KalmanEstimator

• 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:

• 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 is equivalent to KalmanEstimator[{ss, 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:

• 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:

• 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 .