Reset map (ResetMap)

ResetMap{N<:Real, S<:LazySet{N}} <: LazySet{N}

Type that represents a lazy reset map. A reset map is a special case of an affine map $A x + b, x ∈ X$ where the linear map $A$ is the identity matrix with zero entries in all reset dimensions, and the translation vector $b$ is zero in all other dimensions.

Fields

• X – convex set
• resets – resets (a mapping from an index to a new value)

Examples

julia> X = BallInf([2.0, 2.0, 2.0], 1.0);

julia> r = Dict(1 => 4.0, 3 => 0.0);

julia> rm = ResetMap(X, r);

Here rm modifies the set X such that x1 is reset to 4 and x3 is reset to 0, while x2 is not modified. Hence rm is equivalent to the set Hyperrectangle([4.0, 2.0, 0.0], [0.0, 1.0, 0.0]), i.e., an axis-aligned line segment embedded in 3D.

The corresponding affine map $A x + b$ would be:

Use the function get_A (resp. get_b) to create the matrix A (resp. vector b) corresponding to a given reset map.

julia> get_A(rm)
3×3 LinearAlgebra.Diagonal{Float64,Array{Float64,1}}:
 0.0   ⋅    ⋅
  ⋅   1.0   ⋅
  ⋅    ⋅   0.0

julia> get_b(rm)
3-element SparseArrays.SparseVector{Float64,Int64} with 1 stored entry:
  [1]  =  4.0

The application of a ResetMap to a ZeroSet or an EmptySet is simplified automatically.

julia> ResetMap(ZeroSet(3), r)
Singleton{Float64,SparseArrays.SparseVector{Float64,Int64}}(  [1]  =  4.0)

julia> ResetMap(EmptySet(), r)
EmptySet{Float64}()

The (in this case unique) support vector of rm in direction ones(3) is:

julia> σ(ones(3), rm)
3-element Array{Float64,1}:
 4.0
 3.0
 0.0

dim(rm::ResetMap)

Return the dimension of a reset map.

Input

• rm – reset map

Output

The dimension of a reset map.

ρ(d::AbstractVector{N}, rm::ResetMap{N}) where {N<:Real}

Return the support function of a reset map.

Input

• d – direction
• rm – reset map

Output

The support function in the given direction.

Notes

We use the usual dot-product definition, but for unbounded sets we redefine the product between $0$ and $±∞$ as $0$; Julia returns NaN here.

julia> Inf * 0.0
NaN

See the discussion here.

σ(d::AbstractVector{N}, rm::ResetMap{N}) where {N<:Real}

Return the support vector of a reset map.

Input

• d – direction
• rm – reset map

Output

The support vector in the given direction. If the direction has norm zero, the result depends on the wrapped set.

an_element(rm::ResetMap)

Return some element of a reset map.

Input

• rm – reset map

Output

An element in the reset map. It relies on the an_element function of the wrapped set.

isempty(rm::ResetMap)

Return if a reset map is empty or not.

Input

• rm – reset map

Output

true iff the wrapped set is empty.

get_A(rm::ResetMap{N}) where {N<:Real}

Return the $A$ matrix of the affine map $A x + b, x ∈ X$ represented by a reset map.

Input

• rm – reset map

Output

The (diagonal) matrix for the affine map $A x + b, x ∈ X$ represented by the reset map.

Algorithm

We construct the identity matrix and set all entries in the reset dimensions to zero.

get_b(rm::ResetMap{N}) where {N<:Real}

Return the $b$ vector of the affine map $A x + b, x ∈ X$ represented by a reset map.

Input

• rm – reset map

Output

The (sparse) vector for the affine map $A x + b, x ∈ X$ represented by the reset map. The vector contains the reset value for all reset dimensions, and is zero for all other dimensions.

constraints_list(rm::ResetMap{N}) where {N<:Real}

Return the list of constraints of a polytopic reset map.

Input

• rm – reset map of a polytope

Output

The list of constraints of the reset map.

Notes

We assume that the underlying set X is a polytope, i.e., is bounded and offers a method constraints_list(X).

Algorithm

We fall back to constraints_list of a LinearMap of the A-matrix in the affine-map view of a reset map. Each reset dimension $i$ is projected to zero, expressed by two constraints for each reset dimension. Then it remains to shift these constraints to the new value.

For instance, if the dimension $5$ was reset to $4$, then there will be constraints $x₅ ≤ 0$ and $-x₅ ≤ 0$. We then modify the right-hand side of these constraints to $x₅ ≤ 4$ and $-x₅ ≤ -4$, respectively.

constraints_list(rm::ResetMap{N, S}) where
    {N<:Real, S<:AbstractHyperrectangle}

Return the list of constraints of a hyperrectangular reset map.

Input

• rm – reset map of a hyperrectangular set

Output

The list of constraints of the reset map.

Algorithm

We iterate through all dimensions. If there is a reset, we construct the corresponding (flat) constraints. Otherwise, we construct the corresponding constraints of the underlying set.