**2. The Case Study**

The considered case study is quite simple. A vehicle is moving at a velocity *v*ref(*t*) with an acceleration *a*ref(*t*), as shown in Figure 1.

The state equations of the vehicle can be written as:

$$\begin{aligned} \upsilon\_k &= \upsilon\_{k-1} + \tau \cdot a\_{k-1} + w\_k^v \\ a\_k &= a\_{k-1} + w\_k^a \end{aligned} \tag{1}$$


**Figure 1.** Reference values of velocity (blue line) and acceleration (red line) over time.

It is assumed that the noises are random in nature and belong to Gaussian distributions that do not vary with time (Gaussian distributions are considered as in [23], for a direct comparison). So, *w<sup>v</sup> <sup>k</sup>* = *<sup>w</sup><sup>v</sup>* and *<sup>w</sup><sup>a</sup> <sup>k</sup>* = *<sup>w</sup><sup>a</sup>* are the standard deviations of the constant normal distributions with zero mean.

*w<sup>v</sup>* is assumed to be 0.003 m/s. This value has been derived by considering the accuracy of a GPS which has been reported in the official GPS website [24], which is usually quite accurate compared to the speedometer of the vehicle. Whereas, *w<sup>a</sup>* is assumed to be 0.0005 m/s2 and is supposed to be due to some noise in the circuit or to the driver applying force on the accelerator.

The measured values of the velocity and the acceleration are supposed to have been obtained from the on board sensors of the vehicle. The accuracies of the onboard sensors are in general one or two magnitudes less accurate than a GPS based measurement. So, the following is considered:


## **3. Construction of the RFVs and the Possibilistic Kalman Filter**

Although this has been explained in detail in [23], it has been recalled in this paper as the construction of the RFVs is the same also for the alternative possibilisitc KF defined in this paper.

In the possibilistic KF defined in [23], all the states are RFVs and the algorithm is as shown in Figure 2 [23].

$$\text{According to Equation (1): } \mathbf{A}\_k = \mathbf{A} = \begin{bmatrix} 1 & \mathbf{r} \\ 0 & 1 \end{bmatrix} \text{ and } \mathbf{H}\_k = \mathbf{H} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$$

Matrix **Q***POS* considers the model uncertainties and is a matrix of RFVs. According to the assumptions given in Section 2, we define **Q***POS* where:


**Figure 2.** The possibilistic Kalman filter algorithm [23].

As for the initial state vector **X***<sup>a</sup>* 0, it is assumed that there are no systematic contributions to uncertainty. So, the RFV is obtained by just the random PD as follows:


As for the measured values in each step *k*, matrix **Y***<sup>k</sup>* is the matrix of the RFVs of the velocity and acceleration measurements. The RFV associated with the simulated measured velocity is centered on the simulated measured velocity at step k (*vmk*) and


On the other hand, the acceleration has no systematic error. So, the RFV associated to the simulated measured acceleration is centered on the simulated measured acceleration at step k (*amk*) and


Matrix **CX***<sup>f</sup> k* is the noise covariance matrix of the velocity and acceleration RFVs. However, as it is shown in the equations in Figure <sup>2</sup> and explained in [23], **CX***<sup>f</sup> k* = **C***ran* **X***f k* . So, the possibilistic variances and covariances are evaluated from only the random contributions to uncertainty in both the velocity and acceleration RFVs.

Similarly, **CY***<sup>k</sup>* = **CY***ran <sup>k</sup>* which means that the possibilistic variances and covariances of the noise covariance matrix associated with the measurements are evaluated from just the random uncertainty contributions in the velocity and acceleration measurements.

The described KF has been applied to the case study described in Section 2. The results obtained from the simulations are presented in Figures 3 and 4.

The predicted values of the velocity and acceleration from the KF are obtained by evaluating the mean values of the a posteriori RFVs in matrix **X***<sup>a</sup> <sup>k</sup>*. In both Figures 3 and 4, 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 to the state predictions (red lines) are the *α*−cut at *α* = 0.01 of the velocity and acceleration RFVs predicted by the KF. The *α*−cut can be considered as the confidence interval at the confidence level 1-*α* [16]. For *α* = 0.01, these intervals correspond to the 99% confidence interval in the corresponding pdf.

**Figure 3.** Difference in the reference and predicted velocity values (blue line) provided by the possibilistic Kalman filter together with the predicted uncertainty interval (red lines).
