Optimal Kalman gains for combined stochastic and set-membership state estimation

Abstract

In state estimation theory, two directions are mainly followed in order to model disturbances and errors. Either uncertainties are modeled as stochastic quantities or they are characterized by their membership to a set. Both approaches have distinct advantages and disadvantages making each one inherently better suited to model different sources of estimation uncertainty. This paper is dedicated to the task of combining stochastic and set-membership estimation methods. A Kalman gain is derived that minimizes the mean squared error in the presence of both stochastic and additional unknown but bounded uncertainties, which are represented by Gaussian random variables and ellipsoidal sets, respectively. As a result, a generalization of the well-known Kalman filtering scheme is attained that reduces to the standard Kalman filter in the absence of set-membership uncertainty and that otherwise becomes the intersection of sets in case of vanishing stochastic uncertainty. The proposed concept also allows to prioritize either the minimization of the stochastic uncertainty or the minimization of the set-membership uncertainty.

FD-ZKF: A Zonotopic Kalman Filter optimizing fault detection rather than state estimation

“…Additionally, using Kronecker product to introduce matrices M and N in (20), which are independent matrices with respect to G − (for both disturbance and fault cases), the column form of R d and R f can be written as…”

“…In the set-membership approach, the uncertainties are assumed unknown but bounded [4,17,18,19]. In the case of the set-membership-based state estimation [5,20,21], the estimation characterizes a set of possible states. In literature, several families of geometrical structures have been used as e.g.…”

Set-Membership Based Hybrid Kalman Filter for Nonlinear State Estimation under Systematic Uncertainty

“…With the assumption that the statistical characteristics of system noise are known, the issue of nonlinear system state estimation has been extensively studied [6,[8][9][10]. However, in practical applications, the system uncertainty is mixed by stochastic errors as well as unknown but bounded (UBB) errors [11]. The UBB error, as implied by its name, refers to the systematic modelling uncertainty whose probability distribution is difficult to identify or even without the probabilistic nature.…”

Ellipsoidal set filter combined set-membership and statistics uncertainties for bearing-only maneuvering target tracking

“…Hanebeck compares the set-theoretic and statistical information filter with credal state filter [18] and introduces a concept for systematically predicting and updating bounds for the linearization errors within Kalman filtering framework [19]. The optimal Kalman gain for combined stochastic and setmembership estimation has derived by Noack [20].…”