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Address comment about systematic formatting
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henrikt-ma committed Jan 18, 2022
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Expand Up @@ -160,6 +160,6 @@ \chapter{Modelica DAE Representation}\label{modelica-dae-representation}
To summarize, symbolic transformation techniques are needed to transform \eqref{eq:hydrid-dae} into a set of equations which can be numerically solved reliably.
Most important, the algorithm of Pantelides should to be applied to differentiate certain parts of the equations in order to reduce the index.
Note, that also explicit integration methods, such as Runge-Kutta algorithms, can be used to solve \eqref{eq:dae}, after the index of \eqref{eq:dae} has been reduced by the Pantelides algorithm: During continuous integration, the integrator provides $x$ and $t$.
% TODO: Structured formatting of inline "upright 'd' fraction".
Then, \eqref{eq:dae} is a linear or nonlinear system of equations to compute the algebraic variables $y$ and the state derivatives $\mathrm{d}x/\mathrm{d}t$ and the model returns $\mathrm{d}x/\mathrm{d}t$ to the integrator by solving these systems of equations.
Then, \eqref{eq:dae} is a linear or nonlinear system of equations to compute the algebraic variables $y$ and the state derivatives $\udfrac{x}{t}$ and the model returns $\udfrac{x}{t}$ to the integrator by solving these systems of equations.
Often, \eqref{eq:dae} is just a linear system of equations in these unknowns, so that the solution is straightforward.
This procedure is especially useful for real-time simulation where usually explicit one-step methods are used.

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