From 16f659f7b4edba6b05124840beacafbb83b134e7 Mon Sep 17 00:00:00 2001 From: Henrik Tidefelt Date: Wed, 3 Jun 2020 00:31:23 +0200 Subject: [PATCH] Apply some math formatting --- chapters/equations.tex | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/chapters/equations.tex b/chapters/equations.tex index a98fb58b4..c55b397d2 100644 --- a/chapters/equations.tex +++ b/chapters/equations.tex @@ -939,11 +939,11 @@ \subsection{The Number of Equations Needed for Initialization}\doublelabel{the-n \begin{nonnormative} In general, for the case of a pure (first order) ordinary -differential equation (ODE) system with n state variables and m output -variables, we will have n+m unknowns in the simulation problem. The ODE -initialization problem has n additional unknowns corresponding to the +differential equation (ODE) system with $n$ state variables and $m$ output +variables, we will have $n+m$ unknowns in the simulation problem. The ODE +initialization problem has $n$ additional unknowns corresponding to the derivative variables. At initialization of an ODE we will need to find -the values of 2n+m variables, in contrast to just n+m variables to be +the values of $2n+m$ variables, in contrast to just $n+m$ variables to be solved for during simulation. \end{nonnormative} @@ -957,17 +957,17 @@ \subsection{The Number of Equations Needed for Initialization}\doublelabel{the-n Here we have three variables with unknown values: two dynamic variables that also are state variables, \lstinline!x1! and \lstinline!x2!, i.e., -n=2, one output variable \lstinline!y!, i.e., m=1, and one input variable \lstinline!u! with +$n=2$, one output variable \lstinline!y!, i.e., $m=1$, and one input variable \lstinline!u! with known value. A consistent solution of the initial value problem providing initial values for \lstinline!x1!, \lstinline!x2!, \lstinline!der(x1)!, \lstinline!der(x2)!, and \lstinline!y! needs to be found. Two additional initial equations thus need to be provided to solve the initialization problem. -Regarding DAEs, only that at most n additional equations are -needed to arrive at 2n+m equations in the initialization system. The +Regarding DAEs, only that at most $n$ additional equations are +needed to arrive at $2n+m$ equations in the initialization system. The reason is that in a higher index DAE problem the number of dynamic continuous-time state variables might be less than the number of state -variables n. As noted in \autoref{initialization-initial-equation-and-initial-algorithm} a tool may add/remove +variables $n$. As noted in \autoref{initialization-initial-equation-and-initial-algorithm} a tool may add/remove initial equations to fulfill this requirement, if appropriate diagnostics are given. \end{example}