated from data, leading to increased computational complexity and data requirements. The curse of dimensionality can make it challenging to accurately model and control such systems using limited data. Nonlinear Dynamics: Nonlinear systems introduce complexities in modeling and control design, as traditional linear MPC approaches may not be directly applicable. Nonlinearities can lead to non-convex optimization problems, making it harder to guarantee stability and performance. State and Input Constraints: Handling constraints in high-dimensional or nonlinear systems can be more challenging, as the feasible region becomes more complex. Ensuring constraint satisfaction while optimizing control inputs becomes computationally intensive and may require advanced optimization techniques. Model Identification: Estimating accurate models for high-dimensional or nonlinear systems from limited data can be difficult. Nonlinear system identification techniques may be required, adding another layer of complexity to the control design process. Addressing these challenges in the context of high-dimensional or nonlinear systems requires advanced control strategies, robust optimization algorithms, and innovative approaches to data-driven control design.

Q: Can the adaptive data-driven min-max MPC scheme be extended to handle time-varying or switching system dynamics, and how would the theoretical guarantees need to be modified

Extending the adaptive data-driven min-max MPC scheme to handle time-varying or switching system dynamics involves several modifications to the theoretical guarantees and optimization framework: Adaptive Model Updating: In the presence of time-varying dynamics, the system model needs to be continuously updated based on the latest data. Adaptive algorithms that adjust the model parameters in real-time can ensure accurate representation of the system dynamics. Switching Control Strategies: For switching system dynamics, the control strategy needs to adapt to different operating modes or models. Incorporating a switching control framework that selects the appropriate controller based on the system's current mode can enhance the adaptability of the MPC scheme. Online Optimization: To handle dynamic changes, online optimization algorithms can be employed to continuously reoptimize the control inputs based on the latest measurements. Real-time optimization techniques can improve the responsiveness and performance of the adaptive MPC scheme. Robustness Analysis: Theoretical guarantees for stability and constraint satisfaction in the presence of time-varying or switching dynamics need to account for the changing system behavior. Robust stability analysis under varying operating conditions is essential to ensure the effectiveness of the adaptive control scheme. By incorporating these modifications and considering the dynamic nature of the system, the adaptive data-driven min-max MPC scheme can be extended to handle time-varying or switching system dynamics effectively.

核心概念

The core message of this paper is to propose a data-driven min-max model predictive control (MPC) scheme that can robustly stabilize an unknown linear time-invariant (LTI) system and satisfy input and state constraints, even in the presence of process noise. The authors further propose an adaptive data-driven min-max MPC scheme that exploits additional online input-state data to improve closed-loop performance.

摘要

The paper presents a data-driven min-max MPC framework for controlling unknown discrete-time LTI systems with additive process noise. The key aspects are:

Offline Input-State Data Characterization:
- The unknown system matrices are characterized by a set-membership representation using noisy offline input-state data.
- A quadratic matrix inequality is used to describe the set of system matrices consistent with the offline data.
Data-Driven Min-Max MPC Formulation:
- The min-max MPC problem is formulated to minimize the worst-case infinite-horizon cost over the set of consistent system matrices.
- The optimization is restricted to state-feedback control laws, allowing the problem to be reformulated as a semidefinite program (SDP).
Closed-Loop Guarantees:
- Recursive feasibility of the SDP problem is proven, ensuring that the optimization remains feasible over time.
- Robust stability of a robust positive invariant (RPI) set around the origin is established for the closed-loop system.
- Input and state constraints are guaranteed to be satisfied for the closed-loop trajectory.
Adaptive Data-Driven Min-Max MPC:
- An adaptive scheme is proposed that incorporates online input-state data to reduce the parametric uncertainty and improve closed-loop performance.
- The set of consistent system matrices is updated recursively using both offline and online data.

The proposed data-driven min-max MPC schemes provide a principled approach to control unknown LTI systems with process noise, while ensuring robust stability and constraint satisfaction.

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統計資料

The paper does not provide any specific numerical data or metrics to support the key claims. The analysis is primarily theoretical, focusing on establishing theoretical guarantees for the proposed data-driven min-max MPC schemes.

引述

"Data-driven controllers design is an important research problem, in particular when data is corrupted by the noise."
"We prove that the resulting closed-loop system is robustly stabilized and satisfies the input and state constraints."
"Further, we propose an adaptive data-driven min-max MPC scheme which exploits additional online input-state data to improve closed-loop performance."

從以下內容提煉的關鍵洞見

Data-Driven Min-Max MPC for Linear Systems: Robustness and Adaptation

by Yifa... 於 arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19096.pdf

Data-Driven Min-Max MPC for Linear Systems: Robustness and Adaptation

深入探究

How can the conservatism of the proposed data-driven min-max MPC schemes be further reduced, for example, by considering more general control laws or alternative cost upper bounding functions

To reduce the conservatism of the proposed data-driven min-max MPC schemes, several strategies can be considered:

General Control Laws: Instead of restricting the control law to a linear state-feedback form, more general control laws can be explored. Nonlinear or adaptive control laws can provide more flexibility in handling system uncertainties and constraints, potentially leading to less conservative solutions.

Alternative Cost Functions: The choice of the cost function plays a crucial role in determining the performance and conservatism of the MPC scheme. By exploring alternative cost functions that better capture the system dynamics and constraints, it may be possible to design more efficient and less conservative controllers.

Data Augmentation: Incorporating additional information or data sources can help improve the accuracy of the system model and reduce conservatism. This could involve integrating online measurements, sensor fusion techniques, or advanced data processing algorithms to enhance the system representation.

Robust Optimization: Utilizing robust optimization techniques can help account for uncertainties in a more structured manner, leading to controllers that are more resilient to variations in the system dynamics. Robust MPC formulations can provide guarantees under a wider range of operating conditions.

By incorporating these strategies and exploring advanced control design methodologies, the conservatism of the data-driven min-max MPC schemes can be further reduced, leading to more effective and adaptive control strategies.

What are the potential challenges and limitations in applying the proposed data-driven min-max MPC schemes to high-dimensional or nonlinear systems

Applying the proposed data-driven min-max MPC schemes to high-dimensional or nonlinear systems poses several challenges and limitations:

Curse of Dimensionality: High-dimensional systems require a large number of parameters to be estimated from data, leading to increased computational complexity and data requirements. The curse of dimensionality can make it challenging to accurately model and control such systems using limited data.

Nonlinear Dynamics: Nonlinear systems introduce complexities in modeling and control design, as traditional linear MPC approaches may not be directly applicable. Nonlinearities can lead to non-convex optimization problems, making it harder to guarantee stability and performance.

State and Input Constraints: Handling constraints in high-dimensional or nonlinear systems can be more challenging, as the feasible region becomes more complex. Ensuring constraint satisfaction while optimizing control inputs becomes computationally intensive and may require advanced optimization techniques.

Model Identification: Estimating accurate models for high-dimensional or nonlinear systems from limited data can be difficult. Nonlinear system identification techniques may be required, adding another layer of complexity to the control design process.

Addressing these challenges in the context of high-dimensional or nonlinear systems requires advanced control strategies, robust optimization algorithms, and innovative approaches to data-driven control design.

Can the adaptive data-driven min-max MPC scheme be extended to handle time-varying or switching system dynamics, and how would the theoretical guarantees need to be modified

Extending the adaptive data-driven min-max MPC scheme to handle time-varying or switching system dynamics involves several modifications to the theoretical guarantees and optimization framework:

Adaptive Model Updating: In the presence of time-varying dynamics, the system model needs to be continuously updated based on the latest data. Adaptive algorithms that adjust the model parameters in real-time can ensure accurate representation of the system dynamics.

Switching Control Strategies: For switching system dynamics, the control strategy needs to adapt to different operating modes or models. Incorporating a switching control framework that selects the appropriate controller based on the system's current mode can enhance the adaptability of the MPC scheme.

Online Optimization: To handle dynamic changes, online optimization algorithms can be employed to continuously reoptimize the control inputs based on the latest measurements. Real-time optimization techniques can improve the responsiveness and performance of the adaptive MPC scheme.

Robustness Analysis: Theoretical guarantees for stability and constraint satisfaction in the presence of time-varying or switching dynamics need to account for the changing system behavior. Robust stability analysis under varying operating conditions is essential to ensure the effectiveness of the adaptive control scheme.

By incorporating these modifications and considering the dynamic nature of the system, the adaptive data-driven min-max MPC scheme can be extended to handle time-varying or switching system dynamics effectively.