# Why does noise prevent a (kalman) filter from diverging?

I'm using a filter (not exactly kalman) of the following form to estimate angles by fusing gyroscope with accelerometer and gyro with magnetometer:

$(1)\quad \hat{\theta}_k = \hat{\theta}^-_{g,k} + \alpha \left( \theta_{a,k} - \hat{\theta}^-_{g,k} \right)$

$(2)\quad \hat{\theta}_k = \hat{\theta}^-_{g,k} + \beta \left( \theta_{m,k} - \hat{\theta}^-_{g,k} \right)$

The angle $\hat{\theta}^-_{g,k}$ is the predicted angle by integrating the gyro angular velocity and $\theta_{a,k}$ is the angle resulting from the accelerometer data. The second equation is the same but for the magnetometer. In my case the factors $\alpha$ and $\beta$ are simply chosen from the interval $[0,1]$.

I know that in a Kalman-Filter the Gain $K$ is affected by the chosen process $Q$ and measurement $R$ covariance matrices. And if $Q$ is very small compared to $R$ the measurements may be disregarded to much and the filter could diverge.

Now to my specific case. The same gyro is used for the accelerometer and magnetometer. So one would assume to set the gains $\alpha=\beta$. Well, what happens is that due to the more noisy magnetometer data I'm allowed to set $\beta < \alpha$. What I can't find in the literature is a way to explain why the higher measurement noise allows me to choose a smaller gain for the magnetometer without risking divergence.

In my own words I want to say something like: "The cause is the higher fluctuation of the angles coming from the magnetometer, which although weighted less (than those of the acelerometer) still correct the angles strong enough 'to the correct direction' counteracting the drift induced by the gyroscope"

The problem is that I need to write this in a bit more scientific way, without going overboard. Any literature hints or advises would be greatly appreciated.

• Stabilizing noise is better explained using consistency from a statistical sense. – AnkilP May 9 '20 at 17:13