Optimal Control of Multi-Agent Systems with Processing Delays

TL;DR

The paper tackles optimal control of a heterogeneous network of continuous-time LTI agents connected by a fixed directed graph with processing delays. It develops a convex Youla-parameterization-based framework that handles delay through a loop-shifting transformation and yields explicit, closed-form solutions for the $\mathcal{H}_2$-optimal decentralized controller, along with its cost, under $LQG$ assumptions. The results include both delay-free and delayed cases, with an agent-level implementation that exhibits an observer-regulator separation and uses finite-dimensional LTI and FIR components, enabling scalable, low-memory deployments. A thorough cost characterization is provided, decomposing the impact of centralization, decentralization, and delays, and synchronization demonstrations illustrate practical performance trends across network topologies. The framework offers exact solutions, interpretability via an observer-regulator structure, and practical guidelines for distributed deployment in multi-agent systems with known processing delays.

Abstract

In this article, we consider a cooperative control problem involving a heterogeneous network of dynamically decoupled continuous-time linear plants. The (output-feedback) controllers for each plant may communicate with each other according to a fixed and known transitively closed directed graph. Each transmission incurs a fixed and known time delay. We provide an explicit closed-form expression for the optimal decentralized controller and its associated cost under these communication constraints and standard linear quadratic Gaussian (LQG) assumptions for the plants and cost function. We find the exact solution without discretizing or otherwise approximating the delays. We also present an implementation of each sub-controller that is efficiently computable, and is composed of standard finite-dimensional linear time-invariant (LTI) and finite impulse response (FIR) components, and has an intuitive observer-regulator architecture reminiscent of the classical separation principle.

Figures (12)

• Figure 1: Directed graph representing five interconnected systems.
• Figure 2: The loop-shifting approach mirkin2009loopmirkin2012h2mirkin2011dead transforms a loop with adobe input delay (top left) into a modified system involving a rational plant $\tilde{\mathcal{P}}$ and FIR blocks $\Pi_u$ and $\Pi_b$ (right). This transformation $\Gamma$ is defined in \ref{['sec:appendix_gamma']}. We can recover $\mathcal{K}$ from $\tilde{\mathcal{K}}$ via the inverse transformation (bottom left).
• Figure 3: Agent-level implementation of all structured stabilizing controllers, parameterized by $\hat{\mathcal{Q}}\in\mathcal{H}_\infty\cap\mathcal{S}_0$. Here, $F_d$ is any block-diagonal matrix such that $A_{ii} + B_{2_{ii}} F_{d_{ii}}$ is Hurwitz. The $\mathcal{H}_2$-optimal controller is achieved when $\hat{\mathcal{Q}}=0$, and results in the simplified diagram of \ref{['fig:agent-level-opt']}. The blocks that depend on the processing delay $\tau$ are colored in green. All symbols are defined in \ref{['thm:6']}.
• Figure 4: Agent-level implementation of the $\mathcal{H}_2$-optimal controller with processing delays. This is the result of setting $\hat{\mathcal{Q}}=0$ in \ref{['fig:agent-level-subopt']}. The blocks that depend on the processing delay $\tau$ are colored in green. All symbols are defined in \ref{['thm:6']}.
• Figure 5: Four-agent chain graph with standard broadcast (top) and efficient immediate-neighbor implementation (bottom), which is possible because this graph is a multitree.

Theorems & Definitions (22)

• Definition 2: Riccati assumptions
• Lemma 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Theorem 8
• Remark 9
• Remark 10
• Remark 11