Open-loop Estimator

If we have a good model, we can use it to estimate the states:

$$\dot{\hat{x}}=A\hat{x}+Bu$$

by simulating the model given the real input $u(t)$.

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To do this, we need to look at the estimation error, $e=x-\hat{x}$ → $\dot{e}=\dot{x}-\dot{\hat{x}}$

$$\dot{e}=Ax+Bu-(A\hat{x}-Bu)=A(x-\hat{x})=Ae$$

Assuming that the input we give to the system is exactly the same in the model as it is in real life.

We can see that if $A$ has eigenvalue in the left half plane, $\dot{e}\to 0$.

Problems

• Sensitive to modeling errors and disturbances
• Slow convergence rate
• Does not work for unstable systems