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DAE_Initialize.jl
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DAE_Initialize.jl
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### A Pluto.jl notebook ###
# v0.19.43
using Markdown
using InteractiveUtils
# ╔═╡ 0ac4226f-9a06-4378-848e-ccd04f3b1af5
using ModelingToolkit, OrdinaryDiffEq, Plots
# ╔═╡ 9943ce31-d69f-4821-b202-2ae42537aaf1
using ModelingToolkit: t_nounits as t, D_nounits as D
# ╔═╡ 878606f4-3d82-11ef-0653-47b3810fe0c5
md"""
# Initialization of ModelingToolkit DAEs
Now we are deep into ModelingToolkit. We have learned that we can just write out equations, but in general building our model using reusable physical components instead requires acausal modeling, and acausal modeling gives us DAEs. We learned that DAE systems can be rather complicated in terms of how they must be specified in order to be solved, but ModelingToolkit has a deep simplification engine that transforms the equations into a solvable form.
But if I don't even know what my equations or state variables are going to be because MTK can just delete some at will... how do I specify my initial conditions? This is called the initializaiton problem in DAEs, and it's a surprisingly deep topic.
In the context of ODEs, there is no such thing as an initialization step. You give an initial condition, and that is that. Your initial condition is almighty, the solver just uses it. But now let's consider a DAE:
"""
# ╔═╡ d52b1b72-8c28-4551-9f61-7ce60162a7b6
@variables a(t) b(t)
# ╔═╡ 6ab9036b-dad2-4058-b3f1-7c82e771a9bd
[D(a) ~ b
b ~ a]
# ╔═╡ 8cdd3ceb-d670-4586-af21-77eed372f7b2
md"""
Now we give our system the initial conditions `a(0) = 1` and `b(0) = 0`... do you see what just happened? Since `a = b`, we actually need to have `a(0) = b(0)`. So we actually only need one initial condition, even though there's two variables for two equations.
It turns out these redundancies can come from all over, and they can be amplified by things like the dummy derivative transformation. Let's go back over to our trusty old cartesian pendulum which we now know to be much more complicated than one would first think. This system is:
"""
# ╔═╡ 07447ee0-4766-4fd9-8378-43d934c0abb1
@parameters g = 1 #normalize gravity to 1
# ╔═╡ 9d02d920-2cb8-4f73-ae76-293327103a51
@variables x(t) y(t) [state_priority = 10] λ(t)
# ╔═╡ ece89d12-0c7a-4955-bcf6-e7fb49bad2e1
eqs = [D(D(x)) ~ λ * x
D(D(y)) ~ λ * y - g
x^2 + y^2 ~ 1]
# ╔═╡ 6e686160-d3e3-46e3-bad4-85d8fe962271
@mtkbuild pend = ODESystem(eqs, t)
# ╔═╡ 9e343330-3e50-463a-b1b4-f05503aa4d8f
md"""
Notice that because it chose to eliminate the equations for `y`, what we actually need to give are only the initial conditions for `y(0)` and `D(y(0))`, everything else can be found through the implicit constraint equations.
"""
# ╔═╡ f5bd5b0c-2452-4848-b164-7b777276d7b8
prob_yinit = ODEProblem(pend, [y => 0, D(y) => 0], (0.0, 1.5), [g => 1], guesses = [λ => 1, x => 1.0])
# ╔═╡ fa39bbdb-fd60-4e5e-a123-959b0b2f5618
md"""
Notice that for values which are not provided we give a guess so that it knows how to solve for the other variables given the nonlinear systems. This gives a system that can be solved:
"""
# ╔═╡ cc96cca1-fcb7-4849-9812-49de9cb15aa0
# ╠═╡ disabled = true
#=╠═╡
sol = solve(prob_yinit)
╠═╡ =#
# ╔═╡ 9cabe0b7-5b9d-4f98-b304-7d7f72ba6d01
#=╠═╡
plot(sol, idxs = (x,y))
╠═╡ =#
# ╔═╡ 03c0241e-e4d6-4f9b-9dfd-7030c95ff3f0
md"""
But saying "[y => 0, D(y) => 0]" as the initial condition is kind of weird. You don't always know the starting velocity of the system? What if I just know (x,y)? Then you can give that:
"""
# ╔═╡ e8bad433-71d9-427d-92da-961583b72e3f
sol2 = solve(ODEProblem(pend, [x => 1, y => 0], (0.0, 1.5), [g => 1], guesses = [λ => 1]))
# ╔═╡ bc976a5a-7ea4-41e6-9ad4-756a193d7bb0
#=╠═╡
plot(sol, idxs = (x,y))
╠═╡ =#
# ╔═╡ c29dd6c5-f77c-4c00-9309-7131a55e0f9f
md"""
It's the same thing! ModelingToolkit's intiailization system auto-magically builds a system that re-translates the given set of initial conditions into the variables it knows the solver needs. This means you don't need to worry about giving "the right variables", just any consistent set. We can also do more advanced things, like make an initial condition dependent on a parameter or expression:
"""
# ╔═╡ 09be6705-d96b-4476-8158-20711f80c533
sol3 = solve(ODEProblem(pend, [x => 1, D(y) => g], (0.0, 1.5), [g => 1], guesses = [λ => 1, y => 1]))
# ╔═╡ f6ea7b24-494e-489e-b58b-6296c2226376
md"""
You can initialize the derivatives of variables to be values, and create custom equations to be satisfied. There's lots of flexability in what is allowed.
But what happens if we have an incorrect number of conditions? MTK will warn you about that scenario and do a smart thing.
"""
# ╔═╡ 5bc91aae-c603-4fe1-9534-02027143305b
prob_under = ODEProblem(pend, [x => 1], (0.0, 1.5), [g => 1], guesses = [y => 0, λ => 1])
# ╔═╡ 1eab9725-162f-4acf-9ea5-ab55ba93ddb8
md"""
An underdetermined system has a non-unique solution to its initialization, and thus you will get a warning but it will let you solve it and pick a solution of the set of possible solutions that satisfy the given conditions:
"""
# ╔═╡ df9e3e7f-e330-49e0-aadc-c754d4ba2e00
sol_under = solve(prob_under)
# ╔═╡ 79bcda76-9d2a-47bb-a98a-adb5d90170ad
md"""
In the opposite direction, an overdetermined system similarly gives a warning:
"""
# ╔═╡ bba3b975-ae95-4ca8-a8a5-85c21a781e2f
prob_over = ODEProblem(
pend, [x => 1, y => 0.0, D(y) => 0], (0.0, 1.5), [g => 1], guesses = [λ => 1])
# ╔═╡ 2b91406a-0a3f-4821-ae98-d16c3f7ad139
md"""
In this case, the issue is whether there are any initial conditions that satisfy all of the conditions.
"""
# ╔═╡ 8184f96d-4c9e-434f-8650-9abe7fe4f540
sol_over = solve(prob_over, Rodas5P())
# ╔═╡ 050df185-9bf4-4f6e-8adc-d5f949f4a29c
md"""
This worked! But what if we change the conditions?
"""
# ╔═╡ 4aae580f-7d31-44a9-9c09-4eedb684acaa
prob_over2 = ODEProblem(
pend, [x => 1, y => 0.0, D(y) => 2.0, λ => 1], (0.0, 1.5), [g => 1], guesses = [λ => 1])
# ╔═╡ 8cea6063-1335-4e8b-bb45-336e27a74795
sol_over2 = solve(prob_over2, Rodas5P())
# ╔═╡ 18114009-0b4e-419d-9abe-7f5ec3bbf2bd
sol_over2.retcode
# ╔═╡ 6106d073-2974-45b4-8a8a-f68f0a0bbce1
md"""
The system shows that there is an initialization failure, which means it could not find a compatible initial condition.
## Diving Deeper: Investigating the Initialization System
The initialization system is just a NonlinearSystem itself, so you can construct it and investigate it:
"""
# ╔═╡ 13f31239-1d19-4da3-9cdb-fda92e4f7738
isys = generate_initializesystem(pend, u0map = [x => 1.0, y => 0.0], guesses = [λ => 1])
# ╔═╡ a2ad0ca2-581f-432c-b7f3-bc0aba03012b
md"""
You can use this in order to investgate why you have so many/few quesitons, and what equations are missing. Note that if you want to simplify a over/underdetermined system, you need to specify the simplification with `fully_determined=false`:
"""
# ╔═╡ c8d250ea-f065-4502-b61d-e91d5b22b37e
isys_simp = structural_simplify(isys; fully_determined = false)
# ╔═╡ 4bab01f6-b25e-45cb-897d-dc3fcfb90c43
md"""
Using this information, let's find out why our overdetermined system was not initializable:
"""
# ╔═╡ ca18973f-6edb-461e-8d18-0c1926c598e3
isys2 = ModelingToolkit.generate_initializesystem(
pend, u0map = [x => 1, y => 0.0, D(y) => 2.0, λ => 1], guesses = [λ => 1])
# ╔═╡ 64b3447f-cbc2-45bf-89f4-28fb59d7a42e
isys2_simp = structural_simplify(isys2; fully_determined = false)
# ╔═╡ 950cc455-321c-43ed-9296-9b8b6fd2cfdc
unknowns(isys2_simp)
# ╔═╡ 010a19a4-a871-4030-a3d2-9970c6fca7eb
observed(isys2_simp)
# ╔═╡ bf1bd7f7-e291-43e4-a712-434cf5426e15
subdict = Dict(getfield.(observed(isys2_simp),:lhs) .=> getfield.(observed(isys2_simp),:rhs))
# ╔═╡ b71818de-4bad-43af-ab54-01de8b091f3a
ModelingToolkit.fixpoint_sub.(equations(isys2_simp), (subdict,))
# ╔═╡ 71fa338a-f5f2-4bdb-9f04-c915e9aaf4ff
md"""
By substituting the observed variables in, we can see that one of the equations is requiring D(x) = 0, while another is requiring that D(x) is non-zero. That's clearly impossible, and thus it's clear that the numerical solving of this system will not be possible.
While more difficult systems may not be as easy to investigate, this shows how to interactively use the symbolic manipulation system with ModelingToolkit's system transformations in order to really investigate what's going on in the equations!
## Conclusion
We can end here even though there is so much more to show. But this shows how all of the pieces, from symbolic-numeric optimizations to symbolic simplifications, all mixed with the numerical methods, combines to give a system that makes modeling and simulation easier.
"""
# ╔═╡ 00000000-0000-0000-0000-000000000001
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