Building of FESTO servo-motor imitation model using linear ARX-model.

Svetlichny, Pavel ; Kramar, Vadim

1. INTRODUCTION

Research work is often connected with the need of practical verification of theoretical material that frequently leads to a problem of buying expensive equipment. Not every educational or research organization can afford purchasing of such expensive instrumentation for carrying out efforts, especially if there is a risk of the equipment to be damaged.

Building of imitation models with the help of computer for verifying theoretical knowledge is one of the solutions for this problem. Construction of models of real objects gives the possibility of the simultaneous access of several people to the simulated device while only one researcher has access to the one real test bench. Due to working out the imitation model in details it is possible to obtain reliable and durable instrument for research, whose possibilities can be expanded by the addition of new modules (Svetlichny et al., 2008).

There are a lot of methods which allow building of the imitation models of a complicated object. These methods differ of their complexity and field of application. In the context of this article it is decided to use the method of building of linear ARX-model for its simplicity and effectiveness.

2. INFORMATION

The task of imitation modelling can be solved by using the object identification methods, when the mathematical model is created being based on known input and output signals without knowledge of the real object structure.

Servo-controller FESTO SEC-AC is used to control torque, speed and positioning of FESTO MTR-AC asynchronous servo-motor. The current, speed and positioning controllers, which are the basis of servo-controller, are presented as the control cascade. Controllers are adjusted as PI-controllers and can function separately. Thus there are three control modes: torque, speed and positioning control mode.

To build the simplified imitation model of asynchronous motor it is enough to use only one control cascade which is the torque control mode. In this case input of the controller presents a given driving shaft rotation speed and the output is its actual value. Both values are specified in rotations per minute.

2.1 Method

For the sake of simplicity it is better to present an object as the linear model (Eikhoff, 1975). In general the description of model looks like (1):

y(t) = G(q)u(t) + H (q)e(t) (1)

where y(t)--forecast output of the system;

G(q)--transfer operator of the linear system;

u(t)--scalar input signal;

H(q)--filter, which in combination of sequence of stochastic values (white noise) e(t) presents realization of stochastic process v(t) = H(q)e(t);

q--forward shift operator, such that qu(t) = u(t+1).

The dominant factor for the system identification is setting of the coefficients G and H with the help of the finite number of the numerical characteristics. It is often very difficult to predetermine these coefficients from knowing of physical system properties, and so it is necessary to resort to estimation procedures for their determination (or some of their parts). It means that the concerned coefficients are the part of a model (1) as determinate parameter. For the designation of these parameters it is possible to introduce symbol 9 and work with the model description of (2):

y(t) = G(q, [theta])u(t) + H(q, [theta])e(t), (2a)

[f.sub.e] (x, [theta]), (2b)

where (2b) is a probability density e(t), {e(t)}--white noise.

Parameter vector [??] belongs to some area of a real number.

For equation (2) it is possible to compute one-step forecast. It can be marked as (3):

[??](t | [theta]) = [H.sup.-1] (q, [theta])G(q,[theta])u(t)+[1 - [H.sup.-1] (q,[theta])]y(t). (3)

This forecaster is independent of probability density e(t).

The most direct way of the parameterization of G and H is their presentation as rational functions, as the coefficients in numerator and denominator become parameters. Such a model is known as black box model.

The simplest input-output correlation is described as the linear difference equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

The white noise e(t) is presented in this equation as its direct error, so the model (4) is often called a control error model. In this case we have a set of adjustable parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

If we implement

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then we can see that (4) agrees with (2) with choose of

G(q, [theta]) = B(q) / A(q), H(q, [theta]) = 1 / A(q). (6)

Such systems are called ARX-models (Ljung, 1987).

2.2 Instrument and measures

For object identification with the help of building of ARX-model it is good to use the standard MATLAB instruments. One of these tools is "ident" utility, which allows carrying out identification by using input and output data with different methods and models (Ljung, 2008).

In the context of research the white noise, whose means with the reaction of the system are presented on Figure 1, was fed to the input of controller. Input values were chosen in the range of -880 to +844 rotations per minute. It was executed 100 measures with strokes of 0.05 sec.

2.3 Identification

The identification of the system was carried out by ident utility with the use of the first order linear ARX-model with the coefficients na=1, nb=1, and the accuracy of 83% was acquired. Increase in model order to the second order leads to the increase of accuracy to 87.42%. The comparative diagram for the measured and simulated system outputs is presented on Figure 2.

The resulting model can be presented as (7):

A(q) = 1-0,0467[q.sup.-1] -0,03087[q.sup.2] -0,014[q.sup.3] (7) B(q) = 0,9936-0,02649[q.sup.1] -0,02463[q.sup.2]

By using the (7) it is possible to construct Simulink scheme (please see Fig. 3). Coefficients of (7) are parameters of a discrete transfer function by corresponding orders of x.

Values of "input" massive are transmitted to the input of transfer function with transformation from a parallel to a scalar form via m-function, which retrieves elements of massive with indexes from 1 to 100. For the purpose of comparison of simulated and measured data values of massive "vihod" are transmitted to the scope through Zero-Order Hold for their digitization (please see Fig. 4).

3. CONCLUSION

Built model is adequate only in range of the values used during identification. As these values exceed the bounds it is hard to predict behaviour of the system. That is why the wider range of input values is required for more accurate identification. But it leads to the problem of the used method.

By using the method of building of linear ARX-model with values in the range from -1710 to +1660 rotations per minute the accuracy of 48.41% is acquired. By increasing of the model order with coefficients na=16, nb=16--to 54.34%. It tells us about impossibility of using this method of identification object with such input values.

The next step of research is acquiring of more accurate results. This task should be solved by using identification methods, based on building of nonlinear ARX-model with the use of neural network (Kruglov, 2001).

The used method allows acquiring of profitable results in the field of imitation modelling, to obtain reliable instrument for research work. In spite of the presence of some disadvantages connected with complexities of different real objects, these disadvantages could be minimized by using complexer mathematical model or using other identification methods based on a behaviour of a real and modelled object and results of modelling.

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When the exacter model will be built, it will be possible to start creation of 3D-model of the whole servo-system.

4. REFERENCES

Eikhoff, P. (1975). The basics of control system identification, "Mir", Moscow

Kruglov, V. (2001). The artificial neural networks. Theory and practice, Goryachaya liniya--Telekom, Moscow

Ljung, L. (1987). System identification: Theory for the User, PTR Prentice Hall, ISBN 0-13-881640-9, New Jersey.

Ljung, L. (2008). System Identification Toolbox 7. User's Guide, The MathWorks, Inc.

Svetlichny, P. & Alchakov V. & Kramar V. (2008). The conception of the virtual learning centre for the tasks of remote education, Proceeding of the 15th international conference on automatic control, pp 514-516, September 2008, "IzdatInform" ONMA, Odessa