改进的变分贝叶斯自适应噪声卡尔曼滤波器

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本文主要探讨了一种改进的变分贝叶斯自适应卡尔曼滤波器（VB-MAKF）。在传统的卡尔曼滤波器（KF）在处理动态系统中，当噪声统计特性未知时，其性能往往会下降。为了解决这个问题，研究者们引入了变分贝叶斯方法，这是一种强大的工具，它能够联合估计状态和噪声参数，从而提高滤波器的鲁棒性。在VB-AKF的基础上，作者提出了一种新型的自适应策略，即通过设计一个独特的动态模型来跟踪测量噪声方差。这个动态模型的关键创新在于它利用估计的噪声方差变化来控制一个连续且有界的函数，该函数被精心设计用来跟踪实际噪声方差的变化。这种方法旨在实现对噪声特性的实时和自适应估计，从而提升滤波器的性能。与常规的VB-AKF相比，VB-MAKF的优势在于其更高的估计精度。通过对数值模拟的分析，结果表明，当面对未知噪声统计情况时，VB-MAKF能够更有效地减小滤波误差，提高系统的稳定性和准确性。这在诸如信号处理、控制系统、机器人导航等应用领域具有显著的实际意义，因为准确的状态估计对于这些领域的性能至关重要。总结来说，这篇论文提出了一个改进的变分贝叶斯自适应卡尔曼滤波器框架，通过动态地调整对噪声估计的处理，以适应不断变化的系统条件。这种创新的方法有望在实际工程问题中展现出更好的性能，尤其是在需要高精度和鲁棒性的系统中，如复杂环境中的传感器数据融合和动态系统状态估计。

Circuits Syst Signal Process

where Z

k−1

={z

,...,z

k−1

} are the measurement data obtained before the time

instant k. The goal of the Bayesian ﬁltering based on (1) and (2) is to recursively

compute the posterior distribution using

p(x

, Σ

) =

p(z

, Σ

) p(x

, Σ

k−1

)

p(z

k−1

)

. (5)

Notice that the recursions (4) and (5) reduce to the conventional Kalman ﬁlter [14],

when Σ

is known. The closed-form solution to the ﬁrst two moments of the posterior

distribution can be obtained in the form of Kalman ﬁlter. Speciﬁcally, suppose that the

posterior distribution p(x

k−1

, Σ

k−1

) at time step k − 1 can be approximated as

[16]

p(x

k−1

, Σ

k−1

) = N(x

k−1

;

k−1

, P

k−1

)



i=1



k−1,i

;ˆα

k−1,i



, (6)

where the notation N (x

k−1

;

k−1

, P

k−1

) represents a Gaussian probability density

function for the variable x

k−1

with mean

k−1

and covariance matrix P

k−1

.The

abbreviated notation IG(σ

k−1,i

;ˆα

k−1,i

) is deﬁned for the inverse-gamma dis-

tribution about the variable σ

k−1,i

, i.e., the i th component of variable σ

k−1

with the

shape parameter ˆα

k−1,i

and the scale parameter

k−1,i

. Over the support θ>0, we

have the following inverse-gamma distribution [18]

IG(θ; α, β) =

Γ(α)

−α−1

exp



−



, (7)

where the gamma function is Γ(α)=



∞

α−1

exp(−t)dt.

Substituting the obtained posterior distribution (6) into the predictive distribution

(4), we have

p(x

, Σ

k−1

) = N(x

k|k−1

;

k|k−1

, P

k|k−1

)



i=1



k|k−1,i

;ˆα

k|k−1,i



, (8)

where

k|k−1

and P

k|k−1

are the state prediction and its covariance in Kalman ﬁlter,

i.e.,

k|k−1

= A

k−1

, (9)

k|k−1

= AP

k−1

+ Q

k−1

. (10)

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