**1. Introduction**

The Kalman filter (KF) is an algorithm that has long been in existence. It filters the noise on the measured values of the states and provides an estimation of the system states based on the state equations. The classical KF algorithm requires that the states are free from any systematic errors and that the state variables are independent from each other and can be represented by Gaussian distributions [1]. But in most practical situations, the systematic error can not be compensated perfectly and there is a residual systematic error. In this case, the classical formulations of the KF underestimate the uncertainty associated to the state estimates, because the systematic error is not propagated in a correct mathematical way. To deal with this, attempts have been made to develop KF algorithms that are also able to consider systematic contributions to uncertainty [2–5]. For instance, in [5], the authors try to use a Schmidt KF that considers the systematic error as a separate state in the state equations and a noise covariance matrix of the possible systematic errors is built and propagated.

More recently, the theory of possibility has been proposed in the literature to represent and propagate both systematic and random contributions to uncertainty. The theory of possibility has been proven by numerous applications in the literature [6–11] to be an effective alternative to the theory of probability when both random and systematic contributions to uncertainty are present in the measurement procedure.

**Citation:** Jetti, H.V.; Salicone, S. A Possibilistic Kalman Filter for the Reduction of the Final Measurement Uncertainty, in Presence of Unknown Systematic Errors. *Metrology* **2021**, *1*, 39–51. https://doi.org/10.3390/ metrology1010003

Academic Editor: Richard Leach

Received: 1 July 2021 Accepted: 10 August 2021 Published: 17 August 2021

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Some attempts to define KFs based on the theory of possibility are already present in the literature [12,13]. However, in [12,13], as far as understood, they consider uncertainty in a fuzzy way that is not compatible with the recommended guidelines in metrology, as specified in [14,15]. In metrology, uncertainty must be considered according to the definitions given in [15].

Within the framework of the theory of possibility, quantities are represented by possibility distributions [16–21]. In particular, as shown in [16–18], where measurement results are considered to be affected by both random and systematic contributions to uncertainty, measured quantities are represented by random-fuzzy variables (RFVs). RFVs consist of an internal membership function which represents the systematic contribution to uncertainty in the quantity and an external membership function which represents the overall uncertainty due to both the systematic and random contributions. As shown in [16,18], this way of representation is perfectly compatible with the metrological definitions given in [14,15]. So, to be able to utilize all the advantages of RFVs, the KF should be able to process them as well.

Possibilistic KFs based on RFVs are available in the literature [22,23]. In [22], a KF using RFVs is defined but there is a high noise in the state predictions given by the KF. In [23], the authors define a possibilistic KF that also uses RFVs and make a comparison with a few other existing KFs, including the Schmidt KF, clearly showing the advantages of the defined possibilistic KF.

Starting from the possibilistic KF defined in [23], this paper proposes an alternative version, which also allows reducing the effects of the systematic contributions to uncertainty, thereby reducing the overall uncertainty associated to the system state predictions. While the possibilistic KF defined in [23] is useful when we are only interested in propagating the residual systematic uncertainty to evaluate the total uncertainty associated to the state predictions from both the random and systematic contributions, the KF defined in this paper can be used to reduce the effects of the systematic contributions to uncertainty and thereby also reduce the overall uncertainty associated to the state predictions.

The rest of the paper has been organized in six sections. Section 2 describes the case study used for the simulation results for an initial validation of the alternative possibilistic KF. Section 3 describes the construction of the RFVs and the algorithm of the modified possibilistic KF described in [23]. Section 4 describes the algorithm for the alternative possibilistic KF proposed in this paper. Section 5 describes more simulations that have been performed to further validate the alternative possibilistic KF. Section 6 describes the experimental case study that has been performed to prove the effectiveness of the alternative possibilistic KF. Section 7 summarizes the paper and gives a conclusion.

To facilitate an easy comparison between the proposed possibilistic KF and the original one defined in [23], the same simulated case study as in [23] is considered here, as briefly described in Section 2.
