**5. Further Simulations**

Further simulations have been performed in order to verify the effectiveness of the alternative possibilistic KF in all situations. In particular, we want to verify whether the algorithm still works in a good way when it is applied, but no residual systematic error is present.

In fact, the result of the introduction of the "feedback" loop is that the residual systematic error is compensated by the maximum possible value since the uncertainty limit of the RFVs evaluated in each step (which is the value of the *α*-cut at *α* = 0.01 of the RFV) is considered. This means that it is possible that the residual systematic error could be overcompensated as the magnitude of this is unknown.

So, it is important that even if the residual systematic error happens to be zero (which is the limiting case), the overcompensation should not be so high that the predictions of the state variables obtained from the KF fall out of the evaluated uncertainty limits. To verify this, the same example described in the Section 2 is considered except that the systematic error is considered to be zero (instead of 0.3 m/s).

In this case, the results in Figures 8 and 9 are obtained.

**Figure 8.** Difference in the reference and predicted velocity values (blue line) provided by the alternative possibilistic KF, together with the predicted uncertainty interval (red lines) when residual systematic error is zero.

**Figure 9.** Difference in the reference and predicted acceleration values (blue line) provided by the alternative possibilistic KF, together with the predicted uncertainty interval (red lines) when residual systematic error is zero.

As expected, as can be seen in Figure 8, the systematic error in the velocity has been overcompensated, but it is still mostly inside the evaluated uncertainty limits. This demonstrates that the alternative possibilistic KF algorithm successfully decreases the uncertainty associated to the state predictions provided by the KF in all situations. In fact, the average uncertainty in Figure 8 is in any case smaller than the one in Figure 3.
