Detailed Description

Functions
Eigen::MatrixXd	SteadyStateKalmanFilter (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::MatrixXd > &C, const Eigen::Ref< const Eigen::MatrixXd > &W, const Eigen::Ref< const Eigen::MatrixXd > &V)
	Computes the optimal observer gain, L, for the continuous-time linear system defined by \[ \dot{x} = Ax + Bu + w, \] \[ y = Cx + Du + v. \] The resulting observer is of the form \[ \dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x} - Du). \] The process noise, w, and the measurement noise, v, are assumed to be iid mean-zero Gaussian. More...

Eigen::MatrixXd	DiscreteTimeSteadyStateKalmanFilter (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::MatrixXd > &C, const Eigen::Ref< const Eigen::MatrixXd > &W, const Eigen::Ref< const Eigen::MatrixXd > &V)
	Computes the optimal observer gain, L, for the discrete-time linear system defined by \[ x[n+1] = Ax[n] + Bu[n] + w, \] \[ y[n] = Cx[n] + Du[n] + v. \] The resulting observer is of the form \[ \hat{x}[n+1] = A\hat{x}[n] + Bu[n] + L(y - C\hat{x}[n] - Du[n]). \] The process noise, w, and the measurement noise, v, are assumed to be iid mean-zero Gaussian. More...

Function Documentation

◆ DiscreteTimeSteadyStateKalmanFilter()

Eigen::MatrixXd drake::systems::estimators::DiscreteTimeSteadyStateKalmanFilter	(	const Eigen::Ref< const Eigen::MatrixXd > &	A,
		const Eigen::Ref< const Eigen::MatrixXd > &	C,
		const Eigen::Ref< const Eigen::MatrixXd > &	W,
		const Eigen::Ref< const Eigen::MatrixXd > &	V
	)

Computes the optimal observer gain, L, for the discrete-time linear system defined by

\[ x[n+1] = Ax[n] + Bu[n] + w, \]

\[ y[n] = Cx[n] + Du[n] + v. \]

The resulting observer is of the form

\[ \hat{x}[n+1] = A\hat{x}[n] + Bu[n] + L(y - C\hat{x}[n] - Du[n]). \]

The process noise, w, and the measurement noise, v, are assumed to be iid mean-zero Gaussian.

This is a simplified form of the full Kalman filter obtained by assuming that the state-covariance matrix has already converged to its steady-state solution P, and the observer gain L = APC'(CPC' + V)⁻¹.

Parameters

A	The state-space dynamics matrix of size num_states x num_states.
C	The state-space output matrix of size num_outputs x num_states.
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 steady-state observer gain matrix of size num_states x num_outputs.

Exceptions

std::exception if W is not positive semi-definite or if V is not positive definite.

◆ SteadyStateKalmanFilter()

Eigen::MatrixXd drake::systems::estimators::SteadyStateKalmanFilter	(	const Eigen::Ref< const Eigen::MatrixXd > &	A,
		const Eigen::Ref< const Eigen::MatrixXd > &	C,
		const Eigen::Ref< const Eigen::MatrixXd > &	W,
		const Eigen::Ref< const Eigen::MatrixXd > &	V
	)

Computes the optimal observer gain, L, for the continuous-time linear system defined by

\[ \dot{x} = Ax + Bu + w, \]

\[ y = Cx + Du + v. \]

The resulting observer is of the form

\[ \dot{\hat{x}} = A\hat{x} + Bu + L(y - C\hat{x} - Du). \]