This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# KalmanEstimator

 KalmanEstimator constructs the Kalman estimator for the StateSpaceModel object ss with process and measurement noise covariance matrices w and v. KalmanEstimatorincludes 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:
 Out[1]=

The Kalman estimator of a system with one stochastic output:
 Out[1]=

KalmanEstimator recognizes the discrete-time system and assembles a discrete-time Kalman estimator:
 Out[1]=
 Scope   (4)
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:
 Options   (2)
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:
 Applications   (1)
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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