Drake
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Classes

class  LuenbergerObserver< T >
 A simple state observer for a dynamical system of the form:

\[\dot{x} = f(x,u) \]

\[y = g(x,u) \]

the observer dynamics takes the form

\[\dot{\hat{x}} = f(\hat{x},u) + L(y - g(\hat{x},u)) \]

where \(\hat{x}\) is the estimated state of the original system. More...

 

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 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...

 
std::unique_ptr< LuenbergerObserver< double > > SteadyStateKalmanFilter (std::unique_ptr< 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 above. More...
 
std::unique_ptr< LuenbergerObserver< double > > SteadyStateKalmanFilter (std::unique_ptr< System< double >> system, std::unique_ptr< 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. More...
 
Eigen::MatrixXd ObservabilityMatrix (const LinearSystem< double > &sys)
 Returns the observability matrix: O = [ C; CA; ...; CA^{n-1} ]. More...
 
bool IsObservable (const LinearSystem< double > &sys, double threshold)
 Returns true iff the observability matrix is full column rank. More...
 

Detailed Description

Function Documentation

bool IsObservable ( const LinearSystem< double > &  sys,
double  threshold 
)

Returns true iff the observability matrix is full column rank.

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

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

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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 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.

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.

Parameters
AThe state-space dynamics matrix of size num_states x num_states.
CThe state-space output matrix of size num_outputs x num_states.
WThe process noise covariance matrix, E[ww'], of size num_states x num_states.
VThe measurement noise covariance matrix, E[vv'], of size num_.
Returns
The steady-state observer gain matrix of size num_states x num_outputs.
Exceptions
std::runtime_errorif V is not positive definite.

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std::unique_ptr< LuenbergerObserver< double > > SteadyStateKalmanFilter ( std::unique_ptr< 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 above.

Parameters
systemA unique_ptr to a LinearSystem describing the system to be observed. The new observer will take and maintain ownership of this pointer.
WThe process noise covariance matrix, E[ww'], of size num_states x num_states.
VThe measurement noise covariance matrix, E[vv'], of size num_.
Returns
A unique_ptr to the constructed observer system.
Exceptions
std::runtime_errorif V is not positive definite.

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std::unique_ptr< LuenbergerObserver< double > > SteadyStateKalmanFilter ( std::unique_ptr< System< double >>  system,
std::unique_ptr< 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.

Assuming system has the (continuous-time) dynamics: dx/dt = 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.

Parameters
systemA unique_ptr to a System describing the system to be observed. The new observer will take and maintain ownership of this pointer.
contextA unique_ptr to the context describing a fixed-point of the system (plus any additional parameters). The new observer will take and maintain ownership of this pointer for use in its internal forward simulation.
WThe process noise covariance matrix, E[ww'], of size num_states x num_states.
VThe measurement noise covariance matrix, E[vv'], of size num_.
Returns
A unique_ptr to the constructed observer system.
Exceptions
std::runtime_errorif V is not positive definite.

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