PolynomialZonotopes.jl
PolynomialZonotopes is a Julia package that extends LazySets.jl to the non-convex sets representation given by polynomial zonotopes. This library defines the PolynomialZonotope type and methods to manipulate them; see the manual sections Types and Methods in this documentation for further details.
Features
A
PolynomialZonotopetype of arbitrary order.Basic set operations (e.g. linear maps, Minkowski sum) for polynomial zonotopes.
Overapproximation of polynomial zonotopes with usual zonotopes.
Order reduction of polynomial zonotopes.
Definition
Polynomial zonotopes were introduced by M. Althoff in Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets, Hybrid Systems: Computation and Control, 2013, pp. 173–182. They have been applied as a non-convex set representation in the reachability problem if nonlinear ODEs.
Mathematically, a polynomial zonotope is the tuple $(c, E, F, G)$ that defines the set:
where the number of factors in the final product, $β_j β_k ⋯ β_m$, corresponds to the polynomial order $η$, and:
$c ∈ \mathbb{R}^n$ is the starting point (in some particular cases it corresponds to the center of a usual zonotope),
$E = [E^{[1]} ⋯ E^{[η]}]$ is an $n × p × η(η+1)/2$ matrix with column-blocks
called the matrix of multi-indexed generators with equal indices, where each $f^{([i], k_1, k_2, …, k_i)}$ is an $n$-vector,
$F = [F^{[2]} ⋯ F^{[η]}]$ is a matrix with column-blocks
called the matrix of multi-indexed generators with unequal indices (or, more accurately, not-all-equal indices), where each $f^{([i], k_1, k_2, …, k_i)}$ is an $n$-vector,
$G = [G^{[1]} ⋯ G^{[q]}]$ is an $n × q$ matrix with columns
called the matrix of single-indexed generators, where each $g^{(i)}$ is an $n$-vector.