Distributed Output-Feedback LQG Control with Delayed Information Sharing

Technical Summary: Distributed Output-Feedback LQG Control with Delayed Information Sharing

Problem Formulation
This paper addresses the challenge of designing optimal controllers for distributed Linear Quadratic Gaussian (LQG) systems subject to communication delays. While the state-feedback case for specific delay structures has been previously solved, the extension to output-feedback is nontrivial because the classical separation principle (which allows independent design of estimation and control) fails under general information constraints.

The authors focus on a system of three interconnected linear subsystems operating over a strongly connected graph. The information structure is defined such that:

1. Information from immediate neighbors is available with a one-step delay.
2. Global information (from all nodes) is available to all decision-makers with a two-step delay.
   The objective is to minimize a finite-horizon quadratic cost function subject to these delayed information constraints. The system dynamics are linear, and disturbances and measurement noises are modeled as correlated Gaussian white noises.

Methodology
The core methodology involves decomposing the complex distributed control problem into two tractable sub-problems by separating the state and control vectors into independent components based on information availability.

1. Estimation Algorithm: The authors first derive an optimal estimation algorithm for the three-player problem. Unlike the standard Kalman filter, which assumes access to all measurements up to time $k-1$, the proposed estimator utilizes:

   • A common estimate based on global information delayed by two steps ($y(0:k-2)$).
   • Local correction terms based on measurements available with one-step delay ($y_i(k-1)$ and neighbor measurements).
     This results in a modified state estimate $\hat{x}^{[1]}(k)$ that incorporates local innovations.

2. Controller Decomposition: The optimal control input $u(k)$ is decomposed into two independent parts:

   • Centralized Component: A term based on the global information set delayed by two steps, equivalent to a centralized LQG controller with two-step delay.
   • Correction Component: A sum of correction terms based on the additional local information available to each decision-maker (measurements at $k$ and $k-1$).

3. Optimization via Convex Programming: The authors demonstrate that finding the optimal gain matrices for the correction terms can be transformed into an unconstrained convex optimization problem. By utilizing the vectorization operator ($\text{vec}$) and a specific matrix construction to handle sparsity constraints, the problem is reduced to solving a set of recursive equations. This avoids the need for complex semi-definite programming where the controller structure is not explicit.

Key Contributions

• Explicit State-Space Realization: The paper provides an explicit state-space realization for the optimal output-feedback controller in a distributed setting with delays, a problem where the separation principle typically fails.
• Controller Structure: The optimal controller is shown to have the form:
  $$u(k) = F(k)(y(k) - C\hat{x}^{[1]}(k)) + F^{[1]}(k)(y(k-1) - C\hat{x}(k-1|k-2)) + L(k)\hat{x}(k)$$
  Here, $L(k)$ is the standard LQR gain computed via discrete-time Riccati recursion, while $F(k)$ and $F^{[1]}(k)$ are sparse gain matrices computed via convex programming that respect the information constraints.
• Handling Correlated Disturbances: Unlike prior work restricted to independent disturbances, this approach generalizes to correlated disturbances across subsystems.
• Computational Tractability: The solution reduces the design of the optimal controller to solving one standard discrete-time Riccati equation and one recursive equation, making the computation efficient.

Results
Numerical simulations on a three-player system with a time horizon of $N=1000$ compare the proposed controller against three benchmarks:

1. Centralized control with no delay.
2. Centralized control with a two-step delay.
3. A one-step delay sharing information pattern.

The results indicate that the proposed three-player controller significantly outperforms the centralized controller with a two-step delay. Furthermore, it achieves a cost only approximately 1.74% higher than the one-step delay sharing pattern, suggesting that for this specific topology, the benefit of two-way communication is marginal but the proposed method effectively utilizes the available delayed information.

Significance and Claims
The paper claims to provide a fundamental understanding of general delay structures in distributed LQG control. By successfully decomposing the controller into a centralized part and local correction terms, the authors bridge the gap between state-feedback solutions and the more complex output-feedback scenario. The work demonstrates that even with the failure of the separation principle, explicit and computationally efficient solutions can be derived for specific, non-trivial information structures. The authors note that this framework lays the groundwork for extending these results to infinite-horizon problems and more general network topologies.