Types
This section describes systems types implemented in PolynomialZonotopes.jl
.
Polynomial Zonotope
PolynomialZonotope{N} <: LazySet{N}
Type that represents a polynomial zonotope.
Fields
c
– starting pointE
– matrix of multi-indexed generators such that all indices have the same valueF
– matrix of multi-indexed generators such that not all indices have the same valueG
– matrix of single-indexed generators
Notes
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.
Mathematically, it is defined as the tuple $(c, E, F, G)$, where:
$c ∈ \mathbb{R}^n$ 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.
The polynomial zonotope $(c, E, F, G)$ defines the set:
where the number of factors in the final product, $β_j β_k ⋯ β_m$, corresponds to the polynomial order $η$.