Detailed Description

Implementations of estimators that operate as Systems in a block diagram.

Algorithms that synthesize controllers are located in State Estimation.

Classes
class	LuenbergerObserver< T >
	A simple state observer for a continuous-time dynamical system of the form: More...

Functions
std::unique_ptr< LuenbergerObserver< double > >	SteadyStateKalmanFilter (std::shared_ptr< const LinearSystem< double > > system, const Eigen::Ref< const Eigen::MatrixXd > &W, const Eigen::Ref< const Eigen::MatrixXd > &V)
	Creates a Luenberger observer system using the optimal steady-state Kalman filter gain matrix, L, as described in SteadyStateKalmanFilter and DiscreteTimeSteadyStateKalmanFilter.
std::unique_ptr< LuenbergerObserver< double > >	SteadyStateKalmanFilter (std::shared_ptr< const System< double > > system, const Context< double > &context, const Eigen::Ref< const Eigen::MatrixXd > &W, const Eigen::Ref< const Eigen::MatrixXd > &V)
	Creates a Luenberger observer system using the steady-state Kalman filter observer gain.
Eigen::MatrixXd	ObservabilityMatrix (const LinearSystem< double > &sys)
	Returns the observability matrix: O = [ C; CA; ...; CA^{n-1} ].
bool	IsObservable (const LinearSystem< double > &sys, std::optional< double > threshold=std::nullopt)
	Returns true iff the observability matrix is full column rank.
bool	IsDetectable (const LinearSystem< double > &sys, std::optional< double > threshold=std::nullopt)
	Returns true iff the system is detectable.

Function Documentation

◆ IsDetectable()

bool IsDetectable	(	const LinearSystem< double > &	sys,
		std::optional< double >	threshold = std::nullopt )

Returns true iff the system is detectable.

◆ IsObservable()

bool IsObservable	(	const LinearSystem< double > &	sys,
		std::optional< double >	threshold = std::nullopt )

Returns true iff the observability matrix is full column rank.

◆ ObservabilityMatrix()

Eigen::MatrixXd ObservabilityMatrix ( const LinearSystem< double > & sys )

Returns the observability matrix: O = [ C; CA; ...; CA^{n-1} ].

◆ SteadyStateKalmanFilter() [1/2]

std::unique_ptr< LuenbergerObserver< double > > SteadyStateKalmanFilter	(	std::shared_ptr< const LinearSystem< double > >	system,
		const Eigen::Ref< const Eigen::MatrixXd > &	W,
		const Eigen::Ref< const Eigen::MatrixXd > &	V )

Creates a Luenberger observer system using the optimal steady-state Kalman filter gain matrix, L, as described in SteadyStateKalmanFilter and DiscreteTimeSteadyStateKalmanFilter.

Parameters

system	The LinearSystem describing the system to be observed.
W	The process noise covariance matrix, E[ww'], of size num_states x num_states.
V	The measurement noise covariance matrix, E[vv'], of size num_outputs x num_outputs.

Returns: The constructed observer system.

Exceptions

std::exception if V is not positive definite.

◆ SteadyStateKalmanFilter() [2/2]

std::unique_ptr< LuenbergerObserver< double > > SteadyStateKalmanFilter	(	std::shared_ptr< const System< double > >	system,
		const Context< double > &	context,
		const Eigen::Ref< const Eigen::MatrixXd > &	W,
		const Eigen::Ref< const Eigen::MatrixXd > &	V )

Creates a Luenberger observer system using the steady-state Kalman filter observer gain.

If system has continuous-time dynamics: ẋ = f(x,u), and the output: y = g(x,u), then the resulting observer will have the form dx̂/dt = f(x̂,u) + L(y − g(x̂,u)), where x̂ is the estimated state and the gain matrix, L, is designed as a steady-state Kalman filter using a linearization of f(x,u) at context as described above.

If system has discrete-time dynamics: x[n+1] = f(x[n],u[n]), and the output: y[n] = g(x[n],u[n]), then the resulting observer will have the form x̂[n+1] = f(x̂[n],u[n]) + L(y − g(x̂[n],u[n])), where x̂[n+1] is the estimated state and the gain matrix, L, is designed as a steady-state Kalman filter using a linearization of f(x,u) at context as described above.