#512 - Add new option to discretize

[mforets]

Closes #512.

• add tests

[mforets]

    A = sparse([1, 1, 2, 3, 4], [1, 2, 2, 4, 3], [1., 2., 3., 4., 5.], 4, 4)
    X0 = BallInf(zeros(4), 0.1)
    cont_sys_homog = IVP(LCS(A), X0)
    δ = 0.01

julia> discr_sys_homog = discretize(cont_sys_homog, δ, algorithm="forward", set_operations="zonotope")
ERROR: MethodError: no method matching minkowski_sum(::BallInf{Float64}, ::Zonotope{Float64})
Closest candidates are:

  minkowski_sum(::Zonotope{N<:Real}, ::Zonotope{N<:Real}) where N<:Real at /Users/forets/.julia/dev/LazySets/src/Zonotope.jl:330
Stacktrace:
 [1] _discretize_interpolation_homog(::BallInf{Float64}, ::Array{Float64,2}, ::Hyperrectangle{Float64}, ::Val{:zonotope}) at /Users/forets/.julia/dev/Reachability/src/ReachSets/discretize.jl:6
92
 [2] #discretize_interpolation#9(::String, ::String, ::String, ::String, ::Function, ::InitialValueProblem{LinearContinuousSystem{Float64,SparseMatrixCSC{Float64,Int64}},BallInf{Float64}}, ::F
loat64) at /Users/forets/.julia/dev/Reachability/src/ReachSets/discretize.jl:648
 [3] #discretize_interpolation at ./none:0 [inlined]
 [4] #discretize#1(::String, ::String, ::String, ::String, ::Function, ::InitialValueProblem{LinearContinuousSystem{Float64,SparseMatrixCSC{Float64,Int64}},BallInf{Float64}}, ::Float64) at /Us
ers/forets/.julia/dev/Reachability/src/ReachSets/discretize.jl:100
 [5] (::getfield(Reachability.ReachSets, Symbol("#kw##discretize")))(::NamedTuple{(:algorithm, :set_operations),Tuple{String,String}}, ::typeof(discretize), ::InitialValueProblem{LinearContinu
ousSystem{Float64,SparseMatrixCSC{Float64,Int64}},BallInf{Float64}}, ::Float64) at ./none:0
 [6] top-level scope at none:0

[schillic]

I think generally the code cannot assume that the initial states are given as a zonotope or related sets, right? So you need to call overapproximate.

[mforets]

Agreed, i was just about to comment on that. I have been assuming that the initial states are hyperrectangular for which we have a good conversion. Changing to overapproximate doesn't work ATM (we may do this in another PR):

julia> H = rand(Hyperrectangle, dim=10);

julia> convert(Zonotope, H);

julia> using LazySets.Approximations

julia> overapproximate(H, Zonotope);
ERROR: MethodError: no method matching overapproximate(::Hyperrectangle{Float64}, ::Type{Zonotope})
Closest candidates are:
  overapproximate(::LazySet{N<:Real}, ::Type{#s65} where #s65<:HPolygon) where N<:Real at /Users/forets/.julia/dev/LazySets/src/Approximations/overapproximate.jl:42
  overapproximate(::LazySet{N<:Real}, ::Type{#s35} where #s35<:HPolygon, ::Real) where N<:Real at /Users/forets/.julia/dev/LazySets/src/Approximations/overapproximate.jl:42
  overapproximate(::ConvexHull{N<:Real,Zonotope{N<:Real},Zonotope{N<:Real}}, ::Type{#s66} where #s66<:Zonotope) where N<:Real at /Users/forets/.julia/dev/LazySets/src/Approximations/overapproximate.jl:195
  ...
Stacktrace:
 [1] top-level scope at none:0

[schillic]

We can also keep convert here and change that later I think.

[mforets]

I kept convert and added tests for the homogeneous and the constant case.

[mforets]

Opened #530 as follow-up for the cases where the sets are not convertible to Zonotope.

[mforets]

Thanks for the review ♻️