The Kalman Filter Uncertainty Concept in the Possibility Domain

Abstract

The Kalman filter is one of the most important and common optimal recursive data processing algorithm in many applications characterized by linear dynamical behavior and affected by random zero-mean white Gaussian noise. However, when measurement processes are considered, inaccuracy is not only due to noise, but also to several contributions to uncertainty that can be due to both random and uncompensated systematic effects. Therefore, when the Kalman filter is used on experimental data, all uncertainty contributions should be considered.While several proposals are available in the literature to modify the Kalman filter in order to consider different probability distributions and systematic effects, represented both in probability and possibility domains, they are not fully compliant with the uncertainty concept adopted in metrology. The aim of this paper is hence to reformulate the Kalman filter theory within the possibility domain in compliance with the measurement uncertainty concept, in order to be able to consider both the random and systematic contributions to uncertainty (regardless of their distribution) that may affect the measurement process.An experimental set-up is considered and the results obtained under different assumptions are reported.

The Possibilistic Kalman Filter: Definition and Comparison With the Available Methods

“…The authors are with: 1 Politecnico di Milano, Department of Electronics, Information and Bioengineering, P.za Leonardo da Vinci 32, 20133 Milano, Italy (e-mail: alessandro.ferrero@polimi.it, simona.salicone@polimi.it); 2 Politecnico di Milano, Department of Energy, Via Lambruschini 4, 20156 Milano, Italy (e-mail: harshavardhana.jetti@polimi.it) count also systematic uncertainty contributions, as also done with the possibilistic KFs defined in [16] and in this paper. Other contributions can also be found in the literature, to include systematic effects in the Kalman filter, as, for instance [17]- [19].…”

“…A few possibilistic KF have also been defined in the literature [20], [21] but, at the Authors' knowledge, all of them consider uncertainty in a sort of semantic way [16], as typical of the fuzzy applications, and not as a well specified concept in metrology, as recommended by [22], [23]. Since uncertainty, in metrology, must be considered according to the definition given by [23], this paper is hence aimed at proposing a possibilistic KF, whose definition is perfectly framed within the present Standards [22], [23].…”

“…Five different approaches to the KF are applied and compared with each other: two probabilistic approaches (the classical KF and the Schmidt KF [15]), two possibilistic approaches (the possibilistic Kalman filter proposed in [16] and a refinement of it, proposed in this paper) and a hybrid KF, which combines the classical KF and the possibilistic KF defined in this paper, for the random and systematic uncertainty contributions respectively.…”

Adaptive Stochastic Filtration Based on the Estimation of the Covariance Matrix of Measurement Noises Using Irregular Accurate Observations

“…Among the numerous methods of processing dynamic measurements under noisey conditions, focused on various specific features of certain measured processes [3,[11][12][13], the most universal and effective today are the methods of the stochastic filtration theory [4,11,[14][15][16][17][18][19], which provide an optimal evaluation of the measured process by a given criterion.…”

“…The use of filtration theory approaches including the Kalman filter [20], for the assessment of dynamic stochastic processes assumes an accurate initialization of the random noises of these processes [14,16]. At the same time, in real information and control systems exposed to various disturbing effects, the meters' stochastic noises are recognized, as a rule, approximately or fluctuate randomly [10,19,[21][22][23][24][25][26][27]. As a consequence, one of the very critical Kalman filter characteristics is the Covariance Matrix of Measurement Noises (CMMN), which straightforwardly influences the filter gain change and, consequently, the rate of convergence of the filtration process.…”