Maximum Fuzzy Correntropy Kalman Filter and Its Application to Bearings-Only Maneuvering Target Tracking

Abstract

In this paper, a novel maximum fuzzy correntropy Kalman filter (MFC-KF) algorithm is proposed to solve the problem that the effect of different samples on state estimation is uncertain in common correntropy. In the proposed algorithm, a new optimization criterion—the maximum fuzzy correntropy criterion with fuzzy correntropy based on fuzzy information theory—is used to optimize the Kalman filter, by reducing the effect of the common correntropy applying the same weight for all samples. Moreover, to apply the MFC-KF algorithm to bearings-only maneuvering target tracking, it is combined with the least-squares method for measurement conversion. Moreover, the kernel width is set adaptively. Simulations show that the proposed algorithm can track a target more accurately than the interactive multi-model extended Kalman filter (IMMEKF), the interactive multi-model unscented Kalman filter (IMMUKF), or the maximum correntropy Kalman filter (MCKF).

Appendix

Equation (24) is derived as follows:

$${\mathbf{W}}(k) = {\mathbf{B}}^{ - 1} (k)\left[ \begin{aligned} \text{ }{\mathbf{\rm I}} \hfill \\ {\mathbf{H}}(k) \hfill \\ \end{aligned} \right]{ = }\left[ {\begin{array}{*{20}c} {{\mathbf{B}}_{p}^{ - 1} (k|k - 1)} & 0 \\ 0 & {{\mathbf{B}}_{r}^{ - 1} (k)} \\ \end{array} } \right]\left[ \begin{aligned} \text{ }{\mathbf{I}} \hfill \\ {\mathbf{H}}(k) \hfill \\ \end{aligned} \right] = \left[ \begin{aligned} {\mathbf{B}}_{p}^{ - 1} (k|k - 1) \hfill \\ {\mathbf{B}}_{r}^{ - 1} (k){\mathbf{H}}(k) \hfill \\ \end{aligned} \right]$$

(A.1)

$${\mathbf{U}}(k) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\mathbf{U}}_{x} (k)} \\ 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ {{\mathbf{U}}_{y} (k)} \\ \end{array} } \\ \end{array} } \right]$$

(A.2)

$${\mathbf{D}}(k) = {\mathbf{\rm B}}^{ - 1} (k)\left[ {\begin{array}{*{20}c} {{\hat{\mathbf{x}}}(k|k - 1)} \\ {{\mathbf{y}}(k)} \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} {{\mathbf{B}}_{p}^{ - 1} (k|k - 1){\hat{\mathbf{x}}}(k|k - 1)} \\ {{\mathbf{B}}_{r}^{ - 1} (k){\mathbf{y}}(k)} \\ \end{array} } \right]$$

(A.3)

From Eqs. (A.1) and (A.2), we have

$$\left( {{\mathbf{W}}^{T} (k){\mathbf{U}}(k){\mathbf{W}}(k)} \right)^{ - 1} { = [(}{\mathbf{B}}_{p}^{ - 1} )^{T} {\mathbf{U}}_{x} {\mathbf{B}}_{p}^{ - 1} + {\mathbf{H}}^{T} ({\mathbf{B}}_{r}^{ - 1} )^{T} {\mathbf{U}}_{y} {\mathbf{B}}_{r}^{ - 1} {\mathbf{H}} ]^{ - 1}$$

(A.4)

From the inverse formula of a matrix, $(A - BD^{ - 1} C)^{ - 1} = A^{ - 1} + A^{ - 1} B(D - CA^{ - 1} B)^{ - 1} CA^{ - 1}$, we get

$$\begin{aligned} & \left( {{\mathbf{W}}^{T} (k){\mathbf{U}}(k){\mathbf{W}}(k)} \right)^{ - 1} { = (}{\mathbf{B}}_{p} {\mathbf{U}}_{x}^{ - 1} {\mathbf{B}}_{p}^{T} \hfill \\ & - {\mathbf{B}}_{p} {\mathbf{U}}_{x}^{ - 1} {\mathbf{B}}_{p}^{T} {\mathbf{H}}^{T} ({\mathbf{B}}_{r} {\mathbf{U}}_{y}^{ - 1} {\mathbf{B}}_{r}^{T} + {\mathbf{HB}}_{p} {\mathbf{U}}_{x}^{ - 1} {\mathbf{B}}_{p}^{T} {\mathbf{H}}^{T} )^{ - 1} {\mathbf{HB}}_{p} {\mathbf{U}}_{x}^{ - 1} {\mathbf{B}}_{p}^{T} ) \hfill \\ \end{aligned}$$

(A.5)

From Eqs. (A.1)–(A.3), we have

$${\mathbf{W}}^{T} (k){\mathbf{U}}(k){\mathbf{D}}(k) = ({\mathbf{B}}_{p}^{ - 1} )^{T} {\mathbf{U}}_{x} {\mathbf{B}}_{p}^{ - 1} {\hat{\mathbf{x}}}(k|k - 1) + {\mathbf{H}}^{T} ({\mathbf{B}}_{r}^{ - 1} )^{T} {\mathbf{U}}_{y} {\mathbf{B}}_{r}^{ - 1} {\mathbf{y}}(k)$$

(A.6)

Equation (24) can be obtained from Eqs. (23), (A.5), and (A.6).