Attempt at defining initialization, and simulation. (modelica#3146)

* Attempt at defining initialization, simulation, and avoid defining transient analysis.
* Move definition to 'Some definitions'.
* Define transient analysis and initialization.
* Do not use IVP here.
* Also define translation, since that ties together the simulation with the rest.
* Use 'transient analysis' when appropriate.
* Add more detailed description of well-determined initialization problem.
* Apply suggestions from code review
* Split non-normative paragraphs to make them closer to specific definitions.

Co-authored-by: Elena Shmoylova <eshmoylova@users.noreply.github.com>
Co-authored-by: Henrik Tidefelt <henrikt@wolfram.com>

chapters/classes.tex

@@ -351,13 +351,14 @@ \subsection{Acyclic Bindings of Constants and Parameters}\label{acyclic-bindings
 
 \subsection{Component Variability Prefixes}\label{component-variability-prefixes-discrete-parameter-constant}
 
-The prefixes \lstinline!discrete!, \lstinline!parameter!, \lstinline!constant! of a component declaration are called \firstuse[variability!prefix]{variability prefixes}\index{component variability}\index{declared variability}\index{variability!declared|see{declared variability}} and define in which situation the variable values of a component are initialized (see \cref{events-and-synchronization} and \cref{initialization-initial-equation-and-initial-algorithm}) and when they are changed in transient analysis (= solution of initial value problem of the hybrid DAE):
+The prefixes \lstinline!discrete!, \lstinline!parameter!, \lstinline!constant! of a component declaration are called \firstuse[variability!prefix]{variability prefixes}\index{component variability}\index{declared variability}\index{variability!declared|see{declared variability}} and define in which situation the variable values of a component are initialized (see \cref{events-and-synchronization} and \cref{initialization-initial-equation-and-initial-algorithm}) and when they are changed during simulation:
 \begin{itemize}
 \item
-  A variable \lstinline!vc! declared with \lstinline!constant!\indexinline{constant} prefix remains constant during transient analysis, with a value that is unaffected by the initialization problem.
+  A variable \lstinline!vc! declared with \lstinline!constant!\indexinline{constant} prefix remains constant during simulation, with a value that is unaffected even by the initialization problem.
   This is called a \firstuse[---]{constant}, or \firstuse[constant!variable]{constant variable}\index{component variability!constant}.
 \item
-  A variable \lstinline!vc! declared with the \lstinline!parameter!\indexinline{parameter} prefix remains constant during transient analysis, with a value determined by the initialization problem.
+  A variable \lstinline!vc! declared with the \lstinline!parameter!\indexinline{parameter} prefix has a value determined at initialization, and keeps that value during the entire simulation.
+  Thus it is known and non-changing during transient analysis.
   This is called a \firstuse[---]{parameter}, or \firstuse[parameter!variable]{parameter variable}\index{component variability!parameter}.
 \item
   A \firstuse[discrete-time!variable]{discrete-time variable}\index{component variability!discrete-time} \lstinline!vd! is a variable that is discrete-valued (that is, not of \lstinline!Real! type) or assigned in a \lstinline!when!-clause.

chapters/dae.tex

@@ -98,10 +98,12 @@ \chapter{Modelica DAE Representation}\label{modelica-dae-representation}
 \end{example}
 
 The generated set of equations is used for simulation and other analysis activities.
-Simulation means that an initial value problem is solved, i.e., initial values have to be provided for the states $x$, \cref{initialization-initial-equation-and-initial-algorithm}.
+Simulation proceeds as follows.
+First, initialization takes place, during which initial values for the states $x$ are found, \cref{initialization-initial-equation-and-initial-algorithm}.
+Given those initial values the equations are simulated forward in time; this is the transient analysis.
 The equations define a DAE\index{DAE} (Differential Algebraic Equations) which may have discontinuities, a variable structure and/or which are controlled by a discrete-event system.
 Such types of systems are called \firstuse[hybrid DAE]{hybrid DAEs}.
-Simulation is performed in the following way:
+After initialization, simulation proceeds with transient analysis in the following way:
 \begin{enumerate}
 \item\label{perform-simulation-integrate}
   The DAE \eqref{eq:dae} is solved by a numerical integration method.

chapters/equations.tex

@@ -898,9 +898,9 @@ \section{Initialization, initial equation, and initial algorithm}\label{initiali
 \subsection{Equations Needed for Initialization}\label{the-number-of-equations-needed-for-initialization}\label{equations-needed-for-initialization}
 
 \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.
+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 during transient analysis.
 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 solved for during simulation.
+During initialization of an ODE we will need to find the values of $2n+m$ variables, in contrast to just $n+m$ variables to be solved for during transient analysis.
 \end{nonnormative}
 
 \begin{example}
@@ -912,8 +912,9 @@ \subsection{Equations Needed for Initialization}\label{the-number-of-equations-n
 \end{lstlisting}
 
 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 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.
+A consistent solution of the initialization problem requires finding initial values for \lstinline!x1!, \lstinline!x2!, \lstinline!der(x1)!, \lstinline!der(x2)!, and \lstinline!y!.
+Two additional initial equations thus need to be provided to obtain a globally balanced initialization problem.
+Additionally, those two initial equations must be chosen with care to ensure that they, in combination with the dynamic equations, give a well-determined initialization problem.
 
 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 $n$.

chapters/introduction.tex

@@ -68,12 +68,41 @@ \section{Some Definitions}\label{some-definitions}
 \end{definition}
 
 \begin{definition}[Flattening]\index{flattening}
-The translation of a model described in Modelica to the corresponding model described as a hybrid DAE, involving expansion of inherited base classes, parameterization of base classes, local classes
-and components, and generation of connection equations from \lstinline!connect!-equations (basically, mapping the hierarchical structure of a model into a set of differential, algebraic and discrete equations together with the corresponding variable declarations and function definitions from the model).
+The translation of a model described in Modelica to the corresponding model described as a hybrid DAE (see \cref{modelica-dae-representation}), involving expansion of inherited base classes, parameterization of base classes, local classes and components, and generation of connection equations from \lstinline!connect!-equations.
+In other words, mapping the hierarchical structure of a model into a set of differential, algebraic and discrete equations together with the corresponding variable declarations and function definitions from the model.
 \end{definition}
 
+\begin{definition}[Initialization]\index{initialization}
+Simulation starts with solving the initialization problem at the starting time, resulting in values for all variables that are consistent with the result of the flattening.
+\end{definition}
+
+\begin{definition}[Transient analysis]\index{transient analysis}\index{initial value problem!see{transient analysis}}
+Starting from the result of the initialization problem, the model is simulated forward in time.
+This uses numerical methods for handling the hybrid DAE, resulting in solution trajectories for the model's variables, i.e., the value of the variables as a function of time.
+\end{definition}
+
+\begin{nonnormative}
+In the numerical literature transient analysis is often called solving the \firstuse{initial value problem}, but that term is not used here to avoid confusion with the initialization problem.
+\end{nonnormative}
+
+\begin{definition}[Simulation]\index{simulation}
+Simulation is the combination of initialization followed by transient analysis.
+\end{definition}
+
+\begin{nonnormative}
+The model can be analyzed in ways other than simulation, e.g., linearization, and parameter estimation, but they are not described in the specification.
+\end{nonnormative}
+
+\begin{definition}[Translation]\index{translation}
+Translation is the process of preparing a Modelica simulation model for simulation, starting with flattening but not including the simulation itself.
+\end{definition}
+
+\begin{nonnormative}
+Typically, in addition to flattening, translation involves symbolic manipulation of the hybrid DAE and transforming the result into computer code that can simulate the model.
+\end{nonnormative}
+
+
 % More terms that would be useful to define here:
-% - translation (for phrases such as "during translation")
 % - deprecated
 % - quality of implementation