`structural_simplify` broken for some underdetermined `NonlinearSystem`s

[AayushSabharwal]

Describe the bug 🐞

Some underdetermined nonlinear systems when simplified have a variable that only occurs on the RHS of observed equations, but is also not an unknown. Thus, it has no value assigned to it and errors.

Expected behavior

The variable should be on the LHS of an observed equation or be an unknown.

Minimal Reproducible Example 👇

Using the model in https://github.com/SciML/ModelingToolkit.jl/blob/master/test/downstream/inversemodel.jl,

@named model = InverseControlledTank()
get_sensitivity(model, :y)

Error & Stacktrace ⚠️

ERROR: UndefVarError: `filter₊x(t)[1]ˍtt` not defined
Stacktrace:
  [1] macro expansion
    @ ~/.julia/packages/SymbolicUtils/0opve/src/code.jl:375 [inlined]
  [2] macro expansion
    @ ~/.julia/packages/RuntimeGeneratedFunctions/M9ZX8/src/RuntimeGeneratedFunctions.jl:163 [inlined]
  [3] macro expansion
    @ ./none:0 [inlined]
  [4] generated_callfunc
    @ ./none:0 [inlined]
  [5] (::RuntimeGeneratedFunctions.RuntimeGeneratedFunction{…})(::Vector{…}, ::Vector{…}, ::Vector{…})
    @ RuntimeGeneratedFunctions ~/.julia/packages/RuntimeGeneratedFunctions/M9ZX8/src/RuntimeGeneratedFunctions.jl:150
  [6] f
    @ ~/.julia/packages/ModelingToolkit/3Suzf/src/systems/nonlinear/nonlinearsystem.jl:292 [inlined]
  [7] NonlinearFunction
    @ ~/.julia/packages/SciMLBase/roUH9/src/scimlfunctions.jl:2297 [inlined]
  [8] evaluate_f(prob::NonlinearLeastSquaresProblem{…}, u::Vector{…})
    @ NonlinearSolve ~/.julia/packages/NonlinearSolve/m8oNC/src/internal/helpers.jl:7
  [9] __init(::NonlinearLeastSquaresProblem{…}, ::NonlinearSolve.GeneralizedFirstOrderAlgorithm{…}; stats::SciMLBase.NLStats, alias_u0::Bool, maxiters::Int64, abstol::Nothing, reltol::Nothing, maxtime::Nothing, termination_condition::Nothing, internalnorm::Function, linsolve_kwargs::@NamedTuple{}, kwargs::@Kwargs{})
    @ NonlinearSolve ~/.julia/packages/NonlinearSolve/m8oNC/src/core/generalized_first_order.jl:164
 [10] __init
    @ ~/.julia/packages/NonlinearSolve/m8oNC/src/core/generalized_first_order.jl:154 [inlined]
 [11] __solve(::NonlinearLeastSquaresProblem{…}, ::NonlinearSolve.GeneralizedFirstOrderAlgorithm{…}; stats::SciMLBase.NLStats, kwargs::@Kwargs{})
    @ NonlinearSolve ~/.julia/packages/NonlinearSolve/m8oNC/src/core/generic.jl:3
 [12] __solve
    @ ~/.julia/packages/NonlinearSolve/m8oNC/src/core/generic.jl:1 [inlined]

Environment (please complete the following information):

MTK downstream environment

[ChrisRackauckas]

Test this on latest master with the new fixes.

[AayushSabharwal]

Still broken

[ChrisRackauckas]

@AayushSabharwal looks like this issue is back to you now, some issue with the p_getter. Then I think we should add it as a test.

[AayushSabharwal]

The p_getter issue and another one are fixed in #2912 but the original issue here still exists with the same error