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Preliminaries of KF – Asymptotic estimator of deterministic systems

14/05/2011

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:

x_{i+1}=Ax_i+Bu_i
y_i=Cx_i

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.

Observer:

\hat{x}_{i+1}=A\hat{x}_i+Bu_i+K(y_i-\hat{y}_i)
\hat{y}_i=C\hat{x}_i

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

\tilde{x}_{i+1}=(A-KC)\tilde{x}_i

Hence estimation error model is

\tilde{x}_{i}=(A-KC)^i\tilde{x}_0

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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