Online Convex Optimization for Robust Control of Constrained Dynamical Systems with Disturbances and Measurement Noise

Główne pojęcia

This paper presents an online convex optimization algorithm for robust control of linear time-invariant systems that guarantees constraint satisfaction despite time-varying costs, disturbances, and measurement noise, achieving bounded dynamic regret.

Streszczenie

Bibliographic Information:

Nonhoff, M., Dall’Anese, E., & Müller, M. A. (2024). Online convex optimization for robust control of constrained dynamical systems. IEEE Transactions on Automatic Control. (Under Review)

Research Objective:

This paper addresses the challenge of controlling linear time-invariant systems with time-varying and a priori unknown cost functions, subject to state and input constraints, disturbances, and measurement noise. The objective is to design an algorithm that guarantees robust constraint satisfaction and achieves satisfactory performance in terms of dynamic regret.

Methodology:

The authors propose an online convex optimization algorithm that combines elements of robust model predictive control and online gradient descent. The algorithm utilizes a constraint tightening approach based on robust positively invariant sets to ensure robust constraint satisfaction despite uncertainties. Dynamic regret, defined as the cumulative performance difference between the closed-loop trajectory and the optimal steady states, is used to analyze the algorithm's performance.

Key Findings:

The paper proves that the proposed algorithm guarantees recursive feasibility, ensuring that the algorithm's output is well-defined at all times. Moreover, it demonstrates that the algorithm guarantees robust constraint satisfaction for the closed-loop system. Finally, the authors prove that the dynamic regret of the algorithm is bounded linearly by the variation of the cost functions and the magnitude of the disturbances and measurement noise.

Main Conclusions:

The proposed online convex optimization algorithm provides a robust and efficient solution for controlling constrained dynamical systems in the presence of time-varying costs, disturbances, and measurement noise. The algorithm's ability to guarantee constraint satisfaction and achieve bounded dynamic regret makes it suitable for various applications, including robotics, power systems, and autonomous driving.

Significance:

This research contributes to the field of online convex optimization and robust control by providing a novel algorithm that addresses the challenges of time-varying costs, constraints, and uncertainties in dynamical systems. The theoretical guarantees and practical applicability of the proposed approach make it a valuable tool for controlling complex systems in dynamic environments.

Limitations and Future Research:

The paper focuses on linear time-invariant systems. Future research could explore extensions of the proposed framework to nonlinear systems or systems with uncertain dynamics. Additionally, investigating the impact of different online optimization methods on the algorithm's performance could be of interest.

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Kluczowe wnioski z

Online convex optimization for robust control of constrained dynamical systems

by Mark... o arxiv.org 11-25-2024

https://arxiv.org/pdf/2401.04487.pdf

Głębsze pytania

How can the proposed algorithm be adapted for applications with safety-critical constraints, where even small constraint violations are unacceptable?

In safety-critical applications where even minor constraint violations are unacceptable, the proposed algorithm needs modifications to provide rigorous safety guarantees. Here's a breakdown of potential adaptations:

Robust Constraint Tightening:

Increased margins: The current constraint tightening approach using RPI sets provides probabilistic guarantees. For hard guarantees, the tightening margins need to be more conservative. This can be achieved by:

Inflating the disturbance sets (W, V): Instead of using estimates of the disturbance bounds, consider worst-case bounds or add a safety margin to the estimated bounds.
Employing robust optimization techniques: Formulate the constraint tightening problem (15) using robust optimization techniques like scenario optimization or tube-based MPC to explicitly account for the worst-case disturbance realizations.

Verification and Safety Layers:

Formal verification: Employ formal verification methods to rigorously prove that the tightened constraints guarantee safety for all possible system trajectories under the given disturbance bounds.
Safety layer: Implement a separate safety layer on top of the proposed controller. This layer would monitor the system state and intervene with a safe backup strategy if a potential constraint violation is detected, even if it means deviating from the optimal trajectory.

Addressing Model Uncertainty:

Robust MPC techniques: The current algorithm assumes a perfect system model. In reality, model uncertainties exist. Incorporate robust MPC techniques like tube-based MPC or min-max MPC to handle model uncertainties while guaranteeing constraint satisfaction.

Real-Time Considerations:

Computational complexity: The increased conservatism in constraint tightening and the use of robust optimization techniques can increase computational complexity. Explore computationally efficient methods for robust constraint tightening and optimization to ensure real-time feasibility.

By implementing these adaptations, the proposed algorithm can be tailored for safety-critical applications, providing stronger guarantees of constraint satisfaction even in the presence of disturbances and uncertainties.

While the paper focuses on dynamic regret as a performance metric, could alternative metrics like competitive ratio or regret bounds with respect to a dynamic benchmark provide additional insights into the algorithm's performance?

Yes, alternative performance metrics like competitive ratio and regret bounds with respect to a dynamic benchmark can offer complementary insights into the algorithm's performance beyond dynamic regret.

Competitive Ratio:

Definition: Competitive ratio compares the algorithm's performance to that of an optimal offline algorithm that has full knowledge of the future cost functions and disturbances. It measures the factor by which the online algorithm's cost is worse than the optimal offline cost.
Insights: A low competitive ratio indicates that the algorithm performs well even with limited information about the future, suggesting robustness to unpredictable changes in the environment.

Regret Bounds with Respect to a Dynamic Benchmark:

Definition: Instead of comparing to the optimal steady-state trajectory, consider a more dynamic benchmark, such as the trajectory of an optimal online algorithm with a limited prediction horizon or access to a noisy version of the future cost functions.
Insights: This metric can provide a more realistic assessment of the algorithm's performance in dynamic environments, as it acknowledges that achieving the optimal steady-state trajectory might not always be feasible or desirable in rapidly changing conditions.

Benefits of Alternative Metrics:

Robustness evaluation: Competitive ratio specifically quantifies the algorithm's ability to handle uncertainty and adapt to unforeseen changes.
Practical relevance: Dynamic benchmarks can offer a more realistic performance assessment in real-world scenarios where the environment is constantly evolving.
Algorithm comparison: These metrics provide a standardized way to compare the performance of different online optimization algorithms under various uncertainty models.
By incorporating these alternative metrics, a more comprehensive understanding of the algorithm's strengths and limitations in different dynamic and uncertain environments can be obtained.

Considering the increasing prevalence of learning-based control methods, how can the insights from this paper on robust online optimization be leveraged to develop data-driven control algorithms with provable guarantees?

The insights from this paper on robust online optimization can be instrumental in developing data-driven control algorithms with provable guarantees, especially as learning-based methods become increasingly prevalent. Here's how:

Learning for Robust Constraint Tightening:

Data-driven disturbance modeling: Leverage historical data or online measurements to learn accurate and adaptive models of the disturbances (wt, vt). Techniques like Gaussian Processes or Recurrent Neural Networks can capture complex disturbance patterns.
Data-driven RPI set estimation: Use collected data to estimate robust positively invariant (RPI) sets directly from system trajectories, relaxing the need for precise system models. This can be achieved using techniques like set-membership identification or scenario optimization.

Safe Exploration in Online Learning:

Safe policy updates: Integrate the robust online optimization framework into reinforcement learning algorithms to ensure safe exploration during the learning process. The constraint tightening mechanism can prevent the agent from taking actions that could lead to constraint violations, even when the learned model is inaccurate.

Robustness to Model Uncertainty:

Uncertainty-aware learning: Incorporate the concept of robust optimization into the learning process to develop controllers that are robust to model uncertainties. This can involve training the learning agent on a distribution of possible models or using robust loss functions that penalize sensitivity to model errors.

Combining Model-Based and Data-Driven Approaches:

Hybrid control architectures: Develop hybrid control architectures that combine the strengths of model-based and data-driven approaches. For instance, use a model-based controller with robust online optimization for safety-critical aspects and a data-driven controller for performance optimization in less critical regions.

Provable Guarantees for Learned Controllers:

Verification of learned models: Apply formal verification techniques to provide provable guarantees for the learned components of the control system, ensuring that the overall system remains safe and reliable.

By integrating these insights into the development of data-driven control algorithms, it becomes possible to leverage the power of learning while retaining the safety and robustness guarantees offered by model-based techniques. This synergy is crucial for deploying learning-based control systems in real-world applications where safety and reliability are paramount.