State observer - Definition

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

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

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

If this system is observable then the output, <math>\mathbf{y}(k)<math>, can be used to steer the state of another state space model. This observer system is commonly denoted with a "hat": <math>\mathbf{\hat{x}}(k)<math> and <math>\mathbf{\hat{y}}(k)<math>. The output of the observer system is substracted from the output of the plant system; multiplied by a matrix <math>L<math>; and added to the state equation.

<math>\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)<math>
<math>\mathbf{\hat{y}}(k) = C \mathbf{\hat{x}}(k) + D \mathbf{\hat{u}}(k)<math>

The output of the observer system is feedback as the input such that <math>\mathbf{\hat{u}(k)} = -K \mathbf{\hat{x}}(k)<math> for some matrix <math>K<math>.

<math>\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)<math>
<math>\mathbf{\hat{y}}(k) = C \mathbf{\hat{x}}(k) - D K \mathbf{\hat{x}}(k)<math>

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