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KalmanEstimator

KalmanEstimator
constructs the Kalman estimator for the StateSpaceModel object ss with process and measurement noise covariance matrices w and v.
KalmanEstimator
includes the cross-covariance matrix h.
KalmanEstimator
specifies sensors as the noisy measurements of ss.
KalmanEstimator
specifies dinputs as the deterministic inputs of ss.
  • The state-space model ss can be given as StateSpaceModel, where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
continuous-time system
discrete-time system
  • The inputs can include the process noise as well as deterministic inputs .
  • The argument dinputs is a list of integers specifying the positions of in .
  • The outputs consist of the noisy measurements as well as other outputs.
  • The argument sensors is a list of integers specifying the positions of in .
  • The arguments sensors and dinputs can also accept values All and None.
  • The noisy measurements are modeled as , where and are the submatrices of and associated with , and is the noise.
  • The process and measurement noises are assumed to be white and Gaussian:
, process noise
, measurement noise
  • The cross-covariance between the process and measurement noises is given by .
  • By default, the cross-covariance matrix is assumed to be a zero matrix.
"CurrentEstimator"constructs the current estimator
"PredictionEstimator"constructs the prediction estimator
  • The current estimate is based on measurements up to the current instant.
  • The prediction estimate is based on measurements up to the previous instant.
  • For continuous-time systems, the current and prediction estimators are the same, and the estimator dynamics are given by .
  • The optimal gain for continuous-time systems is computed as , where solves the continuous algebraic Riccati equation .
  • Block diagram for the continuous-time system with estimator:
  • The matrices with subscripts , , and are submatrices associated with the deterministic inputs, stochastic inputs, and noisy measurements, respectively.
  • For discrete-time systems, the prediction estimator dynamics is given by . The block diagram of the discrete-time system with prediction estimator is the same as the one above.
  • The estimator dynamics of a current estimator for a discrete-time system are , and the current state estimate is obtained from the current measurement as .
  • The optimal gain for the discrete-time system is computed as , where solves the discrete algebraic Riccati equation .
  • Block diagram for the discrete-time system with current estimator:
  • The inputs to the Kalman estimator model are the deterministic inputs and the noisy measurements .
  • The outputs of the Kalman estimator model consist of the estimated states and estimates of the noisy measurements .
  • The optimal estimator is asymptotically stable if is nonsingular, the pair is detectable, and is stabilizable for any .
The Kalman estimator for a continuous-time system:
The Kalman estimator of a system with one stochastic output:
KalmanEstimator recognizes the discrete-time system and assembles a discrete-time Kalman estimator:
The Kalman estimator for a continuous-time system:
In[1]:=
Click for copyable input
Out[1]=
 
The Kalman estimator of a system with one stochastic output:
In[1]:=
Click for copyable input
Out[1]=
 
KalmanEstimator recognizes the discrete-time system and assembles a discrete-time Kalman estimator:
In[1]:=
Click for copyable input
Out[1]=
The Kalman estimator for a system with one measured output and one stochastic input:
The Kalman estimator of a system with nonzero cross-covariance:
The estimator for a system with one sensor output and two deterministic inputs:
The Kalman estimator for a continuous-time system with cross-correlated process and measurement noise:
By default, the Kalman estimator is based on the current measurements:
The prediction estimator:
For continuous-time systems, the current and prediction estimates are equivalent:
Construct a Kalman filter that smooths the response of a stochastic system:
The response of the system to a sinusoid input in the presence of process and measurement noise:
The filtered response :
KalmanEstimator estimates the states and outputs of a system:
Extract the state estimator:
The output estimator:
Construct a Kalman estimator using StateOutputEstimator:
Use KalmanEstimator directly:
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