A Modified Variational Bayesian Noise Adaptive Kalman Filter

Abstract

Kalman filter suffers from performance degradation when applied to dynamic systems with unknown noise statistics. To address this problem, the variational Bayesian noise adaptive Kalman filter (VB-AKF) jointly estimates the state and noise using the variational Bayesian approximation method. In this paper, a modified variational Bayesian noise adaptive Kalman filter (VB-MAKF) is proposed by designing a novel dynamic model for tracking the variances of measurement noise. In the proposed dynamic model, the change in estimated noise variance is utilized to control a continuous and bounded function, which is specifically designed to follow the change in real noise variance, adaptively. We see from the numerical simulations that, in comparison with VB-AKF, the proposed VB-MAKF can achieve higher estimation accuracy of noise variances and thus provide higher estimation accuracy of states.

Appendix

This appendix shows the convergence feature of sequence $\{{\hat{\alpha }}_{k, i}\}$ and $\rho _i {\hat{\alpha }}_{k, i} \gg 1/2$ when $0.6< \rho _i<1$.

The explicit formula of sequence $\{{\hat{\alpha }}_{k, i}\}$ can be obtained from the following recursive form.

$$\begin{aligned} \alpha _{k+1, i}&=1/2 + \rho _{i} \alpha _{k, i}, \end{aligned}$$

(44)

$$\begin{aligned} \alpha _{k+2, i}&=1/2 + \rho _{i} \alpha _{k+1, i}. \end{aligned}$$

(45)

Subtracting (45) from (44), we have

$$\begin{aligned} \alpha _{k+2, i} - \alpha _{k+1, i} = \rho _{i} (\alpha _{k+1, i} - \alpha _{k, i}). \end{aligned}$$

(46)

Let $b_{k,i} = \alpha _{k+1, i} - \alpha _{k, i}$. We rewrite (46) as

$$\begin{aligned} b_{k+1,i} = \rho _{i} b_{k, i}, \end{aligned}$$

(47)

where $b_{1,i} = \alpha _{2,i} - \alpha _{1,i}=1/2 + \rho _{i} \alpha _{1,i} - \alpha _{1, i}$. Herein, the geometric progression $\{ b_{k,i} \}$ can be written by the explicit form

$$\begin{aligned} b_{k,i} = \rho _{i}^{k-1} b_{1,i}. \end{aligned}$$

(48)

It follows from (46) that

$$\begin{aligned} \alpha _{k+1,i} - \alpha _{k,i} = \rho _{i}^{k-1} b_{1,i}. \end{aligned}$$

(49)

The dislocation subtraction gives

$$\begin{aligned}&\alpha _{k+1,i} - \alpha _{k,i} = \rho _{i}^{k-1} b_{1,i}, \end{aligned}$$

(50)

$$\begin{aligned}&\alpha _{k,i} - \alpha _{k-1,i} = \rho _{i}^{k-2} b_{1,i}, \end{aligned}$$

(51)

$$\begin{aligned}&\cdots \nonumber \\&\alpha _{2,i} - \alpha _{1,i} = b_{1,i}. \end{aligned}$$

(52)

Adding (50) to (52) together, we have

$$\begin{aligned} \alpha _{k+1,i} - \alpha _{1,i}&= b_{1,i} \sum _{j=0}^{k-1} \rho _{i}^{j} \nonumber \\&= \frac{b_{1,i}(1-\rho _{i}^{k})}{1-\rho _{i}}. \end{aligned}$$

(53)

If $0.6 \le \rho _i < 1$, then

$$\begin{aligned} \lim _{k \rightarrow +\infty } \alpha _{k+1,i}&= \frac{b_{1,i}}{1-\rho _{i}} + \alpha _{1,i} \nonumber \\&= \frac{1/2 + \rho _{i} \alpha _{1,i} - \alpha _{1, i}}{1 - \rho _{i}} + \alpha _{1, i} \nonumber \\&= \frac{1/2}{1-\rho _{i}}. \end{aligned}$$

(54)

Thus, the sequence $\{\alpha _{k+1, i}\}$ converges to $\frac{1/2}{1-\rho _i}$. Also, from (54), we have $\rho _{i} {\hat{\alpha }}_{k+1, i} \gg 1/2$.