Application of the theory of systems with distributed parameters for mineral complex facilities management

APPLICATION OF THE THEORY OF SYSTEMS WITH

DISTRIBUTED PARAMETERS FOR MINERAL

COMPLEX FACILITIES MANAGEMENT

Ilyushin Yury V., Pervukhin Dmitry A. and Afanaseva Olga V.

Department of Systems Analysis and Management Information Security, Saint-Petersburg Mining University, Russia E-Mail: [email protected]

ABSTRACT

Practically all real control objects are characterized by a certain spatial extent. In this regard, the controlled values of such objects depend not only on time, but also on the distribution over the spatial domain occupied by the object. The changes of controlled quantities, both in time and in space, are described by partial differential equations, integral, integro-differential equations or systems of equations of the most diverse nature. Compared to systems with lumped parameters, the class of control actions is expanding in principle. The number of control actions can include space-time controls, described by functions of several arguments - time and spatial coordinates. The structural representation of systems and the representation of distributed objects in the form of complex transfer coefficients for eigenfunctions greatly simplifies the problem of distributed systems analyzing, but can be used if there is a solution to a boundary value problem. To solve practical problems, most often, dimensional approximation of distributed systems is used, which is based on finite-dimensional representations of partial derivatives based on the method of “grids” and “direct”, using Fourier series and Taylor series. The representation of discrete control actions in the form of delta functions allows one to investigate the class of systems with distributed parameters, for which there is a fundamental solution in the form of expansion in eigenvector functions of the object operator [1,4,9].The study of the control actions discretization parameters influence on the regulation process allows the regulation of nonlinear discrete systems in the relay mode. The proposed work considers systems with distributed control objects. Three spatially distributed mathematical models are built. A closed-loop control system for the temperature field of a spatially distributed control object was synthesized. The frequency surfaces of all proposed regulators are constructed. A new method for analyzing the stability of spatially distributed control objects has been proposed.

Keywords: control, analysis, synthesis, system with distributed parameters, spatial mode, controller.

INTRODUCTION

In the 21st century, more and more stringent requirements are imposed on industry for the quality of its products, reliability, durability, etc. These requirements force the manufacturer to optimize production processes, choose better, environmentally friendly, raw materials, switch from manual labor to high-tech robotic production, etc. technological solutions: machine tools with digital program control, automated control systems, automated conveyor systems and much more - all that allows you to automate processes.

Management theory began to actively develop since the 50s of the 20th century and during its existence has changed and transformed several times. First, simple automated control systems were created - systems involving human beings, then the time came for automatic relay systems. With the advent of computers, the first digital control systems were created [1-8,10-15].

Early generation control systems provided for the regulation of parameters in fairly wide ranges of variation. However, with the advent of composite and high-tech materials, their qualitative characteristics: accuracy, dynamic, etc., did not sufficiently satisfy the increased requirements of the complex technological control processes. There was an urgent need to ensure, for example, the accuracy of parameters regulation to the hundredth and even thousandths. The mathematical apparatus of control systems with lumped parameters that

existed at that time could not cope with the tasks set. Then in different parts of the world scientific schools began to be created, focused on solving these problems, the theory of systems with distributed parameters began to develop.

The theory of systems with distributed parameters began to develop from the first papers published by A.G. Butkovsky, J.-L. Lyons, S.G. Tzafistasa, V. Wertz, P. Demise, I.S. Meditch. Then the development of the theory was continued in the works of J.S Gibsona, I.G. Rosena, T.K. Sirazitdinova, E.Ya. Rapoport, I.M. Pershina, V.A. Koval, and other authors. At the department in 1960, this work was headed by N.A. Kauzov, and in 1965 under the leadership of Ph.D., associate professor Krizmer L.P. and the future Nobel laureate of the D.Sc. L.V. Kantorovich published the first works on this scientific direction [1,3,4,7,9,11,16,17,21,24].

The description complexity of systems with distributed parameters caused by need to take into account a large number of input effects, located on a spatially distributed object. At the same time, the space-time characteristics require preliminary decomposition in the eigenvector functions of the object operator.

Formulation of the problem

the thermal process. Let there be a control object described by the following mathematical model:

2 2 2

2

2 2 2

( , , , )

( , , , )

( , , , )

( , , , )

( , , , )

дQ x y z t

д Q x y z t

д Q x y z t

д Q x y z t

a

f x y z t

дt

дx

дy

дz







_





_







; (1)

)

,

,

(

)

0

,

,

,

(

x

y

z

u

₀

x

y

z

Q



; (2)

1

(0, , , )

( , , )

Q

y z t



q y z t

;

Q l y z t

( , , , )

₁



q y z t

₂

( , , )

; 3

( , 0, , )

( , , )

Q x

z t



q y z t

;

2 4

( , , , )

( , , )

Q x l z t



q x z t

;

Q x y

( , , 0, )

t



q x y t

₅

( , , )

;

3 6

( , , , )

( , , )

Q x y l t



q x y t

;

1

0

 

x

l

;

0

 

y

l

₂;

0

 

z

l

₃;

t



0

;

a



0

, where f(x,y,z,t) - control temperature field; a2 - given coefficient of thermal diffusivity of the control object material; l1, l2, l3- spatial characteristics of the control

object; u0(x,y,z) - control action, t - time.

This mathematical model can be used as a basis for calculating thermal processes that occurs in steel furnaces, drying furnaces, induction furnaces, furnaces for melting refractory materials, and others.

Methods

The main forms of the distributed objects (systems) representation are: representation in partial differential equations, as transfer functions, as temporal characteristics, as frequency characteristics.

A distinctive feature of systems with distributed parameters is the presence of spatial components in the input and output.

As is well known, in concentrated systems, the impulse transition function characterizes the response of the system to a single ideal impulse δ(t), transient response characterizes the response of the system to a single step function 1(t), and the complex transfer function is the response of the system to the harmonic input. In distributed systems, it is necessary to add a spatial form to the temporary input impacts discussed above [1,3,4,7,9-16,17,21-24].

When forming the spatial form of the “standard” input, two approaches can be distinguished. Determination of the system response to the input signal, presented as a combination of delta functions in the spatial and temporal regions:

 

₁

,





₁



₀

  





₀



x

t



x





t



, where

1

1

D

x



,

2 0



D



; x1,0 – set points of space D1,

D2; τ, t – temporary independent variables.

The response of the object to the input effect ω(x, t) represented as a Green function G(x, t, , τ) or pulse transition function.

The definition of the object’s response to the eigenvector functions of the object operator. In this case,

the distributed object (system) can be structurally represented as an infinite set of independent conditionally concentrated contours. The transfer function of each conditionally concentrated circuit can be represented as a ratio of analytic entire functions.

If the eigenvector functions are presented on the basis of orthogonal expansions in Fourier series in spatial coordinates, the class of spatial-invariant objects and systems can be distinguished, for which a method of synthesis of distributed controllers has been developed. Consider distributed objects whose mathematical models allow decomposition in eigenvector functions of the object operator.

Using such a decomposition, the transfer function of a distributed object can be represented by a set of transfer functions in spatial modes.

Suppose that there is a plate of finite size, in which the process of heat propagation takes place. The mathematical model of the object is described by equation (1).

Decompose the input action

f



x

,

y

,





in a Fourier row. Considering the boundary conditions (2), the input effect can be represented as:



x

y



C

 



x

 

y



f















  

 









_

_

_sin

_sin

~

,

,

1 ,

, , (3)

where

L

х







_





;

L

у







~

_





_.

Find the response of the object to each component of the series (3). We will look for this reaction in the form:



x

y

z



H

 

z



x

 

y



f

_,

,

,

,





_,

,





sin



_





sin



~

_



. (4) Substituting (4) into equation (2) and transforming, we arrive to the following result:

 

 

,



~



 

,

0

,

2 , 2 2 2 ,

,















z

z

H

z

H

a

z

H











_

  

 





 

_{ }

0

,

~

,

, 2 2 2 , 2

















_

_









H

z

s

a

s

z

s

z

H

     

_

_

,





,





1

,





, (6)

where

H

,

 

z

,

s

- Laplace function image

 



 _,

z

,

H

with zero initial conditions





,





1

,





. The solution of equation (6) can be represented as:

 

z s D



z



D



z



H, ,  1,_,_ exp_,_   2,_,_ exp_,_  ,





,





1

,





, (7)

where 2

1 2 2 ,

~













_

_



_{ }  







a

s

,





,





1

,





.

From the boundary conditions (2) we find:

  

, 2, , ,

1

D

D



, (8)

 



z

L





z

L



s

C

D



























_

_

,

,

,

,

,

2

exp

exp

,





,





1

,





, (9)

where

С



_,



 

s

- Laplace function image

C



_,



 



with zero initial conditions





,





1

,





.

Considering together (7), (8), (9), taking into account (4), we get:





















C

 

s



x

 

y



z

z

z

z

s

z

y

x

T

L L





























_{  }          













_~

sin

sin

exp

exp

exp

exp

,

,

,

_, , , , , , ,

where

T

_,



x

,

y

,

z

,

s



- transformed by Laplace function









_,

x

,

y

,

z

,

T

.

The transfer function of the object



,







,





1

,





input exposure mode is:

 

_{ }

_

_{ }

_

              y x s C s z z y x T s W       

 _sin

_

_sin

_

~

, , , , , , , 0



zL





zL



z z                             









, , , , exp exp exp exp ,





,





1

,





.

Thus, the considered distributed object can be represented as a set of transfer functions.

W

_0,,

 

s





,





1

,





.

We synthesize a number of regulators for this system.

Numerical study

Spatially distributed P-controller

Construct the logarithmic amplitude-phase frequency characteristics in the program MathCAD. Define the parameters E1, n1. To do this, we solve the

system of equations:

1 1 1 1 1 1 4 4 1 1 1

( ) [ ] 25.542;

1

( ) [ ] 42.933,

П

П

n G

K G E

n

n G

K G E

n    _{  }    _{ }  _{  }  

where КП(G1), КП(G4)- gain factors. Divide the 1st

equation of the system, on the 2nd, then we get the equation to find n1:

1 1

1

1 ₁

1 4 1

1

1

1

[

]

1 19, 739

0,595;

1

1

493, 48

[

]

n

G

E

n

n

n

G

n

E

n

 



_







_{ }



 



 

 



_











55,22

293.621

+

0.595

-n

0.595

=

18.739

+

n

1



1



.

Then we transform the system of equations (1) to the form [1-17,21]:

1 1 1 1 1

4 1 1 1 4

(

)

1

;

(

)

1

.

K G

n

E n

G

K G

n

E n

G

 

  





_{ }

_{  }



Subtracting the 2nd equation from the 1st equation, we obtain the equality to determine the parameter E1:

)

(

))

(

)

(

Substituting the values, we get:

131

.

14

1



E

.

Then the transfer function of the regulator will take the form:

)

018

.

0

982

.

0

(

131

.

14

)

,

,

(

x

y

z









2

W

.

Figure-1. Phase surfaces. Spatially distributed PI controller

Construct the logarithmic amplitude-phase frequency characteristics in the program MathCAD. Define the parameters E1, n1. To do this, we solve the

system of equations:

1 1

1 1

1

1 4

4 1

1

1

(

)

[

] 15.431;

1

(

)

[

]

37.212,

ПИ

ПИ

n

G

K

G

E

n

n

G

K

G

E

n

 



_

_

_





_{ }



_

_

_



where КП(G1), КП(G4) - gain factors. Divide the 1 st

equation of the system, on the 2nd, then we get the equation to find n1:

415 , 0 48 , 493 1

739 , 19 1 ]

1 [

] 1 [

1 1

1 4 1

1

1 1 1

1

 

       

 

   

 

  

  

n n

n G n

E

n G n

E

;

1 1

1 1

n +18.739=0.415 n -0.415+204.794; 0,585 n 320,952;

n 548, 636.   



Then we transform the system of equations (1) to the form:

1 1 1 1 1

4 1 1 1 4

(

)

1

;

(

)

1

.

K G

n

E

n

G

K G

n

E

n

G







 





_

_

_

_{ }



We divide the 1st equation, systems (2), on the 2nd, then we get the equation to find n1:

)

(

))

(

)

(

(

K

G

₁



K

G

₄



n

₁



E

₁



G

₁



G

₄ . Thence:

) (

)) ( ) ( (

4 1

1 4 1

1

G G

n G K G K E



 

 .

Substituting the values, we get:

274

.

3

1



E

.

To find n4 (weighting factor), we solve the

following system of equations:

1 1

1 1

4

4 4

4 4

4

1 ;

1 .

n G E

n n G E

n





   _{ }



 _{ }

 _{ } 

To determine the value n4 divide the 1st equation

by the 2nd, then we get:

4 4 4

4

4 1 1

1

4 ₁

1

n G n

E

n G n

E n

  

  

For determining E4 we solve the following

equation:

)

(

]

)

[(

1 4

4 1 4 4

G

G

n

E













.

Then the transfer function of the regulator will take the form:

s z

y x

W( , , )_3.274(0.957_0.043_2)_0.001837_(0.995_0.004992_2)_1

.

Figure-2. Phase surfaces.

Figure-3. Phase surfaces of the constructed regulator. Spatially Distributed PID Controller

Construct the logarithmic amplitude-phase frequency characteristics in the program MathCAD.

Parameters E1, n1 for the proportional component,

we take on the assumption that the phase shift introduced into the system by the PID controller is zero.

To determine the parameters E2, E4, n2, n4 it is

necessary to solve the following system of equations:

4 1 2 1

1 4 2

1 2

4 4 2 4

4 4 2

4 2

1 1

lg( ( ) 0.5lg [ ] 0.5lg [ ];

1 1

lg( ( ) 0.5lg [ ] 0.5lg [ ].

n G n G

G E E

n n

n G n G

G E E

n n





   

 _{   }



 _{   }

 _{   }



Subtracting from the 2nd equation of the 1st we get:

)

1

1

(

lg

]

1

1

(

lg

)

lg(

1 2

4 2

1 4

4 4

2

G

n

G

n

G

n

G

n



























.

From the dynamic characteristics we define:

89

.

2

))

(

(

))

(

(

2 1

2 4

2







G

G







.

Because Δω2>0,then n2 will be taken equal ∞.

Substituting n2 into the equation and transforming, we get

the equations to determine n4:

917

.

231

1

)

1

(

2

1 2 4 2

4

_

_























G

G

n

.

We introduce a parameter Δ (G1), the choice of

compensation of parametric disturbances of the control object:

)]

(

[

lg

5

.

0

)]

(

[

lg

5

.

0

))

(

lg(



G

₁





K

₄

G

₁





K

₂

G

₁ . To do this, we will add an equation to this equation, connecting the parameters K4(G1) and R2(G1)

with parameter Δ (G1). The equation of connection can be

represented as:

2 1 1

)

lg

lg

(











G

;

)]

(

lg[

)

(

1

lg

lg

_{4 1}

1 2

1

K

G

G

K







;

4 1 2 1

1 1 4 2

4 2

4 4

1 4

2 1 ₄

2 2

1 1

lg( ( )) 0.5lg [ [ ]] 0.5lg [ [ ]];

1 1

( ) lg [ ] lg [ [ ]].

1

[ ]

n G n G

G E E

n n

n G

G E

n G n

E n

    

 _{     }

 

 

     

 _{ }

 



Solving the resulting system of equations:

2

4

5.623;

0.052.

E

E







_



Thus, the transfer function of the synthesized regulator will take the following form:

s

s

z

y

x

W

(

,

,

)



14

,

131

(

0

,

982



1

,

018





2

)



0

.

052



(

0

,

996



0

,

0044312





2

)



1



5

,

623



.

Figure-4. Phase surfaces of the constructed PID controller Spatially distributed precision controller

It is known that the HTC installation (high-precision temperature controller) created in the industry made it possible to solve a number of temperature control problems at a given point. The regulator in this setup

consists of amplifying, differentiating and integrating links. From the similar distributed links we will form structure of the distributed high-precision regulator (DHR) [1-3,17-24].

The transfer function PBP has the form:



,

,



1

1

1

1

1

1

1

2

.

2 2 2 2 2

4 4 4 4 2 1 1 1

1

s

n

n

n

E

s

n

n

n

E

n

n

n

E

s

y

x

W

_















 





















 



























 

,

1

1

1

1

1

1

1

,

0

.

2

2 2 2 4

4 4 4 1

1 1

1





















 



















 

























G

s

G

n

n

n

E

s

G

n

n

n

E

G

n

n

n

E

s

G

W

For frequency analysis DHR we put in the above equation S=j and define the module (М) and phase ()

function W(G, j):



,



 

 



 



,

2 1 2 1 2 4 2 2

    

  

    



  

 K G K G K G

G M

 

 (10)



,



 

_{ }

 

,

1 4 2 2

























G

K

G

K

G

K

arctg

G









(11)

where



1

,

2

,

4



.

,

1

1

)

(

_



















G

i

n

n

n

E

G

K

i i i i i

As follows from (11), the minimum value of the module will be at:

.

0

)

(

)

(

2 ₄

2

G





K

G



K



(12)

Convert (12) to the form:

2 4

lg

K G

( ) 2lg







lg

K G

( )

Or



lg

(

)

lg

(

)



.

5

,

0

lg







K

4

G



K

2

G

(13)

The minimum value of the module will be

Mmin(G)=K1(G).

Equation (13) defines the inflection line  (G). Considering the solution of the equation (13). Let be

K2(G)=1, then (13) converted to lg ω=0.5*lgK4 (G).

Assuming that K4(G)=1, then (13) converted to:

lg ω=–0,5*lgK2(G).

Converting (13) in the form:

. 1 1 lg 5 , 0

1 1 lg 5 , 0 lg 5 , 0 lg 5 , 0 lg

2 2 2

4 4 4 2

4

   

   _ 



    

 

       

G n n n

G n n n E

E

 ₍₁₄₎

Figure-5. Frequency surfaces DHR. If

n

₄



n

₂

, then (14) is converted to the form

2 4

0

,

5

lg

lg

5

,

0

lg







E





E

. (15) Function graphs lg ω in this case will be presented in the form of straight lines parallel to the axis

G. From (15) it follows that the value of the function

(G), satisfying the equation does not change if K4(G) and

K2(G) multiply by the same constant number



0



~

~

_

K

K

.

Let’s determine how the amplitude characteristics changes. [1-21,24].

Assuming that

K

~



1, then from (10): for





0

will get

M



G

,







K

4

 

G

_

, for







M

(

G

,



)



K

₂

(

G

)





.

Let be

K

~



1, then

for





0

will get





 





K

K

G

G

M

4

~

for







M

(

G

,



)



K

~



K

₂

(

G

)





. Denote:

 



_











1 2 1

lg

1

lg

G

K



,

lg



₂



lg



K

₄

 

G

₁



,

 

₁



lg



₁



lg



₂



G

. (16) Value Δ(G1) related to



as follows:

 

G

₁



2



lg

K

₁

(

G

₁

)







. (17)

if

K

~



1

, then

 





















K

G

K

~

1

lg

~

lg

1 2 1



,

lg



~

2



lg



K

4

 

G

1



K

~



,

 

₁

_

_lg

_

~

₁

_

_lg

_

~

₂



G

. (18)

Subtracting (18) from (16) and transforming, we get:

 

G

 

G 2lgK~

~ ~

1 1  

 

 .

So, by changing the value

K

~

, you can influence the nature of the change module (M(G,)) in the vicinity of the inflection line.

The method of synthesis of a distributed high-precision controller, discussed below, assumes that the values of the inflection points of the amplitude response coincide with the values of the cut-off frequency of the open-loop system. Due to parametric perturbations, it is not the exact values of the cutoff frequencies of the

open-loop system that are determined for each spatial mode, but only some areas of change of the cutoff frequencies of the modulus δv, (=1, 2, …). To account the area δv in the

method of synthesis parameter  is used, which is associated with the parameter



by ratio (17). To select a value



ratio can be used

3

max





, where δmax -

maximum value of the sequence δv, (=1, 2, …).

Let’s check the stability of the resulting system. In [1] - [8] the concept of a generalized coordinate (G) is introduced, the discrete values of which (Gη, γ) determined by the ratio:

,

~

2 2

,  











G





,





1

,





.

The transfer function of the object using the generalized coordinate is written as:







_



_



_



_

) ) ( exp ) ( (exp ) ( ( exp ( exp , , , , * , * , , 0 L

L G z

z G G z G z G s G W                        _{   }   _, 2 1 ,











 



G

_ηγ

a

s

β

.

The transfer function of the distributed correction device recorded using the generalized coordinate (G), has form:

.

1

1

1

1

1

1

1

)

,

(

, 2 2 2 2 , 4 4 4 4 , 1 1 1 1 ,

s

G

n

n

n

E

s

G

n

n

n

E

G

n

n

n

E

s

G

W









































































       

Reaction of the system to a pulse having an area (i·t), determined by the following relationship:

),

(

)

(

)

(

С

вых,,





С

и,,

i



t



W

фп,,





i



t

where W_фп,η,γ(τ-i·t) – weight function of the reduced

continuous part according to the chosen spatial mode (η,γ). Summing up all the reactions to the pulse, we obtain the

output value for the chosen spatial mode:



 











0 , , , , ,

вых,

(

)

(

)

(

),

С

i

фп

и

i

t

W

i

t

С





_{ }

 

lt≤ τ≤ (l+1)t.

The discrete value of the output signal for the selected spatial mode is written as (τ=lt):

вых, , , , , ,

0

С

( )

и

( )

фп

((

) ).

i

С

i t W

l i t

 



   









 

 

*

вых, ,

вых, , в, , , ,

0 0 0

С

( )

(

)

(

)

_фп

((

)

) exp(

).

l l i

s

С

l t

С

l t W

l i t

slt

             







 



 

  



Putting k=l-I and considering W_фп,η,γ(τ)=0, at τ<0, the discrete Laplace transform for the selected spatial mode is:

*

вых, , , , , ,

0 0

С

( ) (

фп

(

) exp(

)) (

B

( ) exp(

)).

k i

s

W

k t

skt

C

i t

sit

     

 

 





 







 



The transfer function of the open-loop system for the selected spatial (W *p,η,γ) mode is written as:

* *

p, , вых, , , ,

0

, , 0

W (s) С ( ) / ( ) exp( )

( ) exp( ).

B i

фп k

s C i t sit

W k t skt

               _    _      





For the Fourier image, using the spectra of continuous and discrete values, we get:

)

)

(

(

1

)

(j

W

*p, ,





_{, ,}





 

r фпηγ

γ

η

W

j

ω

r

t

ω

.

The amplitude and phase characteristics of the considered systems are determined in the range

–ωи/2 ≤ω≤ ωи/2.

For the considered example, the transfer function of the continuous part can be represented as:





















1 4

, 1 , 4 ,

1 1 4 4

* *

, ,

2

2 ,

2 2 , , ,

1 2

,

1

1

1

1

1

,

(

exp

(

exp

(

1

1

)

,

(

) (exp

(

)

exp

(

)

)

.

H

L L

n

n

W

G

s

E

G

E

G

n

n

n

n

s

G

z

G

z

n

E

G

s

n

n

G

G

z

G

z

s

G

a

                   





 























_





_





_





_

 



























_





_

 























_



_





The function, describing the shape of a rectangular pulse, has the form: Wf (s)=(1-exp(-st))/s .

The transfer function of an open-pulse system can be written as:





















,

),

)

(

exp

)

(

(exp

)

(

)

(

exp

)

(

exp

)

1

1

1

1

1

1

1

(

(

1

s

1

exp(-st))

1

(

,

2 1 , , , , * , * , , 2 2 2 2 , 4 4 4 4 , 1 1 1 1 , *















_











































































































                      

















G

a

jr

s

z

G

z

G

G

z

G

z

G

s

G

n

n

n

E

s

G

n

n

n

E

G

n

n

n

E

t

s

G

W

и L L r r Р



η

,

γ



1

,





.

The transfer function of an open-pulse system for each spatial mode can be represented as a ratio of transcendental functions.

The transfer function of a pulsed distributed system

η

,

γ

control loop allows equivalent conversions.

A closed distributed system with a scalar input function can be structurally represented as an infinite set of independent circuits.

The transfer function of a closed system in a selected spatial mode can be represented as:

 



_



_

s

s

s

γ η γ η γ η

,

G

W

1

,

G

W

W

, * , * , * *





.

Let the transfer function

η

,

γ



η

,

γ



1

,





control loop, has the form:

 

)

(

)

(

W

, , , * *

s

М

s

П

s

γ η γ η γ

 

a

_{0, ,}

exp(

)

a

_{1, ,}

exp(

2

)

a

_2,,

exp(

3

)

a

_{3, ,}

exp(

4

)

...

,

s





st





st





st





st



M

_{ηγ ηγ ηγ ηγ ηγ} ,

 

b

0, ,

exp(

)

b

1, ,

exp(

2

)

b

2,,

exp(

3

)

b

3, ,

exp(

4

)

...

,

s





st





st





st





st



П

_{ηγ ηγ ηγ ηγ ηγ} ,

where Мη,γ(s) – characteristic polynomial by η,γ spatial

mode. Assuming that exp(st)=z, characteristic polynomial can be written as:

 

a

_0,,

z

a

_1,, 2

a

_{2, ,} 3

a

_3,, 4

...

,

z









z



z





z



M

_{ηγ ηγ ηγ ηγ ηγ}

It is known, that for system stability by η,γ spatial mode, the roots of the characteristic polynomial must be inside the circle r=1.

Figure-6.Transition from an infinite set of circles to a cylinder. Working with an infinite set of circles is not

always convenient. Let us turn from a set of circles for each spatial mode to a functional dependence (G,s). For this we replace Gη,γby continuous function G with scope



G



G

1,1

...





. In this case, when changing G to G1,1,

all discrete values of the function G_η,γ will be covered, defined for any values

η

,

γ



 

1

,



.

From the set of circles we proceed to the cylinder (6). Construct a spatially distributed hodograph based on the modified Nyquist criterion for the HTC presented in Figure- 4.5.

DISCUSSIONS

The relevance of research conducted by the Department of System Analysis and Management of St. Petersburg Mining University, is determined by the complexity of the nonlinear object management systems with distributed parameters implementation. The controlled values of such systems depend not only on time, but also on the distribution over the spatial domain occupied by the object. In this regard, the class of control actions is fundamentally expanding, primarily due to the possibility of including in their number the space-time controls described by functions of several variables - time and spatial coordinates.

The features of systems with distributed parameters require the creation of an apparatus for their analysis and synthesis based on mathematical tools that are unconventional for the classical theory of control. There are various forms of systems models with distributed parameters: in the form of partial differential equations; structural representation of systems with distributed parameters, which relies on the fundamental solution of the boundary value problem; representation of distributed objects as complex transfer coefficients by eigenvector functions of the object operator.

For the analysis of control objects described by nonlinear partial differential equations, approximation methods are most often used. However, it should be noted that today, the method of distributed systems approximation using a specially selected concentrated system has not been developed. At the same time, in many problems, the approximation process is unstable with respect to errors of intermediate calculations. The development of models of the systems and synthesis methods under consideration has recently been dealt with by a large number of authors, in view of the absolute relevance and high demand for such technical solutions in practice. At the same time, many works end with the stages of system modeling, suggesting further parametric synthesis, the use of which is associated with solving a number of problems. Research conducted at the department compares favorably with the fact that they are brought to a logical conclusion - control algorithms.

The significance of the results described above for science lies in the fact that the methods of distributed systems analysis and synthesis have been further developed in research. The developed methods and the proposed procedures significantly expand the range of tasks to be solved. The proposed methods can be implemented in the form of software products that can be used for the analysis and synthesis of distributed control systems of the mineral complex objects. The results of the work can find application in various industries in solving the problems of the objects with distributed parameters control processes automation.

CONCLUSIONS

This study summarizes the work carried out by the Department of System Analysis and Management at St. Petersburg Mining University, namely: an applied theory and methods for the synthesis of distributed,

nonlinear control objects are proposed using as an example the modeling of thermal processes in metals. The application of these methods depends not only on time, but also on the distribution over the spatial area occupied by the object. In this regard, the class of control actions is fundamentally expanding, primarily due to the possibility of including in their number space-time controls, described by functions of several variables - time and spatial coordinates[1-11,16-24].

All of the above advantages of this approach make it possible to build systems for managing and regulating objects of the mineral resource complex, in which tasks are solved in a comprehensive manner, taking into account the space-time controls occurring in the object under consideration. The effectiveness of regulation is provided by dynamic characteristics and the system response to external disturbances.

The application of these methods and control systems has been tested on conveyor-type tunnel kilns (PPP - confectionery, S-type tunnel kiln for porcelain, and furnaces for rolling and hardening the tubing of oil-producing wells).

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