# Zonotope

LazySets.ZonotopeType
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 set 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 zonotope's center, and each column of the second argument corresponds to a generator. The functions center and genmat return the center and the generator matrix of this zonotope respectively.

We can collect its 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., 1.], 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., 1.], [1., 1.]])
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
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LazySets.centerMethod
center(Z::Zonotope)

Return the center of a zonotope.

Input

• Z – zonotope

Output

The center of the zonotope.

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Base.randMethod
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 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).

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LazySets.generatorsMethod
generators(Z::Zonotope)

Return an iterator over the generators of a zonotope.

Input

• Z – zonotope

Output

An iterator over the generators of Z.

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LazySets.genmatMethod

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.

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LazySets.scaleMethod
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 $Z$.

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LazySets.scale!Method
scale!(α::Real, Z::Zonotope)

Concrete scaling of a zonotope modifing Z in-place

Input

• α – scalar
• Z – zonotope

Output

The zonotope Z after applying the numerical scale α to its center and generators.

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LazySets.ngensMethod
ngens(Z::Zonotope)

Return the number of generators of a zonotope.

Input

• Z – zonotope

Output

Integer representing the number of generators.

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LazySets.togrepMethod
togrep(Z::Zonotope)

Return a generator representation of a zonotope.

Input

• Z – zonotope

Output

The same set Z.

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LazySets.lowMethod
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.

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LazySets.highMethod
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.

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LazySets.remove_zero_generatorsMethod
remove_zero_generators(Z::Zonotope)

Return a new zonotope removing the generators which are zero of the given zonotope.

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.

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LazySets.linear_map!Method
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.

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LazySets._bound_intersect_2DMethod
_bound_intersect_2D(Z::Zonotope, L::Line2D)

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

Input

• Z – zonotope
• L – vertical line 2D

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].

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

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LazySets.remove_redundant_generatorsMethod
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.

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Inherited from ConvexSet:

Inherited from AbstractPolytope:

Inherited from AbstractCentrallySymmetricPolytope:

Inherited from AbstractZonotope: