System identification

System ID ( system identification also ) is the theoretical and / or experimental determination of the quantitative dependence of the output of the input variables of a system. For this, the system is excited with defined test signals (jump, pulse, ramp, or the like. ) And recorded the output. The methods used to mathematical analysis methods can be deterministic or stochastic.

  • 6.1 Software tools for experimental system identification

Theoretical System Identification

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

And, since in this case, the Laplace transform can be carried out, applies for the transfer function

If all the coefficients and known, identification problem is solved. Otherwise, the unknown coefficients to be determined by the experimental system identification.

Experimental system identification

The system is excited with suitable test signals (step, pulse, ramp or other). These signals are applied to a mathematical model, which has free parameters supplied. The model is known from a previous theoretical process identification. The model can be either in the time domain or the frequency domain. From two output signals ( system and model) the deviation (difference ) is calculated and evaluated by a quality criterion in the form of a functional. The result of judgment is used by an algorithm to adjust the parameters of the model. This process is repeated until the desired quality is reached. The iterative adjustment of the model parameters can be shortened by supporting appropriate software tools.

Turning tangent method

A system with compensation and without overshoot has a turning point in the step response. It occurs on systems with the series connection of a plurality of delay elements ( PT1 elements ) on. By applying the tangent at the inflection point, the delay time and the recovery time can be determined. The goal is to identify from these experimental values ​​of the system time constants of the transfer function of the model. The model function must also describe a system with compensation and be the step response function of the model in analytical form.

Principle

Of the transfer function of the model

And the step function in the image domain

Obtained by the inverse Laplace transform of the step response function of the model

The increase of the step response function of the model at time t is

The reversing time is calculated from the condition

With this data

The equation of the inflectional tangent of the model. Continue to apply to the Duch passage point of inflection tangent through the time axis

And by the steady-state value

From the diagram it can be seen that, and is regarded. Thus, the relationship between the parameters of the system and the characteristics of the model and

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

Model transfer functions

Only functions with two time constants, or at n equal time constants, the number n and the time constant are determined using the method described. When transfer functions are used, among other things:

  • Two different time constants and
  • Same time constants
  • Two of the same time constant and a time constant
  • For large dead times or time delays also goes

Example

For the case of equal time constants

Is the step response function

The increase of the inflectional tangent

And from

Follows with

After umindizieren the sums

And

The turnaround time

The increase in the turning point

And

The inflectional tangent construction delivers with the numerically evaluated relationships

And with

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) below) / (n -1) ^ (n- 1);   printf ("% - 5d | % 5.3f |% 5.3f - | % 5.3f \ n", n, fa, fu, fa / fu); end; printf (" ---------------------------- \ n"); produces a table of the values ​​of the relations given above.

N | Tg / T | Tu / T | Tg / Tu ---------------------------- 2 | 2718 | 0282 | 9649 3 | 3695 | 0805 | 4587 4 | 4,463 | 1,425 | 3,131 5 | 5,119 | 2,100 | 2,437 6 | 5699 | 2811 | 2027 7 | 6226 | 3549 | 1754 8 | 6711 | 4307 | 1558 9 | 7164 | 5081 | 1410 10 | 7,590 | 5,869 | 1,293 ---------------------------- From the measured step response values ​​were determined numerically and with a Scilab script. Follows from the table, and. Because the time constants are different and calculated the average time constant was calculated to be. Therefore applies for the transfer function of the model

Applications

An essential tool of the system identification is the linear regression analysis. It is here as a functional dependence on a linear combination of an arbitrarily chosen basis functions. Each trial function is an arithmetic expression of causing physical quantities. The caused physical quantity is calculated by multiplying each trial function with an initially unknown coefficients and added to the result. The coefficients are determined such that the mean square deviation of the measured from the calculated result becomes minimum. This means that the partial derivative of the mean square error according to each coefficient must be zero. This results in an inhomogeneous linear equation system for determining the coefficients. The matrix of the equation system consists of products of two basis functions, averaged over all measurements. The right side of the equation system consists of products caused size, each with a trial function, averaged over all measurements. In the stepwise linear regression analysis carried out is determined iteratively, which series elements most and which have the least impact on the accuracy and the number of limbs without significant influence are omitted.

Instead of stepwise linear regression analysis can ends with many problems alternatively a multi-layer perceptron (English multi- layer perceptron, MLP ) may be used, which is often referred to with the generic term neural networks.

The system identification is used for example in fluid mechanics is used, whether to calculate drag and lift of a profile or be it to simulate maneuvering ships numerically. Another area of ​​application is the vibration control technology, where with transfer functions (English: RAO = response amplification operator) is calculated with which magnification and phase shift reacts an oscillatory system to the individual frequencies of the vibration cause.

In aviation, the system identification is used, for example, to determine the aerodynamic parameters, which are often only imprecisely known from analytical method and wind tunnel tests. Here, the " quad -M" - system use, that the maneuver (ie, the system excitation ), the measurement of the system response, the mathematical model of the system and the methods as indicating the essential part of functions of the system identification for parameter estimation.

LTI system theory Functional (mathematics) Image (mathematics)
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