control//state estimation//Kalman filter//process noise

How much to distrust the model, the number of the filter that is tuned rather than measured. The **process noise** \(w\) is everything the model does not know. A prediction model is written \(x_k=Fx_{k-1}+w_{k-1}\) with \(w\sim\mathcal N(0,Q)\): the model says constant velocity, and \(Q\) says the driver can still press the pedal. \(Q\) is how much the filter distrusts its own physics, and it is the number compared against \(R\), how much it distrusts the sensor (measurement noise).

How much to distrust the model, the number of the filter that is tuned rather than measured. The process noise www is everything the model does not know. A prediction model is written xk=Fxk−1+wk−1x_k=Fx_{k-1}+w_{k-1}xk​=Fxk−1​+wk−1​ with w∼N(0,Q)w\sim\mathcal N(0,Q)w∼N(0,Q): the model says constant velocity, and QQQ says the driver can still press the pedal. QQQ is how much the filter distrusts its own physics, and it is the number compared against RRR, how much it distrusts the sensor (measurement noise).

It enters the prediction of the uncertainty, Pk−=FPk−1FT+QP_k^-=FP_{k-1}F^{\mathsf T}+QPk−​=FPk−1​FT+Q: the old uncertainty carried forward by the physics, plus the model's own. The +Q+Q+Q makes perfect intuitive sense. Every time you predict the future without measuring anything, you become a little less certain. Between two measurements the filter predicts and predicts, and its doubt swells until the next reading deflates it.

RRR is measured, QQQ is tuned. Its structure comes from the physics of the system (a car accelerates, a room warms slowly), its size from trial and error. A small QQQ believes the model blindly: the estimate is smooth but late on every manoeuvre, and the truth leaves its band. A large QQQ is nervous and ends up copying the raw sensor.

Q=0Q=0Q=0 puts the filter to sleep. With no model doubt, each reading shrinks PPP and nothing makes it grow back, so PPP and the gain tend to zero and the filter stops listening to the sensor exactly when the world starts changing. A small positive QQQ, or a floor on PPP, keeps it awake.

Inflating QQQ is also the classic dirty fix for a filter that is wrong for reasons nobody has found yet: it makes it humbler. It works, and it hides the cause (Gaussian assumption).

What FFF and QQQ look like for a vector state is in the state-space model.