Allows to export a tailored implicit Runge-Kutta integrator for fast model predictive control. More...

#include <irk_export.hpp>

Inheritance diagram for ImplicitRungeKuttaExport:

Public Member Functions
virtual returnValue	getCode (ExportStatementBlock &code)
virtual returnValue	getDataDeclarations (ExportStatementBlock &declarations, ExportStruct dataStruct=ACADO_ANY) const
virtual returnValue	getFunctionDeclarations (ExportStatementBlock &declarations) const
	ImplicitRungeKuttaExport (UserInteraction *_userInteraction=0, const std::string &_commonHeaderName="")
	ImplicitRungeKuttaExport (const ImplicitRungeKuttaExport &arg)
ImplicitRungeKuttaExport &	operator= (const ImplicitRungeKuttaExport &arg)
virtual returnValue	setDifferentialEquation (const Expression &rhs)
returnValue	setModel (const std::string &_rhs, const std::string &_diffs_rhs)
virtual returnValue	setup ()
virtual returnValue	setupOutput (const std::vector< Grid > outputGrids_, const std::vector< Expression > rhs)
virtual returnValue	setupOutput (const std::vector< Grid > outputGrids_, const std::vector< std::string > _outputNames, const std::vector< std::string > _diffs_outputNames, const std::vector< uint > _dims_output)
virtual returnValue	setupOutput (const std::vector< Grid > outputGrids_, const std::vector< std::string > _outputNames, const std::vector< std::string > _diffs_outputNames, const std::vector< uint > _dims_output, const std::vector< DMatrix > _outputDependencies)
virtual	~ImplicitRungeKuttaExport ()
Protected Member Functions
DVector	computeCombinations (const DVector &cVec, uint index, uint numEls)
virtual returnValue	copy (const ImplicitRungeKuttaExport &arg)
DVector	divideMeasurements (uint index)
DVector	evaluateDerivedPolynomial (double time)
returnValue	evaluateDerivedPolynomial (ExportStatementBlock &block, const ExportVariable &variable, const ExportVariable &grid)
DMatrix	evaluateDerivedPolynomial (uint index)
virtual returnValue	evaluateMatrix (ExportStatementBlock *block, const ExportIndex &index1, const ExportIndex &index2, const ExportIndex &tmp_index, const ExportVariable &Ah, const ExportVariable &C, bool evaluateB, bool DERIVATIVES)
DVector	evaluatePolynomial (double time)
returnValue	evaluatePolynomial (ExportStatementBlock &block, const ExportVariable &variable, const ExportVariable &grid, const std::string &h)
DMatrix	evaluatePolynomial (uint index)
virtual returnValue	evaluateRhsImplicitSystem (ExportStatementBlock *block, const ExportIndex &stage)
virtual returnValue	evaluateStatesImplicitSystem (ExportStatementBlock *block, const ExportVariable &Ah, const ExportVariable &C, const ExportIndex &stage, const ExportIndex &i, const ExportIndex &j)
returnValue	evaluateStatesOutputSystem (ExportStatementBlock *block, const ExportVariable &Ah, const ExportIndex &stage)
virtual DMatrix	formMatrix (const DMatrix &mass, const DMatrix &jacobian)
returnValue	generateOutput (ExportStatementBlock *block, const ExportIndex &index0, const ExportIndex &index1, const ExportIndex &tmp_index1, const ExportIndex &tmp_index2, const ExportVariable &tmp_meas, const ExportVariable &time_tmp, const uint directions)
ExportVariable	getAuxVariable () const
returnValue	getCRSIndex (uint output, ExportIndex row, ExportIndex col)
uint	getNumIts () const
uint	getNumItsInit () const
returnValue	initializeCoefficients ()
returnValue	initializeDDMatrix ()
virtual returnValue	prepareInputSystem (ExportStatementBlock &code)
returnValue	prepareOutputEvaluation (ExportStatementBlock &code)
virtual returnValue	prepareOutputSystem (ExportStatementBlock &code)
virtual returnValue	solveImplicitSystem (ExportStatementBlock *block, const ExportIndex &index1, const ExportIndex &index2, const ExportIndex &index3, const ExportIndex &tmp_index, const ExportVariable &Ah, const ExportVariable &C, const ExportVariable &det, bool DERIVATIVES=false)
virtual returnValue	solveInputSystem (ExportStatementBlock *block, const ExportIndex &index1, const ExportIndex &index2, const ExportIndex &index3, const ExportIndex &tmp_index, const ExportVariable &Ah)
virtual returnValue	solveOutputSystem (ExportStatementBlock *block, const ExportIndex &index1, const ExportIndex &index2, const ExportIndex &index3, const ExportIndex &tmp_index, const ExportVariable &Ah, bool DERIVATIVES=false)
Protected Attributes
DMatrix	coeffs
bool	CONTINUOUS_OUTPUT
DMatrix	DD
ExportVariable	debug_mat
std::vector< ExportVariable >	gridVariables
ExportAcadoFunction	lin_output
uint	NDX2
DVector	numDX_output
uint	numIts
uint	numItsInit
std::vector< ExportIndex >	numMeas
std::vector< ExportVariable >	numMeasVariables
DVector	numVARS_output
DVector	numXA_output
uint	NVARS2
uint	NVARS3
std::vector< ExportVariable >	polynDerVariables
ExportVariable	polynEvalVar
std::vector< ExportVariable >	polynVariables
bool	REUSE
ExportVariable	rk_A
ExportVariable	rk_auxSolver
ExportVariable	rk_b
ExportVariable	rk_diffK
ExportVariable	rk_diffsOutputTemp
ExportVariable	rk_diffsTemp3
ExportVariable	rk_dk1
ExportVariable	rk_dk3
ExportVariable	rk_mat1
ExportVariable	rk_mat3
ExportVariable	rk_out
ExportVariable	rk_outH
std::vector< ExportVariable >	rk_outputs
ExportVariable	rk_rhsOutputTemp
ExportVariable	rk_rhsTemp
ExportLinearSolver *	solver
ExportVariable	stepsH
std::vector< uint >	totalMeas

Detailed Description

Allows to export a tailored implicit Runge-Kutta integrator for fast model predictive control.

The class ImplicitRungeKuttaExport allows to export a tailored implicit Runge-Kutta integrator for fast model predictive control.

Author:: Rien Quirynen

Definition at line 55 of file irk_export.hpp.

Constructor & Destructor Documentation

BEGIN_NAMESPACE_ACADO ImplicitRungeKuttaExport::ImplicitRungeKuttaExport	(	UserInteraction *	_userInteraction = `0`,
		const std::string &	_commonHeaderName = `""`
	)

Default constructor.

Parameters:

[in]	_userInteraction	Pointer to corresponding user interface.
[in]	_commonHeaderName	Name of common header file to be included.

Definition at line 44 of file irk_export.cpp.

ImplicitRungeKuttaExport::ImplicitRungeKuttaExport ( const ImplicitRungeKuttaExport & arg )

Copy constructor (deep copy).

Parameters:

[in] arg Right-hand side object.

Definition at line 67 of file irk_export.cpp.

ImplicitRungeKuttaExport::~ImplicitRungeKuttaExport ( ) [virtual]

Destructor.

Definition at line 91 of file irk_export.cpp.

Member Function Documentation

DVector ImplicitRungeKuttaExport::computeCombinations	(	const DVector &	cVec,
		uint	index,
		uint	numEls
	)		`[protected]`

Recursive function that helps with the computation of the coefficients of polynomials that are used to evaluate the continuous output (see initializeCoefficients), by computing the correct combinations of elements of the vector cc from the Butcher table.

Parameters:

[in]	cVec	The vector with all the elements of the vector cc from the Butcher table, of which combinations are computed in a recursive way.
[in]	index	An index of the vector cVec which denotes the relevant part for this invocation.
[in]	numEls	The number of elements in the combination.

Returns:: SUCCESSFUL_RETURN

Definition at line 900 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::copy ( const ImplicitRungeKuttaExport & arg ) [protected, virtual]

Copies all class members from given object.

Parameters:

[in] arg Right-hand side object.

Returns:: SUCCESSFUL_RETURN

Definition at line 1413 of file irk_export.cpp.

DVector ImplicitRungeKuttaExport::divideMeasurements ( uint index ) [protected]

Divide the total number of measurements over the different integration steps.

Parameters:

[in] index The index of the continuous output for which the division of measurements is returned.

Returns:: The division of measurements over the integration steps, corresponding the given continuous output.

Definition at line 1041 of file irk_export.cpp.

DVector ImplicitRungeKuttaExport::evaluateDerivedPolynomial ( double time ) [protected]

Returns the coefficients of the derived polynomial, representing the derivative of the continuous output with respect to time.

Parameters:

[in] time The point in the interval (0,1] for which the coefficients are returned.

Returns:: Coefficients of the polynomial, corresponding the given grid point

Definition at line 1002 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::evaluateDerivedPolynomial	(	ExportStatementBlock &	block,
		const ExportVariable &	variable,
		const ExportVariable &	grid
	)		`[protected]`

Exports the evaluation of the coefficients of the derived polynomial, representing the derivative of the continuous output with respect to time.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	variable	The variable containing the coefficients of the polynomial.
[in]	grid	The variable containing the grid point for the specific output.

Returns:: SUCCESSFUL_RETURN

Definition at line 1021 of file irk_export.cpp.

DMatrix ImplicitRungeKuttaExport::evaluateDerivedPolynomial ( uint index ) [protected]

Returns the coefficients of the derived polynomial for the complete grid of the output, corresponding a certain index.

Parameters:

[in] index The index of the continuous output for which the coefficients are returned.

Returns:: Coefficients of the derived polynomial, corresponding the given continuous output

Definition at line 982 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::evaluateMatrix	(	ExportStatementBlock *	block,
		const ExportIndex &	index1,
		const ExportIndex &	index2,
		const ExportIndex &	tmp_index,
		const ExportVariable &	Ah,
		const ExportVariable &	C,
		bool	evaluateB,
		bool	DERIVATIVES
	)		`[protected, virtual]`

Exports the evaluation of the matrix of the linear system.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	index1	The loop index of the outer loop.
[in]	index2	The loop index of the inner loop.
[in]	tmp_index	A temporary index to be used.
[in]	Ah	The matrix A of the IRK method, multiplied by the step size h.
[in]	evaluateB	True if the right-hand side of the linear system should also be evaluated, false otherwise.

Returns:: SUCCESSFUL_RETURN

Reimplemented in DiagonallyImplicitRKExport.

Definition at line 724 of file irk_export.cpp.

DVector ImplicitRungeKuttaExport::evaluatePolynomial ( double time ) [protected]

Returns the coefficients of the polynomial, representing the continuous output of the integrator.

Parameters:

[in] time The point in the interval (0,1] for which the coefficients are returned.

Returns:: Coefficients of the polynomial, corresponding the given grid point

Definition at line 942 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::evaluatePolynomial	(	ExportStatementBlock &	block,
		const ExportVariable &	variable,
		const ExportVariable &	grid,
		const std::string &	h
	)		`[protected]`

Exports the evaluation of the coefficients of the polynomial, representing the continuous output of the integrator.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	variable	The variable containing the coefficients of the polynomial.
[in]	grid	The variable containing the grid point for the specific output.
[in]	h	The integration step size.

Returns:: SUCCESSFUL_RETURN

Definition at line 958 of file irk_export.cpp.

DMatrix ImplicitRungeKuttaExport::evaluatePolynomial ( uint index ) [protected]

Returns the coefficients of the polynomial for the complete grid of the output, corresponding a certain index.

Parameters:

[in] index The index of the continuous output for which the coefficients are returned.

Returns:: Coefficients of the polynomial, corresponding the given continuous output

Definition at line 922 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::evaluateRhsImplicitSystem	(	ExportStatementBlock *	block,
		const ExportIndex &	stage
	)		`[protected, virtual]`

Exports the evaluation of the right-hand side of the linear system at a specific stage.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	index	The loop index, defining the stage.

Returns:: SUCCESSFUL_RETURN

Reimplemented in DiagonallyImplicitRKExport.

Definition at line 705 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::evaluateStatesImplicitSystem	(	ExportStatementBlock *	block,
		const ExportVariable &	Ah,
		const ExportVariable &	C,
		const ExportIndex &	stage,
		const ExportIndex &	i,
		const ExportIndex &	j
	)		`[protected, virtual]`

Exports the evaluation of the states at a specific stage.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	Ah	The matrix A of the IRK method, multiplied by the step size h.
[in]	index	The loop index, defining the stage.

Returns:: SUCCESSFUL_RETURN

Reimplemented in DiagonallyImplicitRKExport.

Definition at line 657 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::evaluateStatesOutputSystem	(	ExportStatementBlock *	block,
		const ExportVariable &	Ah,
		const ExportIndex &	stage
	)		`[protected]`

Exports the evaluation of the states at a specific stage.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	Ah	The matrix A of the IRK method, multiplied by the step size h.
[in]	index	The loop index, defining the stage.

Returns:: SUCCESSFUL_RETURN

Definition at line 682 of file irk_export.cpp.

DMatrix ImplicitRungeKuttaExport::formMatrix	(	const DMatrix &	mass,
		const DMatrix &	jacobian
	)		`[protected, virtual]`

Forms a constant linear system matrix for the collocation equations, given a constant jacobian and mass matrix.

Parameters:

[in]	jacobian	given constant Jacobian matrix
[in]	mass	given constant mass matrix

Returns:: SUCCESSFUL_RETURN

Reimplemented in DiagonallyImplicitRKExport.

Definition at line 529 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::generateOutput	(	ExportStatementBlock *	block,
		const ExportIndex &	index0,
		const ExportIndex &	index1,
		const ExportIndex &	tmp_index1,
		const ExportIndex &	tmp_index2,
		const ExportVariable &	tmp_meas,
		const ExportVariable &	time_tmp,
		const uint	directions
	)		`[protected]`

Exports the necessary code for the computation of the continuous output.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	tmp_meas	The number of measurements in the current integration step (in case of an online grid).
[in]	rk_tPrev	The time point, defining the beginning of the current integration step (in case of an online grid).
[in]	time_tmp	A variable used for time transformations (in case of an online grid).

Returns:: SUCCESSFUL_RETURN

Definition at line 768 of file irk_export.cpp.

ExportVariable ImplicitRungeKuttaExport::getAuxVariable ( ) const [protected, virtual]

Returns the largest global export variable.

Returns:: SUCCESSFUL_RETURN

Implements IntegratorExport.

Reimplemented in ForwardIRKExport.

Definition at line 171 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::getCode ( ExportStatementBlock & code ) [virtual]

Exports source code of the auto-generated integrator into the given directory.

Parameters:

[in] code Code block containing the auto-generated integrator.

Returns:: SUCCESSFUL_RETURN

Implements RungeKuttaExport.

Reimplemented in ForwardIRKExport.

Definition at line 256 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::getCRSIndex	(	uint	output,
		ExportIndex	row,
		ExportIndex	col
	)		`[protected]`

Returns the index corresponding Compressed Row Storage (CRS) for the dependency matrix of a specific output function.

Parameters:

[in]	output	The number of the output function.
[in]	row	The number of the row, corresponding the element of interest.
[in]	col	The number of the column, corresponding the element of interest.

Returns:: The CRS index.

Reimplemented in ForwardIRKExport.

returnValue ImplicitRungeKuttaExport::getDataDeclarations	(	ExportStatementBlock &	declarations,
		ExportStruct	dataStruct = `ACADO_ANY`
	)		const `[virtual]`

Adds all data declarations of the auto-generated integrator to given list of declarations.

Parameters:

[in] declarations List of declarations.

Returns:: SUCCESSFUL_RETURN

Implements RungeKuttaExport.

Reimplemented in ForwardIRKExport.

Definition at line 200 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::getFunctionDeclarations ( ExportStatementBlock & declarations ) const [virtual]

Adds all function (forward) declarations of the auto-generated integrator to given list of declarations.

Parameters:

[in] declarations List of declarations.

Returns:: SUCCESSFUL_RETURN

Implements RungeKuttaExport.

Reimplemented in ForwardIRKExport.

Definition at line 244 of file irk_export.cpp.

uint ImplicitRungeKuttaExport::getNumIts ( ) const [protected]

Returns the performed number of Newton iterations.

Returns:: The performed number of Newton iterations.

Definition at line 1473 of file irk_export.cpp.

uint ImplicitRungeKuttaExport::getNumItsInit ( ) const [protected]

Returns the performed number of Newton iterations for the initialization of the first step.

Returns:: The performed number of Newton iterations for the initialization of the first step.

Definition at line 1479 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::initializeCoefficients ( ) [protected]

Initializes the matrix coeffs, containing coefficients of polynomials that are used to evaluate the continuous output (see evaluatePolynomial).

Returns:: SUCCESSFUL_RETURN

Definition at line 860 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::initializeDDMatrix ( ) [protected]

Initializes the matrix DD, which is used to extrapolate the variables of the IRK method to the next step.

Returns:: SUCCESSFUL_RETURN

Definition at line 841 of file irk_export.cpp.

ImplicitRungeKuttaExport & ImplicitRungeKuttaExport::operator= ( const ImplicitRungeKuttaExport & arg )

Assignment operator (deep copy).

Parameters:

[in] arg Right-hand side object.

Definition at line 101 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::prepareInputSystem ( ExportStatementBlock & code ) [protected, virtual]

Precompute as much as possible for the linear input system and export the resulting definitions.

Parameters:

[in] code The block to which the code will be exported.

Returns:: SUCCESSFUL_RETURN

Reimplemented in ForwardIRKExport, and DiagonallyImplicitRKExport.

Definition at line 501 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::prepareOutputEvaluation ( ExportStatementBlock & code ) [protected]

Prepares the structures to evaluate the continuous output and exports the resulting definitions.

Parameters:

[in] code The block to which the code will be exported.

Returns:: SUCCESSFUL_RETURN

Definition at line 1376 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::prepareOutputSystem ( ExportStatementBlock & code ) [protected, virtual]

Precompute as much as possible for the linear output system and export the resulting definitions.

Parameters:

[in] code The block to which the code will be exported.

Returns:: SUCCESSFUL_RETURN

Reimplemented in DiagonallyImplicitRKExport, and ForwardIRKExport.

Definition at line 515 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::setDifferentialEquation ( const Expression & rhs ) [virtual]

Assigns Differential Equation to be used by the integrator.

Parameters:

[in] rhs Right-hand side expression.

Returns:: SUCCESSFUL_RETURN

Implements RungeKuttaExport.

Definition at line 112 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::setModel	(	const std::string &	_rhs,
		const std::string &	_diffs_rhs
	)		`[virtual]`

Assigns the model to be used by the integrator.

Parameters:

[in]	_rhs	Name of the function, evaluating the right-hand side.
[in]	_diffs_rhs	Name of the function, evaluating the derivatives of the right-hand side.

Returns:: SUCCESSFUL_RETURN

Reimplemented from IntegratorExport.

Definition at line 159 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::setup ( ) [virtual]

Initializes export of a tailored integrator.

Returns:: SUCCESSFUL_RETURN

Implements RungeKuttaExport.

Reimplemented in DiagonallyImplicitRKExport, and ForwardIRKExport.

Definition at line 1055 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::setupOutput	(	const std::vector< Grid >	outputGrids_,
		const std::vector< Expression >	rhs
	)		`[virtual]`

Sets up the output with the grids for the different output functions.

Parameters:

outputGrids_	The vector containing a grid for each output function.
rhs	The expressions corresponding the output functions.

Returns:: SUCCESSFUL_RETURN

Implements RungeKuttaExport.

Definition at line 1185 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::setupOutput	(	const std::vector< Grid >	outputGrids_,
		const std::vector< std::string >	_outputNames,
		const std::vector< std::string >	_diffs_outputNames,
		const std::vector< uint >	_dims_output
	)		`[virtual]`

Sets up the output with the grids for the different output functions.

Parameters:

outputGrids_	The vector containing a grid for each output function.
_outputNames	The names of the output functions.
_diffs_outputNames	The names of the functions, evaluating the derivatives of the outputs.
_dims_output	The dimensions of the output functions.

Returns:: SUCCESSFUL_RETURN

Implements IntegratorExport.

Definition at line 1279 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::setupOutput	(	const std::vector< Grid >	outputGrids_,
		const std::vector< std::string >	_outputNames,
		const std::vector< std::string >	_diffs_outputNames,
		const std::vector< uint >	_dims_output,
		const std::vector< DMatrix >	_outputDependencies
	)		`[virtual]`

Sets up the output with the grids for the different output functions.

Parameters:

outputGrids_	The vector containing a grid for each output function.
_outputNames	The names of the output functions.
_diffs_outputNames	The names of the functions, evaluating the derivatives of the outputs.
_dims_output	The dimensions of the output functions.
_outputDependencies	A separate dependency matrix for each output.

Returns:: SUCCESSFUL_RETURN

Implements IntegratorExport.

Definition at line 1359 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::solveImplicitSystem	(	ExportStatementBlock *	block,
		const ExportIndex &	index1,
		const ExportIndex &	index2,
		const ExportIndex &	index3,
		const ExportIndex &	tmp_index,
		const ExportVariable &	Ah,
		const ExportVariable &	C,
		const ExportVariable &	det,
		bool	DERIVATIVES = `false`
	)		`[protected, virtual]`

Exports the code needed to solve the system of collocation equations for the nonlinear, fully implicit system.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	Ah	The variable containing the internal coefficients of the RK method, multiplied with the step size.
[in]	det	The variable that holds the determinant of the matrix in the linear system.

Returns:: SUCCESSFUL_RETURN

Reimplemented in DiagonallyImplicitRKExport.

Definition at line 583 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::solveInputSystem	(	ExportStatementBlock *	block,
		const ExportIndex &	index1,
		const ExportIndex &	index2,
		const ExportIndex &	index3,
		const ExportIndex &	tmp_index,
		const ExportVariable &	Ah
	)		`[protected, virtual]`

Exports the code needed to solve the system of collocation equations for the linear input system.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	A1	A constant matrix defining the equations of the linear input system.
[in]	B1	A constant matrix defining the equations of the linear input system.
[in]	Ah	The variable containing the internal coefficients of the RK method, multiplied with the step size.

Returns:: SUCCESSFUL_RETURN

Reimplemented in DiagonallyImplicitRKExport.

Definition at line 556 of file irk_export.cpp.

returnValue ImplicitRungeKuttaExport::solveOutputSystem	(	ExportStatementBlock *	block,
		const ExportIndex &	index1,
		const ExportIndex &	index2,
		const ExportIndex &	index3,
		const ExportIndex &	tmp_index,
		const ExportVariable &	Ah,
		bool	DERIVATIVES = `false`
	)		`[protected, virtual]`

Exports the code needed to solve the system of collocation equations for the linear output system.

Parameters:

[in]	block	The block to which the code will be exported.
[in]	Ah	The variable containing the internal coefficients of the RK method, multiplied with the step size.
[in]	A3	A constant matrix defining the equations of the linear output system.