Network-Realised Model Predictive Control Part II: Distributed Constraint Management

1. Problem Statement

The paper addresses the challenge of distributed constraint management for interconnected dynamical systems (networked systems) subject to:

• Exogenous disturbances: Including process noise, measurement noise, and communication errors.
• Hard constraints: Limits on both state variables (e.g., vehicle spacing, speed) and control inputs.
• Scalability: The need for control laws that can be implemented in a distributed manner without requiring a central coordinator, which is often computationally intractable for large networks.

Specific Gap: Existing distributed Model Predictive Control (MPC) approaches often rely on static state-feedback laws with restrictive sparsity structures (block-diagonal), assume specific network topologies (e.g., bipartite), or require complex joint set-invariance calculations that hinder scalability. Furthermore, few methods effectively decouple the tasks of disturbance rejection/stabilization from constraint enforcement in a distributed setting.

2. Methodology: Two-Layer Control Architecture

The authors propose a novel two-layer control architecture that separates the control objectives into two distinct layers, leveraging the Network Realisation Function (NRF) framework introduced in their companion paper.

Layer 1: NRF-Based Distributed Subcontrollers (Stabilization & Disturbance Rejection)

• Role: Acts as the primary feedback controller. It handles reference tracking, disturbance rejection, and ensures closed-loop stability.
• Mechanism: Uses dynamic state-feedback controllers implemented via state-space realizations. These controllers are designed offline using NRF theory to ensure specific closed-loop properties (e.g., string stability) and to attenuate the propagation of disturbances across the network.
• Output: Generates a primary control signal $u_f$ (feedforward/feedback) based on local and neighbor state measurements.

Layer 2: Distributed MPC Subcontrollers (Constraint Management)

• Role: Acts as a Reference Governor and safeguard. It does not replace the first layer but modifies its inputs to ensure constraints are never violated.
• Mechanism:
  • It computes two auxiliary command signals, $u_{s1}$ and $u_{s2}$, which are added to the first layer's signals.
  • $u_{s1}$: Broadcasted to neighbors; primarily ensures the internal states of the first-layer subcontrollers remain within feasible sets.
  • $u_{s2}$: Local signal; primarily ensures the physical plant states and total control inputs satisfy global constraints.
• Optimization: Each area solves a local MPC problem with a receding horizon. The cost function minimizes the deviation from the first layer's nominal trajectory, ensuring the second layer intervenes only when necessary (i.e., when constraints are imminent).

Key Technical Innovations

1. Set-Theoretic Constraint Decomposition: The global constraint satisfaction problem is decomposed into local subproblems using Minkowski sums and Pontryagin differences. This allows each area to solve a local optimization problem while guaranteeing global constraint satisfaction.
2. Handling Uncertainty: The framework explicitly models measurement noise, communication errors, and initial condition inaccuracies as additive disturbances. It constructs robust invariant sets to ensure recursive feasibility despite these uncertainties.
3. Recursive Feasibility: The paper provides rigorous theoretical proofs that if the optimization problem is feasible at the initial time step, it remains feasible for all future steps, provided the first layer is properly tuned.
4. Offline Design vs. Online Execution: The complex set calculations and controller synthesis are performed offline. The online implementation involves only standard Quadratic Programming (QP) and simple matrix-vector operations, making it suitable for real-time embedded systems.

3. Key Contributions

• Decoupling of Control Tasks: Successfully separates the design of stabilizing controllers (Layer 1) from constraint enforcement (Layer 2). This allows the use of advanced NRF-based stabilization techniques without complicating the MPC design.
• Dynamic Feedback in Distributed MPC: Unlike many existing distributed MPC methods that rely on static gains, this approach utilizes dynamic state-feedback subcontrollers, offering greater flexibility and performance.
• Recursive Feasibility Guarantees: Provides a theoretical framework ensuring that the distributed MPC problems remain feasible indefinitely, even in the presence of bounded disturbances and communication noise.
• Scalable Implementation: The architecture allows for parallel design of local controllers. Once the "command budget" (constraint sets) is negotiated, each area's MPC design can proceed independently, avoiding the computational bottlenecks of joint invariance calculations.
• Minimal Intervention: The second layer is designed to be "transparent," intervening only when constraints are at risk, thereby preserving the high-performance characteristics of the underlying NRF-based controller.

4. Results (Numerical Example: Vehicle Platooning)

The methodology was validated using a 10-vehicle platoon simulation.

• Scenario: The platoon tracks a virtual lead vehicle with varying speeds, subject to communication noise and actuator limits.
• Performance:
  • Constraint Satisfaction: The system successfully maintained inter-vehicle distances and speed limits within prescribed bounds, preventing collisions and string instability.
  • Minimal Interference: The second-layer control signals ($u_{s1}, u_{s2}$) remained near zero (magnitude $\approx 10^{-6}$) for the majority of the simulation, only activating significantly when the first layer's nominal trajectory threatened to violate constraints (e.g., during rapid acceleration/deceleration).
  • Computational Efficiency:
    • Synthesis: The offline design time for the 10th vehicle was approximately 71 seconds (on a standard laptop), demonstrating tractability.
    • Runtime: The online MPC execution time was roughly 2 milliseconds per vehicle, well below the 100ms sampling period, confirming real-time viability.
• Comparison: The paper compares the results against other distributed MPC approaches (e.g., [31, 28, 6, 10, 27]), highlighting that those methods often lack robustness to disturbances, require impractical communication topologies (broadcasting to all), or sacrifice string stability to satisfy constraints.

5. Significance

This work represents a significant advancement in distributed control theory by bridging the gap between robust stabilization and constraint management.

• Practicality: It offers a solution that is theoretically rigorous (guaranteeing safety and feasibility) yet computationally lightweight enough for real-world deployment in large-scale networks (e.g., smart grids, autonomous vehicle fleets, industrial automation).
• Modularity: The separation of concerns allows engineers to upgrade the stabilization layer or the constraint layer independently without redesigning the entire system.
• Robustness: By explicitly accounting for communication errors and noise within the set-theoretic framework, the proposed method is more resilient to real-world imperfections than many existing theoretical MPC formulations.

In summary, the paper presents a scalable, robust, and computationally efficient framework for managing constraints in distributed networked systems, utilizing a two-layer architecture that leverages the strengths of both NRF-based control and Model Predictive Control.