Stability of temperature field of the distributed control system

STABILITY OF TEMPERATURE FIELD OF THE DISTRIBUTED

CONTROL SYSTEM

Yury Ilyushin and Ekaterina Golovina

Saint-Petersburg Mining University, Vasilyevsky Island, 21st line, Saint-Petersburg, Russia E-Mail: [email protected]

ABSTRACT

At the present stage of automatic control systems development, the question of maintaining temperature specifications comes up. The authors developed a synthesis technique for nonlinear controllers to stabilize the temperature field of control object. The resulting controller allows creating an adaptive controlled system to maintain the temperature field - for this it is necessary to analyze the control system in order to find optimal number of heating elements to stabilize the temperature field. This article considers the issue of stable occurrence of thermal processes while stabilizing the temperature field. Thermal processes in furnaces for various purposes are considered, in particular in drying and roasting chambers. The control system and stabilization of the temperature field are reviewed.

Keywords: control, stability, temperature field, green's function.

INTRODUCTION

At the present stage of human civilization development, automatic control systems have affected all areas of human society. In recent years, they have been deeply embedded in agricultural systems. If a few decades ago a person collected and dried wheat, he ground flour and baked bread with his own hands. Now, all these procedures are made by combines, drying ovens, baking and confectionery ovens. But process of automation of manual labor leads to a greater complexity of technological process. For example, there was a problem of stabilization of the temperature field during thermal processes in the drying chambers (drying ovens of SZS type), heating, and baking of bread and flour products [1-3]. However, it was solved, but the question of thermal processes stability remained. Stability of the process in this case is especially important, since during drying a large amount of excess moisture appears, which reduces efficiency of the process of raising temperature of the drying chamber.

Formulation of the problem. Problem of stability analysis of a distributed temperature control system of the drying chamber is posed. As an object of control, we consider an isotropic cylindrical wire. The control action is heat flux generated by the sources, in the form of sections of a sectional heater, distributed along the boundary of the side surface of the rod. Inclusion of sources is implemented using pulse elements. At the ends of the rod (wire), zero temperature is maintained. Mathematical model of the heat propagation process will have the form [4]:

2

a

t

T

_





)

(

)

(

2 2



















t

x

x

T

;

0



x



l

;

t



0

;



)

,

0

(

t

T

T

(

l

,

t

)



0

;

)

(

)

0

,

(

x





x





T



(

t

)

.

Block diagram of a closed-loop control system is shown in Figure-1.

Figure-1. Block diagram of a control system.

METHODOLOGY AND ITS SOLUTION

Deviation of the system output function from the set value will be the input signal of a nonlinear link

zad

T

t

x

T

t

x







(

,

)

(

,

)

.

If the condition

T

(



кр

,



1

)



T

zad



0

is

fulfilled at the extreme points of sources



1_and



2_{, at}

some point in time



1_{, at the observation point}

x

H_:

zad

Н

Н

T

x

T

x











(

,

₁

)

(

,

₁

)

, where





























_























зад

d

i

i

кр

lT

l

x

l

a

l

1

2

1

sin

sin

2

ln

, Providing that:

zad i

d

i

кр

lT

l

x

l









_

1

sin

sin

2

. Express value of the signal.

zad i d

i

Н

Н T

l x

l l

a l

x   

    

  

         







1 1

2

1 exp sin sin

2 ) , (

The reaction of a nonlinear element



(



(

x

,

t

))

will be the total value of impulse actions generated at the extreme points and, which can be represented as the Green function [5, 6-12]:





                       



  l n x l n t l na l t x G n sin sin exp 2 ) , , , ( 1 2 .

For the observation point

x

H_{, which is the}

middle of the segment, with a symmetrical arrangement of sources, we can write:

)

,

,

,

(

2

))

,

(

(









₁



x

t

G

x

t

_кр

.

The output function of a nonlinear element can be represented as the sum of two values of the delta function at the observation point

x

. We use the formula [7.8]:





1

1 1 2 sin sin exp 4 )) , ( (                        



  l n x l n t l na l t x H n . The maximum value of these effects at a point

H

x

_{will be observed at a time}

t

max_{, where}

t

max _{is the} time of entry of the maximum signal from the source to observation point, is determined by the formula.



m

t

                           _             _            _  10 0 ; 25 4 10 10 3 10 ; 25 2 10 2 10 3 ; 2 2 1 1 2 2 2 2 1 1 1 2 2 2 2 1 1 1 2 2 1 l при a l a l l l при a l a l l l при a l ;

Thus, expression of the output function of a nonlinear element at the observation point is the function value





1

1

1 max 2

max exp sin sin

4 )) , ( (                        



  l n x l n t l na l t x H n H

These expressions determine implicit dependence between input and output signals of a nonlinear element.

Slope of straight line bounding the sector in which non-linear characteristic is located is defined as

ratio of the value



(



(

x

H

,

t

max

))

_{to the value}



(

x

_Н

,



₁

)

_, i.e.

Figure-2. Angle limiting the sector of nonlinear characteristic.

In systems with distributed parameters, amplifier element can be represented as:

 

_























G

G

n

n

n

E

G

K

1

1

,

0

1 1 1 1

.

Having accepted

E

1



k

_and

n

1



1

_{, we} determine values of angular coefficients for each spatial

mode [12-19].

K

n



E

1

G

n_.

Then, choosing a real number

q

, one can construct the Popov straight line for each spatial mode

passing through the point















1

,

0

n

K

of the real axis and the point of imaginary axis of the complex plane.

Transfer function of the object in

n

mode of input exposure can be represented in the form [9-11]:









 

l



l



x

x

s

W

n n H n H n

n

_

_

_

_











exp

exp

exp

exp

)

(

,

(

n



1

,



)

,

where 2 1 2

_











_

_





_{n n}

a

s

,

x

H _{- observing point. For}

frequency analysis, consider

s



j



. When the frequency

changes



from 0 to



, function

W

n

(

j



)

_{will describe}

the hodograph for each spatial mode. To analyze absolute

stability, a modified frequency response

(

)

*



j

W

is used. It is known that when frequency



changes from

zero to infinity, the vector

))

(

Im(

))

(

Re(

)

(

*













j

W

j

W

j

W

_{will also}

NUMERICAL EXPERIMENT

Consider the control object with the following given parameters: l = 0.45 m - rod length; xН = l/2 - observation point; a = 0.0044 is the coefficient of thermal

diffusivity of the material. Let the number of heater

sections be r = 20, then ξ1=l/20=0, 0225 is the midpoint of the left extreme section, ξ20=l-ξ1=0, 4275 is the midpoint of the right extreme section [17].

0.1 0.05 0 0.05 0.1 0.15 0.2

0.1 0.05 0.05

Y X( )

I()

X R () X00.04

Figure-3. Relative position of the hodograph and the line at r=20; n=1. For the second mode, hodograph intersects the

Popov straight line; therefore, with a number of sections equal to 20, the system is not stable. Let the number of

sections be 23, then analysis of the four spatial modes shows that the system will be stable.

0.1

 0 0.1 0.2 0.3

0.2 

0.1 

0.1 0.2

Y X( )

I()

X R ()

n1

X00.055

Figure-4. Relative position of the hodograph and the Popov straight line with r=23; n=1.

CONCLUSIONS

Dependence of the stability of a nonlinear distributed system on the value of discretization step of control actions is established. The number of sampling points, from a practical point of view, can be interpreted as

REFERENCES

[1] Ilyushin Y. V., Novozhilov I. M. 2017. Analyzing of heating elements location of distributed control objects. Proceedings of 2017 20th IEEE International Conference on Soft Computing and Measurements, SCM 2017, No 7970519, pp. 138-141.

[2] Ilyushin Y., Mokeev A. 2017. Technical realization of the task of controlling the temperature field of a tunnel furnace of a conveyor type. International Journal of Applied Engineering Research. 12(8): 1500-1510.

[3] Kazanin O. I., Sidorenko A. A., Vinogradov E. A. 2018. Assessment of the influence of the first established and identification of critical steps in main roof caving. ARPN Journal of Engineering and Applied Sciences. 13(10): 3350-3354.

[4] Ilyushin Y., Pervukhin D., Afanasieva O., Klavdiev A. & Kolesnichenko S. 2014. The Methods of the Synthesis of the Nonlinear Regulators for the Distributed One-Dimension Control Objects. Modern Applied Science. 9(2): 42-61.

[5] Ivanov V. V., Sidorenko S. A., Sidorenko A. A. 2015. Justification of the method for determination the optimum performance of limestone quarry for steel and cement production. Biosciences Biotechnology Research Asia. 12(2): 1797-1803.

[6] Cherepovitsyn A. E., Ilinova A. A. 2018. Methods and tools of scenario planning in areas of natural resources management. European Research Studies Journal. 21(1): 434-446.

[7] Kukharova T. V., Pershin I. M. 2019. Conditions of Application of Distributed Systems Synthesis Methods to Multidimensional Object/ 2018 International Multi-Conference on Industrial Engineering and Modern Technologies, Far East Con 2018, No 8602749.

[8] Kukharova T. V., Utkin V. A., Boev I. V. 2018. Observation and Prediction Systems Modeling for Human Mental State. 2018 International Multi-Conference on Industrial Engineering and Modern Technologies, Far East Con. 2018. No 8602831.

[9] Martirosyan A. V. 2019. Application of Fourier series in Distributed Control Systems Simulation [Text] / A. V. Martirosyan, K. V. Martirosyan, A. B. Chernyshev. 2019 IEEE Conference of Russian

Young Researchers in Electrical and Electronic Engineering (EIConRus) (January 28-31). - St. Petersburg. pp. 609-613.

[10]Martirosyan A. V. 2016. Methods of distributed systems structured modeling [Text] / A. V. Martirosyan, K. V. Martirosyan, A. B. Chernyshev. 2016 IEEE NW Russia Young Researchers in Electrical and Electronic Engineering Conference (February 2-3). - St. Petersburg. pp. 283-289.

[11]Martirosyan A. V. 2016. Quality improvement information technology for mineral water fields controls [Text] / A. V. Martirosyan, K. V. Martirosyan. 2016 IEEE Conference on Quality Management, Transport and Information Security, Information Technologies (October 4-11). - Nalchik. pp. 147-151.

[12]Meshkov S., Sidorenko A. 2017. Numerical Simulation of Aerogasdynamics Processes in A Longwall Panel for Estimation of Spontaneous Combustion Hazards. E3S Web of Conferences, 21, paper No 01028.

[13]Schipachev A. 2018. Optimum Conditions of Turning and Surface Plastic Defomation Determination Taking into Account Technological Heredity. Journal of Physics: Conference Series. 1118(1): 012036.

[14]Samigullin G., Schipachev A., Samigullina L. 2018. Control of physical and mechanical characteristics of steel by small punch test method. Journal of Physics: Conference Series. 1118(1): 012038.

[15]Verzhbitskiy K., Samigullin G., Schipachev A. 2018. Increasing service life of chuck unit of tank during cyclic loading. Journal of Physics: Conference Series 1118(1): 012040.

[16]Shipachev A. M., Nazarova M. N. 2018. Phenomenon of low-alloy steel parametrization transformation at cyclic loading in low-cyclic area. IOP Conference Series: Earth and Environmental Science. 87(9): 092017.

[18]Lenkovets O. M., Kirsanova N. Y. 2014. Investments in human capital as a contributing factor to Russia's better competitiveness. International Multidisciplinary Scientific Conference on Social Sciences and Arts SGEM 2014, www.sgemsocial.org, SGEM 2014 Conference Proceedings, ISBN 978-619-7105-28-5/ ISSN 2367-5659, September 1-9, 2014, Book 2, 4: 413-420/.

[19]Nikulin A., Ikonnikov D., Nikulina A., Dolzhikov I. 2018. OSH challenges for oil and gas companies in the Arctic Zone of the Russian Federation. Delta Fire protection & safety Scientific Journal. 12(2): 46-55.