pydrake.systems.estimators

pydrake.systems.estimators.DiscreteTimeSteadyStateKalmanFilter(A: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], C: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], W: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], V: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → numpy.ndarray[numpy.float64[m, n]]

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.

Parameter A:

The state-space dynamics matrix of size num_states x num_states.

Parameter C:

The state-space output matrix of size num_outputs x num_states.

Parameter W:

The process noise covariance matrix, E[ww’], of size num_states x num_states.

Parameter 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.

Raises:

• RuntimeError if W is not positive semi-definite or if V is not –

• positive definite. –

class pydrake.systems.estimators.LuenbergerObserver

Bases: pydrake.systems.framework.LeafSystem

A simple state observer for a continuous-time 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. The output of the observer system is \(\hat{x}\).

Or a simple state observer for a discrete-time dynamical system of the form:

\[x[n+1] = f_d(x[n],u[n])\]

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

the observer dynamics takes the form

\[\hat{x}[n+1] = f_d(\hat{x}[n],u[n]) + L(y - g(\hat{x}[n],u[n]))\]

where \(\hat{x}\) is the estimated state of the original system. The output of the observer system is \(\hat{x}[n+1]\).

observed_system_input→ observed_system_output→

LuenbergerObserver

→ estimated_state

Note

This class is templated; see LuenbergerObserver_ for the list of instantiations.

__init__(self: pydrake.systems.estimators.LuenbergerObserver, observed_system: pydrake.systems.framework.System, observed_system_context: pydrake.systems.framework.Context, observer_gain: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → None

Constructs the observer.

Parameter observed_system:

The forward model for the observer. Currently, this system must have a maximum of one input port and exactly one output port.

Parameter observed_system_context:

Required because it may contain parameters which we need to evaluate the system.

Parameter observer_gain:

A m-by-n matrix where m is the number of state variables in observed_system, and n is the dimension of the output port of observed_system.

Precondition:

The observed_system output port must be vector-valued.

get_estimated_state_output_port(self: pydrake.systems.estimators.LuenbergerObserver) → pydrake.systems.framework.OutputPort

get_observed_system_input_input_port(self: pydrake.systems.estimators.LuenbergerObserver) → pydrake.systems.framework.InputPort

get_observed_system_output_input_port(self: pydrake.systems.estimators.LuenbergerObserver) → pydrake.systems.framework.InputPort

L(self: pydrake.systems.estimators.LuenbergerObserver) → numpy.ndarray[numpy.float64[m, n]]

Provides access via the short-hand name, L, too.

observer_gain(self: pydrake.systems.estimators.LuenbergerObserver) → numpy.ndarray[numpy.float64[m, n]]

Provides access to the observer gain.

template pydrake.systems.estimators.LuenbergerObserver_

Instantiations: LuenbergerObserver_[float], LuenbergerObserver_[AutoDiffXd], LuenbergerObserver_[Expression]

class pydrake.systems.estimators.LuenbergerObserver_[AutoDiffXd]

Bases: pydrake.systems.framework.LeafSystem_[AutoDiffXd]

A simple state observer for a continuous-time 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. The output of the observer system is \(\hat{x}\).

Or a simple state observer for a discrete-time dynamical system of the form:

\[x[n+1] = f_d(x[n],u[n])\]

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

the observer dynamics takes the form

\[\hat{x}[n+1] = f_d(\hat{x}[n],u[n]) + L(y - g(\hat{x}[n],u[n]))\]

where \(\hat{x}\) is the estimated state of the original system. The output of the observer system is \(\hat{x}[n+1]\).

observed_system_input→ observed_system_output→

LuenbergerObserver

→ estimated_state

__init__(self: pydrake.systems.estimators.LuenbergerObserver_[AutoDiffXd], observed_system: pydrake.systems.framework.System_[AutoDiffXd], observed_system_context: pydrake.systems.framework.Context_[AutoDiffXd], observer_gain: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → None

Constructs the observer.

Parameter observed_system:

The forward model for the observer. Currently, this system must have a maximum of one input port and exactly one output port.

Parameter observed_system_context:

Required because it may contain parameters which we need to evaluate the system.

Parameter observer_gain:

A m-by-n matrix where m is the number of state variables in observed_system, and n is the dimension of the output port of observed_system.

Precondition:

The observed_system output port must be vector-valued.

get_estimated_state_output_port(self: pydrake.systems.estimators.LuenbergerObserver_[AutoDiffXd]) → pydrake.systems.framework.OutputPort_[AutoDiffXd]

get_observed_system_input_input_port(self: pydrake.systems.estimators.LuenbergerObserver_[AutoDiffXd]) → pydrake.systems.framework.InputPort_[AutoDiffXd]

get_observed_system_output_input_port(self: pydrake.systems.estimators.LuenbergerObserver_[AutoDiffXd]) → pydrake.systems.framework.InputPort_[AutoDiffXd]

L(self: pydrake.systems.estimators.LuenbergerObserver_[AutoDiffXd]) → numpy.ndarray[numpy.float64[m, n]]

Provides access via the short-hand name, L, too.

observer_gain(self: pydrake.systems.estimators.LuenbergerObserver_[AutoDiffXd]) → numpy.ndarray[numpy.float64[m, n]]

Provides access to the observer gain.

class pydrake.systems.estimators.LuenbergerObserver_[Expression]

Bases: pydrake.systems.framework.LeafSystem_[Expression]

A simple state observer for a continuous-time 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. The output of the observer system is \(\hat{x}\).

Or a simple state observer for a discrete-time dynamical system of the form:

\[x[n+1] = f_d(x[n],u[n])\]

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

the observer dynamics takes the form

\[\hat{x}[n+1] = f_d(\hat{x}[n],u[n]) + L(y - g(\hat{x}[n],u[n]))\]

where \(\hat{x}\) is the estimated state of the original system. The output of the observer system is \(\hat{x}[n+1]\).

observed_system_input→ observed_system_output→

LuenbergerObserver

→ estimated_state

__init__(self: pydrake.systems.estimators.LuenbergerObserver_[Expression], observed_system: pydrake.systems.framework.System_[Expression], observed_system_context: pydrake.systems.framework.Context_[Expression], observer_gain: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → None

Constructs the observer.

Parameter observed_system:

The forward model for the observer. Currently, this system must have a maximum of one input port and exactly one output port.

Parameter observed_system_context:

Required because it may contain parameters which we need to evaluate the system.

Parameter observer_gain:

A m-by-n matrix where m is the number of state variables in observed_system, and n is the dimension of the output port of observed_system.

Precondition:

The observed_system output port must be vector-valued.

get_estimated_state_output_port(self: pydrake.systems.estimators.LuenbergerObserver_[Expression]) → pydrake.systems.framework.OutputPort_[Expression]

get_observed_system_input_input_port(self: pydrake.systems.estimators.LuenbergerObserver_[Expression]) → pydrake.systems.framework.InputPort_[Expression]

get_observed_system_output_input_port(self: pydrake.systems.estimators.LuenbergerObserver_[Expression]) → pydrake.systems.framework.InputPort_[Expression]

L(self: pydrake.systems.estimators.LuenbergerObserver_[Expression]) → numpy.ndarray[numpy.float64[m, n]]

Provides access via the short-hand name, L, too.

observer_gain(self: pydrake.systems.estimators.LuenbergerObserver_[Expression]) → numpy.ndarray[numpy.float64[m, n]]

Provides access to the observer gain.

pydrake.systems.estimators.SteadyStateKalmanFilter(*args, **kwargs)

Overloaded function.

1. SteadyStateKalmanFilter(A: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], C: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], W: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], V: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) -> numpy.ndarray[numpy.float64[m, n]]

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.

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.

Parameter A:

The state-space dynamics matrix of size num_states x num_states.

Parameter C:

The state-space output matrix of size num_outputs x num_states.

Parameter W:

The process noise covariance matrix, E[ww’], of size num_states x num_states.

Parameter 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.

Raises:

RuntimeError if V is not positive definite. –

2. SteadyStateKalmanFilter(system: pydrake.systems.primitives.LinearSystem, W: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], V: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) -> pydrake.systems.estimators.LuenbergerObserver

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

Parameter system:

The LinearSystem describing the system to be observed.

Parameter W:

The process noise covariance matrix, E[ww’], of size num_states x num_states.

Parameter V:

The measurement noise covariance matrix, E[vv’], of size num_outputs x num_outputs.

Returns:

The constructed observer system.

Raises:

RuntimeError if V is not positive definite. –

3. SteadyStateKalmanFilter(system: pydrake.systems.framework.System, context: pydrake.systems.framework.Context, W: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], V: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) -> pydrake.systems.estimators.LuenbergerObserver

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.

Parameter system:

The System describing the system to be observed.

Parameter context:

The context describing a fixed-point of the system (plus any additional parameters).

Parameter W:

The process noise covariance matrix, E[ww’], of size num_states x num_states.

Parameter V:

The measurement noise covariance matrix, E[vv’], of size num_outputs x num_outputs.

Returns:

The constructed observer system.

Raises:

RuntimeError if V is not positive definite. –