**4. The Alternative Kalman filter Algorithm**

In this paper, an alternative version of the KF algorithm described in Section 3 is presented, which allows for the reduction of the residual systematic error. As can be seen in the results in Figures 3 and 4, the possibilistic KF algorithm described in Section 3 estimates the uncertainty intervals associated with the predictions very accurately in the presence of a systematic error. However, it does not compensate for the systematic error.

The alternative possibilistic KF which is proposed in this paper makes use of the above uncertainty interval to partially compensate for the systematic error. The new algorithm is synthetically shown in Figure 5. With respect to the algorithm in Figure 2, it can be seen that all steps are equal, except the last one, which corresponds to the "correction of the predicted states".

In particular, a new RFV **Y***comp <sup>k</sup>* is considered, which tries to compensate for the residual systematic error. At each step k, **Y***comp <sup>k</sup>* consists of just the internal PD which is centered at the positive uncertainty limit evaluated by the KF at the previous iteration (step *k* − 1) and with the same width and shape as the internal membership function of the RFVs of the state variables estimated by the KF in the previous iteration (**X***int <sup>k</sup>*−1).

**Y***int*\_*modi fied <sup>k</sup>* is then obtained by adding or subtracting the RFV **<sup>Y</sup>***comp <sup>k</sup>* from **<sup>Y</sup>***int <sup>k</sup>* , depending on if the systematic error is positive or negative:

$$\mathbf{Y}\_k^{int\\_modified} = \begin{cases} \mathbf{Y}\_k^{int} + \mathbf{Y}\_k^{comp} & \text{if systematic error} < 0\\ \mathbf{Y}\_k^{int} - \mathbf{Y}\_k^{comp} & \text{if systematic error} > 0 \end{cases} \tag{2}$$

**Figure 5.** The alternative possibilistic Kalman filter algorithm.

It is exactly like a negative feedback loop: the effects of the systematic contrbutions to uncertainty predicted by the KF is used as a feedback to compensate for a possible systematic error and the systematic error is partially compensated for. The intrinsic requirement for applying this method is that we know the direction of the systematic error i.e., it should be known if the error is positive or negative.

The obtained results are shown in Figures 6 and 7. Again, the predicted values for the velocity and acceleration given by the KF are the mean values of the velocity and acceleration RFVs in matrix **X***<sup>a</sup> k*.

**Figure 6.** Difference in the reference and predicted velocity values (blue line) provided by the possibilistic Kalman filter defined in this paper, together with the predicted uncertainty interval (red lines).

As in Figures 3 and 4, also in Figures 6 and 7 the blue lines represent the differences in the predicted values given by the KF and the true values of the velocity and acceleration respectively. The uncertainty limits associated the state predictions (red lines) are the *α* − *cut* at *α* = 0.01 of the velocity and acceleration RFVs predicted by the KF.

**Figure 7.** Difference in the reference and predicted acceleration values (blue line) provided by the possibilistic Kalman filter defined in this paper, together with the predicted uncertainty interval (red lines).

In Figure 6, with respect to Figure 3, it can be clearly seen that the uncertainty limits have been significantly reduced along with the residual systematic error in the velocity estimate. Table 1 gives a comparison with synthetic indexes for the velocity of the possibilistic KF and the alternative possibilistic KF.


**Table 1.** Comparison of synthetic indexes for the velocity.
