Distributed MPC for Multi-Agent Systems

• Distributed Model Predictive Control (DMPC) is a decentralized approach that partitions a global control problem into local subproblems solved by individual agents with neighbor communication.
• It employs methods like ADMM and dSQP to handle both convex and nonconvex constraints, ensuring real-time performance through efficient local computations.
• Practical implementations, such as multi-robot formations, demonstrate DMPC’s capability for near-zero tracking errors, scalability, and robustness even under complex dynamic constraints.

Distributed Model Predictive Control (DMPC) schemes coordinate the dynamic behavior of spatially or functionally partitioned systems by distributing the solution of an overall optimal control problem among multiple agents or subsystems. The central feature is that coupled, potentially high-dimensional constrained optimal control problems are solved in a manner fully compatible with distributed computation and neighbor-to-neighbor communication, enabling scalable, real-time deployment in networked systems such as multi-robot formations, vehicle platoons, and large-scale industrial processes.

1. Mathematical Formulation and System Structure

DMPC is based on the decomposition of a global optimal control problem for a multi-agent or multi-subsystem network. Each agent $i$ (out of $S$ agents or robots) is described by a local, typically discrete-time, dynamical model: $$x^k_i \in \mathbb{R}^{n_{x,i}}, \qquad u^k_i \in \mathbb{R}^{n_{u,i}}$$ For example, in omnidirectional mobile robot formation control, the dynamics are Euler-forward double-integrators: $x^{k+1}_i = x^k_i + \Delta t\, u^k_i$ Aggregating all local states and inputs yields the global variables $x = \mathrm{col}(x_1, \ldots, x_S)$, $u = \mathrm{col}(u_1, \ldots, u_S)$.

The global finite-horizon optimal control problem (OCP) is typically formulated as: $$\min_{\{x_i^k,\ u_i^k\}}\, \sum_{i=1}^S \left[ \sum_{k=0}^{N-1} \ell_i(x^k, u_i^k) + V_{f,i}(x^N) \right]$$ subject to: $$x_i^{k+1} = f_i^d(x_i^k, u_i^k),\quad x_i^k \in \mathcal{X}_i,\quad u_i^k \in \mathcal{U}_i,\quad (x_i^k, x_j^k) \in \mathcal{X}_{ij}\ \forall j\neq i$$ Here, local costs and stage/terminal penalties can embed agent-to-agent coupling through terms such as quadratic penalties on deviations from neighboring states, and coupling constraints often enforce collision avoidance or formation-keeping.

DMPC reformulates this OCP as a partially separable nonlinear program where each agent maintains local decision variables and copies of relevant neighbors’ trajectories, making neighbor-to-neighbor communication sufficient for distributed solution (Stomberg et al., 2023).

2. Distributed Optimization Algorithms

Two algorithmic frameworks are featured prominently:

ADMM (Alternating Direction Method of Multipliers):

A consensus-form augmented Lagrangian is written so that, after introducing auxiliary variables $\bar z_i$ and consensus constraints, the global problem reads: $$\min_{z_i \in \mathcal{Z}_i,\, \bar z_i} \sum_{i=1}^S f_i(z_i)\quad \text{s.t.}\quad z_i - \bar z_i = 0,\quad \sum_{i=1}^S E_i \bar z_i = 0$$ The distributed ADMM iterations per agent are:

• Solve local subproblem for $z_i^{l+1}$
• Consensus/averaging step for $\bar z_i^{l+1}$ via neighbor broadcasts
• Dual variable update $\gamma_i^{l+1}$ This approach allows all communication and computation to be strictly local.

dSQP (Decentralized Sequential Quadratic Programming):

For nonconvex OCPs (e.g., those with quadratic collision-avoidance penalties), dSQP linearizes constraints at each SQP step and solves the resulting convex QP approximately using a fixed number of inner ADMM iterations. This achieves local convergence to a KKT point under regularity conditions and relies on neighbor trajectory exchanges analogous to ADMM (Stomberg et al., 2023).

The main technical synthesis enabling full distribution is the reformulation of global coupling constraints and cost terms using local copies/estimates and carefully designed communications of primal (trajectory) and dual (multiplier) variables. The information exchange per ADMM or ADMM-in-dSQP inner iteration is only the current state-trajectory over each neighbor link.

3. Theoretical Analysis: Convergence and Stability

Convergence Properties:

• For convex quadratic problems, global convergence of ADMM and dSQP to the centralized OCP optimum is guaranteed.
• For nonconvex formulations, dSQP achieves local q-linear or superlinear convergence under SQP regularity assumptions (LICQ, strict complementarity, positive-definite Hessian on tangent space), given a residual-based stopping rule (Stomberg et al., 2023).

Closed-Loop Feasibility and Stability:

• While standard DMPC theory establishes recursive feasibility and asymptotic or exponential stability using terminal invariant sets and penalties, many practical implementations (including (Stomberg et al., 2023)) use constraint softening to guarantee feasibility, and reference existing stabilization frameworks for rigorous proofs.
• Convergence of distributed optimization within a fixed iteration budget, combined with warm-started variables, delivers practical, real-time instability-resilient performance observed empirically.

4. Practical Implementation and Real-Time Performance

Implementation of DMPC using the above principles was demonstrated for formation control of four omnidirectional mobile robots (Stomberg et al., 2023). Key features and results include:

• Each robot executes the DMPC solver on a workstation (C++ code, Eigen, qpOASES for QPs, CasADi for autodiff, LCM for messaging) at 5 Hz.
• Robot states are updated via pose estimates streamed from a motion-capture system.
• Only neighbor-to-neighbor UDP multicasts of trajectory data are required.
• For linear-quadratic cases (ADMM), each 200 ms sampling step required median 6.6 ms (worst 22.3 ms), dominated by communication overhead; dSQP for nonconvex cases required ~34 ms. Each agent met the real-time deadline.
• Communication cost scales with each agent’s neighbor count, while local QP solution cost scales with prediction horizon and state/input dimension.

Empirical control performance:

• Linear formation-keeping tasks achieved nearly-zero tracking error, with DMPC control input matching centralized reference up to a small residual.
• During nonconvex maneuvers enforcing minimum inter-robot distances, transient DMPC inputs deviated up to 0.05 m/s from centralized offline solutions but converged tightly in steady state.
• Warm-starts were critical for iteration efficiency and deadline adherence.

Scalability:

The implementation shows that communication and computation scale favorably with the locality of coupling, neighbor set size, and prediction horizon, allowing larger swarms with modest hardware resources.

5. DMPC in Broader Context

The distributed approach described enables fully decentralized real-time MPC for tightly-coupled, spatially distributed systems without reliance on a master coordinator. Compared to centralized MPC, this yields:

• Improved scalability, as each agent only solves a local subproblem and exchanges information with neighbors.
• Robustness to agent-wise computation heterogeneity and varying message delays.
• Flexibility to incorporate both convex and nonconvex state and input constraints using appropriate optimization backends.

This agent-level perspective makes DMPC applicable beyond formation control to generic networked systems wherever sparse agent-to-agent coupling and communication locality can be exploited.

6. Limitations and Future Directions

While the described scheme robustly demonstrates real-world real-time DMPC, several practical and theoretical challenges persist:

• The stability and recursive feasibility guarantee depends on integrating rigorous terminal constraint/cost synthesis, which may require nontrivial distributed invariant set computations, especially for nonconvex, nonlinear, or hybrid agent models.
• Scalability to very large networks will benefit from further communication protocol optimization and possibly hierarchical or event-triggered message scheduling.
• Extensions to asynchronous, unreliable, or delayed communication, as well as cyber-physical security and privacy, are further topics for DMPC research adopting similar decentralized optimization architectures.

• Cooperative Distributed MPC via Decentralized Real-Time Optimization: Implementation Results for Robot Formations (Stomberg et al., 2023)