Multi-Agent Model Predictive Control

• Multi-Agent Model Predictive Control is a distributed control method that decomposes centralized problems into local receding-horizon optimizations performed by multiple agents.
• It employs decomposition strategies like top-down and bottom-up approaches to manage inter-agent couplings and reduce computational complexity.
• Cooperative coordination protocols, including iterative schemes and game-theoretic strategies, ensure system-wide feasibility, robustness, and optimal performance.

Multi-Agent Model Predictive Control (MPC) refers to a class of closed-loop control methodologies in which multiple autonomous decision-making agents cooperatively or competitively regulate the behavior of large-scale, interconnected dynamical systems. Each agent, associated with a subsystem or physical actuator, solves a local receding-horizon optimization problem that accounts for the evolution of its own state and input over a prediction window, subject to operational and safety constraints as well as inter-agent coupling. This paradigm generalizes classical single-agent (centralized) MPC by introducing system decomposition, iterative coordination protocols, and distributed computation—enabling scalability, modularity, and robustness in the control of complex engineered networks.

1. Transition from Single-Agent to Multi-Agent MPC

Classical single-agent MPC employs a centralized model encompassing the entire plant, typically with difference or differential equations under full-state observability and global cost minimization $J = \sum_{k=0}^{N_p} \ell(x_k, u_k)$. In contrast, multi-agent MPC decomposes the plant into several subsystems—each modeled as a node in a distributed or hierarchical architecture—where agents have access to partial models and partial state information. Local objectives and constraints are developed to respect both subsystem-level and global requirements, and each agent is responsible for a smaller planning problem whose outcome contributes to the overall system goal.

Key differentiators:

Single-Agent MPC	Multi-Agent MPC
Centralized system model	Distributed/hierarchical model
Monolithic objective function	Aggregated/hierarchical objectives
Single optimization agent	Network of coordinating agents
Global state and constraint access	Local, partial state/constraint info

This structural shift enables computational tractability in large-scale systems but requires additional coordination mechanisms to handle interdependencies.

2. Decomposition Strategies

Two primary decomposition paradigms organize how a centralized control problem is disassembled for multi-agent MPC:

A. Top-Down Decomposition:

The overall system is first modeled in aggregate. Engineering analysis, such as system-theoretic invariances or physical coupling identification, informs partitioning the model into weakly or strongly coupled subsystems. Examples include analytical decomposition of large-scale linear systems seen in the works of Motee/Sayyar-Rodsari and Katebi/Johnson, where physical couplings are explicit.

B. Bottom-Up Decomposition:

Subsystems are modeled independently, with interaction characterized subsequently. For example, water network sections are modeled separately and their compatibility enforced via Lagrangian multipliers (Georges, El Fawal et al.). The distinction between “decentralized” and “hierarchical” decomposition is whether subsystems are independent or interconnected across time scales or abstractions.

These strategies facilitate parallelization and allow each agent to solve a smaller, manageable optimization subproblem, significantly reducing computational burden, especially for real-time operation.

3. Problem Assignment: Agent–Subproblem Allocation

Problem assignment addresses how decomposed subproblems are mapped to specific agents, as well as alignment with agents' capabilities (sensing, actuation, computation):

• Frequently, a one-to-one mapping is pursued: each subsystem is managed by a unique agent controlling its local problem (as in Baglietto et al., Georges). Alternate architectures group agents, each responsible for a group of interlinked subsystems or for distinct hierarchical layers (e.g., Dunbar and Murray’s formation control).
• Offline partitioning is common for scalability, but research into optimal partitioning (e.g., Motee/Sayyar-Rodsari's work on agent grouping) attempts to balance actuation cost and minimize inter-agent communication.
• Coupling between subproblems (e.g., shared state variables or coupling constraints) necessitates that an agent’s optimization must be iteratively updated in response to neighbors’ intended actions, often using iterative best-response or equilibrium concepts.

Assignment complexity grows with system size and coupling tightness—a persistent challenge for multi-agent MPC design.

4. Cooperative Coordination and Decision Protocols

Cooperation among agents is critical for the feasibility, optimality, and stability of multi-agent MPC. The surveyed literature details a wide spectrum of coordination mechanisms:

• Iterative Schemes: Agents iteratively solve their local MPC, update their predicted trajectories, and exchange information (state predictions, future inputs, constraints) with neighbors. This occurs via synchronous (serial, parallel) or asynchronous protocols, each with distinct convergence, responsiveness, and communication implications.
• Game-Theoretic Constructs:

The notion of reaction sets is used to define agent strategies: the intersection of agents' reaction sets characterizes Nash equilibria (mutually consistent, self-optimizing decisions). Pareto sets are also referenced as alternative solution benchmarks for efficiency across agents' objectives.

• Decision Mechanisms:
  • Exclusive (independent) decision-making, where individual agents act on their local solution.
  • Shared/democratic decisions, using voting or consensus to select actions.
  • Auction/bidding mechanisms for resource-constrained, competitive situations.
• Constraint Adaptation and Learning Integration:

Agents adapt their own constraints—tightening or relaxing them—to better align with neighbors and system-wide goals. Some protocols incorporate learning elements (such as adaptive constraint and model update through neural networks, as in Baglietto et al.) or online identification (dynamic adaptation using fresh measurements, per Georges and Katabi et al.).

• Communication Topology:

The underlying communication graph (fully connected, sparse, etc.) dictates the possible coordination mechanisms and rate of convergence toward global objectives.

These coordinated schemes ensure that local actions respect both subsystem objectives and the consistency/integrity of the overall system.

5. Mathematical Formulations for Coupling and Coordination

To address interactions and constraints, multi-agent MPC literature extends classical formulations:

• Lagrangian/Augmented Lagrangian:

Coupling constraints $g(u_1, ..., u_N)$ are enforced with dual variables ($\lambda$), yielding augmented objectives such as:

$$J_\text{total} = \sum_{i=1}^N J_i(u_i) + \lambda^T g(u_1, ..., u_N)$$

where $J_i(u_i)$ is the local cost and $g$ encodes coupling (e.g., shared resources, state consistency).

• Min-Max and Robust Formulations:

Robustness to uncertainty is implemented by designing objectives with a min-max structure (Jia and Krogh), ensuring constraint satisfaction under worst-case disturbance realizations.

• Hierarchical Architectures:

In hierarchical multi-agent MPC (Dunbar and Murray), lower layers handle fast, agent-local dynamics, and upper layers enforce slow-timescale formation or coordination constraints.

These mathematical tools provide a rigorously defined mechanism for encoding and reconciling multi-level objectives and constraints across agents.

6. Real-World Applications and Advancement Trends

Recent literature demonstrates multi-agent MPC in application domains including:

• Large-scale power and water distribution networks, exploiting distributed computation to overcome centralized intractability.
• Multi-vehicle and formation control, realizing coordinated behavior for autonomous fleets using coupled hierarchical objectives.
• Smart building climate control, where independent HVAC units act as agents within building-wide resource constraints.

Key advancement trends include:

• Movement from theoretical, strictly convex/differentiable convergence assumptions toward practical, heuristic, and learning-based strategies suitable for non-convex, uncertain real-world systems.
• Asynchronous and scalable algorithms that trade strict optimality for responsiveness and computational tractability.
• Integration of online adaptation, identification, and reinforcement learning for agents to refine models and constraints during operation.
• Formalization of conditions under which agent coordination guarantees overall system feasibility and stability, with explicit trade-offs between communication, computation, and control performance.

Notably, while mathematical guarantees (e.g., convergence, recursive feasibility) rely on restrictive assumptions, much current research seeks to relax these via novel distributed, adaptive, and learning-enhanced protocols.

7. Summary and Significance

Multi-agent MPC embodies a modular, scalable approach to optimal and resilient control of large-scale, interconnected dynamical systems. Its core distinctive features are:

• Systematic decomposition of centralized MPC problems into distributed, cooperative subproblems.
• Assignment and reconciliation of local and global objectives and constraints via communication and coordination protocols.
• Formal mathematical tools (augmented Lagrangian, game-theoretic equilibria, robust min-max) for encoding inter-agent dependencies.
• Architectural flexibility facilitating adaptation and robustness to uncertainty, communication losses, and agent reconfiguration.

While theoretical guarantees (e.g., convexity-based convergence, Nash optimality) are often strong, practical deployments typically require approximation, heuristic iteration, and adaptive learning. The field remains an active research area, with principal challenges being the design of efficient, robust cooperation mechanisms and assignment strategies under limited communication and uncertain environments (0908.1076).