Preliminaries of KF – Asymptotic estimator of deterministic systems
KF actually is nothing but an observer. But it is not for regular systems, instead it is for systems with noise. We need to understand estimators for deterministic systems first.
are known, are known. The problem is to use to estimate the states .
The estimation error is . Then we have
Hence estimation error model is
Therefore, as long as are observable, we can choose to assign arbitrary eigenvalues to the matrix such that the estimation error will converges to zero with arbitrary speed.
PS: if no feedback is used in the estimator, i.e., , we will have . This estimator obviously is not good.