From 2cedebf023a45600d78122597b1da0a9d23e8b45 Mon Sep 17 00:00:00 2001
From: Henrik Tidefelt <henrikt@wolfram.com>
Date: Wed, 27 Apr 2022 21:03:49 +0200
Subject: [PATCH] Illustrate potential implicit nature of discrete-time Real
 variable equations

---
 chapters/dae.tex | 26 ++++++++++++++++++++++++++
 1 file changed, 26 insertions(+)

diff --git a/chapters/dae.tex b/chapters/dae.tex
index 2d15b6a1c..830c78c88 100644
--- a/chapters/dae.tex
+++ b/chapters/dae.tex
@@ -84,6 +84,32 @@ \chapter{Modelica DAE Representation}\label{modelica-dae-representation}
 Further, each of the original model equations behind \eqref{eq:dae-discrete-valued} must be given in solved form, at most requiring flipping sides of the equation to obtain the used assignment form.
 The interpretation of the non-solved form of \eqref{eq:dae-discrete-real} at events, on the other hand, is that at events, the discrete-time \lstinline!Real! variables $z$ are solved together with the continuous-time variables using \eqref{eq:dae} and \eqref{eq:dae-discrete-real}.
 
+\begin{example}
+The following model demonstrates that \cref{eq:dae-discrete-real} does not imply that all discrete-time \lstinline!Real! variables are given by equations in solved form, as also the discrete-time \lstinline!Real! variables are included in $z$:
+\begin{lstlisting}[language=modelica]
+model M
+  discrete Real x(start = 1.0, fixed = true);
+equation
+  when sample(1.0, 1.0) then
+    x = 3 * pre(x) - x^2; // Valid equation for discrete-time Real variable x.
+  end when;
+end M;
+\end{lstlisting}
+
+Another way that a discrete-time \lstinline!Real! variable can end up becoming determined by a nonlinear equation is through coupling with other variables.
+The other variables may be discrete-time, or continuous-time as in the following example:
+\begin{lstlisting}[language=modelica]
+model M
+  discrete Real x(start = 1.0, fixed = true);
+  Real y;
+equation
+  y = x ^ 2 + 2 * exp(-time);
+  when sample(1.0, 1.0) then
+    x = 3 * pre(x) - y; // OK, forming nonlinear equation system with y.
+  end when;
+end M;\end{lstlisting}
+\end{example}
+
 \begin{example}
 The following model is illegal since there is no equation in solved form that can be used in \eqref{eq:dae-discrete-valued} to solve for the discrete-valued variable \lstinline!y!:
 \begin{lstlisting}[language=modelica]