Merge pull request #1019 from JuliaReach/mforets/1016

#1016 - Add faster zonotope interval box approximation

src/Approximations/overapproximate.jl

@@ -133,6 +133,37 @@ function overapproximate(S::ConvexHull{N, Zonotope{N}, Zonotope{N}},
     return Zonotope(center, generators)
 end
 
+"""
+    overapproximate(Z::Zonotope, ::Type{<:Hyperrectangle})::Hyperrectangle
+
+Return a tight overapproximation of a zonotope with an axis-aligned box.
+
+### Input
+
+- `Z`              -- zonotope
+- `Hyperrectangle` -- type for dispatch
+
+### Output
+
+A hyperrectangle.
+
+### Algorithm
+
+This function implements the method in [Section 5.1.2, 1]. A zonotope
+``Z = ⟨c, G⟩`` can be overapproximated tightly by an axis-aligned box
+(i.e. a `Hyperrectangle`) such that its center is ``c`` and the radius along
+dimension ``i`` is the column-sum of the absolute values of the ``i``-th row
+of ``G`` for ``i = 1,…, p``, where ``p`` is the number of generators of ``Z``.
+
+[1] *Althoff, M., Stursberg, O., & Buss, M. (2010). Computing reachable sets of
+hybrid systems using a combination of zonotopes and polytopes. Nonlinear analysis:
+hybrid systems, 4(2), 233-249.*
+"""
+function overapproximate(Z::Zonotope, ::Type{<:Hyperrectangle})::Hyperrectangle
+    r = Compat.sum(abs.(Z.generators), dims=2)[:]
+    return Hyperrectangle(Z.center, r)
+end
+
 """
     overapproximate(X::LazySet{N}, dir::AbstractDirections{N})::HPolytope{N}
         where {N}

test/unit_overapproximate.jl

@@ -1,4 +1,4 @@
-import LazySets.Approximations.overapproximate
+import LazySets.Approximations: overapproximate, box_approximation
 
 for N in [Float64, Rational{Int}, Float32]
     # overapproximating a set of type T1 with an unsupported type T2 is the
@@ -37,6 +37,12 @@ for N in [Float64, Rational{Int}, Float32]
     for d in [N[1], N[-1]]
         @test σ(d, p)[1] ≈ σ(d, b)[1]
     end
+
+    # approximation with an axis-aligned hyperrectangle
+    Z = Zonotope(N[-1.0, -1.0], N[-1/2 0; -1.2 -1])
+    Zoa = overapproximate(Z, Hyperrectangle) # faster o.a.
+    Zba = box_approximation(Z) # default o.a. implementation that uses supp function
+    @test Zoa.center ≈ Zba.center && Zoa.radius ≈ Zba.radius
 end
 
 # tests that do not work with Rational{Int}