State observer

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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.

[edit] Typical observer model

Two of the basic parts of a control system are the plant and the controller. The "plant" is the black box model of the system that is to be controlled. The name originates from systems used to control factories or "plants." The controller is the subsystem designed to control the plant. The usual state space model for a plant can be written as

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

(Although this is a discrete system a similar equation holds for continuous systems).

If this system is observable then the output, $\mathbf{y}(k)$ , can be used to steer the state of another state space model. This observer system is commonly denoted with a "hat": $\mathbf{\hat{x}}(k)$ and $\mathbf{\hat{y}}(k)$ . 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. This gives the so-called Luenberger observer:

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

For control purposes the output of the observer system is fed back to the input of both the observer and the plant: $\mathbf{\hat{u}(k)} =\mathbf{u(k)}= -K \mathbf{\hat{x}}(k)$ for some matrix $K$ . The observer equations become:

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

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