Observer (control engineering)

Observer (system model) and "real system" (observed real reference system)

In control engineering, an observer is a system that uses known input variables (e.g. manipulated variables or measurable disturbance variables ) and output variables ( measured variables ) of an observed reference system to reconstruct non-measurable variables (states) . To do this, it simulates the observed reference system as a model and uses a controller to track the measurable state variables that are therefore comparable with the reference system. This is to avoid a model, especially in the case of reference systems with integrating behavior, from generating an error that increases over time. Tellingly, it would be a reference-controlled synthesizer (English reference controlled synthesizer ) to speak.

An observer can then accurately designed , if the reference system of the existing measured variables can be observed. However, observability is generally not a necessary condition for observer design. Instead, it is sufficient if the system is detectable.

Observers are used e.g. B.

• with state regulators for the reconstruction of non-measurable state variables
• for time-discrete controls where the measured variable cannot be updated in every cycle,
• in measurement technology as a substitute for not technically or economically possible measurements .

A consistent theory was developed from 1964 by the American control engineer David Luenberger for linear system models and a constant proportional feedback of the error. The method can in principle be extended to non-linear models.

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Luenberger observer

Block diagram of the Luenberger observer: Error correction: The input of L must be formed as. ${\displaystyle y\mathbf {-} {\hat {y}}}$

The idea of ​​Luenberger in 1964 is based on a parallel connection of the observer to the controlled system model. The difference between the measured value of the distance and the "measured value" of the observer , i.e. H. traced back to the model. The observer can thus react to disturbances or his own inaccuracies. The basic equation of the observer is: ${\displaystyle \mathbf {y} }$ ${\displaystyle {\hat {\mathbf {y} }}}$ ${\displaystyle \mathbf {y} (t)-{\hat {\mathbf {y} }}(t)}$

${\displaystyle {\dot {\hat {\mathbf {x} }}}(t)=\mathbf {A} {\hat {\mathbf {x} }}(t)+\mathbf {B} \mathbf {u} (t)+\mathbf {u} _{B}(t)}$ With ${\displaystyle {\hat {\mathbf {x} }}(0)={\hat {\mathbf {x} }}_{0}}$

${\displaystyle {\hat {\mathbf {y} }}(t)=\mathbf {C} {\hat {\mathbf {x} }}(t)}$

thereby determined

${\displaystyle \mathbf {u} _{B}(t)=\mathbf {L} (\mathbf {y} (t)-{\hat {\mathbf {y} }}(t))=\mathbf {L} \mathbf {C} (\mathbf {x} (t)-{\hat {\mathbf {x} }}(t))}$

thus it results for the observer

${\displaystyle {\dot {\hat {\mathbf {x} }}}(t)=(\mathbf {A} -\mathbf {L} \mathbf {C} ){\hat {\mathbf {x} }}(t)+\mathbf {B} \mathbf {u} (t)+\mathbf {L} \mathbf {y} (t)}$

For the observation error of a Luenberger observer, the following applies if all eigenvalues ​​of the matrix have negative real parts. ${\displaystyle \mathbf {e} (t)=\mathbf {x} (t)-{\hat {\mathbf {x} }}(t)}$ ${\displaystyle \lim _{t\to \infty }||\mathbf {e} (t)||=\mathbf {0} }$ ${\displaystyle (\mathbf {A} \ -\ \mathbf {L} \mathbf {C} )}$

The determination of the feedback is carried out analogously to the controller design by specifying the poles by making the following replacements:

${\displaystyle \mathbf {A} ^{T}}$ instead of ${\displaystyle \mathbf {A} }$

${\displaystyle \mathbf {C} ^{T}}$ instead of ${\displaystyle \mathbf {B} }$

${\displaystyle \mathbf {L} ^{T}}$ instead of ${\displaystyle \mathbf {K} }$

The example system has the eigenvalues and . So that the observer can follow the system, its eigenvalues ​​must lie to the left of those of the system. This requirement is met for. The characteristic equation in this case is ${\displaystyle e_{1}=-2}$ ${\displaystyle e_{2}=-4}$ ${\displaystyle \lambda _{1,2}=-8}$

${\displaystyle P(s)=64+16s+s^{2}\,}$

and with it and . The feedback matrix is ​​so ${\displaystyle a_{B0}=64}$ ${\displaystyle a_{B1}=16}$

${\displaystyle \mathbf {L} ^{T}={\begin{pmatrix}56&10\end{pmatrix}}}$.

For the complete observer the differential equation is

${\displaystyle {\dot {\hat {\mathbf {x} }}}(t)={\begin{pmatrix}0&-64\\1&-16\end{pmatrix}}{\hat {\mathbf {x} }}+{\begin{pmatrix}8\\0\end{pmatrix}}u(t)+{\begin{pmatrix}56\\10\end{pmatrix}}y(t)}$.

Structural observability

Systems for two reasons not be observable:

• A specific combination of parameters leads to non-observability.
• The structure of the system means that the system can not be observed or cannot be observed with any combination of parameters, given any occupation of the non- zero elements of the system matrix (which in practice depend on physical parameters) . This is the case when the necessary signal links between state and measured variables are missing. In order to prove that a system is structurally unobservable, graph theoretic methods must be used. Structural observability, on the other hand, is easy to prove if it can be shown that a certain combination of parameters (for example all non-zero elements == 1) describes a completely observable system.
  

Complete observability

Block diagram of state space representation

The state space representation of a linear system is

${\displaystyle \mathbf {\dot {x}} =\mathbf {Ax} +\mathbf {Bu} }$

${\displaystyle \mathbf {y} =\mathbf {Cx} +\mathbf {Du} }$.

The system can be observed if the initial state can be clearly determined with a known control function and known matrices and from the course of the output vector over a finite time interval . ${\displaystyle \mathbf {u(t)} }$ ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {C} }$ ${\displaystyle \mathbf {y(t)} }$ ${\displaystyle t_{0}\leq t\leq t_{1}}$ ${\displaystyle \mathbf {x(t_{0})} }$

In the following, a system with one input and one output ( SISO : Single Input, Single Output) is used as an example . ${\displaystyle u(t)\;}$ ${\displaystyle y(t)\;}$

${\displaystyle \mathbf {\dot {x}} ={\begin{pmatrix}-2&0\\4&-4\end{pmatrix}}\mathbf {x} +{\begin{pmatrix}2\\0\end{pmatrix}}u}$

${\displaystyle y={\begin{pmatrix}0&1\end{pmatrix}}\mathbf {x} }$

It describes the series connection of two PT1 elements with the time constants and . ${\displaystyle T_{1}=0{,}5}$ ${\displaystyle T_{2}=0{,}25}$

proof

Structural observability is a necessary condition for full observability. However, mostly only the following criteria are used to demonstrate complete observability.

The Kalman criterion is relatively easy to determine, but the observability cannot be related to individual intrinsic processes or eigenvalues . This can be done with the help of the Gilbert and the Hautus criteria.

Kalman's criterion

According to Kalman, the system (A, C) is fully observable if and only if the observability matrix has the rank or if its determinant is not equal to 0 in the case of only one measured variable: ${\displaystyle S_{B}}$ ${\displaystyle n}$

${\displaystyle \mathrm {Rang} \ S_{B}{}={}n}$ With

${\displaystyle S_{B}={\begin{pmatrix}C\\CA\\CA^{2}\\...\\CA^{n-1}\end{pmatrix}}}$

The following applies to the example system

${\displaystyle \mathbf {A} ={\begin{pmatrix}-2&0\\4&-4\end{pmatrix}}}$

and

${\displaystyle \mathbf {C} ={\begin{pmatrix}0&1\end{pmatrix}}}$

with the observability matrix

${\displaystyle \mathbf {S_{B}} ={\begin{pmatrix}0&1\\4&-4\end{pmatrix}}}$.

The following applies and the rank is therefore 2. The system is completely observable. ${\displaystyle \mathrm {det} \mathbf {(S_{B})} =-4}$

Gilbert's criterion

If the model is in canonical normal form ( Jordan normal form )

${\displaystyle {\begin{aligned}{\frac {d\mathbf {\tilde {x}} }{dt}}&=\mathrm {diag} (\lambda _{i})\mathbf {\tilde {x}} +\mathbf {\tilde {B}} u,\ \mathbf {\tilde {x}} (0)=V^{-1}\mathbf {x} _{0},\\y&=\mathbf {\tilde {C}} \mathbf {\tilde {x}} \\\end{aligned}}}$

With

${\displaystyle {\begin{aligned}\mathbf {\tilde {B}} &=\mathbf {V} ^{-1}\mathbf {B} ,\\\mathbf {\tilde {C}} &=\mathbf {CV} \end{aligned}}}$

and the matrix of the eigenvectors is available, Gilbert's criterion applies: ${\displaystyle \mathbf {V} \!\,}$

A system whose state space model is in canonical normal form is completely observable if and only if the matrix has no zero column and if the p columns of the matrix , which belong to the canonical state variables of a p -fold eigenvalue, are linearly independent. ${\displaystyle (\mathrm {diag} (\lambda _{i}),\mathbf {\tilde {C}} )}$ ${\displaystyle \mathbf {\tilde {C}} }$ ${\displaystyle \mathbf {\tilde {c}} _{i}}$ ${\displaystyle \mathbf {\tilde {C}} }$

The canonical normal form of the example system is

${\displaystyle {\begin{aligned}{\frac {d\mathbf {\tilde {x}} }{dt}}&={\begin{pmatrix}-2&0\\0&-4\end{pmatrix}}\mathbf {\tilde {x}} +{\begin{pmatrix}4{,}472\\-4\end{pmatrix}}u,\\y&={\begin{pmatrix}0{,}894&1\end{pmatrix}}\mathbf {\tilde {x}} .\end{aligned}}}$

The matrix only has columns (here elements) not equal to 0. The test for linear dependency is omitted here, since the system has simple eigenvalues. ${\displaystyle \mathbf {\tilde {C}} ={\begin{pmatrix}0{,}894&1\end{pmatrix}}}$

The system is fully observable.

Hautus' criterion

The system (A, C) is completely observable according to Hautus if and only if the condition:

${\displaystyle \mathrm {Rang} {\begin{pmatrix}\lambda I-A\\C\end{pmatrix}}=n}$

is satisfied for all eigenvalues ​​of the matrix A. ${\displaystyle \lambda _{i}(i=1,2,...,n)}$

The system matrix of the example has the eigenvalues and . The condition is for both eigenvalues ${\displaystyle \lambda _{1}=-2}$ ${\displaystyle \lambda _{2}=-4}$

${\displaystyle \mathrm {Rang} {\begin{pmatrix}\lambda _{i}+2&0\\-4&\lambda _{i}+4\\0&1\end{pmatrix}}=2}$

Fulfills. The system is completely observable.

Observability of scanning systems

The above relationships also apply to scanning systems when replaced by the transition matrix. According to, the check can be simplified by checking first the conditions for the continuous system and then the additional condition ${\displaystyle \mathbf {A} }$

${\displaystyle e^{s_{i}T_{ab}}\neq e^{s_{j}T_{ab}}}$ For ${\displaystyle s_{i}\neq s_{j}}$

is satisfied.

Observer normal form

For a linear system with one input and one output, the observer normal form can be determined, among other things, from the differential equation equivalent to the transfer function . ${\displaystyle G(s)={\frac {b_{0}+b_{1}s+...+b_{n-1}s^{n-1}+b_{n}s^{n}}{a_{0}+a_{1}s+...+a_{n-1}s^{n-1}+s^{n}}}}$

${\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\...\\{\dot {x}}_{n-1}\\{\dot {x}}_{n}\\\end{bmatrix}}=\underbrace {\begin{pmatrix}0&0&...&0&-a_{0}\\1&0&...&0&-a_{1}\\0&1&...&0&-a_{2}\\...&...&...&...&...\\0&...&...&1&-a_{n-1}\\\end{pmatrix}} _{A_{B}}{\begin{bmatrix}x_{1}\\x_{2}\\...\\x_{n-1}\\x_{n}\\\end{bmatrix}}+\underbrace {\begin{bmatrix}b_{0}-a_{0}b_{n}\\b_{1}-a_{1}b_{n}\\...\\b_{n-2}-a_{n-2}b_{n}\\b_{n-1}-a_{n-1}b_{n}\\\end{bmatrix}} _{b_{B}}u}$

${\displaystyle y={\begin{matrix}\underbrace {(0\ 0\ .\ .\ .\ 1)} \\{\textrm {}}^{\rm {c_{B}^{T}}}\end{matrix}}{\begin{bmatrix}x_{1}\\x_{2}\\...\\x_{n-1}\\x_{n}\\\end{bmatrix}}+{\begin{matrix}\underbrace {b_{n}} \\{\textrm {}}^{\rm {d_{B}}}\end{matrix}}u}$.

The example system has the transfer function

${\displaystyle G(s)={\frac {8}{8+6s+s^{2}}}}$.

It follows with , and ${\displaystyle b_{0}=8}$ ${\displaystyle a_{0}=8}$ ${\displaystyle a_{1}=6}$

${\displaystyle A_{B}={\begin{pmatrix}0&-8\\1&-6\end{pmatrix}}}$

${\displaystyle b_{B}={\begin{pmatrix}8\\0\end{pmatrix}}}$

${\displaystyle {c^{T}}_{B}={\begin{pmatrix}0&1\end{pmatrix}}.}$

Reduced observer

Often some state variables can be measured directly. It is therefore not necessary to reconstruct them. A reduced observer can therefore be derived who only reconstructs the state variables that have not been measured. The order of the reduced observer is reduced by the number of measured variables compared to the full observer. This method can also be expanded in the event that the measured variables are not state variables.

After sorting the matrix lines into measured and observed states, the state space representation of the single variable system is ${\displaystyle \mathbf {x_{M}} }$ ${\displaystyle \mathbf {x_{B}} }$

${\displaystyle {\begin{pmatrix}\mathbf {{\dot {x}}_{M}} \\\mathbf {{\dot {x}}_{B}} \end{pmatrix}}={\begin{pmatrix}\mathbf {A_{11}} &\mathbf {A_{12}} \\\mathbf {A_{21}} &\mathbf {A_{22}} \end{pmatrix}}{\begin{pmatrix}\mathbf {x_{M}} \\\mathbf {x_{B}} \end{pmatrix}}+{\begin{pmatrix}\mathbf {b_{M}} \\\mathbf {b_{B}} \end{pmatrix}}u}$

${\displaystyle \mathbf {y} ={\begin{pmatrix}\mathbf {c_{M}^{T}} &\mathbf {c_{B}^{T}} \end{pmatrix}}{\begin{pmatrix}\mathbf {x_{M}} \\\mathbf {x_{B}} \end{pmatrix}}}$

The equation of state of the full system is

${\displaystyle \mathbf {\dot {x}} =\mathbf {Ax} +\mathbf {b} u}$

and that of the reduced system

${\displaystyle \mathbf {{\dot {x}}_{B}} =\mathbf {A_{22}x_{B}} +\mathbf {A_{21}x_{M}} +\mathbf {b_{B}} u}$.

The measurement equation of the full system is

${\displaystyle \mathbf {y} =\mathbf {c^{T}x} }$

and that of the reduced system

${\displaystyle \mathbf {{\dot {x}}_{M}} -\mathbf {A_{11}x_{M}} -\mathbf {b_{M}} u=\mathbf {A_{12}x_{B}} }$.

The substitution

${\displaystyle \mathbf {x} \leftarrow \mathbf {x_{B}} }$

${\displaystyle \mathbf {A} \leftarrow \mathbf {A_{22}} }$

${\displaystyle \mathbf {b} u\leftarrow \mathbf {A_{21}x_{M}} +\mathbf {b_{B}} u}$

${\displaystyle y\leftarrow \mathbf {{\dot {x}}_{M}} -\mathbf {A_{11}x_{M}} -\mathbf {b_{M}} u}$

${\displaystyle \mathbf {c^{T}} \leftarrow \mathbf {A_{12}} }$

inserted into the equation of the full observer gives

${\displaystyle \mathbf {{\dot {\hat {x}}}_{B}} =(\mathbf {A_{22}} -\mathbf {lA_{12}} )\mathbf {{\hat {x}}_{B}} +(\mathbf {A_{21}} -\mathbf {lA_{11}} )y+(\mathbf {b_{B}} -\mathbf {lb_{M}} )u+\mathbf {l} {\dot {y}}}$.

The time derivative of y is still disturbing in this representation. The transformation

${\displaystyle \mathbf {{\tilde {x}}_{B}} =\mathbf {{\hat {x}}_{B}} -\mathbf {l} y}$

gives the equation

${\displaystyle \mathbf {{\dot {\tilde {x}}}_{B}} =(\mathbf {A_{22}} -\mathbf {lA_{12}} )\mathbf {{\tilde {x}}_{B}} +(\mathbf {A_{21}} -\mathbf {lA_{11}} +\mathbf {A_{22}l} -\mathbf {lA_{12}l} )y+(\mathbf {b_{B}} -\mathbf {lb_{M}} )u}$

and from this the estimated state vector

${\displaystyle \mathbf {{\hat {x}}_{B}} =\mathbf {{\tilde {x}}_{B}} +\mathbf {l} y}$.

See also

• State space representation

swell

Otto Föllinger : Control engineering, introduction to the methods and their application . ISBN 3-7785-2336-8 .

1. ↑ Section 13.7.2 / Formula (13.158)

Otto Föllinger: Non-linear regulations . 7., revised. u. exp. Edition. tape 2 Harmonic balance, Popow and circle criterion, hyperstability, synthesis in the state space. Oldenbourg, Munich 1993.

1. ↑ Section 7.5

Jan Lunze: Control engineering 2: Multi-variable systems, digital control . 5th edition. Springer Verlag, Heidelberg 2008, ISBN 978-3-540-78462-3 .

1. ↑ Section 3.3.2
2. ↑ Section 3.4
3. ↑ Section 3.2.2
4. ↑ ^{a b} Section 3.2.4
5. ↑ Section 11.3.3
6. ↑ Section 8.4

literature

• SDG Cumming: Design of observers of reduced dynamics . In: Electronic Letters . tape 5 , 1969, p. 213-214 .
• DG Luenberger: Observing the state of a linear system . In: IEEE Transaction on Military Electronics . tape 8 , 1964, pp. 74-80 .
• RE Kalman, B. Bucy: New results in linear filtering and prediction theory . In: Trans ASME, Series D, Journal of Basic Engineering (ASME) . 83D, 1961, pp. 98-108 .
• A. Yellow: Applied Optimal Estimation . The MIT press, Massachusetts Institute of Technology, Massachusetts 1974.
• Otto Föllinger: Control engineering, introduction to the methods and their application . ISBN 3-7785-2336-8 .