Transfer function utilities for linear systems and nonlinear elements. More...

Classes
struct	FirstOrderCoeffs
	First-order discrete-time system coefficients. More...
struct	SecondOrderCoeffs
	Second-order discrete-time system coefficients. More...

Functions
template<int N>
Eigen::Matrix< double, N, N >	matrix_exp_pade (const Eigen::Matrix< double, N, N > &A)
	Compute matrix exponential using Padé(13,13) approximation.
template<int N>
void	discretize_zoh (const Eigen::Matrix< double, N, N > &A_c, const Eigen::Matrix< double, N, 1 > &B_c, double dt, Eigen::Matrix< double, N, N > &A_d, Eigen::Matrix< double, N, 1 > &B_d)
	Zero-order hold (ZOH) discretization.
template<int N>
void	discretize_tustin (const Eigen::Matrix< double, N, N > &A_c, const Eigen::Matrix< double, N, 1 > &B_c, double dt, Eigen::Matrix< double, N, N > &A_d, Eigen::Matrix< double, N, 1 > &B_d)
	Tustin (bilinear) discretization.
template<int N>
void	discretize_euler (const Eigen::Matrix< double, N, N > &A_c, const Eigen::Matrix< double, N, 1 > &B_c, double dt, Eigen::Matrix< double, N, N > &A_d, Eigen::Matrix< double, N, 1 > &B_d)
	Forward Euler discretization.
template<int N>
void	discretize_backward_euler (const Eigen::Matrix< double, N, N > &A_c, const Eigen::Matrix< double, N, 1 > &B_c, double dt, Eigen::Matrix< double, N, N > &A_d, Eigen::Matrix< double, N, 1 > &B_d)
	Backward Euler discretization.
template<typename Scalar>
FirstOrderCoeffs< Scalar >	first_order (double tau, double K, double dt)
	Compute first-order discrete-time system coefficients.
template<typename Scalar>
Scalar	first_order_step (const FirstOrderCoeffs< Scalar > &coeffs, const Scalar &state, const Scalar &input)
	Compute next state for first-order system (stateless).
template<typename Scalar>
Scalar	first_order_response (double tau, double K, const Scalar &t)
	Compute analytical step response at time t.
template<typename Scalar>
Scalar	rate_limit (const Scalar &current, const Scalar &commanded, double rate_max, double dt)
	Apply rate limiting to a value change (stateless).
template<typename Scalar>
Scalar	saturate (const Scalar &value, const Scalar &min, const Scalar &max)
	Apply saturation/clamping (stateless).
template<typename Scalar>
Scalar	deadband (const Scalar &input, double deadband_width)
	Apply deadband (stateless).
template<typename Scalar>
Scalar	hysteresis (const Scalar &input, const Scalar &last_output, double hysteresis_width)
	Apply hysteresis (stateless computation).
template<typename Scalar>
SecondOrderCoeffs< Scalar >	second_order (double omega_n, double zeta, double K, double dt)
	Compute second-order discrete-time system coefficients.
template<typename Scalar>
Eigen::Matrix< Scalar, 2, 1 >	second_order_step (const SecondOrderCoeffs< Scalar > &coeffs, const Eigen::Matrix< Scalar, 2, 1 > &state, const Scalar &input)
	Compute next state for second-order system (stateless).
template<typename Scalar>
Scalar	second_order_output (const SecondOrderCoeffs< Scalar > &coeffs, const Eigen::Matrix< Scalar, 2, 1 > &state, const Scalar &input)
	Extract output from state (stateless).
std::tuple< double, double, double >	second_order_characteristics (double omega_n, double zeta)
	Compute theoretical system characteristics.

Detailed Description

Transfer function utilities for linear systems and nonlinear elements.

Provides symbolic-compatible, stateless functions for:

First-order systems (time constant filters)
Second-order systems (natural frequency, damping)
Discretization methods (ZOH, Tustin, Euler)
Nonlinear elements (rate limiting, saturation, deadband, hysteresis)

All functions are pure and stateless. State management is the responsibility of the calling simulation framework (e.g., Icarus).

Function Documentation

◆ deadband()

template<typename Scalar>

Scalar vulcan::tf::deadband	(	const Scalar &	input,
		double	deadband_width )

Apply deadband (stateless).

Returns zero for inputs within deadband, otherwise input minus deadband.

Template Parameters

Scalar Variable type

Parameters

input	Input value
deadband_width	Deadband half-width (symmetric around zero)

Returns: Scalar Output with deadband applied

◆ discretize_backward_euler()

template<int N>

void vulcan::tf::discretize_backward_euler	(	const Eigen::Matrix< double, N, N > &	A_c,
		const Eigen::Matrix< double, N, 1 > &	B_c,
		double	dt,
		Eigen::Matrix< double, N, N > &	A_d,
		Eigen::Matrix< double, N, 1 > &	B_d )

Backward Euler discretization.

Template Parameters

N	State dimension

◆ discretize_euler()

template<int N>

void vulcan::tf::discretize_euler	(	const Eigen::Matrix< double, N, N > &	A_c,
		const Eigen::Matrix< double, N, 1 > &	B_c,
		double	dt,
		Eigen::Matrix< double, N, N > &	A_d,
		Eigen::Matrix< double, N, 1 > &	B_d )

Forward Euler discretization.

Template Parameters

N	State dimension

◆ discretize_tustin()

template<int N>

void vulcan::tf::discretize_tustin	(	const Eigen::Matrix< double, N, N > &	A_c,
		const Eigen::Matrix< double, N, 1 > &	B_c,
		double	dt,
		Eigen::Matrix< double, N, N > &	A_d,
		Eigen::Matrix< double, N, 1 > &	B_d )

Tustin (bilinear) discretization.

Template Parameters

N	State dimension

◆ discretize_zoh()

template<int N>

void vulcan::tf::discretize_zoh	(	const Eigen::Matrix< double, N, N > &	A_c,
		const Eigen::Matrix< double, N, 1 > &	B_c,
		double	dt,
		Eigen::Matrix< double, N, N > &	A_d,
		Eigen::Matrix< double, N, 1 > &	B_d )

Zero-order hold (ZOH) discretization.

Template Parameters

N	State dimension

Parameters

A_c	Continuous state matrix
B_c	Continuous input matrix
dt	Sample time [s]
A_d	Output discrete state matrix
B_d	Output discrete input matrix

◆ first_order()

template<typename Scalar>

FirstOrderCoeffs< Scalar > vulcan::tf::first_order	(	double	tau,
		double	K,
		double	dt )

Compute first-order discrete-time system coefficients.

Template Parameters

Scalar Variable type

Parameters

tau	Time constant [s] (must be positive)
K	DC gain (default: 1.0)
dt	Sample time [s] (must be positive)

Returns: FirstOrderCoeffs<Scalar> The discretized coefficients

◆ first_order_response()

template<typename Scalar>

Scalar vulcan::tf::first_order_response	(	double	tau,
		double	K,
		const Scalar &	t )

Compute analytical step response at time t.

y(t) = K * (1 - exp(-t/tau))

Template Parameters

Scalar Variable type

Parameters

tau	Time constant [s]
K	DC gain
t	Time [s]

Returns: Scalar Output at time t for unit step input

◆ first_order_step()

template<typename Scalar>

Scalar vulcan::tf::first_order_step	(	const FirstOrderCoeffs< Scalar > &	coeffs,
		const Scalar &	state,
		const Scalar &	input )

Compute next state for first-order system (stateless).

Computes: y[k+1] = a*y[k] + b*u[k]

Template Parameters

Scalar Variable type

Parameters

coeffs	The first-order coefficients
state	Current state y[k]
input	Current input u[k]

Returns: Scalar Next state y[k+1]

◆ hysteresis()

template<typename Scalar>

Scalar vulcan::tf::hysteresis	(	const Scalar &	input,
		const Scalar &	last_output,
		double	hysteresis_width )

Apply hysteresis (stateless computation).

Given input and last output, computes new output with hysteresis.

Template Parameters

Scalar Variable type

Parameters

input	Current input command
last_output	Previous output
hysteresis_width	Hysteresis half-width

Returns: Scalar New output

◆ matrix_exp_pade()

template<int N>

Eigen::Matrix< double, N, N > vulcan::tf::matrix_exp_pade ( const Eigen::Matrix< double, N, N > & A )

Compute matrix exponential using Padé(13,13) approximation.

Template Parameters

N	Matrix dimension

Parameters

A	Input matrix

Returns: Eigen::Matrix<double, N, N> exp(A)

◆ rate_limit()

template<typename Scalar>

Scalar vulcan::tf::rate_limit	(	const Scalar &	current,
		const Scalar &	commanded,
		double	rate_max,
		double	dt )

Apply rate limiting to a value change (stateless).

Limits the delta from current to commanded value.

Template Parameters

Scalar Variable type

Parameters

current	Current value
commanded	Commanded value
rate_max	Maximum rate magnitude [units/s]
dt	Time step [s]

Returns: Scalar Rate-limited next value

◆ saturate()

template<typename Scalar>

Scalar vulcan::tf::saturate	(	const Scalar &	value,
		const Scalar &	min,
		const Scalar &	max )

Apply saturation/clamping (stateless).

Template Parameters

Scalar Variable type

Parameters

value	Value to saturate
min	Minimum allowed value
max	Maximum allowed value

Returns: Scalar Saturated value

◆ second_order()

template<typename Scalar>

SecondOrderCoeffs< Scalar > vulcan::tf::second_order	(	double	omega_n,
		double	zeta,
		double	K,
		double	dt )

Compute second-order discrete-time system coefficients.

Uses ZOH discretization via closed-form solution.

Template Parameters

Scalar Variable type

Parameters

omega_n	Natural frequency [rad/s] (must be positive)
zeta	Damping ratio (0 < zeta for stability)
K	DC gain
dt	Sample time [s] (must be positive)

Returns: SecondOrderCoeffs<Scalar> The discretized coefficients

◆ second_order_characteristics()

std::tuple< double, double, double > vulcan::tf::second_order_characteristics	(	double	omega_n,
		double	zeta )

inline

Compute theoretical system characteristics.

Parameters

omega_n	Natural frequency [rad/s]
zeta	Damping ratio

Returns: std::tuple<double, double, double> {rise_time, settling_time, overshoot_percent}

◆ second_order_output()

template<typename Scalar>

Scalar vulcan::tf::second_order_output	(	const SecondOrderCoeffs< Scalar > &	coeffs,
		const Eigen::Matrix< Scalar, 2, 1 > &	state,
		const Scalar &	input )

Extract output from state (stateless).

Computes: y = C*x + D*u

Template Parameters

Scalar Variable type

Parameters

coeffs	The second-order coefficients
state	Current state [y, ẏ]ᵀ
input	Current input u[k]

Returns: Scalar Output y

◆ second_order_step()

template<typename Scalar>

Eigen::Matrix< Scalar, 2, 1 > vulcan::tf::second_order_step	(	const SecondOrderCoeffs< Scalar > &	coeffs,
		const Eigen::Matrix< Scalar, 2, 1 > &	state,
		const Scalar &	input )

Compute next state for second-order system (stateless).

Computes: x[k+1] = A*x[k] + B*u[k]

Template Parameters

Scalar Variable type

Parameters

coeffs	The second-order coefficients
state	Current state [y, ẏ]ᵀ
input	Current input u[k]

Returns: Eigen::Matrix<Scalar, 2, 1> Next state

Classes

Functions

Detailed Description

Function Documentation

◆ deadband()

◆ discretize_backward_euler()

◆ discretize_euler()

◆ discretize_tustin()

◆ discretize_zoh()

◆ first_order()

◆ first_order_response()

◆ first_order_step()

◆ hysteresis()

◆ matrix_exp_pade()

◆ rate_limit()

◆ saturate()

◆ second_order()

◆ second_order_characteristics()

◆ second_order_output()

◆ second_order_step()