Kalman gain matrix

The general form of the Kalman filter usually eontains a diserete model of the system together with a set of reeursive equations that eontinuously update the Kalman gain matrix K and the system eovarianee matrix P. 

The Kalman gain matrix K is obtained from a set of reeursive equations that eommenee from some initial eovarianee matrix P(/c//c)... 

The reeursive equations (9.74)-(9.76) that ealeulate the Kalman gain matrix and eovarianee matrix for a Kalman filter are similar to equations (9.29) and (9.30) that... 

Before equations (9.99) can be run, and initial value of P(/c//c) is required. Ideally, they should not be close to the final value, so that convergence can be seen to have taken place. In this instance, P(/c//c) was set to an identity matrix. Figure 9.16 shows the diagonal elements of the Kalman gain matrix during the first 20 steps of the recursive equation (9.99). 

Fig. 9.16 Convergence of diagonal elements of Kalman gain matrix.

The final values of the Kalman Gain matrix K and eovarianee matrix P were... 

Disturbance noise covariance matrix %Kalman gain matrix... 

Step 2. Update of the Kalman gain vector k(l) and variance-covariance matrix P(l)... 

Equation (6) updates the current state vector by adding the correction factor, which is givCT by the difference between the actual measurranent z(k) and the predicted state vector, weighted by the Kalman gain vector. Similarly, the Kalman gain is used to estimate the updated current error covariance matrix, as in Eqn.(7). 

R(k) is the covariance of the measurement errors. (They are assumed white, this is not valid for SA-dominated pseudo-range errors, which are correlated over minutes. Strictly speaking this correlation should be modeled via additional state variables in system model, but normally it is not.) H (fc) Matrix of direction cosines and ones (as A above) that relate pseudo-range or TOA errors to positions and clock bias and Doppler s to velocity and frequency errors. z(fc) Pseudo-range (or TOA) and Doppler measurements K(k) Kalman gains d>(fc) State transition matrix... 

A typical state space model for stand-alone GPS would have 8 states, the spatial coordinates and their velocities, and the clock offset and frequency. The individual pseudo-range measurements can be processed sequentially, which means that the Kalman gains can be calculated as scalars without the need for matrix inversions. There is no minimum number of measurements required to obtain an updated position estimate. The measurements are processed in an optimum fashion and if not enough for good geometry, the estimate of state error variance [P (fc)] will grow. If two sateUites are available, the clock bias terms are just propagated forward via the state transition matrix. 

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