SOLUTIONS

BUILTIN MATHEMATICA SYMBOL
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 crosscovariance 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 statespace 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 continuoustime or a discretetime system:

continuoustime system discretetime system  The descriptor statespace model ssm can be given as StateSpaceModel[{a, b, c, d, e}] in either continuous time or discrete time:

continuoustime system discretetime 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 crosscovariance between the process and measurement noises is given by .
 By default, the crosscovariance 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 continuoustime systems, the current and prediction estimators are the same, and the estimator dynamics are given by .
 The optimal gain for continuoustime systems is computed as , where solves the continuous algebraic Riccati equation .
 Block diagram for the continuoustime system with estimator:
 The matrices with subscripts , , and are submatrices associated with the deterministic inputs, stochastic inputs, and noisy measurements, respectively.
 For discretetime systems, the prediction estimator dynamics are given by . The block diagram of the discretetime system with prediction estimator is the same as the one above.
 The estimator dynamics of a current estimator for a discretetime system are , and the current state estimate is obtained from the current measurement as .
 The optimal gain for the discretetime system is computed as , where solves the discrete algebraic Riccati equation .
 Block diagram for the discretetime 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 .
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