Observer of state

In Automatic and Information theory, an observant of state is an extension of a model represented in the form of Représentation of state. When the state of a system is not measurable, one builds an observer which makes it possible to rebuild the state starting from a model of the dynamic Système and measurements of other sizes.

The theory of the observant of state Déterministe was introduced into the Sixties by Luenberger for the linear systems. $kalman also formulated an observer by considering a linear system Stochastique. For the non-linear systems, the observation remains a field where research is very active, but the most common use is the use of a wide Kalman filter (EKF).

Structure with an observer

The notation " is used; chapeau" to express an estimate. If X represents the real state of the system not measured, $\ hat X$ represents the estimate of the state made by the observer.

The estimate of the state is done by recopying in a virtual way dynamics of the system by taking of account the order U but also the exits of the systems (measurements) Y with an aim of correcting the possible variations.

Observer of $kalman

That is to say the following linear system:

\ begin {boxes} \ dowry X = HAS X + B U \ \ Y = C X \ end {boxes}

A dynamic observer with the following form:

\ begin {boxes} \ dowry \ hat X = has \ hat X + B U + L (Y - \ hat Y) \ \ \ hat Y = C \ hat X \ end {boxes}

One comes to correct the evolution of the state thanks to the model according to the variation noted between the exit observed and the exit rebuilt by the observer: $(Y - \ hat Y)$ .

One can rewrite the observer in the following way:

\ dowry \ hat X = (- LLC has) \ hat X + B U + L Y

it is checked well that the observer rebuilt the state X according to the order U and of measurements Y as on the diagram above.

The matrix L is called matrix of profit and must be selected so that the error on the state converges exponentially towards 0, that is to say $\ tilde X= (\ hat X - X) \ to 0$ . For that, it is enough to choose L such that the matrix (ALE) is a matrix Hurtwitz, i.e. its eigenvalues is with negative real parts in the continuous case or has a module lower than 1 in the discrete case.

Order by return of state rebuilt by an observer of $kalman

The linear observer of $kalman - Luenberger has a known interesting characteristic under the name of principle of separation: in the case of a linear order by return of state, work of synthesis of ordering and synthesis of observer can be done independently. Indeed, if the ordered system is stable, and if the observer thus designed is stable (i.e the matrices $A-BK$ and $A-LC$ are Hurtwitz) then the system ordered by return of the rebuilt state is stable.

Indeed, let us consider the linear system observable following invariant, and commandable, provided with an observer with $kalman - Luenberger:

\ begin {boxes} \ dowry X = HAS X + B U \ \ Y = C X \ end {boxes}
\ begin {boxes} \ dowry \ hat X = has \ hat X + B U + L (Y - \ hat Y) \ \ \ hat Y = C \ hat X \ end {boxes}

By carrying out a looping by return of state $U = - K \ hat X$ , the dynamics of the buckled system is written then:

\ begin {boxes} \ dowry X = HAS X - B K \ hat X \ \ \ dowry \ hat X = (has - BK) \ hat X + L (Y - C \ hat X) \ end {boxes}

One can make the change of variable according to, to write the error of rebuilding:

\ tilde X = X - \ hat X

from where, while replacing,

\ dowry \ tilde X = (has - LLC) \ X

tilde

By writing a new system increased, made up of the state and error of rebuilding, one obtains:

\ begin {pmatrix} \ dowry X \ \ \ dowry \ tilde X \ end {pmatrix} = \ begin {pmatrix} A-BK & BK \ \ 0 & ALE \ end {pmatrix} \ begin {pmatrix} X \ \ \ tilde X \ end {pmatrix}

This matrix has the good taste to be triangular per blocks, and consequently the spectrum of the buckled system is consisted of the union of the spectra of the diagonal blocks, i.e. the union of the spectra of the ordered initial system, and the initial system observed. Thus the synthesis of a system ordered by a return of state rebuilt by an observer is particularly simple for the linear systems invariants, since one can synthesize the two functions separately.

Some note:

For the placement of poles, one has any interest so that the observer is faster than the dynamic system, so that it can continue the system in question. Thus it will be necessary that the spectral X-coordinate of the observer is more negative than that of the ordered system.
Because of its dynamic nature (integration of the signals of measurement) the observer is also used in treatment of the signal to filter measurements. It is in this context that $kalman with published the filter which bears from now on its name.
the spectral X-coordinate should not be too negative, to limit the sensitivity to the noise of the observer. In practice, one uses values ranging between 2 and 5 times that of the spectral X-coordinate of the ordered system.
an order based on a return of rebuilt state is not robust with the errors of modeling. This fact is rather intuitive because one passes by an approach based model to rebuild our state, therefore the precision of the rebuilt state depends on the pertinance of the model used.

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