State observer

In control theory, a state observer is a system that models a real system in order to provide an estimate of its internal state, given measurements of the input and output of the real system. It is typically a computer-implemented mathematical model.

Knowing the system state is necessary to solve many control theory problems; for example, stabilizing a system using state feedback. In most practical cases, the physical state of the system cannot be determined by direct observation. Instead, indirect effects of the internal state are observed by way of the system outputs. A simple example is that of vehicles in a tunnel: the rates and velocities at which vehicles enter and leave the tunnel can be observed directly, but the exact state inside the tunnel can only be estimated. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer.

Typical observer model

The state of a physical system, or plant, is assumed to satisfy

:$mathbf\{x\}(k+1)\; =\; A\; mathbf\{x\}(k)\; +\; B\; mathbf\{u\}(k)$
$mathbf\{y\}(k)\; =\; C\; mathbf\{x\}(k)\; +\; D\; mathbf\{u\}(k)$

where, at time $k,\; mathbf\{x\}(k)$ is the plant's state; $mathbf\{u\}(k)$ is its inputs; and $mathbf\{y\}(k)$ is its outputs. These equations simply say that the plant's current outputs and its future state are both determined solely by its current state and the current inputs. (Although these equations are expressed in terms of discrete time steps, very similar equations hold for continuous systems). If this system is observable then the output of the plant, $mathbf\{y\}(k)$, 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 $L$; this is then added to the equations for the state of the observer to produce a so-called "Luenberger observer", defined by the equations below. Note that the variables of a state observer are commonly denoted by a "hat": $mathbf\{hat\{x(k)$ and $mathbf\{hat\{y(k)$ to distinguish them from the variables of the equations satisfied by the physical system.

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

The observer is called asymptotically stable if the observer error $mathbf\{e\}(k)\; =\; mathbf\{hat\{x(k)\; -\; mathbf\{x\}(k)$ converges to zero when $k\; ightarrow\; infty$. For a Luenberger observer, the observer error satisfies $mathbf\{e\}(k+1)\; =\; (A\; -\; LC)\; mathbf\{e\}(k)$. The Luenberger observer is therefore asymptotically stable when the matrix $A\; -\; LC$ has all the eigenvalues with strictly negative real part (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 through the gains matrix $K$.

$mathbf\{u(k)\}=\; -K\; mathbf\{hat\{x(k)$

The observer equations then become:

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

or, more simply,

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

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

ee also

* Kalman filter
* Extended Kalman filter

References

*cite book
last = Sontag
first = Eduardo
authorlink = Eduardo D. Sontag
year = 1998
title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
publisher = Springer
id = ISBN 0-387-984895