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Asynchronous Push-sum Dual Gradient Algorithm in Distributed Model Predictive Control

Pengbiao Wang, Xuemei Ren, Dongdong Zheng

TL;DR

This work tackles distributed model predictive control for networks of discrete-time linear subsystems with both local and global constraints on directed communication graphs. It introduces an asynchronous push-sum dual gradient (APDG) algorithm with an adaptive step-size and a distributed termination criterion, achieving $R$-linear convergence to the dual optimum and ensuring recursive feasibility and closed-loop stability. The method reformulates the global constraint through dual decomposition and uses gradient-tracking and augmented-graph techniques to handle delays and asynchrony without central coordination. Numerical results on a four-subsystem water-tank example show faster dual convergence and reliable constraint satisfaction, illustrating the approach's practicality for scalable DMPC on directed networks.

Abstract

This paper studies the distributed model predictive control (DMPC) problem for distributed discrete-time linear systems with both local and global constraints over directed communication networks. We establish an optimization problem to formulate the DMPC policy, including the design of terminal ingredients. To cope with the global constraint, we transform the primal optimization problem into its dual problem. Then, we propose a novel asynchronous push-sum dual gradient (APDG) algorithm with an adaptive step-size scheme to solve this dual problem in a fully asynchronous distributed manner. The proposed algorithm does not require synchronous waiting and any form of coordination, which greatly improves solving efficiency. We prove that the APDG algorithm converges at an R-linear rate as long as the step-size does not exceed the designed upper bound. Furthermore, we develop a distributed termination criterion to terminate the APDG algorithm when its output solution satisfies the specified suboptimality and the global constraint, thereby avoiding an infinite number of iterations. The recursive feasibility and the stability of the closed-loop system are also established. Finally, a numerical example is provided to clarify and validate our theoretical findings.

Asynchronous Push-sum Dual Gradient Algorithm in Distributed Model Predictive Control

TL;DR

This work tackles distributed model predictive control for networks of discrete-time linear subsystems with both local and global constraints on directed communication graphs. It introduces an asynchronous push-sum dual gradient (APDG) algorithm with an adaptive step-size and a distributed termination criterion, achieving -linear convergence to the dual optimum and ensuring recursive feasibility and closed-loop stability. The method reformulates the global constraint through dual decomposition and uses gradient-tracking and augmented-graph techniques to handle delays and asynchrony without central coordination. Numerical results on a four-subsystem water-tank example show faster dual convergence and reliable constraint satisfaction, illustrating the approach's practicality for scalable DMPC on directed networks.

Abstract

This paper studies the distributed model predictive control (DMPC) problem for distributed discrete-time linear systems with both local and global constraints over directed communication networks. We establish an optimization problem to formulate the DMPC policy, including the design of terminal ingredients. To cope with the global constraint, we transform the primal optimization problem into its dual problem. Then, we propose a novel asynchronous push-sum dual gradient (APDG) algorithm with an adaptive step-size scheme to solve this dual problem in a fully asynchronous distributed manner. The proposed algorithm does not require synchronous waiting and any form of coordination, which greatly improves solving efficiency. We prove that the APDG algorithm converges at an R-linear rate as long as the step-size does not exceed the designed upper bound. Furthermore, we develop a distributed termination criterion to terminate the APDG algorithm when its output solution satisfies the specified suboptimality and the global constraint, thereby avoiding an infinite number of iterations. The recursive feasibility and the stability of the closed-loop system are also established. Finally, a numerical example is provided to clarify and validate our theoretical findings.
Paper Structure (15 sections, 14 theorems, 62 equations, 4 figures, 2 algorithms)

This paper contains 15 sections, 14 theorems, 62 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System and Constraints
  4. Network Topology: A Digraph
  5. Optimization Problem
  6. Dual Problem
  7. APDG Algorithm
  8. APDG Algorithm Design
  9. Reformulation of The APDG Algorithm
  10. Convergence Analysis and Distributed Termination Criterion Design
  11. Convergence Analysis
  12. Distributed Termination Criterion
  13. The Iterative DMPC Approach
  14. Numerical Simulation
  15. Conclusion

Key Result

Lemma 1

Zhang2019You Under Assumptions assumption 3 and assumption 5, the following statements hold. (1) Each subsystem is activated at least once during $(t(k), t(k+\eta_{1}^{})]$, where $\eta_{1}^{}=(M-1)\lfloor\bar{\tau}/\underline{\tau}\rfloor+1$. Moreover, for any $k$, the union of graphs $\bigcup_{t=k

Figures (4)

  • Figure 1: Iteration counts and activation times of three subsystems in the APDG algorithm.
  • Figure 2: Solving time of the dual problem at each time $t$.
  • Figure 3: (a) Evolution of the global constraint. (b) Input trajectories of four subsystems.
  • Figure 4: State trajectories of four subsystems.

Theorems & Definitions (30)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Proposition 1
  • proof
  • ...and 20 more