Editing State observer (section)

=== Discrete-time case ===
The state of a linear, time-invariant discrete-time system is assumed to satisfy

: <math>x(k+1) = A x(k) + B u(k)</math>

: <math>y(k) = C x(k) + D u(k)</math>

where, at time <math>k</math>, <math>x(k)</math> is the plant's state; <math>u(k)</math> is its inputs; and <math>y(k)</math> is its outputs.  These equations simply say that the plant's current outputs and its future state are both determined solely by its current states and the current inputs.  (Although these equations are expressed in terms of [[discrete mathematics|discrete]] time steps, very similar equations hold for [[continuous function|continuous]] systems).  If this system is [[Observability|observable]] then the output of the plant, <math>y(k)</math>, can be used to steer the state of the state observer.

The observer model of the physical system is then typically derived from the above equations.  Additional terms may be included in order to ensure that, on receiving successive measured values of the plant's inputs and outputs, the model's state converges to that of the plant.  In particular, the output of the observer may be subtracted from the output of the plant and then multiplied by a matrix <math>L</math>; this is then added to the equations for the state of the observer to produce a so-called ''[[David Luenberger|Luenberger]] observer'', defined by the equations below.  Note that the variables of a state observer are commonly denoted by a "hat": <math>\hat{x}(k)</math> and <math>\hat{y}(k)</math> to distinguish them from the variables of the equations satisfied by the physical system.
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: <math>\hat{x}(k+1) = A \hat{x}(k) + L \left[y(k) - \hat{y}(k)\right] + B u(k)</math>

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

The observer is called asymptotically stable if the observer error <math>e(k) = \hat{x}(k) - x(k)</math> converges to zero when <math> k \to \infty </math>. For a Luenberger observer, the observer error satisfies <math> e(k+1) = (A - LC) e(k)</math>.  The Luenberger observer for this discrete-time system is therefore asymptotically stable when the matrix <math> A - LC </math> has all the eigenvalues inside the unit circle.

For control purposes the output of the observer system is fed back to the input of both the observer and the plant through the gains matrix <math>K</math>.

: <math>u(k)= -K \hat{x}(k)</math>

The observer equations then become:

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

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

or, more simply,

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

: <math>\hat{y}(k) = \left(C - D K\right) \hat{x}(k)</math><!--

Substituting in the equation for <math>y(k)</math> from the plant system

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

: <math>\hat{y}(k) = \left(C - D K\right) \hat{x}(k)</math>

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Due to the [[separation principle]] we know that we can choose <math>K</math> and <math>L</math> independently without harm to the overall stability of the systems.  As a rule of thumb, the poles of the observer <math>A-LC</math> are usually chosen to converge 10 times faster than the poles of the system <math>A-BK</math>.