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Distributed Model Predictive Control for Piecewise Affine Systems Based on Switching ADMM

Samuel Mallick, Azita Dabiri, Bart De Schutter

TL;DR

The paper tackles distributed model predictive control for large-scale piecewise affine systems, where non-convexity from PWA dynamics makes conventional solutions computationally expensive. It introduces a switching ADMM-based framework that decomposes the global non-convex problem into a sequence of convex subproblems, while enforcing consensus on shared states to maintain coordination among subsystems. The authors prove stability and recursive feasibility under well-defined assumptions and demonstrate substantial reductions in online computation time with competitive closed-loop performance in numerical examples, including a hybrid vehicle platoon. The approach enables scalable, privacy-preserving distributed control for strongly coupled PWA networks and provides avenues for guaranteed performance via terminal sets and Lyapunov-based analysis.

Abstract

This paper presents a novel approach for distributed model predictive control (MPC) for piecewise affine (PWA) systems. Existing approaches rely on solving mixed-integer optimization problems, requiring significant computation power or time. We propose a distributed MPC scheme that requires solving only convex optimization problems. The key contribution is a novel method, based on the alternating direction method of multipliers, for solving the non-convex optimal control problem that arises due to the PWA dynamics. We present a distributed MPC scheme, leveraging this method, that explicitly accounts for the coupling between subsystems by reaching agreement on the values of coupled states. Stability and recursive feasibility are shown under additional assumptions on the underlying system. Two numerical examples are provided, in which the proposed controller is shown to significantly improve the CPU time and closed-loop performance over existing state-of-the-art approaches.

Distributed Model Predictive Control for Piecewise Affine Systems Based on Switching ADMM

TL;DR

The paper tackles distributed model predictive control for large-scale piecewise affine systems, where non-convexity from PWA dynamics makes conventional solutions computationally expensive. It introduces a switching ADMM-based framework that decomposes the global non-convex problem into a sequence of convex subproblems, while enforcing consensus on shared states to maintain coordination among subsystems. The authors prove stability and recursive feasibility under well-defined assumptions and demonstrate substantial reductions in online computation time with competitive closed-loop performance in numerical examples, including a hybrid vehicle platoon. The approach enables scalable, privacy-preserving distributed control for strongly coupled PWA networks and provides avenues for guaranteed performance via terminal sets and Lyapunov-based analysis.

Abstract

This paper presents a novel approach for distributed model predictive control (MPC) for piecewise affine (PWA) systems. Existing approaches rely on solving mixed-integer optimization problems, requiring significant computation power or time. We propose a distributed MPC scheme that requires solving only convex optimization problems. The key contribution is a novel method, based on the alternating direction method of multipliers, for solving the non-convex optimal control problem that arises due to the PWA dynamics. We present a distributed MPC scheme, leveraging this method, that explicitly accounts for the coupling between subsystems by reaching agreement on the values of coupled states. Stability and recursive feasibility are shown under additional assumptions on the underlying system. Two numerical examples are provided, in which the proposed controller is shown to significantly improve the CPU time and closed-loop performance over existing state-of-the-art approaches.
Paper Structure (20 sections, 7 theorems, 50 equations, 9 figures, 6 algorithms)

This paper contains 20 sections, 7 theorems, 50 equations, 9 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Distributed Piecewise Affine Systems
  5. Centralized MPC
  6. Distributed MPC
  7. Switching ADMM-based Distributed MPC
  8. Structure of $\mathcal{P}_\text{d}$
  9. Switching ADMM Procedure
  10. Distributed MPC Algorithm
  11. Stability and Recursive Feasibility
  12. Assumptions
  13. Stabilizing Distributed MPC Algorithm
  14. Illustrative Examples
  15. Stabilizing Example
  16. ...and 5 more sections

Key Result

Lemma 1

Suppose $x$ and $s$ are given such that $\mathcal{Y}_s(x) \neq \emptyset$. Then for all $\textbf{y} = (\Tilde{\textbf{x}}^\top, \textbf{u}^\top)^\top \in \mathcal{Y}_s(x)$: $\forall k \in \mathcal{K},\forall i \in \mathcal{M}$.

Figures (9)

  • Figure 1: Visualization of switching ADMM procedure.
  • Figure 2: State trajectories for SwA and RCS with weak coupling.
  • Figure 3: Performance drop with respect to the centralized controller for 100 different initial conditions.
  • Figure 4: Computation times with different solvers over 100 different initial conditions. The quadratic programming solvers are osqposqp, qpOASESFerreau2014, and qrqpAndersson2019. The MIQP solvers are CPLEXcplex2009v12, Mosekaps2019mosek, and Gurobigurobi.
  • Figure 5: State trajectories for SwA with strong coupling.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Definition 1: Globally feasible local control sequences
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Definition 2: Weakly feasible local control sequences
  • Proposition 2
  • proof
  • Remark 2
  • ...and 10 more