Decentralized Feedback Optimization for Networked Systems: Stability and Sub-optimality Analysis

Core Concepts

The core message of this article is to propose a fully decentralized feedback optimization controller for networked systems that approximates the overall input-output sensitivity matrix through its diagonal elements, and to characterize the stability and sub-optimality of the closed-loop system.

Abstract

The article presents a decentralized feedback optimization approach for networked systems, where communication between agents is limited or undesirable. The key idea is to approximate the overall input-output sensitivity matrix by its diagonal elements, allowing each agent to update its control input based only on local information.
The authors first show that the stationary points of the proposed decentralized controller coincide with the Nash equilibria of an underlying convex game. They then conduct a comprehensive analysis of the closed-loop stability and sub-optimality, deriving sufficient conditions for stability and bounding the distance between the globally optimal solution and the stationary point to which the decentralized controller converges.
The analysis is performed for two cases: when the plant is represented by its steady-state input-output map, and when the plant is modeled as a linear time-invariant (LTI) system. In the former case, the sub-optimality bound depends on the degree of diagonal dominance of the sensitivity matrix and the properties of the objective function. In the latter case, the authors show that the coupled errors involving the distance to the plant's steady state and the distance to the controller's equilibrium point decay to zero with a linear rate, provided that the steady-state map of the LTI plant satisfies the diagonal dominance condition.
The theoretical results are illustrated through numerical simulations of a voltage control problem in a direct current (DC) power grid, demonstrating the effectiveness of the decentralized controller and the tightness of the derived sub-optimality bound.

Stats

The sensitivity matrix H = C(I - A)^-1 B + D.
The global objective function Φ(u, y) = Σ_i Φ_i(u_i, y_i), where Φ_i(u_i, y_i) is the local objective function for agent i.
The decentralized controller update is u_{k+1} = u_k - η (∇_u Φ(u_k, y_k) + H_diag ∇_y Φ(u_k, y_k)).

Quotes

"The stationary points of the decentralized controller (5), if they exist, equal to the Nash equilibria of the following convex game: ∀i, min_u_i Φ̃_i(u)."
"The sub-optimality of the controller (5) depends on the coupling degree H^T - H_diag and the properties of objective functions (e.g., ∇Φ^(2)(y) and m). The bound decreases as m increases."

Key Insights Distilled From

Decentralized Feedback Optimization via Sensitivity Decoupling

by Wenb... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2311.09408.pdf

Deeper Inquiries

How can the decentralized controller be extended to handle more general plant dynamics, such as nonlinear or time-varying systems

To extend the decentralized controller to handle more general plant dynamics, such as nonlinear or time-varying systems, several approaches can be considered. One option is to incorporate adaptive techniques that can adjust the decentralized control strategy based on the observed system behavior. This adaptation can involve updating the local sensitivity information dynamically or adjusting the decentralized control law to account for nonlinearities or time-varying dynamics.
Another approach is to introduce learning mechanisms within each agent to improve the approximation of the sensitivity matrix or to adapt the control inputs based on past experiences. Reinforcement learning or online learning algorithms can be utilized to enhance the performance of the decentralized controller in handling more complex plant dynamics. By continuously updating the local models and control strategies based on real-time data, the decentralized controller can become more robust and effective in optimizing the system's performance.
Furthermore, exploring distributed optimization techniques that can handle non-convex objective functions or constraints can also enhance the capabilities of the decentralized controller in dealing with general plant dynamics. By incorporating advanced optimization algorithms that can handle nonlinearity and time-varying dynamics, the decentralized controller can achieve better performance in optimizing the system's behavior.

What are the implications of the sub-optimality bound in terms of the practical trade-offs between centralized and decentralized control approaches

The sub-optimality bound derived in the context of decentralized control approaches has significant implications for practical trade-offs between centralized and decentralized control strategies.

Trade-off between Communication and Performance: The sub-optimality bound quantifies the cost of decentralization in terms of the distance between the solution obtained by the decentralized controller and the globally optimal solution. This provides insights into the trade-off between communication requirements for centralized control and the performance achieved by decentralized approaches. By understanding the sub-optimality bound, system operators can make informed decisions on the level of decentralization based on the acceptable performance trade-offs.

Scalability and Robustness: Decentralized control approaches are often more scalable and robust in large-scale systems where communication overhead or network constraints limit centralized control. The sub-optimality bound helps in evaluating the robustness of decentralized strategies and provides a measure of the performance degradation compared to centralized approaches.

Flexibility and Adaptability: Decentralized controllers offer flexibility and adaptability in dynamic environments where plant dynamics may change over time. The sub-optimality bound guides the design of decentralized strategies to balance adaptability with performance, enabling systems to operate efficiently under varying conditions.

In essence, the sub-optimality bound serves as a crucial metric for system designers and operators to assess the practical implications of choosing decentralized control approaches over centralized ones, considering factors such as communication costs, system complexity, and performance requirements.

Can the decentralized controller be further improved by incorporating additional local information or coordination mechanisms between agents

The decentralized controller can be further improved by incorporating additional local information or coordination mechanisms between agents to enhance its performance and efficiency. Some strategies to enhance the decentralized controller include:

Local Model Fusion: Agents can share summarized local model information to improve the accuracy of the sensitivity matrix approximation. By fusing local models or sharing learned information, agents can collectively build a more accurate representation of the system dynamics, leading to better control decisions.

Dynamic Communication: Introducing dynamic communication mechanisms where agents selectively exchange critical information based on the current system state can enhance the decentralized controller's performance. Adaptive communication strategies can optimize the information flow between agents, improving coordination without excessive communication overhead.

Distributed Learning: Incorporating distributed learning algorithms within the decentralized controller can enable agents to adapt their control strategies based on local observations and interactions. By allowing agents to learn from their experiences and adjust their behavior autonomously, the decentralized controller can become more adaptive and responsive to changing system conditions.

Hierarchical Control: Implementing hierarchical control structures where agents operate at different levels of decision-making can improve coordination and efficiency. By organizing agents into hierarchies based on their roles and responsibilities, the decentralized controller can achieve better coordination and optimization of the overall system.

By integrating these additional elements into the decentralized controller, the system can benefit from enhanced coordination, improved decision-making, and increased adaptability, leading to better overall performance in optimizing the system's behavior.