Zonotope

Zonotope{N, VN<:AbstractVector{N}, MN<:AbstractMatrix{N}} <: AbstractZonotope{N}

Type that represents a zonotope.

Fields

• center – center of the zonotope
• generators – matrix; each column is a generator of the zonotope

Notes

Mathematically, a zonotope is defined as the set

\[Z = \left\{ x ∈ \mathbb{R}^n : x = c + ∑_{i=1}^p ξ_i g_i,~~ ξ_i \in [-1, 1]~~ ∀ i = 1,…, p \right\},\]

where $c \in \mathbb{R}^n$ is its center and $\{g_i\}_{i=1}^p$, $g_i \in \mathbb{R}^n$, is the set of generators. This characterization defines a zonotope as the finite Minkowski sum of line segments. Zonotopes can be equivalently described as the image of a unit infinity-norm ball in $\mathbb{R}^n$ by an affine transformation.

Zonotopes can be constructed in two different ways: either passing the generators as a matrix, where each column represents a generator, or passing a list of vectors, where each vector represents a generator. Below we illustrate both ways.

Examples

A two-dimensional zonotope with given center and matrix of generators:

julia> Z = Zonotope([1.0, 0.0], [0.1 0.0; 0.0 0.1])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.0, 0.0], [0.1 0.0; 0.0 0.1])

julia> dim(Z)
2

julia> center(Z)
2-element Vector{Float64}:
 1.0
 0.0

julia> genmat(Z)
2×2 Matrix{Float64}:
 0.1  0.0
 0.0  0.1

Here, the first vector in the Zonotope constructor corresponds to the center and each column of the second argument corresponds to a generator. The functions center and genmat respectively return the center and the generator matrix of a zonotope.

We can collect the vertices using vertices_list:

julia> vertices_list(Z)
4-element Vector{Vector{Float64}}:
 [1.1, 0.1]
 [0.9, 0.1]
 [0.9, -0.1]
 [1.1, -0.1]

The support vector along a given direction can be computed using σ (resp. the support function can be computed using ρ):

julia> σ([1.0, 1.0], Z)
2-element Vector{Float64}:
 1.1
 0.1

Zonotopes admit an alternative constructor that receives a list of vectors, each vector representing a generator:

julia> Z = Zonotope(ones(2), [[1.0, 0.0], [0.0, 1.0], [1.0, 1.0]])
Zonotope{Float64, Vector{Float64}, Matrix{Float64}}([1.0, 1.0], [1.0 0.0 1.0; 0.0 1.0 1.0])

julia> genmat(Z)
2×3 Matrix{Float64}:
 1.0  0.0  1.0
 0.0  1.0  1.0

center(Z::Zonotope)

Return the center of a zonotope.

Input

• Z – zonotope

Output

The center of the zonotope.

rand(::Type{Zonotope}; [N]::Type{<:Real}=Float64, [dim]::Int=2,
     [rng]::AbstractRNG=GLOBAL_RNG, [seed]::Union{Int, Nothing}=nothing)

Create a random zonotope.

Input

• Zonotope – type for dispatch
• N – (optional, default: Float64) numeric type
• dim – (optional, default: 2) dimension
• rng – (optional, default: GLOBAL_RNG) random number generator
• seed – (optional, default: nothing) seed for reseeding
• num_generators – (optional, default: -1) number of generators of the zonotope (see the comment below)

Output

A random zonotope.

Algorithm

All numbers are normally distributed with mean 0 and standard deviation 1.

The number of generators can be controlled with the argument num_generators. For a negative value we choose a random number in the range dim:2*dim (except if dim == 1, in which case we only create a single generator).

generators(Z::Zonotope)

Return an iterator over the generators of a zonotope.

Input

• Z – zonotope

Output

An iterator over the generators of Z.

genmat(Z::Zonotope)

Return the generator matrix of a zonotope.

Input

• Z – zonotope

Output

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

scale(α::Real, Z::Zonotope)

Concrete scaling of a zonotope.

Input

• α – scalar
• Z – zonotope

Output

The zonotope obtained by applying the numerical scale to the center and generators of $Z$.

scale!(α::Real, Z::Zonotope)

Concrete scaling of a zonotope modifing Z in-place.

Input

• α – scalar
• Z – zonotope

Output

The zonotope obtained by applying the numerical scale to the center and generators of $Z$.

ngens(Z::Zonotope)

Return the number of generators of a zonotope.

Input

• Z – zonotope

Output

An integer representing the number of generators.

togrep(Z::Zonotope)

Return a generator representation of a zonotope.

Input

• Z – zonotope

Output

The same set Z.

low(Z::Zonotope, i::Int)

Return the lower coordinate of a zonotope in a given dimension.

Input

• Z – zonotope
• i – dimension of interest

Output

The lower coordinate of the zonotope in the given dimension.

high(Z::Zonotope, i::Int)

Return the higher coordinate of a zonotope in a given dimension.

Input

• Z – zonotope
• i – dimension of interest

Output

The higher coordinate of the zonotope in the given dimension.

remove_zero_generators(Z::Zonotope)

Return a new zonotope removing the generators that are zero.

Input

• Z – zonotope

Output

If there are no zero generators, the result is the original zonotope Z. Otherwise the result is a new zonotope that has the center and generators as Z except for those generators that are zero.

linear_map!(Zout::Zonotope, M::AbstractMatrix, Z::Zonotope)

Compute the concrete linear map of a zonotope, storing the result in Zout.

Input

• Zout – zonotope (output)
• M – matrix
• Z – zonotope

Output

The zonotope Zout, which is modified in-place.

_bound_intersect_2D(Z::Zonotope, L::Line2D)

Evaluate the support function in the direction [0, 1] of the intersection between the given zonotope and line.

Input

• Z – zonotope
• L – vertical 2D line

Output

The support function in the direction [0, 1] of the intersection between the given zonotope and line.

Notes

The algorithm assumes that the given line is vertical and that the intersection between the given sets is not empty.

Algorithm

This function implements [Algorithm 8.2, 1].

[1] Colas Le Guernic. Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics. Computer Science [cs]. Université Joseph-Fourier - Grenoble I, 2009. English. fftel-00422569v2f

remove_redundant_generators(Z::Zonotope{N}) where {N}

Remove all redundant (pairwise linearly dependent) generators of a zonotope.

Input

• Z – zonotope

Output

A new zonotope with fewer generators, or the same zonotope if no generator could be removed.

Algorithm

For each generator $g_j$ that has not been checked yet, we find all other generators that are linearly dependent with $g_j$. Then we combine those generators into a single generator.

For one-dimensional zonotopes we use a more efficient implementation where we just take the absolute sum of all generators.