State Estimation

Given a system on the State Space Representation, where you know $y(t)$ and $u(t)$, what is $x(t)$?

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When is it used?

• Good for noisy measurements
• State feedback without direct measurements etc
• Disturbance estimation

Notation

x&: \ce{ state to be estimated } \\ \hat{x}&: \ce{ estimation } \\ y &: \ce{ Output (no noise) } \\ y_{m} = y+n &: \ce{ measured output } \\ n &: \ce{ noise } \end{align}

Estimation Error

The estimation error is defined as

$$e=x-\hat{x}$$

And the error is dynamic and given by

$$\dot{e}=Ax+Bu-\dot{\hat{x}}$$

And we want to choose the Estimate Update Law in such a way that

$$e\to 0,t\to \infty$$Link to original

Methods

Measurement Equation Inversion Open-loop Estimator Closed-loop Estimator - Luenberger Observer

Examples Using Observers

Noise Supression Band-limited Differentiation Conveyor Belt Case Study