vulcan::markov Namespace Reference

Classes
struct	State
	First-order Gauss-Markov process state. More...
struct	Coeffs
	Discretized first-order Markov process coefficients. More...

Functions
template<typename Scalar>
State< Scalar >	init_state ()
	Initialize state to zero.
template<typename Scalar>
State< Scalar >	init_state (const Scalar &initial_value)
	Initialize state to specific value.
Coeffs	discretize (double tau, double sigma, double dt)
	Discretize first-order Markov process.
double	process_noise_psd (double tau, double sigma)
	Compute continuous-time process noise PSD.
double	process_noise_intensity (double tau, double sigma)
	Compute continuous-time process noise intensity.
template<typename Scalar>
Scalar	step (State< Scalar > &state, const Coeffs &coeffs, const Scalar &noise_input)
	Step the first-order Markov process.
template<typename Scalar>
Scalar	step (State< Scalar > &state, double tau, double sigma, double dt, const Scalar &noise_input)
	Step with inline discretization.
double	autocorrelation (double tau, double sigma, double tau_lag)
	Compute autocorrelation at lag τ_lag.
double	psd_at_frequency (double tau, double sigma, double f)
	Compute power spectral density at frequency f.

Function Documentation

autocorrelation()

double vulcan::markov::autocorrelation ( double tau, double sigma, double tau_lag )

inline

Compute autocorrelation at lag τ_lag.

For first-order Markov process: R(τ_lag) = σ² · exp(-|τ_lag|/τ)

Parameters

tau	Correlation time [s]
sigma	Steady-state standard deviation
tau_lag	Time lag [s]

Returns

Autocorrelation value

discretize()

Coeffs vulcan::markov::discretize ( double tau, double sigma, double dt )

inline

Discretize first-order Markov process.

Uses exact discretization that preserves the steady-state variance regardless of the time step.

Parameters

tau	Correlation time [s]
sigma	Steady-state standard deviation
dt	Time step [s]

Returns

Discretized coefficients

init_state() [1/2]

template<typename Scalar>

State< Scalar > vulcan::markov::init_state

(

)

Initialize state to zero.

Template Parameters

Scalar

double or casadi::MX

Returns

State initialized to zero

init_state() [2/2]

template<typename Scalar>

State< Scalar > vulcan::markov::init_state

(

const Scalar &

initial_value

)

Initialize state to specific value.

Useful for starting with a non-zero initial bias.

Template Parameters

Scalar

double or casadi::MX

Parameters

initial_value

Initial process value

Returns

State initialized to given value

process_noise_intensity()

double vulcan::markov::process_noise_intensity ( double tau, double sigma )

inline

Compute continuous-time process noise intensity.

The process noise intensity q_c relates to PSD by: q_c² = S_w

Parameters

tau	Correlation time [s]
sigma	Steady-state standard deviation

Returns

Process noise intensity q_c

process_noise_psd()

double vulcan::markov::process_noise_psd ( double tau, double sigma )

inline

Compute continuous-time process noise PSD.

For a first-order Markov process with correlation time τ and steady-state variance σ², the continuous-time process noise PSD is: S_w = 2σ²/τ

Parameters

tau	Correlation time [s]
sigma	Steady-state standard deviation

Returns

Process noise power spectral density

psd_at_frequency()

double vulcan::markov::psd_at_frequency ( double tau, double sigma, double f )

inline

Compute power spectral density at frequency f.

For first-order Markov: S(f) = σ²·(2τ)/(1 + (2πfτ)²)

This is a Lorentzian spectrum.

Parameters

tau	Correlation time [s]
sigma	Steady-state standard deviation
f	Frequency [Hz]

Returns

Power spectral density at frequency f

step() [1/2]

template<typename Scalar>

Scalar vulcan::markov::step	(	State< Scalar > &	state,
		const Coeffs &	coeffs,
		const Scalar &	noise_input )

Step the first-order Markov process.

Updates the process state: x[k+1] = φ·x[k] + q·w[k]

Template Parameters

Scalar

double or casadi::MX

Parameters

state	Current state (updated in-place)
coeffs	Discretized coefficients
noise_input	White noise sample with unit variance

Returns

Current process output after update

step() [2/2]

template<typename Scalar>

Scalar vulcan::markov::step	(	State< Scalar > &	state,
		double	tau,
		double	sigma,
		double	dt,
		const Scalar &	noise_input )

Step with inline discretization.

Convenience function when coefficients don't need to be pre-computed.

Template Parameters

Scalar

double or casadi::MX

Parameters

state	Current state (updated in-place)
tau	Correlation time [s]
sigma	Steady-state standard deviation
dt	Time step [s]
noise_input	White noise sample with unit variance

Returns

Current process output after update