A state observer is an extension to a state space model that provides feedback to control a system. A state observer is used on a system where direct access to the state is not possible. If the system is observable, then state observers can be designed to estimate the signals that cannot be measured. Such a system would be on a moving object where only velocity is measured but access to position is necessary. A state observer can then be used to estimate the position to provide full state access for feedback control.

Typical observer model

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The usual state space model for a (plant) system can be written as

$\backslash mathbf\{x\}(k+1)\; =\; A\; \backslash mathbf\{x\}(k)\; +\; B\; \backslash mathbf\{u\}(k)$
$\backslash mathbf\{y\}(k)\; =\; C\; \backslash mathbf\{x\}(k)\; +\; D\; \backslash mathbf\{u\}(k)$

Although this is a discrete system it holds for continuous systems.

If this system is observable then the output, $\backslash mathbf\{y\}(k)$, can be used to steer the state of another state space model. This observer system is commonly denoted with a "hat": $\backslash mathbf\{\backslash hat\{x\}\}(k)$ and $\backslash mathbf\{\backslash hat\{y\}\}(k)$. The output of the observer system is subtracted from the output of the plant system; multiplied by a matrix $L$; and added to the state equation.

$\backslash mathbf\{\backslash hat\{x\}\}(k+1)\; =\; A\; \backslash mathbf\{\backslash hat\{x\}\}(k)\; -\; L\; \backslash left-\; \backslash mathbf\{\backslash hat\{y\}\}(k)\backslash right+\; B\; \backslash mathbf\{\backslash hat\{u\}\}(k)$
$\backslash mathbf\{\backslash hat\{y\}\}(k)\; =\; C\; \backslash mathbf\{\backslash hat\{x\}\}(k)\; +\; D\; \backslash mathbf\{\backslash hat\{u\}\}(k)$

The output of the observer system is feedback as the input such that $\backslash mathbf\{\backslash hat\{u\}(k)\}\; =\; -K\; \backslash mathbf\{\backslash hat\{x\}\}(k)$ for some matrix $K$.

$\backslash mathbf\{\backslash hat\{x\}\}(k+1)\; =\; A\; \backslash mathbf\{\backslash hat\{x\}\}(k)\; -\; L\; \backslash left(\backslash mathbf\{y\}(k)\; -\; \backslash mathbf\{\backslash hat\{y\}\}(k)\backslash right)\; -\; B\; K\; \backslash mathbf\{\backslash hat\{x\}\}(k)$
$\backslash mathbf\{\backslash hat\{y\}\}(k)\; =\; C\; \backslash mathbf\{\backslash hat\{x\}\}(k)\; -\; D\; K\; \backslash mathbf\{\backslash hat\{x\}\}(k)$

$\backslash mathbf\{\backslash hat\{x\}\}(k+1)\; =\; \backslash left(A\; -\; B\; K)\; \backslash right)\; \backslash mathbf\{\backslash hat\{x\}\}(k)\; -\; L\; \backslash left(\backslash mathbf\{y\}(k)\; -\; \backslash mathbf\{\backslash hat\{y\}\}(k)\backslash right)$
$\backslash mathbf\{\backslash hat\{y\}\}(k)\; =\; \backslash left(C\; -\; D\; K\backslash right)\; \backslash mathbf\{\backslash hat\{x\}\}(k)$

Control theory

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