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

TL;DR

This paper tackles the scalability problem of explicit MPC by reformulating constraint sets as constrained zonotopes and solving the multi-parametric problem in a lifted generator space. It leverages second-order optimality conditions to cope with rank-deficient Hessians and introduces low-rank, iterative updates to efficiently handle active-set changes, producing an explicit solution in tree form through analytic enumeration of critical regions. The combination of constrained-zonotope geometry, lifted MPQP, and iterative updates yields faster computations and better scalability than traditional polyhedral approaches, as evidenced by numerical simulations. The approach offers a practical route to real-time explicit MPC for higher-dimensional systems with box or zonotopic constraints, with the code openly available.

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.

Figures (6)

• Figure 1: (a) Domain of $\xi\in\mathbb{R}^{n_g}$, $n_g=3$ and hyperplane defined by $F\xi= \theta$; (b) Zonotope (black contour) and constrained zonotope (blue) in $\mathbb{R}^n$, $n= 2$.
• Figure 2: Illustration of the update of matrices $Z_{(\mathbb A)}$, $K_{(\mathbb A)}$.
• Figure 3: Tree structure obtained from Alg. \ref{['alg:mpQP_CZ']}, applied to dynamics \ref{['eq:example_dynamics']}, for prediction horizon $N=4$.
• Figure 4: Computation time vs. prediction horizon $N$.
• Figure 5: Number of critical regions vs. prediction horizon $N$.