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docs/make.jl

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@@ -22,7 +22,8 @@ mathengine = MathJax3(Dict(:loader => Dict("load" => ["[tex]/require", "[tex]/ma
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makedocs(sitename = "ModelingToolkit.jl",
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authors = "Chris Rackauckas",
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modules = [ModelingToolkit],
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clean = true, doctest = false,
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clean = true, doctest = false, linkcheck = true,
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linkcheck_ignore = ["https://epubs.siam.org/doi/10.1137/0903023"],
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strict = [
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:doctest,
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:linkcheck,

docs/src/basics/Linearization.md

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This means that to simulate this system, some order of derivatives of the input is required. To allow `linearize` to proceed in this situation, one may pass the keyword argument `allow_input_derivatives = true`, in which case the resulting model will have twice as many inputs, ``2n_u``, where the last ``n_u`` inputs correspond to ``\dot u``.
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If the modeled system is actually proper (but MTK failed to find a proper realization), further numerical simplification can be applied to the resulting statespace system to obtain a proper form. Such simplification is currently available in the experimental package [ControlSystemsMTK](https://github.com/baggepinnen/ControlSystemsMTK.jl#internals-transformation-of-non-proper-models-to-proper-statespace-form).
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If the modeled system is actually proper (but MTK failed to find a proper realization), further numerical simplification can be applied to the resulting statespace system to obtain a proper form. Such simplification is currently available in the package [ControlSystemsMTK](https://juliacontrol.github.io/ControlSystemsMTK.jl/dev/#Internals:-Transformation-of-non-proper-models-to-proper-statespace-form).
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## Tools for linear analysis
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docs/src/examples/modelingtoolkitize_index_reduction.md

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It turns out that higher order DAEs can be transformed into lower order DAEs.
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[If you differentiate the last equation two times and perform a substitution,
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you can arrive at the following set of equations](https://courses.seas.harvard.edu/courses/am205/g_act/dae_notes.pdf):
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you can arrive at the following set of equations](https://people.math.wisc.edu/~chr/am205/g_act/DAE_slides.pdf):
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```math
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\begin{aligned}

docs/src/examples/perturbation.md

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## Prelims
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In the previous tutorial, [Mixed Symbolic-Numeric Perturbation Theory](https://symbolics.juliasymbolics.org/stable/examples/perturbation), we discussed how to solve algebraic equations using **Symbolics.jl**. Here, our goal is to extend the method to differential equations. First, we import the following helper functions that were introduced in [Mixed Symbolic/Numerical Methods for Perturbation Theory - Algebraic Equations](@ref perturb_alg):
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In the previous tutorial, [Mixed Symbolic-Numeric Perturbation Theory](https://symbolics.juliasymbolics.org/stable/examples/perturbation/), we discussed how to solve algebraic equations using **Symbolics.jl**. Here, our goal is to extend the method to differential equations. First, we import the following helper functions that were introduced in [Mixed Symbolic/Numerical Methods for Perturbation Theory - Algebraic Equations](@ref perturb_alg):
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```julia
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using Symbolics, SymbolicUtils

docs/src/tutorials/acausal_components.md

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circuit component. At every input and output pin, a circuit component has
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two values: the current at the pin and the voltage. Thus we define the `Pin`
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component (connector) to simply be the values there. Whenever two `Pin`s in a
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circuit are connected together, the system satisfies [Kirchhoff's laws](https: //en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws),
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circuit are connected together, the system satisfies [Kirchhoff's laws](https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws),
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i.e. that currents sum to zero and voltages across the pins are equal.
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`[connect = Flow]` informs MTK that currents ought to sum to zero, and by
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default, variables are equal in a connection.

docs/src/tutorials/parameter_identifiability.md

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Both parameters `b, c` are globally identifiable with probability `0.9` in this case.
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[^1]: > R. Munoz-Tamayo, L. Puillet, J.B. Daniel, D. Sauvant, O. Martin, M. Taghipoor, P. Blavy [*Review: To be or not to be an identifiable model. Is this a relevant question in animal science modelling?*](https://doi.org/10.1017/S1751731117002774), Animal, Vol 12 (4), 701-712, 2018. The model is the ODE system (3) in Supplementary Material 2, initial conditions are assumed to be unknown.
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[^2]: > Moate P.J., Boston R.C., Jenkins T.C. and Lean I.J., [*Kinetics of Ruminal Lipolysis of Triacylglycerol and Biohydrogenationof Long-Chain Fatty Acids: New Insights from Old Data*](doi:10.3168/jds.2007-0398), Journal of Dairy Science 91, 731–742, 2008
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[^2]: > Moate P.J., Boston R.C., Jenkins T.C. and Lean I.J., [*Kinetics of Ruminal Lipolysis of Triacylglycerol and Biohydrogenationof Long-Chain Fatty Acids: New Insights from Old Data*](https://doi.org/10.3168/jds.2007-0398), Journal of Dairy Science 91, 731–742, 2008
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[^3]: > Goodwin, B.C. [*Oscillatory behavior in enzymatic control processes*](https://doi.org/10.1016/0065-2571(65)90067-1), Advances in Enzyme Regulation, Vol 3 (C), 425-437, 1965
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[^4]: > Dong, R., Goodbrake, C., Harrington, H. A., & Pogudin, G. [*Computing input-output projections of dynamical models with applications to structural identifiability*](https://arxiv.org/pdf/2111.00991). arXiv preprint arXiv:2111.00991.

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