Add analysis points

[baggepinnen]

This PR adds something called an AnalysisPoint. These are used for linear analyses, such as

• Computing sensitivity functions.
• Computing complementary sensitivity functions.
• Computing loop-transfer functions.
• Linearizing systems between predefined analysis points.
• Open loops

An AnalysisPoint can be created explicitly, but also automatically when connecting an output to an input. Analysis points are causal, and are thus defined in the Blocks submodule. They thus expect connections of type Blocks.RealInput and Blocks.ReaOutput.

Notes to reviewers

• The current state of the PR does not handle namespaces and subsystems properly. In particular, I'm unsure about the proper way of modifying equations and namespaces when changing the equations in a system with subsystems. Namespaces are now handled.
• The user must currently call expand_analysis_points manually before simulating a system containing analysis points. It would be good if this would be handled automatically by structural_simplify. The current implementation requires expand_analysis_points to run before expand_connections, but I could not find a clean way of extending this functionality from outside of ModelingToolkit. One approach would be to define AnalysisPoint in MTK, but keep the functions related to sensitivity functions here (they require components from MTKStdlib)
• The last commit depends on the PR below
• Add namespace_expr for Connection ModelingToolkit.jl#1827

[codecov]

Codecov Report

Merging #110 (c90780a) into main (a8bb688) will increase coverage by 0.53%.
The diff coverage is 70.86%.

@@            Coverage Diff             @@
##             main     #110      +/-   ##
==========================================
+ Coverage   66.92%   67.45%   +0.53%     
==========================================
  Files          24       25       +1     
  Lines        1052     1177     +125     
==========================================
+ Hits          704      794      +90     
- Misses        348      383      +35     

Impacted Files	Coverage Δ
src/Blocks/analysis_points.jl	70.86% <70.86%> (ø)
src/Blocks/utils.jl	61.53% <0.00%> (+4.39%)	⬆️

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[baggepinnen]

Unrelated change made by the formatter

[YingboMa]

Remember compat.

[baggepinnen]

This is in docs/Project.toml where no packages have compat so far :P
https://github.com/SciML/ModelingToolkitStandardLibrary.jl/blob/de1151f3381b6f4c9a3205c767a076d1627308e6/docs/Project.toml

[YingboMa]

flatten doesn't work with connect, as mentioned in SciML/ModelingToolkit.jl#1826 (comment) . It should be deprecated and removed. Could you take a look at the generate_connection_* functions in MTK? https://github.com/SciML/ModelingToolkit.jl/blob/644d76c73cbeb17458f606c7b299bd450edb0c44/src/systems/connectors.jl Also, it's expensive to traverse the model. It may make sense to add this in MTK. Alternatively, we should provide interfaces that makes expanding connection equations generic.

[YingboMa]

This is roughly how connection generation works in MTK.

julia> sys_without_connections, connection_sets = ModelingToolkit.generate_connection_set(rc_model);

julia> connection_sets
7-element Vector{ModelingToolkit.ConnectionSet}:
 <constant₊output.u(t)::inner, source₊V.u(t)::inner>
 <source₊p.v(t)::inner, resistor₊p.v(t)::inner>
 <source₊p.i(t)::inner, resistor₊p.i(t)::inner>
 <resistor₊n.v(t)::inner, capacitor₊p.v(t)::inner>
 <capacitor₊p.i(t)::inner, resistor₊n.i(t)::inner>
 <capacitor₊n.v(t)::inner, source₊n.v(t)::inner, ground₊g.v(t)::inner>
 <capacitor₊n.i(t)::inner, ground₊g.i(t)::inner, source₊n.i(t)::inner>

julia> connection_equations, stream_connection_sets = ModelingToolkit.generate_connection_equations_and_stream_connections(connection_sets);

julia> connection_equations
8-element Vector{Equation}:
 constant₊output₊u(t) ~ source₊V₊u(t)
 source₊p₊v(t) ~ resistor₊p₊v(t)
 0 ~ resistor₊p₊i(t) + source₊p₊i(t)
 resistor₊n₊v(t) ~ capacitor₊p₊v(t)
 0 ~ capacitor₊p₊i(t) + resistor₊n₊i(t)
 capacitor₊n₊v(t) ~ source₊n₊v(t)
 capacitor₊n₊v(t) ~ ground₊g₊v(t)
 0 ~ capacitor₊n₊i(t) + ground₊g₊i(t) + source₊n₊i(t)