# Adaptive Unscented Kalman Filter for Target Tracking with Unknown Time-Varying Noise Covariance

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The UKF Algorithm for Nonlinear State Estimation

#### 2.1. Standard UKF

^{2}(n + κ) − n is the composite scaling factor. α and κ are tuning parameters. The parameter α is set to 0 ≤ α ≤ 1 and a good default setting on κ is κ = 0 [35].

#### 2.2. Problem Description of UKF for Time-Varying Noise Covariance

## 3. An Innovative Adaptive UKF Scheme

**Q**and the redundant measurement difference sequences are exploited to estimate the measurement noise covariance

**R**.

#### 3.1. Adaptive **Q** Estimation

**Theorem.**

**Proof.**

**Remark.**

**Q**-estimation, both the innovations and the residuals are used [37]. Taking the expectation of the difference between innovation and residual follows that

#### 3.2. Adaptive **R** Estimation

**R**is closely related to the performance of the radar. Due to different external and internal time varying disturbances,

**R**is also time varying and should be estimated adaptively.

#### 3.3. Adaptive UKF Scheme

**Q**by solving the equation.

**R**via Equations (30) and (31).

## 4. Simulation Results and Discussion

#### 4.1. Simulation Parameter and Cases

**Simulation Case 1:**The measurement noise covariance matrix

**R**= diag[100 0.001

^{2}] is known and the process noise covariance matrix

**Q**varies over time. During the period 200–350 s, the process noise covariance matrix is assigned to be

**Q**= diag[0.015 0.015].

**Simulation Case 2:**The measurement noise covariance matrix

**R**is uncertain, and the process noise covariance matrix

**Q**is known. The measurement noise covariance matrix is taken as

**R**= diag[20 × 100 20 × 0.001

^{2}] during the period 200–350 s, and it is assigned to be

**R**= diag[100 0.001

^{2}] for the remaining periods.

**Simulation Case 3:**Both the measurement noise covariance matrix

**R**and the process noise covariance matrix

**Q**are uncertain. In this case, the changes in Case 1 and Case 2 are implemented simultaneously.

**X**

_{0}= [1000 m, 5000 m, 10 m/s, 50 m/s, 2 m/s

^{2}, −4 m/s

^{2}], and the process noise covariance matrix is

**R**= diag[100 0.001

^{2}]. The noise covariance of the redundant measurement is unknown, which can be estimated with the RMNCE algorithm.

#### 4.2. Simulation Results

**Q**will lead to a divergence of the filter since the mismatches. Under the constraints, the position tracking error of the adaptive fading UKF algorithm is decreased compared with the standard UKF. Comparing the means and variances of the position errors during [200 s, 550 s] and [550 s, 1400 s] intervals, it is clear that the filtering performances of the used methods except for the standard UKF are all robust. As shown in Table 1, the tracking accuracy of our proposed Q-adaptive UKF scheme is almost the same as that of the IMM-UKF method, which demonstrates that both algorithms can resist the uncertainty of process noise. However, the computational load of IMM-UKF method is approximately two times higher than that of our proposed adaptive UKF scheme. In addition, although the N-UKF algorithm resists the disturbance of the changing statistics properties of states, its accuracy is not optimal due to the neglect of the correlativity between the innovation and residual sequences. In this case, the simulation results demonstrate that our proposed method is affected by neither the time-varying process noise nor the maneuvering motion models.

^{2}and 1.0507 × 10

^{−6}rad

^{2}, and the reference variances are 100 m

^{2}and 1.0 × 10

^{−6}rad

^{2}. It is clear that the RMNEC algorithm can provide a reliable estimation for the redundant measurement variances.

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix

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Algorithm | 200–550 s | 550–1400 s | ||
---|---|---|---|---|

Mean (m) | Variance (m^{2}) | Mean (m) | Variance (m^{2}) | |

Standard UKF | 1.4549 | 0.3783 | 25.7565 | 341.6688 |

Adaptive fading UKF | 2.1264 | 0.2519 | 2.8608 | 0.2411 |

IMM-UKF | 1.3258 | 0.1141 | 2.7311 | 0.5510 |

N-UKF | 3.7332 | 0.4344 | 4.5229 | 0.6996 |

Our proposed scheme | 1.3398 | 0.1080 | 2.7165 | 0.5497 |

Algorithm | 200–350 s | 600–1400 s | ||
---|---|---|---|---|

Mean (m) | Variance (m^{2}) | Mean (m) | Variance (m^{2}) | |

Standard UKF | 2.9130 | 0.7260 | 26.7439 | 337.3589 |

Improved Sage-Husa UKF | 2.4260 | 0.8123 | 359.2692 | 2.1492 × 10^{5} |

IMM-UKF | 3.0745 | 2.2106 | 2.8958 | 1.3549 |

N-UKF | 7.9900 | 27.0631 | 3.5731 | 0.9831 |

Our proposed scheme | 1.9730 | 0.1264 | 2.9107 | 1.0564 |

Algorithm | 200–550 s | 550–1400 s | ||
---|---|---|---|---|

Mean (m) | Variance (m^{2}) | Mean (m) | Variance (m^{2}) | |

Standard UKF | 4.8845 | 4.0442 | 26.8136 | 316.1329 |

Robust adaptive UKF | 4.6399 | 5.0780 | 3.0834 | 0.2204 |

IMM-UKF | 3.9900 | 1.3483 | 4.6487 | 2.3042 |

N-UKF | 4.7748 | 5.6900 | 3.3517 | 0.3042 |

Our proposed scheme | 3.0623 | 1.0426 | 3.7313 | 0.7709 |

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**MDPI and ACS Style**

Ge, B.; Zhang, H.; Jiang, L.; Li, Z.; Butt, M.M.
Adaptive Unscented Kalman Filter for Target Tracking with Unknown Time-Varying Noise Covariance. *Sensors* **2019**, *19*, 1371.
https://doi.org/10.3390/s19061371

**AMA Style**

Ge B, Zhang H, Jiang L, Li Z, Butt MM.
Adaptive Unscented Kalman Filter for Target Tracking with Unknown Time-Varying Noise Covariance. *Sensors*. 2019; 19(6):1371.
https://doi.org/10.3390/s19061371

**Chicago/Turabian Style**

Ge, Baoshuang, Hai Zhang, Liuyang Jiang, Zheng Li, and Maaz Mohammed Butt.
2019. "Adaptive Unscented Kalman Filter for Target Tracking with Unknown Time-Varying Noise Covariance" *Sensors* 19, no. 6: 1371.
https://doi.org/10.3390/s19061371