State observer
An observer in control theory is a system consisting of known input variables ( eg, control variables or measurable disturbances ) and output variables ( measured variables ) of an observed reference system reconstructs non-measurable variables ( states). To it is according to the observed reference system as a model, and includes a controller which adjusts the measurable state variables. Tellingly it, to speak of a reference- controlled synthesizer (English reference controlled synthesizer ) would be.
An observer can only be created if the reference system of the available measurements is observed. Therefore, the determination of observability is based on criteria necessary condition for the observer design.
Observers are used on the one hand with state regulators to reconstruct non-measurable state variables, on the other hand, in the measurement technique as a replacement not technically or economically possible measurements.
A consistent theory was developed in 1964 by the American control engineer David Luenberger for linear system models and a constant proportional feedback of the error. The method can be extended in principle to non-linear models. [ FOE: NL2 1]
Luenberger observer
The idea of Luenberger 1964 is based on a parallel connection of the observer for the plant model [ LUN: RT2 1]. Here is the difference between the measured value of the route and the " reading " of the observer, ie fed back to the model. This allows the observer to respond to disturbances or own inaccuracies. The fundamental equation of the observer is:
Determined to
Thus results for the observer
Therefore applies to the observation error of a Luenberger observer if all the eigenvalues of the matrix have negative real parts.
The determination of the return procedure is analogous to the controller design by pole placement by the following replacements shall be made [ FOE: RT 1]:
The example system, the eigenvalues and. Thus, the observer can follow the system whose eigenvalues must be from those of the system left. This requirement is satisfied for. The characteristic equation is in this case
And thus and. The feedback matrix so that
For the full observer is the differential equation
Structural observability [ LUN: RT2 2]
Systems can not be observed for two reasons:
- A specific combination of parameters leads to Nichtbeobachtbarkeit.
- The structure of the system means that the system in any occupation of the non-zero elements of the matrix system ( which depend in practice of physical parameters ) is not observable or unobservable in any combination of parameters. This is the case if necessary signal couplings between state metrics and missing. To prove that a system is not structurally observable, graph theoretical techniques must be used. The structural observability, however, is easy to prove, if it can be shown that a particular combination of parameters ( for example, all non-zero elements == 1) describes a completely observable system.
Full observability
The state space representation of a linear system is
The system is observable if it can be clearly determined at known control function and known matrices and from the variation of the output vector over a finite time interval of the initial state.
Below is an example of a system with one input and one output: used ( SISO Single Input, Single Output ).
It describes the series connection of two PT1 - elements and with the time constants.
Proof
Structural observability is a necessary condition for the complete observability. However, usually only the following criteria will be used to demonstrate a full observability.
The criterion according to Kalman 's relatively easy to determine, but you can here the observability do not refer to individual own operations or eigenvalues. This can be done by means of the Gilbert and Hautus criterion.
Criterion of Kalman
The system (A, C) if and only fully observable by Kalman [ LUN: RT2 3] if the observability matrix has rank or whose determinant in the case is only one measured variable equal to 0:
For the example system
And
With the observability
It is, thus, the rank is equal to 2, the system is fully observable.
Criterion of Gilbert
If the model in canonical normal form ( Jordan normal form)
With
[: RT2 LUN 4 ] and present as a matrix of eigenvectors, the criterion of Gilbert applies:
A system whose state space model is present in a canonical normal form, if and only fully observable when the matrix does not have a zero column and when the p columns of the matrix associated with the canonical variables for a p- times the natural value, are linearly independent.
The canonical normal form of the example system is
The matrix has only the columns ( here elements ) equal to 0, the test in a linear dependence is omitted here, since the system has a simple eigenvalues.
The system is fully observable.
Criterion of Hautus
The system (A, C) if and only fully observable by Hautus [ LUN: RT2 4], if the condition:
The matrix system of the example, the eigenvalues and. For both eigenvalues is the condition
Met. Thus the system is fully observable.
Observability of scanning systems
The above relationships are also valid for scanning, if is replaced by the transition matrix. According to [ LUN: RT2 5], the check can be simplified by first the conditions for the continuous system are checked and then the additional condition
Is satisfied.
Observer normal form
For a system having an input and an output, the observer canonical form may be determined, inter alia, from the equivalent to the transfer function equation.
The example system has the transfer function
It follows with, and
Reduced observer
Often some state variables can be measured directly. Thus it is not necessary to reconstruct them. A reduced observer can therefore be deduced that only reconstructs the unmeasured state variables. The order of the reduced observer is reduced compared to the full observer to the number of measured variables. This method can also be the case extend that the measured variables are not state variables. [ LUN: RT2 6]
After resorting the Matrizenzeilen in measured and observed states is the state space representation of the Eingrößensystems
The equation of state of the full system
And the reduced system
The measurement equation of the full system
And the reduced system
The substitution
Used in the equation of the observer provides full
In this illustration, not interfere with the time derivative of y. The transformation
, the equation
And from the estimated state vector