A New Derivation of the Cubature Kalman Filters

Abstract

This paper investigates the cubature Kalman filtering (CKF) for nonlinear dynamic systems. This third-degree rule based filter employs a spherical-radial cubature rule to numerically compute the integrals encountered in nonlinear filtering problems, thereby removing the requirements of explicitly computing the Jacobians. The cubature rule, however, requires computing the intractable integrals over a high-dimensional spherical region for multidimensional applications. Moreover, the cubature formula that has been used to construct the spherical cubature formula has some demerits, most notably its inconvenient properties in computation and low estimation accuracy. Aimed at these issues, a general class of CKFs that uses only cubature rules is derived in this paper. It can be shown that the conventional CKF is a special case of the proposed algorithm. The paper also includes higher-degree CKFs, especially two representative types of the fifth-degree CKFs. Performance of the proposed algorithms is demonstrated via two target tracking problems. The experimental results, presented herein, illustrate the superior performance of higher-degree CKFs to conventional nonlinear filters.

Gaussian approximate filter for stochastic dynamic systems with randomly delayed measurements and colored measurement noises

“…Using Gaussian approximations of p(x k−1 , v k−1 |Y k−1 ) and according to marginal integral formula in Appendix A, P xv k|k−1 can be computed as (24), wherev x,k−1|k−1 is given in (25). With Gaussian approximations of p(x k |Y k−1 ) and p(v k |Y k−1 ) in (12) and (29), the joint PDF of x k and v k conditioned on Y k−1 is also Gaussian, i.e…”

“…Similarly, according to the definition ofẑ k−1|k−1 in (18), we have (43). With Gaussian approximation of p(x k−1 , v k−1 |Y k−1 ) and marginal integral formula in Appendix A, and according to the definition of P zz k−1|k−1 in (18), we have (44), wherev x,k−1|k−1 is given by (25) and Ω k−1|k−1 is given by (45). According to the definitions of P xz k,k−1|k−1 and P vz k,k−1|k−1 in (18) and considering that both w k−1 and ξ k−1 are independent of Y k−1 , we have…”

Particle Smoother for Nonlinear Systems With One‐Step Randomly Delayed Measurements

“…Consider the following discrete‐time nonlinear stochastic system as shown by the state‐space model : and the one‐step randomly delayed measurement model where z k is the ideal (undelayed) measurement vector, y k is the actual (available) measurement vector, process noise n k and measurement noise v k are independent white processes with arbitrary PDFs. γ 1 = 0 and γ k ( k ⩾2) is the Bernoulli random variable taking the value of zero or one with known probability p ( γ k = 1) = p k ( k ⩾2), and p k denotes the latency probability and p k ∈[0,1].…”

“…Consider the following discrete-time nonlinear stochastic system as shown by the state-space model [13,14]:…”

Mixed‐Degree Spherical Simplex‐Radial Cubature Kalman Filter

“…An efficient approximation of the Gaussian pdf is therefore required. A series of nonlinear Kalman filters based on deterministic sampling strategy are proposed, for example, the unscented Kalman filter (UKF) [10,15,16], the cubature Kalman filter (CKF) [4,[17][18][19], the Gauss-Hermite quadrature filter (GHQF) [3], and the sparse grid quadrature filter (SGQF) [20]. For non-Gaussian noise environment, for example, impulse noise, the maximum correntropy Kalman filter (MCKF) [8] utilizes the maximum correntropy criterion (MCC) to combat heavy-tailed noises.…”