System identification

System identification (also system identification ) is the theoretical and / or experimental determination of the quantitative dependence of the output on the input variables of a system. For this purpose, the system is stimulated with defined test signals (jump, pulse, ramp, etc.) and the output is recorded. The methods used for the mathematical evaluation can be deterministic or stochastic.

Theoretical system identification

In the case of theoretical system identification, the model is created on the basis of balance equations, taking conservation laws into account . The result is a system of differential equations describing the relationship between input and output variables . In the case of a linear, time-invariant system :

${\displaystyle y^{(n)}+\ldots +a_{1}y^{(1)}+a_{0}y=b_{m}u^{(m)}+\ldots +b_{1}u^{(1)}+b_{0}u}$

and, since the Laplace transform is feasible in this case , applies to the transfer function

${\displaystyle G(s)={\frac {b_{m}s^{m}+\ldots +b_{1}s+b_{0}}{s^{n}+a_{n-1}s^{n-1}+\ldots +a_{1}s+a_{0}}}}$.

If all coefficients and are known, the identification task is solved. Otherwise the unknown coefficients must be determined by the experimental system identification. ${\displaystyle a_{n}}$ ${\displaystyle b_{m}}$

Experimental system identification

Signal flow diagram of the experimental system identification

The system is stimulated with suitable test signals ( jump , pulse , ramp, etc.) . These signals are also fed to a mathematical model which has free parameters. The model is known from a previous theoretical process identification. The model can either be in the time domain or in the frequency domain . The deviation (difference) is calculated from both output signals (system and model) and evaluated using a quality criterion in the form of a functional . The result of the evaluation is used by an algorithm to adapt the parameters of the model. This process is repeated until the desired quality is achieved. ${\displaystyle u(t)}$

The iterative adaptation of the model parameters can be shortened with the support of appropriate software tools.

Turning tangent method

Step response of a controlled system with turning tangent to determine and . ${\displaystyle T_{u}}$ ${\displaystyle T_{g}}$

A system with compensation and without overshoot has a turning point in the step response . It occurs in systems with several delay elements ( PT1 elements ) connected in series. By placing the tangent at the turning point, the delay time and the compensation time can be determined. The aim is to determine the time constants of the transfer function of the model from these experimentally determined values ​​of the system. The model function must also describe a system with compensation and the step response function of the model must be available in analytical form. ${\displaystyle T_{u}\;}$ ${\displaystyle T_{g}\;}$

principle

From the transfer function of the model

${\displaystyle G(s)={\frac {x_{a}(s)}{x_{e}(s)}}}$

and the jump function in the image area

${\displaystyle x_{e}(s)={\frac {K}{s}}}$

the step response function of the model is obtained by inverse Laplace transformation

${\displaystyle x_{a}(t)=L^{-1}\{G(s)\cdot x_{e}(s)\}}$.

The increase in the step response function of the model at time t is ${\displaystyle m(t)\;}$

${\displaystyle m(t)={\frac {dx_{a}(t)}{dt}}}$.

The turning point is calculated from the condition ${\displaystyle t_{w}\;}$

${\displaystyle {\frac {d^{2}x_{a}(t)}{dt^{2}}}=0}$.

With this data is

${\displaystyle x_{T}(t)=m(t_{w})\cdot (t-t_{w})+x_{a}(t_{w})}$

the equation of the turning tangent of the model. Furthermore, the turning tangent through the time axis applies to the point of passage

${\displaystyle x_{T}(t_{0})=0=m(t_{w})\cdot (t_{0}-t_{w})+x_{a}(t_{w})}$

and by the stationary value

${\displaystyle x_{T}(t_{1})=K=m(t_{w})\cdot (t_{1}-t_{w})+x_{a}(t_{w})}$.

It can be seen from the sketch that , and applies. Thus the relationship between the parameters of the system , and the properties of the model reads and ${\displaystyle m(t_{w})={\frac {K}{T_{g}}}}$ ${\displaystyle t_{0}=T_{u}\;}$ ${\displaystyle t_{1}=T_{u}+T_{g}\;}$ ${\displaystyle T_{u}\;}$ ${\displaystyle T_{g}\;}$ ${\displaystyle t_{w}\;}$ ${\displaystyle x_{a}(t_{w})\;}$

${\displaystyle T_{u}=t_{w}-T_{g}\cdot {\frac {x_{a}(t_{w})}{K}}}$.

These relationships are available as tables and nomograms for specific models.

Model transfer functions

With the method described, only functions with two time constants or, with n equal time constants, the number n and the time constant can be determined. The following are used as transfer functions:

• Two different time constants and ${\displaystyle T_{1}\;}$ ${\displaystyle T_{2}\;}$

${\displaystyle G(s)={\frac {K}{(1+T_{1}\cdot s)\cdot (1+T_{2}\cdot s)}}}$

• ${\displaystyle n\;}$ same time constants ${\displaystyle T_{1}\;}$

${\displaystyle G(s)={\frac {K}{(1+T_{1}\cdot s)^{n}}}}$

• Two equal time constants and one time constant ${\displaystyle T_{1}\;}$ ${\displaystyle T_{2}\;}$

${\displaystyle G(s)={\frac {K}{(1+T_{1}\cdot s)^{2}\cdot (1+T_{2}\cdot s)}}}$

• In the case of long dead times or time delays, this is also possible

${\displaystyle G(s)={\frac {K\cdot e^{-T_{t}\cdot s}}{(1+T_{1}\cdot s)}}}$

example

Comparison of measured and identified step response.

In the event that the time constants are the same ${\displaystyle n\;}$ ${\displaystyle T\;}$

${\displaystyle G(s)={\frac {K}{(1+T\cdot s)^{n}}}}$

is the step response function

${\displaystyle x_{a}(t)=K\cdot \left\{1-\left[\sum _{m=0}^{n-1}{\frac {({\frac {t}{T}})^{m}}{m!}}\right]\cdot e^{-t/T}\right\}}$

The rise in the tangent is

${\displaystyle m(t)={\frac {dx_{a}(t)}{dt}}=\left(\sum _{m=0}^{n-1}{\frac {t^{m}}{T^{m+1}\cdot m!}}-\sum _{m=1}^{n-1}{\frac {m\cdot t^{m-1}}{T^{m}\cdot m!}}\right)\cdot e^{-{\frac {t}{T}}}}$

and from

${\displaystyle {\frac {d^{2}x_{a}(t)}{dt^{2}}}=\left(-\sum _{m=2}^{n-1}{\frac {m\cdot (m-1)\cdot t^{m-2}}{T^{m}\cdot m!}}+2\cdot \sum _{m=1}^{n-1}{\frac {m\cdot t^{m-1}}{T^{m+1}\cdot m!}}-\sum _{m=0}^{n-1}{\frac {t^{m}}{T^{m+2}\cdot m!}}\right){\dot {e}}^{-{\frac {t}{T}}}}$

follows with

${\displaystyle {\frac {d^{2}x_{a}(t)}{dt^{2}}}=0}$

after re-indexing the sums

${\displaystyle {-\sum _{m=0}^{n-3}{\frac {({\frac {t}{T}})^{m}}{m!}}+2\cdot \sum _{m=0}^{n-3}{\frac {({\frac {t}{T}})^{m}}{m!}}+2\cdot {\frac {({\frac {t}{T}})^{n-2}}{(n-2)!}}-\sum _{m=0}^{n-3}{\frac {({\frac {t}{T}})^{m-1}}{(m-1)!}}-{\frac {({\frac {t}{T}})^{n-2}}{(n-2)!}}-{\frac {({\frac {t}{T}})^{n-1}}{(n-1)!}}=0}}$

and

${\displaystyle {\frac {({\frac {t}{T}})^{n-2}}{(n-2)!}}={\frac {({\frac {t}{T}})^{n-1}}{(n-1)!}}}$

the turning point

${\displaystyle t_{w}=(n-1)\cdot T}$.

The rise is at the turning point

${\displaystyle m(t_{w})=K\cdot {\frac {e^{-(n-1)}\cdot (n-1)^{n-2}}{T\cdot (n-2)!}}}$

and

${\displaystyle x_{a}(t_{w})=K\cdot \left\{1-\left[\sum _{m=0}^{n-1}{\frac {(n-1)^{m}}{m!}}\right]\cdot e^{-(n-1)}\right\}}$

The turning tangent construction supplies the relationships to be evaluated numerically ${\displaystyle T_{g}={\frac {K}{m(t_{w})}}}$

${\displaystyle {\frac {T_{g}}{T}}={\frac {(n-1)!}{(n-1)^{n-1}}}\cdot e^{n-1}}$

and with ${\displaystyle T_{u}=t_{w}-T_{g}\cdot {\frac {x_{a}(t_{w})}{K}}}$

${\displaystyle {\frac {T_{u}}{T}}=n-1-{\frac {(n-1)!}{(n-1)^{n-1}}}\cdot \left[e^{n-1}-\sum _{m=0}^{n-1}{\frac {(n-1)^{m}}{m!}}\right]}$.

The Scilab script

N=10;
printf("\n");
printf(" n     |Tg/T  |Tu/T  |Tg/Tu\n");
printf("----------------------------\n");
for n=2:N,
 su=0;
 for i=0:n-1,su=su+(n-1)^i/factorial(i);end;
 fa=factorial(n-1)*exp(n-1)/(n-1)^(n-1);
 fu=n-1-factorial(n-1)*(exp(n-1)-su)/(n-1)^(n-1);
 printf(" %-5d |%-5.3f |%-5.3f |%-5.3f\n",n,fa,fu,fa/fu);
end;
printf("----------------------------\n");

creates a table of values ​​according to the relationships given above.

n     |Tg/T  |Tu/T  |Tg/Tu
----------------------------
2     |2.718 |0.282 |9.649
3     |3.695 |0.805 |4.587
4     |4.463 |1.425 |3.131
5     |5.119 |2.100 |2.437
6     |5.699 |2.811 |2.027
7     |6.226 |3.549 |1.754
8     |6.711 |4.307 |1.558
9     |7.164 |5.081 |1.410
10    |7.590 |5.869 |1.293
----------------------------

From the measured step response of the values , and determined with a Scilab script numerically. From the table follows , and . Because the time constants calculated from and are different, the mean time constant was also calculated. This applies to the transfer function of the model ${\displaystyle T_{u}=1{,}030}$ ${\displaystyle T_{g}=5{,}183}$ ${\displaystyle T_{g}/T_{u}=5{,}031}$ ${\displaystyle n=3}$ ${\displaystyle T_{g}/T=3{,}695}$ ${\displaystyle T_{u}/T=0{,}805}$ ${\displaystyle T_{g}}$ ${\displaystyle T_{u}}$ ${\displaystyle T=1{,}341}$

${\displaystyle G(s)={\frac {1}{(1+1{,}341s)^{3}}}}$.

Applications

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An essential aid in system identification is linear regression analysis . A linear combination of arbitrarily chosen approach functions is used as the functional dependency. Every approach function is an arithmetic expression of the physical quantities causing it. The physical quantity caused is calculated by multiplying each approach function by an initially unknown coefficient and adding it to the result. The coefficients are determined in such a way that the mean square deviation of the measured from the calculated result is minimal. This means: the partial derivative of the mean square deviation for each individual coefficient must be zero. This results in a linear, inhomogeneous system of equations for determining the coefficients. The matrix of the system of equations consists of products of two approach functions each, averaged over all measurements. The right-hand side of the system of equations consists of products of the caused variable, each with an approach function, averaged over all measurements. In the linear regression analysis carried out step by step, iteratively determines which members of the series have the most and which of the least influence on the accuracy, and the members of the series that do not have a significant influence are omitted.

Instead of the pitch sending linear regression analysis of a multi-layer can in many problems alternatively perceptron (engl. Multi-layer perceptron , MLP) may be used, often with the preamble neural networks is referred to.

System identification is used, for example, in fluid mechanics , be it to calculate the resistance and buoyancy of a profile or to numerically simulate maneuvering ships. Another area of ​​application is vibration engineering , where transfer functions (RAO = response amplification operator) are used to calculate the magnification and phase shift with which a vibratory system reacts to the individual frequencies of the vibration cause.

In aviation, for example, system identification is used to determine aerodynamic parameters, which are often only imprecisely known from analytical methods and wind tunnel tests. The "Quad-M" system is used here, which specifies the maneuvers (i.e. the system excitation), the measurement of the system reaction, the mathematical model of the system and the methods for parameter estimation as the essential subtasks of system identification.

literature

• Heinz Unbehauen: Control Systems I . Vieweg, 1997, ISBN 3-528-83332-7 .
• Holger Lutz, Wolfgang Wendt: Pocket book of control engineering with MATLAB and Simulink . 11th edition. Europa-Lehrmittel, 2019, ISBN 978-3-8085-5869-0 .
• Lennart Ljung: System Identification: Theory for the User . 2nd Edition. Prentice Hall, Upper Saddle River 2006, ISBN 978-0-13-656695-3 .
• Oliver Nelles: Nonlinear System Identification . 1st edition. Springer, 2000, ISBN 978-3-540-67369-9 .
• Rolf Isermann, Marco Münchhof: Identification of Dynamic Systems - An Introduction with Applications . 1st edition. Springer, 2010, ISBN 978-3-540-78878-2 .