Follow-up of @system macro

[mforets]

Originally posted by @ueliwechsler in #125 (comment)

The following things can be done in a later PR (in approximately increasing order of work needed):

• add template f_(x_,u_) for allowing to write @system(x' = f(x,u_12) (add one line) (@system: add template f_(x_,u_) to allow for arbitrary input #154)
• ensure that state≠input≠noise (@system: error handling for variable names #155 Error handling for repeated variable names #164)
• add docstring to all functions
• make f_(x_,u_) and f_(x_,w_) distinguishable
• correct handling for arbitrary order of input set definition + unit_tests (Handle arbitrary order of set / rhs term specifcation in @system #176)
• make @system(x' = x + Bu) work by extracting state_dim from B
• add LazySets.Universe(n) or nothing as default cases for undefined sets (a straightforward option would be to handle every set as a Constrained set with all set as Universe(n) by default and if a set definition is provided specify the corresponding set)
• Make @system macro work without parantheses, e.g. @system x' = A*x (@system: test macro calls without parentheses #156 )
• Add support for remaining types, e.g. polynomial
• Add support for SystemWithOutput, e.g. @system(x' = Ax + Bu, y=Bx + Du)
• Allow arbitrary input for dynamic equations without *, e.g. @system(x' = Ax + Bu1)
• Major revision by using type specific "templates" for each AbstractSystem type

[ueliwechsler]

from #137:

• allow property initial_state as state(0) ∈ set to create IVP

[ueliwechsler]

from #139

• consistent type conversion in x' = -x and x' = 2x

[ueliwechsler]

Additionally, we should try to remove the explicit dependency on MathematicalSystems in these lines

MathematicalSystems.jl/src/macros.jl

Line 535 in 641d4f5

push!(params, (:(IdentityMultiple(1.0*MathematicalSystems.I,$state_dim)), :B))

MathematicalSystems.jl/src/macros.jl

Line 538 in 641d4f5

push!(params, (:(IdentityMultiple(1.0*MathematicalSystems.I,$state_dim)), :D))

by writing push!(params, I($state_dim), :B) or something similar.