Explicit MPC for the constrained zonotope case with low-rank matrix updates

TL;DR

This paper introduces a novel approach to accelerate explicit MPC solutions for constrained zonotopes by leveraging geometric properties, low-rank updates, and analytic enumeration, reducing computational complexity.

Contribution

It presents a new method combining geometric reformulation, low-rank matrix updates, and analytic enumeration to efficiently compute explicit MPC solutions for constrained zonotope cases.

Findings

Reduces computation time for explicit MPC with zonotope constraints.

Provides a tree-based explicit solution enumeration method.

Demonstrates improved scalability over traditional approaches.

Abstract

Solving the explicit Model Predictive Control (MPC) problem requires enumerating all critical regions and their associated feedback laws, a task that scales exponentially with the system dimension and the prediction horizon, as well. When the problem's constraints are boxes or zonotopes, the feasible domain admits a compact constrained-zonotope representation. Building on this insight, we exploit the geometric properties of the equivalent constrained-zonotope reformulation to accelerate the computation of the explicit solution. Specifically, we formulate the multi-parametric problem in the lifted generator space and solve it using second-order optimality conditions, employ low-rank matrix updates to reduce computation time, and introduce an analytic enumeration of candidate active sets that yields the explicit solution in tree form.