Modelica® - A Unified Object-Oriented Language for Systems Modeling Language Specification Version 3.5-dev

Chapter 8 Equations

8.1 Equation Categories

Equations in Modelica can be classified into different categories depending on the syntactic context in which they occur:

  • Normal equality equations occurring in equation sections, including connect-equations and other equation types of special syntactic form (section 8.3)

  • Declaration equations, which are part of variable, parameter, or constant declarations (section 4.4.2.1).

  • Modification equations, which are commonly used to modify attributes of classes (section 7.2)

  • Binding equations, include both declaration equations and modification equations (for the value itself).

  • Initial equations, which are used to express equations for solving initialization problems (section 8.6)

8.2 Flattening and Lookup in Equations

A flattened equation is identical to the corresponding nonflattened equation.

Names in an equation shall be found by looking up in the partially flattened enclosing class of the equation.

8.3 Equations in Equation Sections

The following kinds of equations may occur in equation sections. The syntax is defined as follows:

equation :
( simple-expression "=" expression
   | if-equation
   | for-equation
   | connect-clause
   | when-equation
   | component-reference function-call-args )
   comment

No statements are allowed in equation sections, including the assignment statement using the := operator.

8.3.1 Simple Equality Equations

Simple equality equations are the traditional kinds of equations known from mathematics that express an equality relation between two expressions. There are two syntactic forms of such equations in Modelica. The first form below is equality equations between two expressions, whereas the second form is used when calling a function with several results. The syntax for simple equality equations is as follows:

simple-expression "=" expression

The types of the left-hand-side and the right-hand-side of an equation need to be compatible in the same way as two arguments of binary operators (section 6.6).

Three examples:

  • simple_expr1 = expr2;

  • (if pred then alt1 else alt2) = expr2;

  • (out1, out2, out3) = function_name(inexpr1, inexpr2);

[Note: According to the grammar the if-then-else expression in the second example needs to be enclosed in parentheses to avoid parsing ambiguities. Also compare with section 11.2.1.1 about calling functions with several results in assignment statements.]

8.3.2 For-Equations – Repetitive Equation Structures

The syntax of a for-equation is as follows:

for for-indices loop
   { equation ";" }
   end for ";"

For-equations may optionally use several iterators (for-indices), see section 11.2.2.3 for more information:

for-indices:
   for-index {"," for-index}
for-index:
   IDENT [ in expression ]

The following is one example of a prefix of a for-equation:

for IDENT in expression loop

The expression of a for-equation shall be a vector expression. It is evaluated once for each for-equation, and is evaluated in the scope immediately enclosing the for-equation. The expression of a for-equation shall be a parameter expression. The iteration range of a for-equation can also be specified as Boolean or as an enumeration type, see section 11.2.2.2 for more information. The loop-variable (IDENT) is in scope inside the loop-construct and shall not be assigned to. The loop-variable has the same type as the type of the elements of the vector expression.

[Example:

for i in 1:10 loop             // i takes the values 1,2,3,…,10
for r in 1.0 : 1.5 : 5.5 loop  // r takes the values 1.0, 2.5, 4.0, 5.5
for i in {1,3,6,7} loop        // i takes the values 1, 3, 6, 7
for i in TwoEnums loop         // i takes the values TwoEnums.one, TwoEnums.two
                               // for TwoEnums = enumeration(one,two)

The loop-variable may hide other variables as in the following example. Using another name for the loop-variable is, however, strongly recommended.

  constant Integer j=4;
  Real x[j]
equation
  for j in 1:j loop  // The loop-variable j takes the values 1,2,3,4
    x[j]=j;          // Uses the loop-variable j
  end for;

]

8.3.2.1 Implicit Iteration Ranges of For-Equations

The iteration range of a loop variable may sometimes be inferred from its use as an array index. See section 11.2.2.1 for more information.

[Example:

  Real x[n],y[n];
for i loop          // Same as:  for i in 1:size(x ,1) loop
  x[i] = 2*y[i];
end for;

]

8.3.3 Connect-Equations

A connect-equation has the following syntax:

connect "(" component-reference "," component-reference ")" ";"

These can be placed inside for-equations and if-equations; provided the indices of the for-loop and conditions of the if-clause are parameter expressions that do not depend on cardinality, rooted, Connections.rooted, or Connections.isRoot. The for-equations/if-equations are expanded. Connect-equations are described in detail in section 9.1.

The same restrictions apply to Connections.branch, Connections.root, and Connections.potentialRoot; which after expansion are handled according to section 8.3.9.

8.3.4 If-Equations

If-equations have the following syntax:

if expression then
   { equation ";" }
{ elseif expression then
   { equation ";" }    }
[ else
   { equation ";" }
]
end if ";"

The expression of an if- or elseif-clause must be a scalar Boolean expression. One if-clause, and zero or more elseif-clauses, and an optional else-clause together form a list of branches. One or zero of the bodies of these if-, elseif- and else-clauses is selected, by evaluating the conditions of the if- and elseif-clauses sequentially until a condition that evaluates to true is found. If none of the conditions evaluate to true the body of the else-clause is selected (if an else-clause exists, otherwise no body is selected). In an equation section, the equations in the body are seen as equations that must be satisfied. The bodies that are not selected have no effect on that model evaluation.

If-equations in equation sections which do not have exclusively parameter expressions as switching conditions shall have the same number of equations in each branch (a missing else is counted as zero equations and the number of equations is defined after expanding the equations to scalar equations). [If this condition is violated, the single assignment rule would not hold, because the number of equations may change during simulation although the number of unknowns remains the same].

8.3.5 When-Equations

When-equations have the following syntax:

when expression then
   { equation ";" }
{ elsewhen expression then
   { equation ";" } }
end when ";"

The expression of a when-equation shall be a discrete-time Boolean scalar or vector expression. The statements within a when-equation are activated when the scalar expression or any of the elements of the vector expression becomes true.

[Example:

The order between the equations in a when-equation does not matter, e.g.

equation
  when x > 2 then
    y3 = 2*x +y1+y2; // Order of y1 and y3 equations does not matter
    y1 = sin(x);
  end when;
  y2 = sin(y1);

]

8.3.5.1 Defining When-Equations by If-Expressions in Equality Equations

A when-equation:

equation
  when x>2 then
    v1 = expr1;
    v2 = expr2;
  end when;

is conceptually equivalent to the following equations containing special if-expressions

  // Not correct Modelica
  Boolean b(start=x.start>2);
equation
  b  = x>2;
  v1 = if edge(b) then expr1 else pre(v1);
  v2 = if edge(b) then expr2 else pre(v2);

[The equivalence is conceptual since pre() of a non discrete-time Real variable or expression can only be used within a when-clause. Example:

  /* discrete*/ Real x;
  input Real u;
  output Real y;
equation
  when sample() then
    x = a*pre(x)+b*pre(u);
  end when;
  y = x;

In this example x is a discrete-time variable (whether it is declared with the discrete prefix or not), but u and y cannot be discrete-time variables (since they are not assigned in when-clauses). However, pre(u) is legal within the when-clause, since the body of the when-clause is only evaluated at events, and thus all expressions are discrete-time expressions.]

The start-values of the introduced Boolean variables are defined by the taking the start-value of the when-condition, as above where b is a parameter variable. The start-values of the special functions initial, terminal, and sample is false.

8.3.5.2 Restrictions on Equations within When-Equations

  • When-statements may not occur inside initial equations.

  • When-equations cannot be nested.

[Example:

The following when-equation is invalid:

when x > 2 then
  when y1 > 3 then
    y2 = sin(x);
  end when;
end when;

]

The equations within the when-equation must have one of the following forms:

  • v = expr;

  • (out1, out2, out3, ...) = function_call_name(in1, in2, ...);

  • operators assert(), terminate(), reinit()

  • For- and if-equations if the equations within the for- and if-equations satisfy these requirements.

  • The different branches of when/elsewhen must have the same set of component references on the left-hand side.

  • The branches of an if-then-else clause inside when-equations must have the same set of component references on the left-hand side, unless the if-then-else have exclusively parameter expressions as switching conditions.

Any left hand side reference, (v, out1, …), in a when-clause must be a component reference, and any indices must be parameter expressions.

[The needed restrictions on equations within a when-equation becomes apparent with the following example:

  Real x, y;
equation
  x + y = 5;
  when condition then
    2*x + y = 7; // error: not valid Modelica
  end when;

When the equations of the when-equation are not activated it is not clear which variable to hold constant, either x or y. A corrected version of this example is:

  Real x,y;
equation
  x + y = 5;
  when condition then
    y = 7 - 2*x;       // fine
  end when;

2 Here, variable yis held constant when the when-equation is deactivated and x is computed from the first equation using the value of y from the previous event instant.

]

8.3.5.3 Application of the Single-assignment Rule to When-Equations

The Modelica single-assignment rule (section 8.4) has implications for when-equations:

  • Two when-equations may not define the same variable.

    [Without this rule this may actually happen for the erroneous model DoubleWhenConflict below, since there are two equations (close = true; close = false;) defining the same variable close. A conflict between the equations will occur if both conditions would become true at the same time instant.

    model DoubleWhenConflict
      Boolean close;   // Erroneous model: close defined by two equations!
    equation
      ...
      when condition1 then
        ...
        close = true;
      end when;
      when condition2 then
        close = false;
      end when;
      ...
    end DoubleWhenConflict;

    One way to resolve the conflict would be to give one of the two when-equations higher priority. This is possible by rewriting the when-equation using elsewhen, as in the WhenPriority model below or using the statement version of the when-construct, see section 11.2.7.]

  • When-equations involving elsewhen-parts can be used to resolve assignment conflicts since the first of the when/elsewhen parts are given higher priority than later ones:

    [Below it is well defined what happens if both conditions become true at the same time instant since condition1 with associated conditional equations has a higher priority than condition2.

    model WhenPriority
      Boolean close;   // Correct model: close defined by two equations!
    algorithm
      ...
      when condition1 then
        close = true;
      elsewhen condition2 then
        close = false;
      end when;
      ...
    end WhenPriority;

    ]

8.3.6 reinit

The reinit operator can only be used in the body of a when-equation. It has the following syntax:

reinit(x, expr);

The operator reinitializes x with expr at an event instant. x is a Real variable (or an array of Real variables) that must be selected as a state (resp., states) , i.e. reinit on x implies stateSelect=StateSelect.always on x. expr needs to be type-compatible with x. The reinit operator can for the same variable (resp. array of variables) only be applied (either as an individual variable or as part of an array of variables) in one equation (having reinit of the same variable in when and else-when of the same variable is allowed). In case of reinit active during initialization (due to when initial), see section 8.6.

The reinit operator does not break the single assignment rule, because reinit(x,expr) in equations evaluates expr to a value (value), then at the end of the current event iteration step it assigns this value to x (this copying from values to reinitialized state(s) is done after all other evaluations of the model and before copying x to pre(x)).

[If a higher index system is present, i.e., constraints between state variables, some state variables need to be redefined to non-state variables. During simulation, non-state variables should be chosen in such a way that variables with an applied reinit operator are selected as states at least when the corresponding when-clauses become active. If this is not possible, an error occurs, since otherwise the reinit operator would be applied on a non-state variable.

Example for the usage of the reinit operator:

Bouncing ball:

der(h) = v;
der(v) = if flying then -g else 0;
flying = not(h<=0 and v<=0);
when h < 0 then
  reinit(v, -e*pre(v));
end when

]

8.3.7 assert

An equation or statement of one of the following forms:

assert(condition, message); // Uses level=AssertionLevel.error
assert(condition, message, assertionLevel);
assert(condition, message, level = assertionLevel);

is an assertion, where condition is a Boolean expression, message is a string expression, and level is a built-in enumeration with a default value. It can be used in equation sections or algorithm sections. [This means that assert can be called as if it were a function with three formal parameters, the third formal parameter has the name ’level’ and the default value AssertionLevel.error.]

If the condition of an assertion is true, message is not evaluated and the procedure call is ignored. If the condition evaluates to false different actions are taken depending on the level input:

  • level = AssertionLevel.error: The current evaluation is aborted. The simulation may continue with another evaluation [e.g., with a shorter step-size, or by changing the values of iteration variables]. If the simulation is aborted, message indicates the cause of the error.
    Failed assertions takes precedence over successful termination, such that if the model first triggers the end of successful analysis by reaching the stop-time or explicitly with terminate(), but the evaluation with terminal()=true triggers an assert, the analysis failed.

  • level = AssertionLevel.warning: The current evaluation is not aborted. message indicates the cause of the warning [It is recommended to report the warning only once when the condition becomes false, and it is reported that the condition is no longer violated when the condition returns to true. The assert(..) statement shall have no influence on the behavior of the model. For example, by evaluating the condition and reporting the message only after accepted integrator steps. condition needs to be implicitly treated with noEvent(..) since otherwise events might be triggered that can lead to slightly changed simulation results].

[The AssertionLevel.error case can be used to avoid evaluating a model outside its limits of validity; for instance, a function to compute the saturated liquid temperature cannot be called with a pressure lower than the triple point value.

The AssertionLevel.warning case can be used when the boundary of validity is not hard: for instance, a fluid property model based on a polynomial interpolation curve might give accurate results between temperatures of 250 K and 400 K, but still give reasonable results in the range 200 K and 500 K. When the temperature gets out of the smaller interval, but still stays in the largest one, the user should be warned, but the simulation should continue without any further action. The corresponding code would be

assert(T > 250 and T < 400, "Medium model outside full accuracy range",        AssertionLevel.warning);
assert(T > 200 and T < 500, "Medium model outside feasible region");

]

8.3.8 terminate

The terminate(...) equation or statement [using function syntax] successfully terminates the analysis which was carried out, see also section 8.3.7. The termination is not immediate at the place where it is defined since not all variable results might be available that are necessary for a successful stop. Instead, the termination actually takes place when the current integrator step is successfully finalized or at an event instant after the event handling has been completed before restarting the integration.

The terminate clause has a string argument indicating the reason for the success. [The intention is to give more complex stopping criteria than a fixed point in time. Example:

model ThrowingBall
  Real x(start=0);
  Real y(start=1);
equation
  der(x)=...
  der(y)=...
algorithm
  when y<0 then
    terminate("The ball touches the ground");
  end when;
end ThrowingBall;

]

8.3.9 Equation Operators for Overconstrained Connection-Based Equation Systems

See section 8.3.9 for a description of this topic.

8.4 Synchronous Data-flow Principle and Single Assignment Rule

Modelica is based on the synchronous data flow principle and the single assignment rule, which are defined in the following way:

  1. 1.

    Discrete variables keep their actual values until these variables are explicitly changed. Differentiated variables have der(x) corresponding to the time-derivative of x, and x is continuous, except when reinit is triggered - see section 8.3.6. Variable values can be accessed at any time instant during continuous integration and at event instants.

  2. 2.

    At every time instant, during continuous integration and at event instants, the equations express relations between variables which have to be fulfilled concurrently..

  3. 3.

    Computation and communication at an event instant does not take time. [If computation or communication time has to be simulated, this property has to be explicitly modeled].

  4. 4.

    There must exist a perfect matching of variables to equations after flattening (perfect matching rule - previously called single assignment rule); see also globally balanced section 4.7.

8.5 Events and Synchronization

The integration is halted and an event occurs whenever an event generation expression, e.g. x > 2 o or floor(x), changes its value. An event generating expression has an internal buffer, and the value of the expression can only be changed at event instants. If the evaluated expression is inconsistent with the buffer, that will trigger an event and the buffer will be updated with a new value at the event instant [in other words, a root finding mechanism is needed which determines a small time interval in which the expression changes its value; the event occurs at the right side of this interval]. During continuous integration event generation expression has the constant value of the expression from the last event instant.

[Example:

y = if u > uMax then uMax else if u < uMin then uMin else u;

During continuous integration always the same if-branch is evaluated. The integration is halted wheneveru-uMax or u-uMin crosses zero. At the event instant, the correct if-branch is selected and the integration is restarted.

Numerical integration methods of order n (n>=1) require continuous model equations which are differentiable up to order n. This requirement can be fulfilled if Real elementary relations are not treated literally but as defined above, because discontinuous changes can only occur at event instants and no longer during continuous integration.

]

[It is a quality of implementation issue that the following special relations

time >= discrete expression
time < discrete expression

trigger a time event at “time = discrete expression”, i.e., the event instant is known in advance and no iteration is needed to find the exact event instant.

]

Relations are taken literally also during continuous integration, if the relation or the expression in which the relation is present, are the argument of the noEvent(..) function. The smooth(p,x) operator also allows relations used as argument to be taken literally. The noEvent feature is propagated to all subrelations in the scope of the noEvent function. For smooth the liberty to not allow literal evaluation is propagated to all subrelations, but the smooth-property itself is not propagated.

[Example:

x = if noEvent(u > uMax) then uMax elseif noEvent(u < uMin) then uMin else u;
y = noEvent(  if u > uMax then uMax elseif u < uMin then uMin else u);
z = smooth(0, if u > uMax then uMax elseif u < uMin then uMin else u);

In this case x=y=z, but a tool might generate events for z. The if-expression is taken literally without inducing state events.

The smooth function is useful, if e.g. the modeler can guarantee that the used if-clauses fulfill at least the continuity requirement of integrators. In this case the simulation speed is improved, since no state event iterations occur during integration. The noEvent function is used to guard against “outside domain” errors, e.g. y = if noEvent(x>= 0) then sqrt(x) else 0.]

All equations and assignment statements within when-clauses and all assignment statements within function classes are implicitly treated with the noEvent function, i.e., relations within the scope of these operators never induce state or time events. [Using state events in when-clauses is unnecessary because the body of a when-clause is not evaluated during continuous integration.]

[Example:

Limit1 = noEvent(x1 > 1);  // Error since Limit1 is a discrete-time variable
when noEvent(x1>1) or x2>10 then // error, when-conditions is not a discrete-time expression
  Close = true;
end when;

]

Modelica is based on the synchronous data flow principle (section 8.4).

[The rules for the synchronous data flow principle guarantee that variables are always defined by a unique set of equations. It is not possible that a variable is e.g. defined by two equations, which would give rise to conflicts or non-deterministic behavior. Furthermore, the continuous and the discrete parts of a model are always automatically “synchronized”. Example:

equation // Illegal example
  when condition1 then
    close = true;
  end when;
  when condition2 then
    close = false;
  end when;

This is not a valid model because rule 4 is violated since there are two equations for the single unknown variable close. If this would be a valid model, a conflict occurs when both conditions become true at the same time instant, since no priorities between the two equations are assigned. To become valid, the model has to be changed to:

equation
  when condition1 then
    close = true;
  elsewhen condition2 then
    close = false;
  end when;

Here, it is well-defined if both conditions become true at the same time instant (condition1 has a higher priority than condition2).

]

There is no guarantee that two different events occur at the same time instant.

[As a consequence, synchronization of events has to be explicitly programmed in the model, e.g. via counters. Example:

  Boolean fastSample, slowSample;
  Integer ticks(start=0);
equation
  fastSample = sample(0,1);
algorithm
  when fastSample then
    ticks      := if pre(ticks) < 5 then pre(ticks)+1 else 0;
        slowSample := pre(ticks) == 0;
  end when;
algorithm
  when fastSample then   // fast sampling
    ...
  end when;
algorithm
  when slowSample then   // slow sampling (5-times slower)
    ...
  end when;

The slowSample when-clause is evaluated at every 5th occurrence of the fastSample when-clause.

]

[The single assignment rule and the requirement to explicitly program the synchronization of events allow a certain degree of model verification already at compile time.]

8.6 Initialization, initial equation, and initial algorithm

Before any operation is carried out with a Modelica model [e.g., simulation or linearization], initialization takes place to assign consistent values for all variables present in the model. During this phase, also the derivatives, der(..), and the pre-variables, pre(..), are interpreted as unknown algebraic variables. The initialization uses all equations and algorithms that are utilized in the intended operation [such as simulation or linearization]. The equations of a when-clause are active during initialization, if and only if they are explicitly enabled with the initial() operator; and only in one of the two forms when initial() then or when {...,initial(),...} then. In this case, the when-clause equations remain active during the whole initialization phase. [If a when-clause equation v = expr; is not active during the initialization phase, the equation v = pre(v) is added for initialization. This follows from the mapping rule of when-clause equations. If the condition of the when-clause contains initial(), but not in one of the specific forms, the when-clause is not active during initialization: when not initial() then print(”simulation started”);end when; ]. In case of a reinit(x,expr) being active during initialization (due to being inside when initial()) this is interpreted as adding x=expr (the reinit-equation) as an initial equation.

Further constraints, necessary to determine the initial values of all variables, can be defined in the following ways:

  1. 1.

    As equations in an initial equation section or as assignments in an initial algorithm section. The equations and assignments in these initial sections are purely algebraic, stating constraints between the variables at the initial time instant. It is not allowed to use when-clauses in these sections.

  2. 2.

    For all non-discrete (that is continuous-time) Real variables vc, the equation pre(vc) = vc is added to the initialization equations. [If pre(vc) is not present in the flattened model, a tool may choose not to introduce this equation, or if it was introduced it can eliminate it (to avoid the introduction of many dummy variables pre(vc)).]

  3. 3.

    Implicitly by using the attributes start=value and fixed=true in the declaration of variables:

    • For all non-discrete (that is continuous-time) Real variables vc, the equation vc = startExpression is added to the initialization equations, if start = startExpression and fixed = true.

    • For all discrete variables vd, the equation pre(vd) = startExpression is added to the initialization equations, if start = startExpression and fixed = true.

    • For all variables declared as constant and parameter, with fixed = true; no equation is added to the initialization equations.

For constants and parameters, the attribute fixed is by default true. For other variables fixed is by default false. For all variables declared as constant it is an error to have fixed = false.

Start-values of variables having fixed = false can be used as initial guesses, in case iterative solvers are used in the initialization phase. [In case of iterative solver failure, it is recommended to specially report those variables for which the solver needs an initial guess, but which only have the default value of the start attribute as defined in section 4.8, since the lack of appropriate initial guesses is a likely cause of the solver failure.]

If a parameter has a modifier for the start-attribute, does not have fixed=false, and neither has a binding equation nor is part of a record having a binding equation, the modifier for the start-attribute can be used to add a parameter binding equation assigning the parameter to that start-modifier. In this case a diagnostic message is recommended in a simulation model. [This is used in libraries to give non-zero defaults so that users can quickly combine models and simulate without setting parameters; but still easily find the parameters that need to be set.]

All variables declared as parameter having fixed = false are treated as unknowns during the initialization phase, i.e. there must be additional equations for them – and the start-value can be used as a guess-value during initialization.

[In the case a parameter has both a binding equation and fixed = false a diagnostics is recommended, but the parameter should be solved from the binding equation.

Non-discrete (that is continuous-time) Real variables vc have exactly one initialization value since the rules above assure that during initialization vc = pre(vc) = vc.startExpression (if fixed= true).

Before the start of the integration, it must be guaranteed that for all variables v, v = pre(v). If this is not the case for some variables vi, “pre(vi) := vi” must be set and an event iteration at the initial time must follow, so the model is re-evaluated, until this condition is fulfilled.

A Modelica translator may first transform the continuous equations of a model, at least conceptually, to state space form. This may require to differentiate equations for index reduction, i.e., additional equations and, in some cases, additional unknown variables are introduced. This whole set of equations, together with the additional constraints defined above, should lead to an algebraic system of equations where the number of equations and the number of all variables (including der(..) and pre(..) variables) is equal. Often, this is a nonlinear system of equations and therefore it may be necessary to provide appropriate guess values (i.e., start values and fixed=false) in order to compute a solution numerically.

It may be difficult for a user to figure out how many initial equations have to be added, especially if the system has a higher index. A tool may add or remove initial equations automatically such that the resulting system is structurally nonsingular. In these cases diagnostics are appropriate since the result is not unique and may not be what the user expects. A missing initial value of a discrete variable which does not influence the simulation result, may be automatically set to the start value or its default without informing the user. For example, variables assigned in a when-clause which are not accessed outside of the when-clause and where the pre() operator is not explicitly used on these variables, do not have an effect on the simulation.

Examples:

Continuous time controller initialized in steady-state:

  Real y(fixed = false);  // fixed=false is redundant
equation
  der(y) = a*y + b*u;
initial equation
  der(y) = 0;

This has the following solution at initialization:

der(y) = 0;
y = -b/a *u;

Continuous time controller initialized either in steady-state or by providing a start value for state y:

  parameter Boolean steadyState = true;
  parameter Real y0 = 0 "start value for y, if not steadyState";
  Real y;
equation
  der(y) = a*y + b*u;
initial equation
  if steadyState then
    der(y)=0;
  else
    y = y0;
  end if;

This can also be written as follows (this form is less clear):

  parameter Boolean steadyState=true;
  Real y    (start=0, fixed=not steadyState);
  Real der_y(start=0, fixed=steadyState) = der(y);
equation
  der(y) = a*y + b*u;

Discrete time controller initialized in steady-state:

  discrete Real y;
equation
  when {initial(), sampleTrigger} then
    y = a*pre(y) + b*u;
  end when;
initial equation
  y = pre(y);

This leads to the following equations during initialization:

y = a*pre(y) + b*u;
y = pre(y);

With the solution:

y := (b*u)/(1-a)
pre(y) := y;

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8.6.1 The Number of Equations Needed for Initialization

[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 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.

Example: Consider the following simple equation system:

der(x1) = f1(x1);
der(x2) = f2(x2);
y = x1+x2+u;

Here we have three variables with unknown values: two dynamic variables that also are state variables, x1 and x2, i.e., n=2, one output variable y, i.e., m=1, and one input variable u with known value. A consistent solution of the initial value problem providing initial values for x1, x2, der(x1), der(x2), and 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 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 section 8.6 a tool may add/remove initial equations to fulfill this requirement, if appropriate diagnostics are given.

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