# Weighted Maximum Correntropy Criterion-Based Interacting Multiple-Model Filter for Maneuvering Target Tracking

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Different from the IMM-MCC filter, the proposed filter adopts another Gaussian kernel function [20] to construct the cost function. Based on this Gaussian kernel function, the weighted maximum correntropy criterion is defined, and a new cost function is given.
- (2)
- The sub-model state update and state fusion steps are derived under the framework of WMCC, and the previous fusion state is used as the input of each sub-model to simplify the state interaction step.
- (3)
- The difference between the IMM-MCC filter and WMCC-IMM filter is discussed, which reveals the superiority of the proposed filter in fusion strategy.
- (4)
- To solve the state estimation problem of a maneuvering target in a radar system, the specific steps of the WMCC-IMM filter in radar maneuvering target tracking is given.
- (5)
- The effectiveness of the WMCC-IMM filter is tested using a set of simulation data and a set of actual test data. Simulation results show that the WMCC-IMM filter has better state estimation performance than the IMM-MCC filter. The great feedback from actual test data indicates that the proposed filter has potential value for practical engineering applications.

## 2. Problem Formulation

#### 2.1. State Interaction

#### 2.2. Sub-Model State Update

#### 2.3. Model Probability Update

#### 2.4. State Fusion

## 3. Main Results

#### 3.1. Weighted Maximum Correntropy Criterion

#### 3.2. WMCC-IMM Filter

#### 3.2.1. State Interaction

#### 3.2.2. Sub-Model State Update

**Remark**

**1.**

- (a)
- The proposed filter is equivalent to the KF with Gaussian measurement noise covariance $a{\mathit{R}}_{k+1}^{i}/({G}_{3}(1-a))$. When outlier interference occurs, the measurement usually deviates from the measurement prediction center ${\mathit{H}}_{k+1}^{i}{\mathit{x}}_{k+1\left|k\right.}^{i}$. At this time, the proposed filter will reduce the trust degree of measurement and increase the measurement noise covariance. This is the main reason why the WMCC-based filter has better robustness than the KF under outlier interference. However, if the prediction center has bad estimation performance, the ${G}_{3}$ will be incorrect, which leads to further increase in the estimation error of state. Therefore, before using the WMCC-IMM filter, the target motion characteristics need to be accurately modeled.

- (b)
- The different weight parameter $a$ means that the proposed filter works in different modes. When $a=0.5$, the WMCC-based filter becomes the MCC-based filter. When $a=1$, the WMCC-based filter only implements the state prediction step. When $a=0$, the WMCC-based filter only uses the measurement to realize the sub-model state update.

- (c)
- In terms of parameter setting, since ${G}_{3}$ is still less than 1 when the measurement is not disturbed by outliers, a value slightly less than 0.5 can be assigned to $a$ to compensate for the measurement noise covariance.

#### 3.2.3. Model probability update

#### 3.2.4. State Fusion

**Remark**

**2.**

Algorithm 1: The pseudo-code of the WMCC-IMM filter |

for k = 0:L − 1Input: ${\mathit{x}}_{k\left|k\right.},{\mathit{P}}_{k\left|k\right.},{\mathit{F}}_{k}^{i},{\mathit{Q}}_{k}^{i},{\mathit{H}}_{k+1}^{i},{\mathit{z}}_{k+1},{\mathit{R}}_{k+1}^{i},a$Using Equations (4) and (5) to calculate ${\mathit{P}}_{k\left|k\right.}^{i,0}$ Using Equation (22) to calculate ${\mathit{x}}_{k+1\left|k\right.}^{i}\mathrm{and}{\mathit{P}}_{k+1\left|k\right.}^{i}$ Using Equations (26)–(28) and (30) to calculate ${\mathit{x}}_{k+1\left|k+1\right.}^{i}\mathrm{and}{\mathit{P}}_{k+1\left|k+1\right.}^{i}$ Using Equations (11), (13), (14) and (31) to calculate ${\mu}_{k+1\left|k+1\right.}^{i}$ Using Equations (35), (36), and (41) to calculate ${\mathit{x}}_{k+1\left|k+1\right.}^{}\mathrm{and}{\mathit{P}}_{k+1\left|k\right.+1}^{}$ Output:${\mathit{x}}_{k+1\left|k+1\right.}^{},{\mathit{P}}_{k+1\left|k\right.+1}^{}$end |

#### 3.3. Principle Analysis

**Theorem**

**1.**

**Proof**

**.**

**Theorem**

**2.**

**Proof**

**.**

**Remark**

**3.**

#### 3.4. WMCC-IMM Filter Applied to Radar Maneuvering Target Tracking

Algorithm 2: The pseudo-code of the WMCC-IMM filter in a radar system |

for k = 0:L − 1Input: ${\mathit{x}}_{k\left|k\right.},{\mathit{P}}_{k\left|k\right.},{\mathit{F}}_{k}^{i},{\mathit{Q}}_{k}^{i},{\mathit{z}}_{k+1},{\mathit{R}}_{k+1}^{i},a$Using Equations (4) and (5) to calculate ${\mathit{P}}_{k\left|k\right.}^{i,0}$ Using Equation (22) to calculate ${\mathit{x}}_{k+1\left|k\right.}^{i}\mathrm{and}{\mathit{P}}_{k+1\left|k\right.}^{i}$ Using Equation (57) to calculate ${\mathit{H}}_{k+1\left|k\right.}^{i}$ Using Equations (60) and (61) to calculate ${\mathit{K}}_{k+1\left|k\right.}^{i}$ Using Equations (59) and (63) ${\mathit{x}}_{k+1\left|k\right.+1}^{i}\mathrm{and}{\mathit{P}}_{k+1\left|k\right.+1}^{i}$ Using Equations (64), (65), (13), and (14) (13), (14), (64) and (65) to calculate ${\mu}_{k+1\left|k+1\right.}^{i}$ Using Equations (35), (36), and (41) to calculate ${\mathit{x}}_{k+1\left|k+1\right.}^{}\mathrm{and}{\mathit{P}}_{k+1\left|k+1\right.}^{}$ Output:${\mathit{x}}_{k+1\left|k+1\right.}^{},{\mathit{P}}_{k+1\left|k\right.+1}^{}$end |

## 4. Experimental Results

#### 4.1. Case 1. Maneuvering Target Tracking Simulation Experiment

^{2}, 25 m

^{2}/s

^{2}, 100 m

^{2}, 25 m

^{2}/s

^{2}]), respectively. The actual motion of the target follows the CT model. For 1–50 s, the turning rate is ${w}_{1}$. For 51–100 s, the turning rate is ${w}_{2}$. The initial state ${\mathit{x}}_{0\left|0\right.}^{i}$($i=1,2$) of each sub-model is chosen from $\mathcal{N}({\mathit{x}}_{0\left|0\right.}^{i};{\mathit{x}}_{0},{\mathit{P}}_{0\left|0\right.}^{i})$. The total time $L$ is 100 s, and the number of Monte Carlo runs is 100. The Markov transition matrix $\mathsf{\pi}$ is set as $\left[\begin{array}{cc}0.95& 0.05\\ 0.05& 0.95\end{array}\right]$. The initial weight of each sub-model is 0.5. The process noise follows the Gaussian distribution, while the measurement noise follows the Gaussian mixture distribution [12]. At this time, ${\mathit{w}}_{k+1}^{i}$ and ${\mathit{v}}_{k+1}^{i}$ ($i=1,2$) are formulated as

#### 4.2. Case 2. Maneuvering Target Tracking Actual Test Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Derivation of ${\mathit{K}}_{k+1}^{i}$

## References

- Wu, P.L.; Li, X.X.; Zhang, L.Z. Tracking algorithm with radar and IR sensors using a novel adaptive grid interacting multiple model. IET Sci. Meas. Technol.
**2014**, 8, 270–276. [Google Scholar] [CrossRef] - Gustafsson, F.; Gunnarsson, F. Particle filters for positioning, navigation, and tracking. IEEE Trans. Signal Process.
**2002**, 50, 425–437. [Google Scholar] [CrossRef] - Jordi, V.V.; Damien, V. Recursive linearly constrained Wiener filter for robust multi-channel signal processing. Signal Process.
**2020**, 167, 107291. [Google Scholar] - Blair, W.D.; Bar-Shalom, Y. MSE Design of Nearly Constant Velocity Kalman Filters for Tracking Targets with Deterministic Maneuvers. IEEE Trans. Aerosp. Electron. Syst.
**2023**, 59, 4180–4191. [Google Scholar] [CrossRef] - Montañez, O.J.; Suarez, M.J.; Fernandez, E.A. Application of Data Sensor Fusion Using Extended Kalman Filter Algorithm for Identification and Tracking of Moving Targets from LiDAR-Radar Data. Remote Sens.
**2023**, 15, 3396. [Google Scholar] [CrossRef] - Kalman, R.E. A New Approach to Linear Filtering and Prediction Problems. J. Basic Eng.
**1960**, 82, 35–45. [Google Scholar] [CrossRef] - Wang, J.L.; Wang, J.H.; Zhang, D. Kalman Filtering through the Feedback Adaption of Prior Error Covariance. Signal Process.
**2018**, 152, 47–53. [Google Scholar] [CrossRef] - Liu, Y.; Ning, X.; Li, J. Adaptive Central Difference Kalman Filter with Unknown Measurement Noise Covariance and Its Application to Airborne POS. IEEE Sens. J.
**2021**, 21, 9927–9936. [Google Scholar] [CrossRef] - Mazor, E.; Averbuch, A.; Bar-Shalom, Y. Interacting multiple model methods in target tracking: A survey. IEEE Trans. Aerosp. Electron. Syst.
**1998**, 34, 103–123. [Google Scholar] [CrossRef] - Qu, H.Q.; Pang, L.P.; Li, S.H. A novel interacting multiple model algorithm. Signal Process.
**2009**, 89, 2171–2177. [Google Scholar] [CrossRef] - Ruan, Y.; Hong, L. Use of the interacting multiple model algorithm with multiple sensors. Math. Comput. Model.
**2006**, 44, 332–341. [Google Scholar] [CrossRef] - Yun, P.; Wu, P.L.; He, S. An IMM-VB Algorithm for Hypersonic Vehicle Tracking with Heavy Tailed Measurement Noise. In Proceedings of the 2018 International Conference on Control, Automation and Information Sciences (ICCAIS), Hangzhou, China, 24–27 October 2018; pp. 169–174. [Google Scholar]
- Yun, P.; Wu, P.L.; He, S. Pearson type VII distribution-based robust Kalman filter under outliers interference. IET Radar Sonar Navig.
**2019**, 13, 1389–1399. [Google Scholar] - Yun, P.; Wu, P.L.; He, S. Robust Kalman Filter with Fading Factor under State Transition Model Mismatch and Outliers Interference. Circuits Syst. Signal Process.
**2021**, 40, 2443–2463. [Google Scholar] [CrossRef] - Chen, B.D.; Xing, L.; Liang, J. Steady-state mean square error analysis for adaptive filtering under the maximum correntropy criterion. IEEE Signal Process. Lett.
**2014**, 21, 880–884. [Google Scholar] - Chen, B.D.; Liu, X.; Zhao, H.Q. Maximum correntropy Kalman filter. Automatica
**2017**, 76, 70–77. [Google Scholar] [CrossRef] - He, J.J.; Sun, C.K.; Zhang, B.S. Maximum Correntropy Square-Root Cubature Kalman Filter for Non-Gaussian Measurement Noise. IEEE Access
**2020**, 8, 70162–70170. [Google Scholar] [CrossRef] - Liu, X.; Qu, H.; Zhao, J. Maximum Correntropy Unscented Kalman Filter for Spacecraft Relative State Estimation. Sensors
**2016**, 16, 1530. [Google Scholar] [CrossRef] [PubMed] - Li, S.X.; Xu, B.; Wang, L.Z. Improved Maximum Correntropy Cubature Kalman Filter for Cooperative Localization. IEEE Sens. J.
**2020**, 20, 13585–13595. [Google Scholar] [CrossRef] - Wang, G.; Li, N.; Zhang, Y. Maximum correntropy unscented Kalman and information filters for non-Gaussian measurement noise. J. Frankl. Inst.
**2017**, 354, 8659–8677. [Google Scholar] [CrossRef] - Wang, G.; Xue, R.; Wang, J.X. A distributed maximum correntropy Kalman filter. Signal Process.
**2019**, 160, 247–251. [Google Scholar] [CrossRef] - Wang, G.Q.; Zhang, Y.G.; Wang, X.D. Maximum Correntropy Rauch-Tung-Striebel Smoother for Nonlinear and Non Gaussian Systems. IEEE Trans. Autom. Control
**2021**, 66, 1270–1277. [Google Scholar] [CrossRef] - Bao, Z.C.; Jiang, Q.X.; Liu, F.Z. Multiple model efficient particle filter based track-before-detect for maneuvering weak targets. J. Syst. Eng. Electron.
**2020**, 31, 647–656. [Google Scholar] - Ebrahimi, M.; Ardeshiri, M.; Khanghah, S.A. Bearing-only 2D maneuvering target tracking using smart interacting multiple model filter. Digit. Signal Process.
**2022**, 126, 103497. [Google Scholar] [CrossRef] - Yang, B.; Zhu, S.Q.; He, X.P. IMM Robust Cardinality Balance Multi-Bernoulli Filter for Multiple Maneuvering Target Tracking with Interval Measurement. Chin. J. Electron.
**2021**, 30, 1141–1151. [Google Scholar] - Fan, X.X.; Wang, G.; Han, J.C. Interacting Multiple Model Based on Maximum Correntropy Kalman Filter. IEEE Trans. Circuits Syst. II Express Briefs
**2021**, 68, 3017–3021. [Google Scholar] [CrossRef] - Yun, P.; Wu, P.L.; Li, X.X. Variational Bayesian Probabilistic Data Association Algorithm. Acta Autom. Sin.
**2022**, 48, 2486–2495. [Google Scholar] [CrossRef] - Yi, S.; Jin, X.; Su, T. An improved online denoising algorithm based on Kalman filter and adaptive current statistics model. In Proceedings of the 2017 9th International Conference on Modelling, Identification and Control (ICMIC), Kunming, China, 10–12 July 2017; pp. 782–786. [Google Scholar]
- Guo, Q.; He, S.; Cheng, J. A Threat Evaluation Method of Autonomous UAV Avoidance Missile. Aeronaut. Sci. Technol.
**2022**, 33, 8–14. [Google Scholar]

**Table 1.**The computational time in the one-step state update process and the TRMSEs from different filters.

Filters | Pos/m | Vel/(m/s) | Time/ms |
---|---|---|---|

IMM | 18.19 | 5.26 | 0.308 |

IMM-VB | 10.28 | 3.74 | 0.392 |

IMM-MCC | 16.46 | 4.71 | 0.318 |

WMCC-IMM ($a=0.6$) | 11.88 | 4.27 | 0.329 |

WMCC-IMM ($a=0.5$) | 10.89 | 3.98 | 0.329 |

WMCC-IMM ($a=0.4$) | 10.28 | 3.78 | 0.329 |

Kernel Bandwidth | IMM-MCC | WMCC-IMM ($\mathit{a}=\mathbf{0.5}$) | ||
---|---|---|---|---|

Pos/m | Vel/(m/s) | Pos/m | Vel/(m/s) | |

$\sigma =1$ | NaN | NaN | 13.22 | 4.46 |

$\sigma =3$ | 18.06 | 5.08 | 9.67 | 3.77 |

$\sigma =5$ | 16.46 | 4.71 | 10.89 | 3.98 |

$\sigma =7$ | 17.28 | 4.88 | 11.58 | 4.08 |

Filters | PosN/m | PosE/m | PosD/(m) | VelN/(m/s) | VelE/(m/s) | VelD/(m/s) |
---|---|---|---|---|---|---|

IMM | 311.34 | 82.32 | 152.22 | 26.65 | 1.42 | 5.53 |

IMM-VB | 355.36 | 82.86 | 123.18 | 30.33 | 1.4 | 4.55 |

IMM-MCC | 314.62 | 81.11 | 174.55 | 27.02 | 1.46 | 4.69 |

WMCC-IMM ($a=0.6$) | 401.77 | 83.47 | 143.54 | 41.14 | 0.93 | 6.42 |

WMCC-IMM ($a=0.5$) | 329.91 | 82.14 | 169.97 | 28.12 | 1.43 | 4.81 |

WMCC-IMM ($a=0.4$) | 273.84 | 80.53 | 191.82 | 20.84 | 1.74 | 3.19 |

Filters | Entry Angle/° |
---|---|

IMM | 4.37 |

IMM-VB | 5.06 |

IMM-MCC | 4.45 |

WMCC-IMM ($a=0.6$) | 6.97 |

WMCC-IMM ($a=0.5$) | 4.65 |

WMCC-IMM ($a=0.4$) | 3.41 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Huai, L.; Li, B.; Yun, P.; Song, C.; Wang, J.
Weighted Maximum Correntropy Criterion-Based Interacting Multiple-Model Filter for Maneuvering Target Tracking. *Remote Sens.* **2023**, *15*, 4513.
https://doi.org/10.3390/rs15184513

**AMA Style**

Huai L, Li B, Yun P, Song C, Wang J.
Weighted Maximum Correntropy Criterion-Based Interacting Multiple-Model Filter for Maneuvering Target Tracking. *Remote Sensing*. 2023; 15(18):4513.
https://doi.org/10.3390/rs15184513

**Chicago/Turabian Style**

Huai, Liangliang, Bo Li, Peng Yun, Chao Song, and Jiayuan Wang.
2023. "Weighted Maximum Correntropy Criterion-Based Interacting Multiple-Model Filter for Maneuvering Target Tracking" *Remote Sensing* 15, no. 18: 4513.
https://doi.org/10.3390/rs15184513