Regularization Algorithms of Synthesis Actuation Devices in Control Systems of Polynomial Objects

International Journal of Emerging Technology and Advanced Engineering

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457

Regularization Algorithms of Synthesis Actuation Devices in

Control Systems of Polynomial Objects

I. Yu. Abdurakhmanov

1

, O. O. Zaripov

2

, H. Z. Igamberdiyev

3

1,2,3

Department of Electronic and Automatic, Tashkent State Technical University, University st.2, 100095, Tashkent city, Republic of Uzbekistan

Abstract – Questions of construction the regularization algorithms of synthesis actuation devices in the control systems of polynomial objects considered described by multidimensional functional ranks of Voltaire. On the basis of methods of regularization of the solution of the operator equations with positively certain matrixes and approximate the set right part of the given regular algorithms of search of pulse characteristics of the actuation device. On the basis of numerous model examples is shown high efficiency of the given synthesis algorithms of actuation devices in the considered class of systems.

Keywords – Polynominal system, actuation device, regularization, regularization parameter.

I. INTRODUCTION

The theory of optimum stochastic systems reached now very high level of development. Especially it belongs to the theory of optimum linear stochastic systems. The aspiration to increase of efficiency of systems which is observed in various areas of equipment, brings inevitably to that nonlinearity should consider real in system, or to enter into it additional nonlinear elements. Thus the studied system gains qualitatively new properties which studying by methods of the theory of linear stochastic systems often is impossible. Therefore there is a need for development of the special methods adapted for optimization of nonlinear systems.

Quite large number of various methods [1-7] useful and the results deserving attention connected with the solution of many tasks arising by optimization of nonlinear systems is so far received. Methods solutions of problems of optimization of nonlinear stochastic systems on the basis of polynomial representation of models of the operated objects are proposed [1,2,8-10]. The wide circulation of polynomial models for the description of processes of the most various nature caused emergence and formation of the theory of polynomial systems [1,2,6,8,9]. The major achievement of the specified theory consists in development for polynomial systems of the mathematical description like "entrance exit" by means of functional ranks of Voltaire.

Direct functional link between an entrance and an exit at known kernels of Voltaire provides rather simple analytical solution of a problem of definition of the movement of nonlinear object under the influence of any entrance signal.

Thus practical realization of the developed methods of identification and optimization of nonlinear stochastic systems faces need of their consideration from positions of the return problems of dynamics of the operated objects. It is connected by that tasks of this kind, in essence, are badly caused. They belong to a class incorrectly of objectives [11-16]. In such situation it is expedient to consider a problem of synthesis of methods and algorithms of identification and optimization of polynomial systems from the point of view of the theory of regular estimation defining methodology of creation of steady synthesis algorithms of the specified class of dynamic systems.

II. PROBLEM DEFINITION

The solution of a problem of synthesis of polynomial control systems, as we know, is consolidated to determination of the parameters of the actuation device of the chosen structure providing the best value of criterion of an optimality when performing certain restrictions [1,2].

We will write down the equation of the actuation device in a look:

,

...

)

(

)...

(

)

,...,

(

...

)

(

1 1

1 2

1 ,...,

1 ,..., 0

1 1

1

v v

N

v

d

d

t

y

t

y

g

t

u

v v

v













 

    











   

 



(1)

Where









0 2

2

(

t

)

q

(



)

f

(

t



)

d



y

.

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458

,

...

...

)

(

....

)

(

)

(

)...

(

)

,...,

,

,...,

(

...

)

(

1 1 2 1 2 1 2 1 2 1 1 1 , 0 , 1 1 2 1 , 1 2 0 1 2 1 m i m i s s m i m i m i

d

d

d

d

t

f

t

f

t

f

t

f

a

t

y

























     



















 

(2) Where

).

,...,

(

)

,...,

(

)

,...,

,

,...,

(

1 1 2 1 1 2 1 1 2 1 , 1 2 m m i i m i m i

b

k

a

















   





The final number of members of the corresponding functional ranks enters expressions (1) and (2). From (1) and (2) follows that the problem of optimization can be presented in the form

min

]

,...,

0

;

,...,

0

;

,...,

0

|

)

,...,

(

);

,...,

,

,...,

(

[

2 1 1 ,..., 1 1 2 1 , 1 2 1









 

N

v

s

m

s

i

g

a

Q

v m i m i

v















 (3)

Under conditions

.

,...,

1

,

]

,...,

0

;

,...,

0

;

,...,

0

|

)

,...,

(

);

,...,

,

,...,

(

[

;

,...,

1

,

]

,...,

0

;

,...,

0

;

,...,

0

|

)

,...,

(

);

,...,

,

,...,

(

[

0 2 1 1 ,..., 1 1 2 1 , 1 2 0 2 1 1 ,..., 1 1 2 1 , 1 2 1 1



































       























   

N

v

s

m

s

i

g

a

c

N

v

s

m

s

i

g

a

C

v m i m i v m i m i v v (4)

The problem of synthesis is usually solved by means of two-stage procedure [1]. The first stage of this procedure consists that the task (3), (4) is solved for the opened system (2) in the beginning. Thus such set of pulse characteristics

a

_2i1,m

(



₁

,...,



_2i1

,



₁

,...,



_m

)

in (2) is defined, at which

.

,...,

1

,

,

,...,

1

,

min,

0 0











   













c

C

Q

(5)

Set of the pulse characteristics

)

,...,

,

,...,

(

1 2 1 1

, 1

2i m i m

a

_





_





which are the solution of a task (5) describes the optimum opened system.

The second stage consists in that, using communication between characteristics of the opened and closed systems, to determine parameters of the actuation device, i.e. its pulse characteristics



v

N



g

_v

v

(

1

,...,

)

0

,...,

,...,

1 







 .

Following [1,2] it is possible to show that look ratios are fair:

),

,

(

)

,

,

,

,

(

)

,

(

...

1 2 , 1 2 1 2 1 2 1 2 , , 1 2 , 1 2 , 0 , 1 2 0 , 1 2 0 2 1 l q l q m i m l i q m l i q s s m i m i m i

R

d

d

K

a























          









 

(6) At

.

,...,

0

,

,

,...

0

,

0

,

0

2 1 1 2

s

l

s

q

m q















The exact decision of this system of the linear integrated equations of rather pulse characteristics

a

₂0_i1,m

(



_2i1

,



_m

)

at any moment functions

K

_{2q1,2i1,l,m}

 



and

R

_2q1,l

(



)

is difficult. One of ways of the approximate decision of this system consists in that to replace all continuous functions entering the equations (6) step [1].

We will present:

,

,

,

,

2 1 2 1

1 2 1 2

T

v

T

r

T

p

T

n

l l m m q q i i









_{  } 









(7) Where

.

,...,

1

,...;

2

,

1

,

0

);

,...,

(

;

1

2

,..,

1

,....;

2

,

1

,

0

);

,...,

(

;

,...,

1

,...;

2

,

1

,

0

);

,...,

(

;

1

2

,...,

1

,...;

2

,

1

,

0

);

,...,

(

1 1 2 1 1 2 1 1 2 1 1 2

l

j

v

q

j

p

p

p

p

m

j

r

r

r

r

i

j

n

n

n

n

j l l j q q j m m j i i





























   







(8)

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459

,

)

,...

;

,...,

(

)

,

(

, 1

1 2 1 0

, 1 2

1 2 0

, 1 2

1 2i rm

n m i

m i

m i m i

a

T

r

T

r

T

n

T

n

a

a









 









),

(

)

,

,...,

;

,...,

,

,...,

;

,...,

(

)

,

,

,

(

, , 1 2 , 1 2

1 1

1 2 1 1 2 1

1 2 1 2 , , 1 2 , 1 2



















m l r

v i q

m l

i q

m l i q m l i q

K

T

r

T

r

T

T

T

n

T

n

T

p

T

p

K

K

 

 

  











(9)

,

)

,...,

;

,...,

(

)

,

(

, 1 2

1 1 2 1

, 1 2

1 2 , 1 2

l

v q p

l q

l q

l q l q

R

T

T

T

p

T

p

R

R

  

 

















Where

n

2_i1

, p

2_q1

, r

_m

,v

_l are determined by ratios (8).

Taking into account (9) each equation in (6) will turn into the algebraic

,

,...,

0

;

,...,

0

,

,

0

,

)

(

2 1

1 2

, ,

0

, , 0

, , ,

, _{2 1}

2 1

1 2

1 2 1 2 1 2

s

l

s

q

L

v

p

R

K

a

l q

v p s

s

m i

L

r n

r v n p r

n _{q l}

m i

m l i q m i













   

  

 



(10)

Where size L defines maximum of values

p

2q1

,

or

v

l,

at which

R

_2q1,l

(



_2q1

,



_l

)

or

K

_{2q1,2i1,l,m}

(



)



0

. Thus, the system (10) consists of the linear algebraic equations of various orders. The system of the equations (10) can be badly caused, i.e. big changes of the decision can answer small changes of basic data. Noted circumstance at the solution of the equation (10) results in need of application of methods of regularization.

III. SOLUTION OF THE TASK

For regularization of the solution of the equation (10) we will use regular methods of the solution incorrectly of objectives. For brevity we will write down the equation (10) in a look:

,...

2

,

1

,





R

j

a

K

_{j j} _j , (11)

Where

K

_j – the linear operator acting from Gilbert space of H in H;

R

_j – a vector of the right part with a condition of approximation of a look

R

_j



R

_j





, j

characterizes an order of the pulse characteristic of the actuation device.

In view of that the matrix operator

K

_j in the considered task is symmetric and positively certain, we will determine family of operators

R

_ by a formula

)

,

(









K

j

R

, (12)

Where



(

K

_j

,



)

– operator function,





0

– regularization parameter.

Following [14] it is possible to show that the family of operators

R

_ determined by a formula (12) generates limited approximation for a task (11) i.e. that the ratio was executed

lim

_,1

0

0





 

K

j

a

j

a

j



R

,

a

j,1



ker

K

j – it

is necessary and enough that look conditions were satisfied

0

,

|

)

,

(

|

sup

}

{E_{ }









K











vrai

,

|)

|

1

(

|

)

,

(

|











c





,

0

1

)

,

(

lim

0















 ,

where

a

_j,1 – any solution of the equation (11),

E

_ – spectral family of the operator

K

_j,





s

(

K

_j

)

– the operator's range,

c

– some constant which isn't depending from



.

In the case under consideration it is possible to use approximation of a look

1

)

(









K

j



R

, (13)

Which is called M.M.Lavrentyev's approximation. Size



K

(13) for this approximation is equal

K

_



1

/



, if

)

(

0



s

K

j . Thus, that Lavrentyev's approximations

generated regularization algorithm for a task (11) of a formula

R

_



R

_(), it is necessary and to choose enough

)

(



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460

0

)

(

lim

0











 .

On the basis of a look inequality



 

  



 



K





























,

, , ,

,

, ,

,

~

~

~

j j

j j j

j j j

j j j

j

a

a

a

a

a

a

a

a

a

a

a

a

(14)

it is possible to show [14] that if

lim

0

0



 





K

,

0

~

lim

_,

0







a

j

a

j

 . In (14)

a

j is determined by a

formula







0



  j j

R

dE

a

, and

a

_j, and

~

a

_j, on the basis

of expressions

a

_j,_





(

K

_j

,



)

R

_j and







j



j

j

K

R

a

~

_,



(

,

)

~

.

At estimation of pulse characteristics of the actuation device of high orders on the basis of (10) dimension of system of the equations strongly increases. In such cases for providing significantly expediently following the best conditionality of systems of high dimension [16] to consider system of a look:

 

 j



j

j

a

R

K

, ,



, (15)

Where

)]

)(

1

(

)

[(

1

, K K j K

j

G

G

K

G

K

_

















,

K

G

– diagonal of a matrix

K

_j,

2 , ,

~

)

~

,

(

E j j j

j

j

K

a

K

a

R

 _{ }











,



₂



1/2 2

/ 1 2 2

1

1

E K j E

K E K

K

G

G

K

G

G















_

_





_





,



,

~

j

a

– decision of system

K

_j,_

a

~

_j,_



R

_j .

Determining a vector

a

~

~

_j by a ratio

2 min 2 ,

~

~

_





_

_

E j j j

K

a

R

it is possible

a

~

~

_j to find a vector

j

a

ˆ

which meets the following conditions on a vector:

,

0

)

ˆ

,

ˆ

(

,

ˆ

2 2











j j j j j

E j j j

a

K

R

a

K

a

K

R

 

Where

(

2

)

/

2

max 2

min

2











, thus sizes



and



are on the basis of the chosen value of parameter



at the solution of the equation (15),

2 2

E j

R







,

2 max 2 2

min











E j

R

.

IV. CONCLUSION

The regularization synthesis algorithms of actuation devices given above in control systems of polynomial objects can be easily realized by means of modern computer technology. Thus for increase of accuracy of procedure of synthesis of the actuation device are very effective concepts and methods of regularization. On the basis of numerous model examples is shown high efficiency of the given synthesis algorithms of actuation devices in the considered class of systems.

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461

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