Optimal Distributionally Robust Control for Linear Quadratic Gaussian Systems under Distributed Uncertainty

Основные понятия

The paper proposes a new paradigm for the robustification of the LQG controller against distributional uncertainties on the noise process. The controller optimizes the closed-loop performance in the worst possible scenario under the constraint that the noise distributional aberrance does not exceed a certain threshold limiting the relative entropy between the actual noise distribution and the nominal one. The key novelty is that the bounds on the distributional aberrance can be arbitrarily distributed along the whole disturbance trajectory.

Аннотация

The paper introduces the Distributed uncertainty Distributionally robust LQG (D2-LQG) control problem, where the distributional uncertainty is modeled using relative entropy constraints at each time step. This is in contrast to previous work that considered a single constraint on the overall noise distribution.

The authors first analyze the worst-case performance problem in the absence of control input. They show that the solution takes the form of a risk-sensitive cost with a time-varying risk-sensitive parameter. This provides valuable insight into the nature of the worst-case model.

The authors then derive the solution to the D2-LQG control problem using dynamic programming and Lagrange duality. The optimal control policy is shown to have a structure similar to the standard linear quadratic regulator, but with a time-varying risk-sensitive parameter. A coordinate descent algorithm is proposed to numerically compute the optimal control gains.

Finally, the authors provide an explicit characterization of the worst-case noise distribution that the D2-LQG controller must counteract.

Статистика

The paper does not contain any explicit numerical data or statistics. The results are presented in a theoretical framework.

Цитаты

"The main novelty is that the bounds on the distributional aberrance can be arbitrarily distributed along the whole disturbance trajectory."
"The structure of the resulting D2-LQG controller does resemble the one of the standard linear quadratic regulator obtained in absence of noise and uncertainty. While the structure of the controllers may appear similar, the arguments needed to derive the solution in the D2-LQG case are much more complex."

Ключевые выводы из

Distributionally Robust LQG control under Distributed Uncertainty

by Lucia Falcon... в arxiv.org 09-18-2024

https://arxiv.org/pdf/2306.05227.pdf

Дополнительные вопросы

How can the proposed D2-LQG framework be extended to handle model uncertainties in the system dynamics in addition to the noise distribution?

The D2-LQG framework can be extended to accommodate model uncertainties in the system dynamics by incorporating additional ambiguity sets that account for variations in the system parameters. This can be achieved by defining a robust control policy that not only considers the distributional uncertainties of the noise but also includes uncertainties in the system matrices (A and B). One approach is to introduce a set of perturbed system dynamics, where the actual system dynamics are modeled as a function of the nominal dynamics plus a bounded perturbation.
To implement this, one could define a new ambiguity set for the system dynamics, similar to the noise distribution ambiguity set, which constrains the deviations of the system matrices from their nominal values. This would involve formulating a dual optimization problem that simultaneously minimizes the worst-case cost over both the noise distributions and the model uncertainties. By applying techniques from robust control theory, such as structured uncertainty modeling and robust optimization, the D2-LQG controller can be designed to maintain performance even when the system dynamics are subject to significant variations.

What are the potential applications of the D2-LQG controller in real-world control systems beyond the linear quadratic Gaussian setting?

The D2-LQG controller has a wide range of potential applications in real-world control systems beyond the traditional linear quadratic Gaussian setting. Some notable applications include:

Aerospace Systems: The D2-LQG framework can be utilized in the control of aircraft and spacecraft, where uncertainties in wind disturbances and model parameters are prevalent. The ability to distribute uncertainty across the trajectory allows for more realistic and robust control strategies in dynamic flight conditions.

Robotics: In robotic systems, the D2-LQG controller can enhance performance in environments with uncertain dynamics, such as varying terrain or unexpected obstacles. The distributed uncertainty approach allows robots to adapt their control strategies in real-time, improving navigation and task execution.

Automotive Control: The D2-LQG framework can be applied to advanced driver-assistance systems (ADAS) and autonomous vehicles, where uncertainties in sensor measurements and environmental conditions (e.g., road surface variations, weather) can significantly impact vehicle dynamics and safety.

Manufacturing Systems: In automated manufacturing processes, the D2-LQG controller can optimize the performance of multi-input and multi-output systems under uncertainties in machine behavior and external disturbances, leading to improved efficiency and reduced downtime.

Energy Systems: The D2-LQG controller can be employed in smart grid applications, where uncertainties in energy demand and supply (e.g., renewable energy sources) need to be managed effectively to ensure stability and reliability in power distribution.

Can the ideas developed in this paper be applied to other robust control problems beyond the LQG framework, such as robust model predictive control?

Yes, the ideas developed in the D2-LQG framework can be effectively applied to other robust control problems, including robust model predictive control (MPC). The core principles of distributionally robust optimization and the use of ambiguity sets based on relative entropy can be integrated into the MPC framework to enhance its robustness against uncertainties.
In robust MPC, the optimization problem typically involves predicting future states and control actions while accounting for uncertainties in the system dynamics and disturbances. By adopting a similar approach to the D2-LQG framework, one can define ambiguity sets for both the noise distributions and the model parameters within the MPC context. This allows for the formulation of a robust optimization problem that minimizes the worst-case cost over a range of possible scenarios.
Furthermore, the dynamic programming techniques and duality principles utilized in the D2-LQG framework can be adapted to derive efficient algorithms for solving robust MPC problems. This would enable the design of control policies that are not only optimal under nominal conditions but also resilient to a variety of uncertainties, thereby improving the overall performance and reliability of control systems in practical applications.