Fix crashes in Schur transformation

[schillic]

My test below shows that there is still something missing to back-transform the result. But I think this PR still improves the state by not crashing anymore.

using SparseArrays, MathematicalSystems, Reachability

A = sparse([1, 1, 2, 2], [1, 2, 1, 2], [3., -9., 4., -3.])
B = reshape([0.0, -1.0], (2, 1))
U = Singleton([1.0])
inv = Universe(2)
model = ConstrainedLinearControlContinuousSystem(A, B, inv, ConstantInput(U))
X0 = Hyperrectangle(low=[10.0, 0.0], high=[10.2, 0.0])
problem = InitialValueProblem(model, X0)

𝑂_original = Options(:T => 20.0, :mode => "reach")
𝑂_op_original = Options(:vars => [1, 2], :partition => [[1], [2]], :δ => 0.003)
sol_original = solve(problem, 𝑂_original, op=BFFPSV18(𝑂_op_original))

𝑂_transform = merge(𝑂_original, Options(:coordinate_transformation => "schur"))
𝑂_op_transform = merge(𝑂_op_original, Options())
sol_transform = solve(problem, 𝑂_transform, op=BFFPSV18(𝑂_op_transform))


using Plots
plot(sol_original)
plot(sol_transform)

EDIT: side note: There is something very wrong with our axis labels 😄

[schillic]

The helper function _wrap_system is necessary because the type parameters of PT may change. For instance, B could be an IdentityMultiple, but then its transformation is a proper Matrix.

[schillic]

The second commit adds a backtransformation. We actually never had that functionality, but thanks to a recent refactoring it was very simple.

Note that the transformed flowpipe is the orange one, so for this system it is more precise to do the computations in the transformed system.

[schillic]

The type of Rsets should be ReachSolution, but that type is not available in this module because it is only defined later in the ReachSets module.