Let $\tilde{\mathbf{x}}=\hat{\mathbf{x}}-\mathbf{x}$ be the estimation error. Then the error dynamics are defined by

with the initial estimation error as $\displaystyle \tilde{\mathbf{x}}(0)=\hat{\mathbf{x}}(0)-\mathbf{x}(0)$ If the eigenvalues of A are inside the unit circle then $\tilde{\mathbf{x}}$ will converge to 0 . But we have no control over the convergence rate.

Moreover, A may have eigenvalues outside the unit circle. In that case $\tilde{\mathbf{x}}$ will diverge from 0 . Thus the open loop estimator is impractical.

2.2 Luenberger State Observer

Consider the system $\mathbf{x}(k+1)=A\mathbf{x}(k)+B\mathbf{u}(k)$ . Luenberger observer is shown in Figure 2. The observer dynamics can be expressed as:

$\displaystyle \hat{\mathbf{x}}(k+1)=A\hat{\mathbf{x}}(k)+B\mathbf{u}(k)+L(\mathbf{y}(k)-\hat{\mathbf{y}}(k))$

(1)

$\begin{figure}\centering \begin{pspicture}(-2,0)(12,8) \pnode(-1,6){u}\rput(-1... ...eA=180,angleB=-90]{->}{G7}{S2}\ncline{->}{G5}{yh} \end{pspicture} \end{figure}$

Figure 2: Luenberger observer

The closed loop error dynamics can be derived as:

It can be seen that $\tilde{\mathbf{x}}\rightarrow 0$ , if L can be designed such that has eigenvalues inside the unit circle of z -plane.

The convergence rate can also be controlled by properly choosing the closed loop eigenvalues.