State observer

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In control theory, a state observer is a system that provides an estimate of the internal state of the observed system given its input and output measurements.

Knowing the system state is a requirement in many control theory problems, for example stabilizing the system using the state feedback. The problem is that in almost every practical case the state of the system can not be accessed directly. Indirect effects of the state can then be observed indirectly through system outputs. If the system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer.

[edit] Typical observer model

Two of the basic parts of a control system are the plant and the controller. The plant is the black box model of the system that is to be controlled. The name originates from systems used to control factories or plants. The controller is the subsystem designed to control the plant. The usual state space model for a plant 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 a similar equation 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. This gives the so-called Luenberger observer:

$\mathbf{\hat{x}}(k+1) = A \mathbf{\hat{x}}(k) - L \left[\mathbf{y}(k) - \mathbf{\hat{y}}(k)\right] + B \mathbf{\hat{u}}(k)$
$\mathbf{\hat{y}}(k) = C \mathbf{\hat{x}}(k) + D \mathbf{\hat{u}}(k)$

The observer is called asymptotically stable if the observer error $\mathbf{e}(k) = \mathbf{\hat{x}}(k) - \mathbf{x}(k)$ converges to zero when $k \rightarrow \infty$ . For Luenberger observer, the following equation holds for the observer error:

$\mathbf{e}(k+1) = (A - LC) \mathbf{e}(k).$

The Luenberger observer is therefore asymptotically stable when the matrix $A - L C$ has all the eigenvalues with absolute values strictly less than unity (is Hurwitz in the continuous case).

For control purposes the output of the observer system is fed back to the input of both the observer and the plant: $\mathbf{\hat{u}(k)} =\mathbf{u(k)}= -K \mathbf{\hat{x}}(k)$ for some matrix $K$ . The observer equations become:

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

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