Georgia, Tbilisi.

Georgian Technical University.

Email: alik@ gtu.edu.ge

This article discusses questions of combined application of the two-step identification algorithm with the methods of the catastrophe theory makes it possible to forecast unstable areas of control, to make necessary corrections of the impact of control and to avoid intrusion into restricted control areas (of defective product manufacturing).

Грузия, Тбилиси

Грузинский технический университет

Email: alik@ gtu.edu.ge

В статье рассматриваются вопросы совместного применения двухшагового алгоритма идентификации и методов теории катастроф для прогнозирования неустойчивых областей управления, осуществления необходимой корректировки управляющего воздействия и исключение попадания в запретные зоны управления (выпуска бракованной продукции).

Adaptive methods are widely used for the creation of automated control systems for complex technological processes in order to improve the quality of the manufactured goods. This required introduction of adaptive control systems (ACS) with an identificator in the chain of reverse connection (ACS) [1].

The main element of the ACS is an identificator, the purpose of which is to create a mathematical model of the automated object and to forecast its output value. The purpose of identification with a given structure of the object is to evaluate the parameters of the model. Various identification algorithms have been developed in this direction.

In the ACS introduced at Pervouralsk, Dnepropetrovsk and Rustavi steelworks a single-step identification algorithm -"Kachmazha" has been used [1].

Perfection and latest developments of computer technology used in ACS enable us to utilize more complex but improved parameters of identification algorithm.

Recently, considerable attention has been paid to the development of multi-step algorithms having much better characteristics in comparison with the single-step algorithms.

The present paper considers the issues in connection with the utilization of two-step algorithm in order to extend the ACS functions. As the ACS provides improvement of the qualitative data of the system. (Considerable decrease of the controlled value dispersion) we face a problem to considerably decrease production of defective goods.

In this regard, methods of catastrophe theory are proposed to be used together with the identification algorithm. As is well known, the methods of disaster theory enable us to determine unstable zones of the object (catastrophe zones) which correspond to the emergency situations.

Combined application of the two-step identification algorithm with the methods of the catastrophe theory makes it possible to forecast unstable areas of control, to make necessary corrections of the impact of control and to avoid intrusion into restricted control areas (of defective product manufacturing).

Multidimensional linear object comprising a wide class of technological procedures including the pipe-rolling processes has been considered:

(1) Where - is output of the object

- Vector of the parameters of the n dimension object

- Transposition sign

-Input vector of the object, containing n-1 occasional disturbance

-Controlling variable

The identificator on the basis of input and output values makes a model of the object according to the equation

(2)

Where - is output; -vector of the parameter values of the object at N -1 step.

Parameters are evaluated by two-step algorithm, using the information of "input and output" on two phases of discrete time. Analitical output of the algorithm ireceived from the so called"basic allgorithm [2].

Let's mark N and N-1 time vectors of input of the object by and and corresponding outputs by and .

Model of the object this time will be described by the system of these equations

Where is a parameter of the algorithm .

Evaluation of average input and output values around which the initial parameters are centered, centering and normalization is done by special algorithms [1]

Dispersion of variables is determined by these values - .

Output value is defined by factor and measured input values. By means of the discovered model of the object the control .impact is calculated. Implementation of it guarantees equality of output value with the given value of

All the identification procedures like determination of control impact and its implementation is carried out by the ACS. Average values of output and input as well as their dispersions calculated during the process of identification can be used as initial data to determine the disaster zones.

Experiments on the majority of linear objects, namely on pipe rolling devices have shown that input values are distributed in accordance with the regular rule.

Curve of possibility distributions of the pipe-rolling device output value of the pipe-wall thickness is given on fig.l. Curve 1 corresponds to the manual control, while curve. 2 corresponds to the automated control, which is carried out by means of ACS. and minimum and maximum values of the pipe wall, according to the Soviet standards;

- is a given value of the wall thickness; - new given value.

A and В zones are restricted areas or disaster zones in accordance with the terminology used in theory of disasters.

Control system should function so that values of the wall thickness should not exceed the limits, that means to meet strictly these conditions

(5)

Application of ACS for controlling pipe-rolling device has decreased the dispersion of the average wall thickness of pipes by 35-40%[1]. Utilization of two step algorithm increases accuracy of forecast in comparison with one step algorithm; thus ACS provides more decrease of dispersion [4]. Decrease of this dispersion excludes violation of conditions; namely possibility of defect occurrence is almost zero.

Practically, in order to obtain greater effect, given value of is substituted by a new value that enables us to produce pipes in the area of minus possibilities. Meanwhile curve 2 moves to the minimum possible values (curve 3). This results in considerable increase of pipe- rolling effectiveness, significant decrease of metal usage due to the decrease of the weight of each linear meter of the pipe.

However the possibility of intrusion into A zone increases and the production of defect goods is relatively higher, namely pipes with walls thinner than limited.

In addition to this, in order to obtain greater effectiveness of the ASC, zone can be attained to the restricted zone in addition to the В area. Though the inrusion into this zone is not a disaster and doesn't cause production defects, it is not economically reasonable.

Different methods can be used to avoid intrusions into the restricted control zones. However, for the objects ran by the ACS, using two-step identification algorithm together with the method of the catastrophe theory can solve problems of the intrusion forecast into the restricted zone.

R.Tom's theory of catastrophe is based on the investigation of the local behavior of the stationary distribution probability to which the system moves when . Stationary distribution can be viewed with a certain possible surface in the space with parameters and investigate this surface by means of the theory of catastrophe. Sharp changes in the shape of the surface correspond to the sharp qualitative changes in the behavior of the present system- catastrophe [5].

Probability of density distribution is determined as follows:

(6) Where - is an output value, and - are parameters dependent on the average value of dispersion.

This distribution achieves maximum at point у =0, which corresponds to the position of balanced equilibrium. It is known, that potential function is the reverse function of the distribution density, namely

(7)

This equation, according to. Tom, corresponds to the diaster of assembly type. Catastrophe occur, when

(8)

(10)

(11)

If , equation (10) has three different substantial roots, when, such root will be one, if, then multiple roots. That's why the equation (9) gives on the surface which is called bifurcation curve. This curve breaks the surface of parameter control into five regions (I, II, III, IV and point 0). Critical points are located in the III and IV regions. That's why, if the parameters get into restricted zone, it is necessary to correct in order to get out of the critical areas.

When, and parameters are on semicubic parable and are critical. When, and parameters get into region I where the values of and parameters are more or less acceptable.

When, and parameters get into region II and values of the parameters from this region give good results.

And factors are determined as follows:

(12)

(13)

We can conclude that combined usage of two-step identification algorithm and catastrophe theory makes it possible to form the conditions for correcting control impact, namely, correction is done when and are in regions III and IV. Correction is not required when and are in region II. As for the region I and point 0, final decision is .made by the operator of the system.

DETERMINATION OF THE CORRECTED CONTROL IMPACT

In order to avoid intrusion into the restricted control zone, it is necessary to correct control impact determined by equation (4).

Corrected control inpact is determined by equation

(14)

Where -is correcting factor, dependent on the parameters of and potential function. Implementation of the calculated control impact excludes probability of getting into restricted zone, namely production of defective goods.

Chart of corrected control impact algorithm implementation

The chart envisages to get information about input and output values, evaluation of model parameters, forecast of output value, determination of correcting factor and calculation of corrected control impact. All the procedures are carried out by ACS.

References:

1. Danilov F.A., Raibman N.S., Rurua A.A., Chadeev V.M., and others. Adaptive Control of Pipe-rolling Accuracy. M.; Metalurgia, 1980-280 p.
2. Raibman N.S., Chadeev V.M. Modelling of production processes. M.; Energia, 1975, - 375р.
3. Rurua A.A. Two-stem Adaptive Algorithm for the Evaluation of Parameters // Reports of the Academy of Sciences of Georgian SSR,No3,1980, p. 669-672.
4. Bodnia V.G., Rurua A.A., Chadeev V.M. Two-step Identification Algorithm for Linear Objects // Avtomatika I Telemekhanika, №8, 1982, p. 77-84.
5. Poston Т., Stewart I. Theory of Disasters. M.; Mir, 1980. - 607 p.
6. Principles of Symmetry in Identification of Non-linear Control Objects. By Gugushvili A.Sh., Tbilisi; Technical University, 1998. - 272 p.
7. Rurua A.A., Guguhvili A.Sh., and others. Control System of the Pipe-rolling Procedure at the Automated Mills. A.s. № 1357102. BI, 1987, №45.