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State observer
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. 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
The usual state space model for a (plant) system can be written as
Although this is a discrete system it holds for continuous systems.
If this system is observable then the output, 
, can be used to steer the state of another state space model.
This observer system is commonly denoted with a "hat": 
and 
.
The output of the observer system is substracted from the output of the plant system; multiplied by a matrix L; and added to the state equation.
![\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)](a/../upload/math/7352ec95f5f10bd786b4636431e859a8.png)
The output of the observer system is feedback as the input such that 
for some matrix K.
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