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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

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 subtracted 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) ight + 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) ight) - 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) ight) \mathbf{\hat{x}}(k) - L \left(\mathbf{y}(k) - \mathbf{\hat{y}}(k) ight)
\mathbf{\hat{y}}(k) = \left(C - D K ight) \mathbf{\hat{x}}(k)