You can understand in this easy way: u is control input, which can be designed by you. Thus, the control input u is known to you inbetween sampling times: means THE CONTROL U IS CONSTANT at every sampling time.

e.g. probably,

Linear Discrete Kalman Filter in your books:

system equation: x(k+1) = F*x(k)+Gamma*w(k)

output equation: y(k+1) = H*x(k+1)+nu(k)

the first equation of the prediction step of Kalman filter in your textbook:

xhat(k+1|k) = F*xhat(k|k)

(because E[w(k)]=0 if you take expectation for both sides of the system eq above as Rudolf Kalman did for his derivation)

Now let us suppose you have a nonzero control input u at time k (that is designed by you and known to you!). the above system equation changes to:

system equation: x(k+1) = F*x(k)+G*u(k)+Gamma*w(k)

the first equation of the prediction step simply changes to:

xhat(k+1|k) = F*xhat(k|k) + G*u(k)

(because E[G*u(k)]=G*u(k), like E[const] = const)

In this reason, all the textbooks about the Kalman filter ignores the control input u for brevity of equations.

If the control input is dependent on the state x, e.g. u = -Kx, F changes to (F-GK). Except this, everything is the same to Kalman filter.

Hope my explanation be helpful to you.

The above answer given by Kim is very easier and sufficient.

If can consider u as input signal , we can deal with with minimum variance state estimator with Unknown input ;

As the input u is known, similar to Kalman filter, it should appear in the filter in the same way as it appears in the system model in order to cancel its effect in the dynamic equation of the state estimation error, i.e., in the equation of e'= x' - xe (in continuous-time), or e(k+1) = x(k+1) - xe(k+1) (in discrete-time), where xe is the estimate of x. For more detail, see, e.g., "Optimal Filtering" by B.D.O. Anderson and J.B. Moore, 2005.

If you do not know u, you can assume u is part of the process noise, hence completely remove it from you model.

Thanks for these helpful answers!

Jihong Yu

no you can't put Q to zeros even if the system doesn't have a process noise.

you will have to tune it until you get the best performance.

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Can you please explain what is the dimension of 'gamma and G matrix' in your answer?