Distributed Model Predictive Control for Piecewise Affine Systems Using a Switching ADMM Approach

Core Concepts

This paper presents a novel distributed model predictive control (MPC) approach for piecewise affine (PWA) systems that leverages a switching alternating direction method of multipliers (ADMM) procedure to solve the non-convex optimal control problem arising from the PWA dynamics. The proposed method requires solving only convex optimization problems and explicitly accounts for the coupling between subsystems.

Abstract

The paper presents a novel distributed MPC approach for piecewise affine (PWA) systems. Existing approaches for distributed MPC of PWA systems rely on solving mixed-integer optimization problems, which can be computationally expensive.
The key contribution of this paper is a novel method based on the alternating direction method of multipliers (ADMM) for solving the non-convex optimal control problem that arises due to the PWA dynamics. The proposed distributed MPC scheme leverages this ADMM-based method and explicitly accounts for the coupling between subsystems by reaching agreement on the values of coupled states.
The paper first analyzes the structure of the distributed MPC problem for PWA systems, showing that it is piecewise-convex over a finite collection of polytopes. This structure is then leveraged to formulate the switching ADMM procedure, where subsystems can change their local switching sequences during the ADMM iterations, effectively moving between different convex pieces of the non-convex problem.
The stability and recursive feasibility of the proposed distributed MPC algorithm are proven under additional assumptions on the underlying system. Two numerical examples are provided, demonstrating that the proposed controller can significantly improve the CPU time and closed-loop performance over existing state-of-the-art approaches.

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Key Insights Distilled From

Distributed MPC for PWA Systems Based on Switching ADMM

by Samuel Malli... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16712.pdf

Deeper Inquiries

How can the conditions for Assumption 2 be verified more easily for practical applications

To verify the conditions for Assumption 2 more easily in practical applications, one approach could be to leverage system identification techniques. By using data-driven methods to analyze the system dynamics and behavior, it may be possible to determine the robustness of the terminal sets and their ability to handle coupling effects. Additionally, simulation studies and sensitivity analyses can be conducted to assess the performance of the terminal sets under different scenarios and perturbations. By systematically testing the terminal sets in various conditions, it becomes feasible to validate the assumptions and ensure their applicability in real-world systems.

What are the potential trade-offs between the size of the terminal set, the length of the prediction horizon, and the closed-loop performance in the proposed approach

The trade-offs between the size of the terminal set, the length of the prediction horizon, and the closed-loop performance in the proposed approach are crucial considerations in designing an effective control strategy.

Terminal Set Size: A larger terminal set increases the region where linear control laws are applied, potentially sacrificing performance as the system operates in a more conservative manner. On the other hand, a smaller terminal set allows for more aggressive control actions but may require a longer prediction horizon to satisfy the terminal constraint.

Prediction Horizon Length: A longer prediction horizon enables the system to anticipate future states and make more informed control decisions. However, this comes at the cost of increased computational complexity and may lead to slower response times in the control loop.

Closed-Loop Performance: The closed-loop performance is influenced by the balance between the terminal set size, prediction horizon length, and the control strategy employed. Optimal performance is achieved when these factors are carefully tuned to meet the system's requirements while ensuring stability and efficiency.
Balancing these trade-offs requires a thorough understanding of the system dynamics, performance objectives, and computational constraints to design a control strategy that optimally addresses the specific requirements of the application.

Can the switching ADMM procedure be extended to handle more general non-convex optimization problems beyond the PWA systems considered in this paper

The switching ADMM procedure can be extended to handle more general non-convex optimization problems beyond the PWA systems considered in the paper by adapting the algorithm to accommodate the specific characteristics of the new problem class.

Problem Formulation: The optimization problem needs to be reformulated to capture the non-convex nature of the new system dynamics. This may involve introducing additional constraints, objective functions, or variables to represent the non-convex elements accurately.

Algorithm Modification: The ADMM algorithm may need to be adjusted to handle the non-convexities present in the problem. This could involve developing new update rules, convergence criteria, or termination conditions tailored to the specific characteristics of the non-convex optimization problem.

Convergence Analysis: Extending the switching ADMM procedure to non-convex problems requires a thorough analysis of the algorithm's convergence properties. Special attention should be given to ensuring convergence to a feasible solution and addressing potential issues such as local minima or oscillations.
By customizing the switching ADMM procedure to suit the requirements of the new non-convex optimization problem, it is possible to leverage the algorithm's distributed optimization capabilities for a broader range of applications and system models.