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Decentralized Real-Time Iterations for Distributed NMPC

Gösta Stomberg, Alexander Engelmann, Moritz Diehl, Timm Faulwasser

TL;DR

The paper addresses real-time feasibility in distributed NMPC for networks of coupled subsystems by introducing a decentralized RTI scheme based on a bi-level dSQP (outer inequality-constrained SQP with inner ADMM). It proves local exponential stability of the closed-loop system under standard MPC assumptions when the sampling interval is below a computed bound ${\bar{\delta}}$, and provides a quantitative inner-outer convergence framework for fixed inner iterations ${l_{\max}}$. The approach requires only neighbor-to-neighbor communication and avoids a central coordinator, and is validated on a chain of coupled inverted pendulums showing real-time capable performance. This work advances distributed MPC by linking rigorous RTI stability guarantees with decentralized optimization, enabling scalable, real-time cooperative control in networked systems.

Abstract

This article presents a Real-Time Iteration (RTI) scheme for distributed Nonlinear Model Predictive Control (NMPC). The scheme transfers the well-known RTI approach, a key enabler for many industrial real-time NMPC implementations, to the setting of cooperative distributed control. At each sampling instant, one outer iteration of a bi-level decentralized Sequential Quadratic Programming (dSQP) method is applied to a centralized optimal control problem. This ensures that real-time requirements are met and it facilitates cooperation between subsystems. Combining novel dSQP convergence results with RTI stability guarantees, we prove local exponential stability under standard assumptions on the MPC design with and without terminal constraints. The proposed scheme only requires neighbor-to-neighbor communication and avoids a central coordinator. A numerical example with coupled inverted pendulums demonstrates the efficacy of the approach.

Decentralized Real-Time Iterations for Distributed NMPC

TL;DR

The paper addresses real-time feasibility in distributed NMPC for networks of coupled subsystems by introducing a decentralized RTI scheme based on a bi-level dSQP (outer inequality-constrained SQP with inner ADMM). It proves local exponential stability of the closed-loop system under standard MPC assumptions when the sampling interval is below a computed bound , and provides a quantitative inner-outer convergence framework for fixed inner iterations . The approach requires only neighbor-to-neighbor communication and avoids a central coordinator, and is validated on a chain of coupled inverted pendulums showing real-time capable performance. This work advances distributed MPC by linking rigorous RTI stability guarantees with decentralized optimization, enabling scalable, real-time cooperative control in networked systems.

Abstract

This article presents a Real-Time Iteration (RTI) scheme for distributed Nonlinear Model Predictive Control (NMPC). The scheme transfers the well-known RTI approach, a key enabler for many industrial real-time NMPC implementations, to the setting of cooperative distributed control. At each sampling instant, one outer iteration of a bi-level decentralized Sequential Quadratic Programming (dSQP) method is applied to a centralized optimal control problem. This ensures that real-time requirements are met and it facilitates cooperation between subsystems. Combining novel dSQP convergence results with RTI stability guarantees, we prove local exponential stability under standard assumptions on the MPC design with and without terminal constraints. The proposed scheme only requires neighbor-to-neighbor communication and avoids a central coordinator. A numerical example with coupled inverted pendulums demonstrates the efficacy of the approach.
Paper Structure (16 sections, 11 theorems, 77 equations, 4 figures, 2 tables, 2 algorithms)

This paper contains 16 sections, 11 theorems, 77 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Centralized Real-Time Iterations
  4. Decentralized Sequential Quadratic Programming
  5. Outer Convergence
  6. Inner Convergence
  7. Optimizer Convergence with Limited Inner Iterations
  8. Decentralized Real-Time Iterations
  9. Implementation Aspects
  10. Numerical Results
  11. Optimal Control Problem Design
  12. Swing-up Simulation
  13. Parameter Estimation and Validity of Assumptions
  14. Conclusion
  15. Centralized RTI Stability
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose that As-sumptions ass:Lyapunov--ass:Lipschitz_sys hold and consider the sufficient sampling interval $\bar{\delta}$ and optimizer initialization radius $\tilde{r}_p$ defined in eq:RTIsuff in Appendix sec:app-rti. If $\delta \leq \bar{\delta}$, then the origin is a locally exponentially stabl for the closed-loop system-optimizer dynamics eq:sysoptdyn. $\square$

Figures (4)

  • Figure 1: Coupled inverted pendulums on carts.
  • Figure 2: Test case 1: closed-loop system and optimizer convergence with 6 ADMM iterations per NMPC step.
  • Figure 3: Test case 2: closed-loop system and optimizer convergence for a challenging initial condition with 18 ADMM iterations per NMPC step.
  • Figure 4: Test case 3: closed-loop system and optimizer convergence with 6 ADMM iterations per NMPC step.

Theorems & Definitions (35)

  • Definition 1
  • Lemma 1: Centralized RTI stability
  • proof
  • Remark 1: Relation to Zanelli2021
  • Remark 2: Decentralized ADMM
  • Remark 3: Communication requirements
  • Definition 2: Convergence rates
  • Lemma 2: Exact SQP convergence Boggs1995Nocedal2006
  • Lemma 3: Inexact SQP convergence
  • proof
  • ...and 25 more