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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.

Deterministic system:


A, B, C are known, u_i, y_i are known. The problem is to use u_i, y_i, A, B, C to estimate the states x_i.



The estimation error is \tilde{x_i}=x_i-\hat{x_i}. Then we have


Hence estimation error model is


Therefore, as long as \{A,C\} are observable, we can choose K to assign arbitrary eigenvalues to the matrix A-KC such that the estimation error will converges to zero with arbitrary speed.

PS: if no feedback is used in the estimator, i.e., \hat{x}_{i+1}=A\hat{x}_i+Bu_i, we will have \tilde{x}_{i+1}=A\tilde{x}_i. This estimator obviously is not good.

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