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.

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 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 .