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# kf_predict

## PURPOSE KF_PREDICT Perform Kalman Filter prediction step

## SYNOPSIS function [x,P] = kf_predict(x,P,A,Q,B,u)

## DESCRIPTION ```KF_PREDICT  Perform Kalman Filter prediction step

Syntax:
[X,P] = KF_PREDICT(X,P,A,Q,B,U)

In:
X - Nx1 mean state estimate of previous step
P - NxN state covariance of previous step
A - Transition matrix of discrete model (optional, default identity)
Q - Process noise of discrete model     (optional, default zero)
B - Input effect matrix                 (optional, default identity)
U - Constant input                      (optional, default empty)

Out:
X - Predicted state mean
P - Predicted state covariance

Description:
Perform Kalman Filter prediction step. The model is

x[k] = A*x[k-1] + B*u[k-1] + q,  q ~ N(0,Q).

The predicted state is distributed as follows:

p(x[k] | x[k-1]) = N(x[k] | A*x[k-1] + B*u[k-1], Q[k-1])

The predicted mean x-[k] and covariance P-[k] are calculated
with the following equations:

m-[k] = A*x[k-1] + B*u[k-1]
P-[k] = A*P[k-1]*A' + Q.

If there is no input u present then the first equation reduces to
m-[k] = A*x[k-1]

History:

26.2.2007 JH Added the distribution model for the predicted state
and equations for calculating the predicted state mean and
covariance to the description section.

KF_UPDATE, LTI_DISC, EKF_PREDICT, EKF_UPDATE```

## CROSS-REFERENCE INFORMATION This function calls:
This function is called by:
• imm_filter IMM_FILTER Interacting Multiple Model (IMM) Filter prediction and update steps
• imm_predict IMM_PREDICT Interacting Multiple Model (IMM) Filter prediction step
• imm_smooth IMM_SMOOTH Fixed-interval IMM smoother using two IMM-filters.
• kf_loop KF_LOOP Performs the prediction and update steps of the Kalman filter
• tf_smooth TF_SMOOTH Two filter based Smoother
• uimm_predict IMM_PREDICT UKF based Interacting Multiple Model (IMM) Filter prediction step
• uimm_smooth UIMM_SMOOTH UKF based Fixed-interval IMM smoother using two IMM-UKF filters.

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