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.

Typical observer model

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

[\mathbf(k+1) = A \mathbf(k) + B \mathbf(k)]
[\mathbf(k) = C \mathbf(k) + D \mathbf(k)]

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

If this system is observable then the output, [\mathbf(k)], can be used to steer the state of another state space model. This observer system is commonly denoted with a "hat": [\mathbf}(k)] and [\mathbf}(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}(k+1) = A \mathbf}(k) - L \left[mathbf(k) - mathbf}(k)right] + B \mathbf}(k)]
[\mathbf}(k) = C \mathbf}(k) + D \mathbf}(k)]

The output of the observer system is feedback as the input such that [\mathbf(k)} = -K \mathbf}(k)] for some matrix [K].

[\mathbf}(k+1) = A \mathbf}(k) - L \left(\mathbf(k) - \mathbf}(k)\right) - B K \mathbf}(k)]
[\mathbf}(k) = C \mathbf}(k) - D K \mathbf}(k)]

[\mathbf}(k+1) = \left(A - B K) \right) \mathbf}(k) - L \left(\mathbf(k) - \mathbf}(k)\right)]
[\mathbf}(k) = \left(C - D K\right) \mathbf}(k)]

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