added: NonLinModelDAE for differential and algebraic equations + sim! support

[franckgaga]

The new SimModel type introduces the support of plant model expressed as differential and algebraic equation (DAE) systems. The API and the code base rely on the following notation (to avoid any confusions with the decision variable $\mathbf{Z}$ and the nonlinear inequality constraint $\mathbf{g}$):

$$\begin{aligned} \mathbf{\dot{x}}(t) &= \mathbf{f}\Big( \mathbf{x}(t), \mathbf{a}(t), \mathbf{u}(t), \mathbf{d}(t), \mathbf{p} \Big) \\\ \mathbf{0} &= \mathbf{q}\Big( \mathbf{x}(t), \mathbf{a}(t), \mathbf{u}(t), \mathbf{d}(t), \mathbf{p} \Big) \\\ \mathbf{y}(t) &= \mathbf{h}\Big( \mathbf{x}(t), \mathbf{a}(t), \mathbf{d}(t), \mathbf{p} \Big) \end{aligned}$$

Pure discrete-time DAEs are not supported for now. This PR added the functionalities for open-loop simulations of DAEs using collocation methods (TrapezoidalCollocation or OrthogonalCollocation). The following generic functions now support NonLinModelDAE objects:

• evaloutput(model::NonLinModelDAE, d) -> y : solve the algebraic equation (if required, see below) and compute the current output $\mathbf{y}(k)$
• updatestate!(model::NonLinModelDAE, u, d) -> xnext : solve the collocation problem and compute the next state $\mathbf{x}(k+1)$
• getinfo(model::NonLinModelDAE) -> info : return information post-solving for troubleshooting, for example $\mathbf{a}(k)$
• sim!(model::NonLinMidelDAE, N, u, <kw args>) -> res: quick open-loop simulations, similar other model types.

As described in #332 and the docstring, the open-loop simulation code base uses JuMP.jl, MOI.VectorNonlinearOracle and DifferentiationInterface.jl to solve the two nonlinear root problems of evalouput and updatestate!. They both use Ipopt.jl and dense ForwardDiff.jl Jacobians by default. Note that SparseConnectivityTracer.jl will analyze the system and return an error if it is not strictly proper. Also note that evaloutput will skip the algebraic equation solving if it is not required, according to SparseConnectivityTracer.jl.

To global goal here is to support plant model generated by acausal modeling tools like ModelingToolkit.jl, since they typically produces DAEs instead of ODEs. I still want to mention that ODEs are insanely simpler in terms of algorithmic complexity, so NonLinModel objects for ODEs should be used when possible to reduce the dimensions and complexity (thanks to multiple dispatch!).

Important

NonLinMPC and MovingHorizonEstimator do not support NonLinModelDAE objects for now.

Related to #332

[codecov-commenter]

Codecov Report

❌ Patch coverage is 93.07159% with 30 lines in your changes missing coverage. Please review.
✅ Project coverage is 96.93%. Comparing base (4275fa7) to head (fad8cf5).
⚠️ Report is 81 commits behind head on main.

Files with missing lines	Patch %	Lines
src/model/nonlinmodeldae.jl	92.94%	23 Missing ⚠️
src/estimator/execute.jl	50.00%	2 Missing ⚠️
src/controller/execute.jl	80.00%	1 Missing ⚠️
src/model/linearization.jl	50.00%	1 Missing ⚠️
src/model/linmodel.jl	75.00%	1 Missing ⚠️
src/model/nonlinmodel.jl	66.66%	1 Missing ⚠️
src/sim_model.jl	83.33%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #442      +/-   ##
==========================================
- Coverage   97.17%   96.93%   -0.24%     
==========================================
  Files          27       28       +1     
  Lines        6270     6600     +330     
==========================================
+ Hits         6093     6398     +305     
- Misses        177      202      +25     

Flag	Coverage Δ
unittests	96.93% <93.07%> (-0.24%)	⬇️

Flags with carried forward coverage won't be shown. Click here to find out more.

☔ View full report in Codecov by Harness.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.

[franckgaga]

The failling test on Win11 and LTS is unrelated. Seems to be an issue with and old manifest file and the cache on GitHub actions.