ZonotopeMD

struct ZonotopeMD{N, VN<:AbstractVector{N}, MN<:AbstractMatrix{N}, DN<:AbstractVector{N}} <: AbstractZonotope{N}

Type that represents a zonotope of order k in normal form.

Fields

• center::VN — the center of the zonotope
• M::MN — matrix of general (non-axis-aligned) generators
• d::DN — vector of axis-aligned (diagonal) generators

Notes

A zonotope is of order k if it has n * k generators in ℝⁿ, where n is the ambient dimension.

A zonotope of order k in normal form is defined as the set

\[Z = \left\{ x ∈ ℝ^n : x = c + Mξ + d ⊙ η, ~~ ξ ∈ [-1, 1]^m, ~~ η ∈ [-1, 1]^n \right\},\]

where M ∈ ℝ^{n×m} is a matrix of general generators with m = n*(k -1) and d ∈ ℝⁿ is a vector of axis-aligned generators. Equivalently, this can be seen as a zonotope with generator matrix [M D], where D is the diagonal matrix formed from the vector d. ZonotopeMD can be constructed in two ways: by passing the full generator matrix [M D] in normal form or by passing M and a vector d separately.

Examples

Constructing a zonotope in normal form from a center, general generator matrix M, and diagonal vector d:

julia> c = [0.0, 0.0];

julia> M = [1.0 2.0; 3.0 1.0];

julia> d = [0.1, 0.2];

julia> Z = ZonotopeMD(c, M, d)
ZonotopeMD{Float64, Vector{Float64}, Matrix{Float64}, Vector{Float64}}([0.0, 0.0], [1.0 2.0; 3.0 1.0], [0.1, 0.2])

julia> center(Z)
2-element Vector{Float64}:
 0.0
 0.0

julia> genmat(Z)
2×4 SparseArrays.SparseMatrixCSC{Float64, Int64} with 6 stored entries:
 1.0  2.0  0.1   ⋅ 
 3.0  1.0   ⋅   0.2

The generator matrix returned by genmat is the concatenation [M D], where D is the diagonal matrix formed from d. THe resulting matrix is stored as a sparse matrix. Constructing the same zonotope by passing the full generator matrix [M D] directly:

julia> G = [1.0 2.0 0.1 0.0;
            3.0 1.0 0.0 0.2];

julia> Z2 = ZonotopeMD([0.0, 0.0], G)
ZonotopeMD{Float64, Vector{Float64}, Matrix{Float64}, Vector{Float64}}([0.0, 0.0], [1.0 2.0; 3.0 1.0], [0.1, 0.2])

julia> genmat(Z2) == G
true

You can also convert back to a standard Zonotope if needed:

julia> Zstd = Zonotope(Z)
Zonotope{Float64, Vector{Float64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}([0.0, 0.0], sparse([1, 2, 1, 2, 1, 2], [1, 1, 2, 2, 3, 4], [1.0, 3.0, 2.0, 1.0, 0.1, 0.2], 2, 4))

genmat(Z::ZonotopeMD)

Return the generator matrix of a ZonotopeMD.

Input

• Z – zonotope in normal form

Output

A matrix where each column represents one generator of the zonotope Z.

cartesian_product(Z1::ZonotopeMD, Z2::ZonotopeMD)

Return the Cartesian product of two zonotopes in normal form (ZonotopeMD).

Input

• Z1, Z2 – zonotopes in normal form (ZonotopeMD)

Output

A new ZonotopeMD representing the Cartesian product Z1 × Z2.