Sources and Effects of Noise in Kalman Filter
In this post, the theoretical hypothesis, engineering acquisition and actual impact of Kalman filter process noise and observation noise are discussed.
This post is synchronously published on my CSDN blog: https://blog.csdn.net/weixin_45766278/article/details/128426596
1. Theoretical assumptions of two noises in the description of state space
First put forward the basic formula: Equation of state: x(k) = Ax(k-1) +Bu(k-1) +w(k-1) Observation equation: y(k)=Cx(k)+v(k) Where w(k-1) is process noise, which is usually recorded as Q, v(k) as observation noise and R. There are four main requirements of standard Kalman filter for Q and R:
- Uncorrelated
- Zero Mean
- Gauss white noise sequence
- Q, R are nonnegative and positive fixed matrices of known values, respectively
i.e.
Where the $\delta_{kj}$ is the Kronecker $\delta$ function,
2. How to obtain two noises from engineering application
Process noise Q: Construct the “ideal state” of the research question, and compare it with the actual situation by using the sample variance as Q
For example, when studying the motion of sliders, it is possible to compare the motion data of relatively smooth surfaces with that of actual rough surfaces. Or control an unmanned car, which actually travels near an arc in DT time, but we often approximate it as a linear model when we study it, so the resulting system error can calculate a range threshold
Observation noise R: This noise is relatively easy to obtain, usually based on the accuracy of the sensor, directly conducting observation experiments, using the sample variance as Q
For example, the error of a thermometer is +0.5, and the observation noise R=0.5^2=0.25
3. Effect of two noise sources on Kalman filter estimation error
In fact, I think it is difficult to directly construct the linear relationship between the specific values of process noise Q and process noise R and the error of Kalman filter estimation at this stage. Current research mainly focuses on finding the optimal combination of Q and R.
For example, genetic algorithm is used to find the best combination of variances in this paper:
[1] Guo Ying Shi, Wang Chang, Zhang Yaqi. The influence of noise variance on Kalman filter result analysis [J]. Computer Engineering and Design, 2014,35(02): 641-645.DOI:10.16208/j.issn1000-7024.02.016.
However, it should be noted that Q and R must exist objectively in practical engineering problems. We try our best to reduce Q and R, which usually helps us to obtain better estimation results.
