Methods for order reduction of zonotopes

Zonotopes are a special subclass of polytopes, which have several favorable properties: They can be represented in a compact way and they are closed under the Minkowski sum as well as under linear transformations. Zono-topes are a popular set representation used e.g. for reachability analysis of dynamic systems, set-based observers and robust control. The complexity of algorithms that work on zonotopes strongly depends on their order (i.e. their number of generators and dimensions), which is often increased by operations like the Minkowski sum. Thus, to keep computations efficient, zonotopes of high orders are often over-approximated as tight as possible by zonotopes of smaller order. This paper has two main contributions: First, we propose new methods based on principle component analysis (PCA), clustering and constrained optimization for tight over-approximation of zonotopes. Second, we provide an overview of the most important known methods for order reduction and compare the performance of new and known methods in low- and high-dimensional spaces.