Luo Yarong, Guo Chi, You Shengyong, Liu Jingnan
Global Navigation Satellite System Research Center, Wuhan University, Wuhan 430079, China.
Artificial Intelligence Institute, Wuhan University, Wuhan 430079, China.
Entropy (Basel). 2020 Sep 3;22(9):982. doi: 10.3390/e22090982.
Rényi entropy as a generalization of the Shannon entropy allows for different averaging of probabilities of a control parameter α. This paper gives a new perspective of the Kalman filter from the Rényi entropy. Firstly, the Rényi entropy is employed to measure the uncertainty of the multivariate Gaussian probability density function. Then, we calculate the temporal derivative of the Rényi entropy of the Kalman filter's mean square error matrix, which will be minimized to obtain the Kalman filter's gain. Moreover, the continuous Kalman filter approaches a steady state when the temporal derivative of the Rényi entropy is equal to zero, which means that the Rényi entropy will keep stable. As the temporal derivative of the Rényi entropy is independent of parameter α and is the same as the temporal derivative of the Shannon entropy, the result is the same as for Shannon entropy. Finally, an example of an experiment of falling body tracking by radar using an unscented Kalman filter (UKF) in noisy conditions and a loosely coupled navigation experiment are performed to demonstrate the effectiveness of the conclusion.
作为香农熵的推广，雷尼熵允许对控制参数α的概率进行不同的平均。本文从雷尼熵的角度给出了卡尔曼滤波器的一个新视角。首先，利用雷尼熵来度量多元高斯概率密度函数的不确定性。然后，我们计算卡尔曼滤波器均方误差矩阵的雷尼熵的时间导数，该导数将被最小化以获得卡尔曼滤波器的增益。此外，当雷尼熵的时间导数等于零时，连续卡尔曼滤波器趋近于稳态，这意味着雷尼熵将保持稳定。由于雷尼熵的时间导数与参数α无关且与香农熵的时间导数相同，结果与香农熵相同。最后，进行了一个在噪声条件下使用无迹卡尔曼滤波器（UKF）进行落体跟踪的实验示例以及一个松耦合导航实验，以证明该结论的有效性。
