BUILT-IN MATHEMATICA SYMBOL

# KalmanEstimator

KalmanEstimator[ssm, {w, v}]
constructs the Kalman estimator for the StateSpaceModel ssm with process and measurement noise covariance matrices w and v.

KalmanEstimator[ssm, {w, v, h}]
includes the cross-covariance matrix h.

KalmanEstimator[{ssm, sensors}, {...}]
specifies sensors as the noisy measurements of ssm.

KalmanEstimator[{ssm, sensors, dinputs}, {...}]
specifies dinputs as the deterministic inputs of ssm.

## Details and OptionsDetails and Options

• The standard state-space model ssm can be given as StateSpaceModel[{a, b, c, d}], 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 descriptor state-space model ssm can be given as StateSpaceModel[{a, b, c, d, e}] in either continuous time or discrete time:
•  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.
• KalmanEstimator[ssm, {...}] is equivalent to KalmanEstimator[{ssm, All, 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.
• KalmanEstimator supports a Method option. The following explicit settings can be given:
•  "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 are 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 .

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

The Kalman estimator for a continuous-time system:

 Out[1]=

The Kalman estimator of a system with one stochastic output:

 Out[1]=

A discrete-time Kalman estimator:

 Out[1]=