Numerical integration, equilibrium finding, and discretization utilities. More...

#include "../../Constants.h"

Include dependency graph for Differential_equations.h:

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Classes
struct	CVODEConfig
	Tolerance and step-size settings for CVODE integration. More...

struct	CVODEResult
	Time histories returned by CVODE integration. More...

struct	KINSOLConfig
	Convergence and scaling settings for KINSOL. More...

Typedefs
using	RHSFunc = std::function< Eigen::VectorXd(double t, const Eigen::VectorXd &x, const Eigen::VectorXd &u)>
	Right-hand side of dx/dt = f(t, x, u).

using	JacFunc = std::function< Eigen::MatrixXd(double t, const Eigen::VectorXd &x, const Eigen::VectorXd &u)>
	Jacobian J = ∂f/∂x(t, x, u).

Enumerations
enum class	KINSOLStrategy { Newton , LineSearch , Picard , FixedPoint }
	Nonlinear solver strategy for KINSOL equilibrium finding. More...

Functions
CVODEResult	integrate (const RHSFunc &rhs, const Eigen::VectorXd &x0, const std::function< Eigen::VectorXd(double t)> &inputFn, double t0, double tEnd, double dt_output, const CVODEConfig &cfg={}, const JacFunc &jac=nullptr)
	Integrates an ODE system using CVODE (BDF, stiff).

Eigen::VectorXd	findEquilibrium (const RHSFunc &rhs, const Eigen::VectorXd &x0, const Eigen::VectorXd &u, const KINSOLConfig &cfg={}, const JacFunc &jac=nullptr)
	Finds a steady-state equilibrium dx/dt = 0 for fixed input u.

Eigen::VectorXd	findEquilibriumRobust (const RHSFunc &rhs, const Eigen::VectorXd &x0, const Eigen::VectorXd &u, const JacFunc &jac=nullptr)
	Robust equilibrium finder with a cascade of solver strategies.

std::pair< Eigen::MatrixXd, Eigen::MatrixXd >	computeJacobians (const RHSFunc &rhs, const Eigen::VectorXd &x, const Eigen::VectorXd &u, double t=0.0, double eps=1e-8)
	Computes ∂f/∂x and ∂f/∂u by central finite differences.

void	padeDelaySystem3 (double tdelay, Eigen::MatrixXd &A, Eigen::MatrixXd &B, Eigen::MatrixXd &C, Eigen::MatrixXd &D)
	First-order Padé delay approximation (single signal, order 3).

void	padeDelaySystemMulti3 (double tdelay, Eigen::MatrixXd &A, Eigen::MatrixXd &B, Eigen::MatrixXd &C, Eigen::MatrixXd &D, int num_signals)
	First-order Padé delay approximation for multiple independent signals.

void	padeDelaySystem2 (double tdelay, Eigen::MatrixXd &A, Eigen::MatrixXd &B, Eigen::MatrixXd &C, Eigen::MatrixXd &D)
	Second-order Padé delay approximation (single signal, order 2).

void	padeDelaySystemMulti2 (double tdelay, Eigen::MatrixXd &A, Eigen::MatrixXd &B, Eigen::MatrixXd &C, Eigen::MatrixXd &D, int num_signals)
	Second-order Padé delay approximation for multiple independent signals.

void	discretizeABCD (const Eigen::MatrixXd &A, const Eigen::MatrixXd &B, const Eigen::MatrixXd &C, const Eigen::MatrixXd &D, double Ts, Eigen::MatrixXd &Ad, Eigen::MatrixXd &Bd, Eigen::MatrixXd &Cd, Eigen::MatrixXd &Dd)
	Discretizes a continuous-time state-space model using the Tustin (bilinear) method.

Detailed Description

Numerical integration, equilibrium finding, and discretization utilities.

Wraps CVODE (stiff ODE integration), KINSOL (nonlinear equilibrium solvers), finite-difference Jacobian computation, Padé delay approximations, and Tustin (bilinear) discretization of state-space models.

Typedef Documentation

◆ JacFunc

using JacFunc = std::function<Eigen::MatrixXd(double t, const Eigen::VectorXd& x, const Eigen::VectorXd& u)>

Jacobian J = ∂f/∂x(t, x, u).

◆ RHSFunc

using RHSFunc = std::function<Eigen::VectorXd(double t, const Eigen::VectorXd& x, const Eigen::VectorXd& u)>

Right-hand side of dx/dt = f(t, x, u).

Enumeration Type Documentation

◆ KINSOLStrategy

enum class KINSOLStrategy

strong

Nonlinear solver strategy for KINSOL equilibrium finding.

Enumerator
Newton
LineSearch
Picard
FixedPoint

Function Documentation

◆ computeJacobians()

std::pair< Eigen::MatrixXd, Eigen::MatrixXd > computeJacobians	(	const RHSFunc &	rhs,
		const Eigen::VectorXd &	x,
		const Eigen::VectorXd &	u,
		double	t = `0.0`,
		double	eps = `1e-8`
	)

Computes ∂f/∂x and ∂f/∂u by central finite differences.

Parameters

rhs	ODE right-hand side.
x	State at which to evaluate.
u	Input at which to evaluate.
t	Evaluation time (default 0).
eps	Finite-difference step size (default 1e-8).

Returns: Pair (df/dx, df/du).

References j.

Referenced by MMC::computeABCD(), MMC::computeABCD_analytical(), and kinsol_fd_jac_cb().

◆ discretizeABCD()

void discretizeABCD	(	const Eigen::MatrixXd &	A,
		const Eigen::MatrixXd &	B,
		const Eigen::MatrixXd &	C,
		const Eigen::MatrixXd &	D,
		double	Ts,
		Eigen::MatrixXd &	Ad,
		Eigen::MatrixXd &	Bd,
		Eigen::MatrixXd &	Cd,
		Eigen::MatrixXd &	Dd
	)

Discretizes a continuous-time state-space model using the Tustin (bilinear) method.

Parameters

A	Continuous-time state matrix.
B	Continuous-time input matrix.
C	Continuous-time output matrix.
D	Continuous-time feed-through matrix.
Ts	Sample period (s).
Ad	Output discrete-time state matrix.
Bd	Output discrete-time input matrix.
Cd	Output discrete-time output matrix.
Dd	Output discrete-time feed-through matrix.

Referenced by Converter::discretize(), and DQsym::run().

◆ findEquilibrium()

Eigen::VectorXd findEquilibrium	(	const RHSFunc &	rhs,
		const Eigen::VectorXd &	x0,
		const Eigen::VectorXd &	u,
		const KINSOLConfig &	cfg = `{}`,
		const JacFunc &	jac = `nullptr`
	)

Finds a steady-state equilibrium dx/dt = 0 for fixed input u.

Parameters

rhs	ODE right-hand side.
x0	Initial guess for the state vector.
u	Fixed input vector.
cfg	Solver strategy and tolerance settings.
jac	Optional analytical Jacobian.

Returns: Equilibrium state vector.

References kinsolSolve().

◆ findEquilibriumRobust()

Eigen::VectorXd findEquilibriumRobust	(	const RHSFunc &	rhs,
		const Eigen::VectorXd &	x0,
		const Eigen::VectorXd &	u,
		const JacFunc &	jac = `nullptr`
	)

Robust equilibrium finder with a cascade of solver strategies.

Attempts LineSearch, then Newton, relaxed warmup, and Picard + Newton until convergence or all strategies are exhausted.

Parameters

rhs	ODE right-hand side.
x0	Initial guess for the state vector.
u	Fixed input vector.
jac	Optional analytical Jacobian.

Returns: Equilibrium state vector.

References KINSOLConfig::damping, KINSOLConfig::ftol, kinsolSolve(), LineSearch, KINSOLConfig::max_iter, Newton, Picard, KINSOLConfig::stol, KINSOLConfig::strategy, and withAutoScaling().

Referenced by MMC::solveEquilibrium().

◆ integrate()

CVODEResult integrate	(	const RHSFunc &	rhs,
		const Eigen::VectorXd &	x0,
		const std::function< Eigen::VectorXd(double t)> &	inputFn,
		double	t0,
		double	tEnd,
		double	dt_output,
		const CVODEConfig &	cfg = `{}`,
		const JacFunc &	jac = `nullptr`
	)

Integrates an ODE system using CVODE (BDF, stiff).

Parameters

rhs	Right-hand side f(t, x, u).
x0	Initial state vector.
inputFn	Time-varying input u(t).
t0	Start time.
tEnd	End time.
dt_output	Output sampling interval.
cfg	Tolerance and Jacobian settings.
jac	Optional analytical Jacobian (nullptr → finite differences).

Returns: Sampled time and state histories.

References CVODEConfig::atol, cvode_jac_cb(), cvode_rhs_cb(), CVODEConfig::dt_max, eigen2nv(), nv2eigen(), CVODEConfig::rtol, CVODEResult::states, CVODEResult::time, and CVODEConfig::use_analytical_jac.

◆ padeDelaySystem2()

void padeDelaySystem2	(	double	tdelay,
		Eigen::MatrixXd &	A,
		Eigen::MatrixXd &	B,
		Eigen::MatrixXd &	C,
		Eigen::MatrixXd &	D
	)

Second-order Padé delay approximation (single signal, order 2).

Parameters

tdelay	Transport delay (s).
A	Output state matrix.
B	Output input matrix.
C	Output output matrix.
D	Output feed-through matrix.

◆ padeDelaySystem3()

void padeDelaySystem3	(	double	tdelay,
		Eigen::MatrixXd &	A,
		Eigen::MatrixXd &	B,
		Eigen::MatrixXd &	C,
		Eigen::MatrixXd &	D
	)

First-order Padé delay approximation (single signal, order 3).

Parameters

tdelay	Transport delay (s).
A	Output state matrix.
B	Output input matrix.
C	Output output matrix.
D	Output feed-through matrix.

◆ padeDelaySystemMulti2()

void padeDelaySystemMulti2	(	double	tdelay,
		Eigen::MatrixXd &	A,
		Eigen::MatrixXd &	B,
		Eigen::MatrixXd &	C,
		Eigen::MatrixXd &	D,
		int	num_signals
	)

Second-order Padé delay approximation for multiple independent signals.

Parameters

tdelay	Transport delay (s).
A	Output state matrix.
B	Output input matrix.
C	Output output matrix.
D	Output feed-through matrix.
num_signals	Number of delayed input channels.

◆ padeDelaySystemMulti3()

void padeDelaySystemMulti3	(	double	tdelay,
		Eigen::MatrixXd &	A,
		Eigen::MatrixXd &	B,
		Eigen::MatrixXd &	C,
		Eigen::MatrixXd &	D,
		int	num_signals
	)

First-order Padé delay approximation for multiple independent signals.

Parameters

tdelay	Transport delay (s).
A	Output state matrix.
B	Output input matrix.
C	Output output matrix.
D	Output feed-through matrix.
num_signals	Number of delayed input channels.

Classes

Typedefs

Enumerations

Functions

Detailed Description

Typedef Documentation

◆ JacFunc

◆ RHSFunc

Enumeration Type Documentation

◆ KINSOLStrategy

Function Documentation

◆ computeJacobians()

◆ discretizeABCD()

◆ findEquilibrium()

◆ findEquilibriumRobust()

◆ integrate()

◆ padeDelaySystem2()

◆ padeDelaySystem3()

◆ padeDelaySystemMulti2()

◆ padeDelaySystemMulti3()