Distributed Nonlinear MPC (DNMPC)

• Distributed Nonlinear Model Predictive Control (DNMPC) is a decentralized strategy that decomposes complex nonlinear control problems into locally solvable optimization tasks with coordinated neighbor interactions.
• It employs methodologies like ADMM-based decomposition, sensitivity analysis, and sampling-based approaches to handle coupling constraints and ensure real-time performance.
• DNMPC has practical applications in automotive platooning, UAV formations, and satellite constellation control, offering enhanced robustness and scalability.

Distributed Nonlinear Model Predictive Control (DNMPC) encompasses a family of control methodologies for coordinating multiple agents with nonlinear dynamics under constraints, in a distributed manner that avoids centralized computation and single-point failure. DNMPC schemes address the synthesis of locally executable, coordinated control actions through repeated, local nonlinear optimal control problems, often with communication and algebraic coupling to neighbor agents. The design and analysis of DNMPC leverage constrained optimization, nonlinear systems theory, distributed algorithms, and networked systems analysis, with robustness, real-time feasibility, and scalability as central objectives.

1. Fundamental Problem Structure

A generic DNMPC problem considers a network of $N$ agents, each described by (possibly nonlinear) discrete or continuous-time dynamics $$x_i(t+1) = f_i(x_i(t), u_i(t), x_{\mathcal{N}_i}(t))$$ where $x_i$ is the state, $u_i$ is the control, and $x_{\mathcal{N}_i}$ are the states of agent $i$'s neighbors in a communication or interaction graph. Each agent is subject to individual state and input constraints, and possibly coupling constraints such as collision-avoidance, network connectivity, or consensus.

At each control step, each agent $i$ solves a local finite-horizon optimal control problem: $$\min_{u_i(\cdot), z_i(\cdot)} \quad J_i(x_i(\cdot), u_i(\cdot), z_i(\cdot), x_{\mathcal{N}_i}(\cdot), u_{\mathcal{N}_i}(\cdot), t)$$ subject to

$$\begin{aligned} \ x_i(k+1) &= f_i(x_i(k), u_i(k), x_{\mathcal{N}_i}(k)), \ \ (x_i(k), u_i(k)) &\in \mathcal{Z}_i, \quad \forall k, \ \ z_i(\cdot)&\in \mathcal{C}_i(x_{\mathcal{N}_i}(\cdot), u_{\mathcal{N}_i}(\cdot)), \end{aligned}$$

where $z_i$ collects auxiliary decision variables or artificial references, and $\mathcal{C}_i$ encodes coupling constraints.

Coordination is achieved by iterative optimization and (local) exchange of planned trajectories or references. Multiple architectures exist for decomposing and solving the coupled multiple shooting or direct transcription problems induced by multi-agent dynamics and constraints, including parallel, sequential, and hierarchical schemes (Bestler et al., 2017, Köhler et al., 2023, Köhler et al., 31 Mar 2025, Zhang et al., 2020, Yoon et al., 21 Oct 2025).

2. Architectures and Algorithmic Paradigms

2.1 ADMM-Based Primal/Dual Decomposition

A dominant paradigm for distributized solution of DNMPC is the use of the alternating direction method of multipliers (ADMM) for decomposing the coupled trajectory optimization (Burk et al., 2020, Bestler et al., 2017, Yoon et al., 21 Oct 2025). Each agent introduces local copies of its neighbors' trajectories or decision variables and enforces consensus through penalized quadratic terms and dual variable updates. The distributed ADMM iteration per sampling step includes:

• Local nonlinear OCP solves with augmented Lagrangian penalties,
• Closed-form or averaged updates of consensus (global) variables,
• Dual (multiplier) ascent enforcing agreement between local and global variables.

ADMM iterations can be terminated suboptimally (bounded maximal number or residual-based stopping), yielding a practical tradeoff between solution quality, communication, and computation (Bestler et al., 2017).

2.2 Sensitivity-Based and Sequential Coordination

Sequential schemes use Gauss–Seidel or sensitivity-based updates, where agents solve their OCPs in a predetermined order, using the latest plans or linearizations received from their neighbors (e.g., as in tractor-trailer DNMPC (Kayacan et al., 2021)). Sensitivity-based parallel algorithms leverage first-order sensitivities of neighbors' cost or Hamiltonian to inform augmented terms in an agent's OCP, achieving contraction under small horizon assumptions (Esch et al., 5 Jun 2024).

2.3 Sampling-Based and Stochastic Search

Non-convex DNMPC problems, particularly at scale, may leverage sampling-based (stochastic search) optimization subroutines to navigate highly non-convex, multi-modal landscapes (such as multi-robot formation with collision avoidance) (Yoon et al., 21 Oct 2025). Techniques include variants of model predictive path integral control (MPPI), cross entropy method (CEM), and Tsallis (q-exponential) sampling, parameterized via shape functions and importance weights.

2.4 Hierarchical and Barrier-Augmented Schemes

For nonconvex and highly coupled DNMPCs, hierarchical ADMM (combining outer Augmented Lagrangian and inner multi-block proximal ADMM) can be used. Outer loops enforce primal feasibility, while inner ADMM solves three-block decomposed updates (decision, bound-projection, slack variable). Barrier methods accelerate inner solves and ensure fast, scalable (to $N = 8+$ agents) real-time feasibility for tasks like UAV formation with collision and box constraints (Zhang et al., 2020).

3. Artificial Reference, Cooperative Task Encoding, and Stability Guarantees

A central recent trend is the use of artificial reference variables ("editor's term")—auxiliary, typically $T$-periodic, agent-level outputs (such as $y_{T,i}$) which agents optimize in addition to their own input trajectories (Köhler et al., 31 Mar 2025, Köhler et al., 2023). These references encode the cooperative goal (e.g., periodic synchronization, formation, consensus) and are penalized for deviation from neighbor references in the cost function.

The DNMPC problem then contains:

• Local tracking cost for following the artificial reference,
• Penalties for the change (Δ) of artificial targets between steps,
• Cooperative costs coupling agent artificial outputs (such as in consensus, formation shape, periodicity).

Stability, feasibility, and performance guarantees are established under:

• Assumptions of convexity, regularity, and existence of local terminal ingredients (terminal cost, set, and feedback to ensure contraction or dissipation),
• Recursive feasibility via tail concatenation or convex combination with previous feasible plans (Chanfreut et al., 14 Oct 2024, Köhler et al., 31 Mar 2025),
• Monotonic decrease of global cost across the network (Chanfreut et al., 14 Oct 2024),
• Performance bounds showing closed-loop cost approaches the centralized open-loop solution with appropriate horizon scaling (Köhler et al., 31 Mar 2025).

4. Coupling Constraints, Robustness, and Real-World Integration

4.1 Nonconvex and State/State-Coupling Constraints

DNMPC encompasses systems with state and input constraints local to agents, and global constraints such as collision avoidance (distance, position overlap), communication connectivity, and formation shape. Techniques for handling nonconvexities include indicator penalties, barrier functions, and explicit projection (Zhang et al., 2020, Nguyen et al., 23 Mar 2024, Katriniok et al., 2019).

Robustness to bounded disturbances and uncertainties is addressed via tube-based schemes (with invariant sets), constraint tightening, and Moving Horizon Estimation (MHE) for online parameter adaptation, as in tractor-trailer systems subject to wheel slip (Kayacan et al., 2021).

4.2 Control Barrier Functions (CBF) and Connectivity Maintenance

DNMPC has been augmented with control barrier functions for safety set enforcement and control Lyapunov functions for stability constraints, notably in underwater multi-agent formation with explicit network connectivity constraints (Nguyen et al., 23 Mar 2024). The resultant nonlinear OCPs encode exponential decrease of Lyapunov-like functions and safety set maintenance.

4.3 Fault-Resilient DNMPC

Adaptive distributed observers, which integrate estimation of leader agent states and dynamics as well as relative formation offsets, are used to decouple global formation tasks into fully local control subproblems and provide resilience to communication link faults (Xu et al., 31 Oct 2024). The observer layer guarantees that local reference trajectories converge to the desired collective trajectory, even as communication topology experiences faults or drops.

5. Practical Implementation and Real-Time Feasibility

5.1 Parallelism, Communication, and Embedded Deployment

Practical, real-time implementation of DNMPC is documented for automotive, robotic, power grid, and aerospace systems, with distributed agents deployed across parallel CPUs or embedded systems (Raspberry Pi, PLCs) (Burk et al., 2020, Kayacan et al., 2021, Ozkan et al., 2021). Communication architectures typically require only neighbor-to-neighbor exchange per iteration. Plug-and-play features allow agents to join and leave dynamically in frameworks such as GRAMPC-D (Burk et al., 2020).

5.2 Scalability and Complexity

Empirical results show that distributed primal/dual methods (ADMM, sensitivity-based) and sampling-based solvers scale near-linearly in $N$ (number of agents), with local per-agent optimization time remaining in the millisecond range for up to tens or hundreds of agents, conditional on the complexity of local dynamics and constraint sets (Yoon et al., 21 Oct 2025). Warm-starting and neighbor approximation further reduce iteration count and computation time.

Sampling-based distributed DNMPC demonstrably succeeds on strongly nonconvex tasks (e.g., 64-agent Dubins vehicle formation) where local optimization solvers fail to find feasible (collision-free) solutions or get stuck in local minima (Yoon et al., 21 Oct 2025).

5.3 Performance Benchmarks

Documented tracking errors for DNMPC can reach sub-decimeter accuracy in fielded vehicles (autonomous tractor-trailer: ≈3 cm straight-line, ≈30 cm curved) (Kayacan et al., 2021), fuel efficiency gains of up to 18% in heavy-duty truck platooning compared to DMPC baselines (Ozkan et al., 2021), and string-stability for platoons and formation flying (Ozkan et al., 2021, Basiri et al., 2020).

6. Extensions: Heterogeneity, Dissimilar Horizons, and Learning

6.1 Heterogeneous and Time-Varying Horizons

Dissimilar and time-varying prediction horizons across agents enable adaptation to heterogeneous computation resource or plant timescale, reducing per-agent computation time while maintaining recursive feasibility and cost decrease (Chanfreut et al., 14 Oct 2024). Agents may prune their horizon during runtime with negligible performance loss, which is particularly relevant in large-scale or resource-constrained deployments.

6.2 Incorporation of Learning

Formulations enabling distributed metric learning of the weight matrices in local DNMPC cost functions, via ADMM consensus constraints, allow the distributed co-design of not just control actions but also the tradeoff structure in the multi-objective cost, improving both fuel economy and comfort in vehicle platooning tasks (Basiri et al., 2020).

6.3 Artificial Reference Synthesis

Artificial reference trajectories synthesizing cooperative goals (periodic behaviors, dynamic formations, flocking) are now used both for constraint separation and to support hierarchical or emergent cooperative tasks. DNMPC schemes dynamically generate and update these references, rather than track externally provided setpoints, enabling emergent cooperative behavior without a master coordinator (Köhler et al., 31 Mar 2025, Köhler et al., 2023).

7. Application Domains and Case Studies

DNMPC methodologies are applied across domains:

• Automotive: heavy-truck and heterogeneous vehicle platooning, robust to cut-in/cut-out maneuvers, with dynamic topology adaptation (Ozkan et al., 2021, Basiri et al., 2020).
• Robotics and UAVs: multi-agent formation, obstacle avoidance, narrow-passage traversal, and reconfiguration under connectivity and collision constraints (Yoon et al., 21 Oct 2025, Nguyen et al., 23 Mar 2024, Zhang et al., 2020).
• Aerospace: satellite constellation formation and reconfiguration on discrete orbital graphs (Köhler et al., 31 Mar 2025).
• Traffic: cooperative intersection crossing under collision avoidance and logical (stop/wait) constraints, solved via fast solvers such as PANOC (Katriniok et al., 2019).
• Power and process systems: dynamic traffic management, smart grid control, and water network operations with heterogeneous agent horizons and plug-and-play support (Chanfreut et al., 14 Oct 2024, Burk et al., 2020).

In all cases, DNMPC frameworks support decentralized, scalable, constraint-aware, and often real-time control under nonlinear multi-agent dynamics, accommodating both cooperative goals and hard operational limits. The field continues to evolve towards higher scalability, less restrictive inter-agent communication, integration of uncertainty and learning, and resilience to network and estimation faults.