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
\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 it 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 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{\hat{y}}(k)\right + B \mathbf{\hat{u}}(k)
\mathbf{\hat{y}}(k) = C \mathbf{\hat{x}}(k) + D \mathbf{\hat{u}}(k)
The output of the observer system is feedback as the input such that \mathbf{\hat{u}(k)} = -K \mathbf{\hat{x}}(k) for some matrix K.
\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)

 

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