Regular Algorithm of Adaptive Estimation of the State of a Managed Object with Correlated Noise Object and Interference Measurements

(1)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 9, September 2014)

797 Regular Algorithm of Adaptive Estimation of the State of a

Managed Object with Correlated Noise Object and Interference

Measurements

H. Z. Igamberdiyev

1

, O. O. Zaripov

2

, A. X. Abduganiyev

3

1,2,3_{Department of Electronic and Automatic, Tashkent State Technical University} 1,2,3

Address: Universitetskaya st.2, 100095, Tashkent city, Republic of Uzbekistan

Abstract — The paper deals with the construction of the regular adaptive estimation algorithms with auto and cross-correlated noise object and interference measurements. In the synthesis of regular estimation algorithms in the case of autocorrelated noise and interference measurements of the object describes how to expand the state vector and difference measurements. As a regular procedure uses statistical form discrepancy principle. When constructing regular estimation algorithms in the case of cross-correlated noise and interference measurements of the object as a regularizing procedures used regularization methods M.M.Lavrentev and V.N.Strahov. These expressions allow for convergence of the estimates and to improve the accuracy of their evaluation.

Keywords — Adaptive filtering, correlated noise and interference measurements of the object, regularization, the regularization parameter.

I. INTRODUCTION

In solving various problems of estimation and filtering may be situations where noise and interference measurements of the object are auto-and cross-correlated. In these cases, for random processes due to the time dependence must be considered dynamic transformations, ie transformations, which are described by differential or difference equations.

II. TASK STATEMENT -I

We first consider the algorithms of state estimation of dynamic systems in the presence of autocorrelated noise object and interference measurements. We assume that the system model and measurements are of the form:

i i i i i i

i

A

x

Г

x

_₁



_₁_|



_₁_|



,



_i_₁



A

~

_i_₁_|_i



_i





~

_i

w

_i_₁, (1)

1 1 1

1   





i i



i

H

x

z



,



_i_₁



H

~

_i_₁



_i





~

_i

v

_i_₁. (2) A priori data set as follows:











,



,

~



,



,

~

,

0 ~

,

0 ~

) ( 0 0 0

0 0 0





N

P

x

N

x

R

N

v

Q

N

w

i i i i















,



cov



,



0 .

cov

,

cov

,

cov

,

cov

0

0 0



i i i

i i

w

v

w

x



Algorithms Of The Decision

Then, following the methods of the theory of optimal filtering and dynamic evaluation [1-4], we can show that the estimation algorithms are given by:



1

~

1 1 1|

~

1 |



,

1 | 1 1 | 1

i i i i i i i i i i

i i i i i

x

H

x

H

z

H

z

K

x

 



   











(3)

i i i i i i i i i

i

A

x

Г

x

_₁_|



_₁_| _|



_₁_|



_| ,



_i_₁_|_i



A

~

_i_₁_|_i



_i_|_i, (4)



₁

~

₁ ₁ ₁_|

~

₁ _|



,

) (

1 | 1 1 | 1

i i i i i i i i i i

i i i i i

x

H

x

H

z

H

z

K

 



 











_



(5)

.

~

1 /

| 1

1 |

| 1 |

) 1 (

1

 







i T i T

i i i i

i T i i i i i T i i i i

H

W

Г

H

P

A

H

P

(6)

1 | | | 1 |

) 2 (

1

~

 







i

T i i T i i i i T i T i i

i

W

H

A

W

H

P

,

i i i

i

W

x

,

)

_|

cov(





. (7)

Thus gains

K

_i_₁ and

K

_i(_₁) are determined based on the expressions:

) 1 (

1 * ) 22 (

| 1

1  

 i i



i

P

K

, (8)

) 2 (

1 * ) 22 (

| 1 ) (

1  

 i i



i

P

(2)

International Journal of Emerging Technology and Advanced Engineering

798

where









,

~

1 1 | 1 1 | 1 | | 1 | 1 1 | | 1 | | 1 1 1 | 1 1 * ) 22 ( | 1 T i i i T i T i i i i i T i T i i i i T i i i i i T i T i T i T i i i i i i i i i T i i i i i i

B

R

B

H

P

H

Г

C

A

P

H

C

Г

P

A

H

P

H

P

              













.

| 1 | | 1 | 1 | | 1 | 1 | | 1 | 1 | | 1 | 1 T i i i i i i T i i T i i i i T i i i i i i T i i i i i i i i

Г

D

Г

A

C

Г

C

A

P

A

P

        





) 1 ( 1 1 | 1 1 |

1   

i



i i



i i

i

P

K

P

, T i i i i i i T i i i i i i i

i

A

C

A

Г

D

A

C

1| 1| | 1| 1| | 1|

~

    





, T i i i T i i i i i i i

i

A

D

A

b

Q

b

D

_₁_|



~

_₁_| _|

~

_₁_|



_₁ ,

) 2 ( 1 1 | 1 1 |

1   

i



i i



i i

i

C

K

P

C

,

D

_i_₁_|_i_₁



D

_i_₁_|_i



K

_i(_₁)

P

_i_(2₁),

1 ,

,

cov

| | | |

1























































i

k

D

C

P

Z

x

i k T i k i k i k i T k k k k



.

In solving the problem under consideration in the evaluation of noise



_i model system should be assessed by the measurement information, and the noise in the measurement model

v

_i is the nuisance parameters and the evaluation can not be. Here it is advisable to use two methods [1,5]: the expansion of the state vector method and the method of difference measurements Bryson-Henriksson.

The method of expansion of the state vector, in which the state vector

x

_i is supplemented by the vector



_i. The new state vector is

x

_i*





x

_iT

,



_iT



T. Relations (1) - (2) take the form:

1 * * * | 1 *

1  





i i i



i i

i

A

x

b

w

x

, (10)

1 *

1

1   





i i



i

H

x

z



, (11) where















i i i

x



* ,















    i i i i i i i i

A

Г

A

| 1 | 1 | 1 * | 1

~

0

,















i i

b

*

0

,



1

0



*

1 





i

H

.

The method of difference measurement eliminates noise

i

v

, forming a new dimension

y

_i_₁ as a linear combination of two consecutive measurements

z

_i and

z

_i_₁:

i i i

i

z

H

z

y

1 1 1

~

 







. (12)

Using (10) - (12), we describe a model for measuring:





.

~

1 1 * * 1 * * 1 | 1 * 1 1       







i i i i i i i i i i i i

v

w

b

H

x

H

A

H

y



(13)

Priori data are given by:











,



,

0 ,

~

,

0 ~

,

0 ~

0 0 0 0 1





z

P

x

N

x

R

N

v

Q

N

w

_i _i _i _i

0 )

,

(

cov

)

,

(

cov

)

,

(

cov

w

_i

v

_j



w

_i

x

₀



v

_i

x

₀



. For the relations (10) and (13) it is advisable to use a filter-Glonti Liptser [1]. Then:



i i i



i i i i

i

x

K

y

x

1 1 1|

* | 1 * 1 |

1    









,





* | * | 1 * 1 | 1 * 1 * | 1 | 1 * | 1

~

ii

i i i i i i i i i i i i i

x

A

H

A

H

A

y

x





























       ,

 

1

| 1 | 1 1    



yy i i xy i i i

P

K

, xy i i i i i i

i

P

K

P

_₁_|_₁



_₁_|



_₁ _₁_| ,

T i i i T i i i i i i i

i

A

P

A

b

Q

b

P

_₁_|



*_₁_| _| *_₁_|



* _₁ * ,



*₁ ₁_|

~

₁ *



* ₁



*₁ *



,

| 1 | 1 T i i i i T i i i i i i i xy i

i

P

H

A

H

b

Q

H

b

P





  





 

(3)

International Journal of Emerging Technology and Advanced Engineering

799

Efficiency of estimation algorithms (3) - (7) substantially depends on the accuracy of calculating the gain matrix

K

_i_₁ and

K

_i(_₁) based on the expressions (8) and (9) [1,6]. Matrix

P

_i_(22₁_|_i)* in these expressions may be degenerate or poorly conditioned. To compute

K

_i_₁ and

) (

1





i

K

, it is expedient to use a regular procedure [7-11]. Regular algorithm solutions will be considered with respect to the equation (8). The resulting algorithm may also be used in solving equation (9).

Equation (8) can be written as follows:

) 1 (

, 1 ,

1 * ) 22 (

|

1 i j i j

T i

i

k

p

P

_ _



_ , (14)

Where

k

_i_₁_,_j- j-th column of the matrix

n

j

K

iT1

,



1 ,

2 ,...,

;



j i

d

_, - j-th row of the matrix

P

_i(11)T,

n

j



1 ,

2 ,...,

.

In equation (14) the vector elements

p

_i(_1₁)_,_j known from errors, i.e. instead of

p

_i(1_)₁_,_j has its random realization

) 1 (

, 1 )

1 (

, 1 )

1 (

,

1j i j i j

i

p

_



_





_ . To solve it, we will use the least squares method, whereby estimates of the unknown

k

ˆ

_i_₁_,_j

normal pseudo solution

k

_i*_₁_,_j



(

P

_i_(₁22_|_i)*T

)



p

_i(_1)₁_,_j system (14) are determined as a solution of

2

, 1

inf

k

_i_ _j to

}

)

(

:

{

₁_,

Q

₁_,

Q

_min

K



k

i_ j

k

i_ j



[11,12];

)

(

inf

₁_, min

, 1

j i R k

k

m j i

 





Q

here, where

2 ) 1 (

, 1 , 1 * ) 22 (

| 1 ,

1

)

(

k

_i_ _j



P

_i_ _i T

k

_i_ _j



p

_i__j

Q

,

||



||

- Euclidean norm of

(

P

_i_(22₁_|_i)*T

)

 - to

P

_i(_22₁_|_i)*T pseudo-inverse matrix.

Ratings

k

ˆ

_i_₁_,_j are the best in the class of linear unbiased estimators. However, in the case of a degenerate or ill-conditioned matrix of

P

_i(_22₁_|_i)*T they are unstable [13], ie,

2 *

, 1 2

, 1

ˆ

j i j

i

k

M

_



_ .

Taking into account that the main error of unbiased estimators

2 *

, 1 , 1

ˆ

j i j i

k

M

_



_ caused by the displacement

of a square length

k

ˆ

_i1,_j, ie value of

2 *

, 1 2

, 1

ˆ

j i j

i

k

M

_



_

naturally in solving degenerate or ill-conditioned systems of equations of the approximate (8), (9) instead of the class of unbiased estimators consider the class of unbiased estimates with the square of the length, ie class

}

:

{

*1, 2

2

, 1 ,

1j i j i j

i

M

k

_ _



_



K

ratings. To do this, use the fact [14] that:

2 *

, 1

, 1 2

min

)

(

)

(

inf

)

(

, 1



n

k

M

k

M

l

n

M

j i

j i R

k_i _j m











 



Q

(15)

and

ˆ

_



*_₁_, 2



2

(

(_22₁_|)*

)

 2

, 1

T i i j

i j

i

k

tr

P

k

M



, where

)

(

),

ˆ

(

₁_, (22₁_|)*

min

T i i j

i

l

rank

P

k

_



_



Q

.

Inequality (15) shows that using the method of least-squares leads to the "defects" average value of the sum of squared residuals. Natural to require that the regularized estimates eliminate this defect. Then, on the basis of (15) is natural to define the regularized estimates

k

~

_i_₁_,_j so as to satisfy the condition:

.

)

~

(

k

1,

n



2

M

Q

i j



(16)

The parameter



can be found from the equation residual [11] of the form:

0 ,

)

(

_₁_,



Q

_min







Q

k

i j . (17)

Then the corresponding regularized estimates of the solutions will be determined from the joint solution of the equations:

,

~

(1)

, 1 * ) 22 (

| 1 ) (

, 1 )

( , 1 * ) 22 (

|

1 i j i j i i i j

T i

i

k

P

p

P

_ _





_



_ _ (18)





_

_

_

 

 min

2 ) 1 (

, 1 )

( , 1 * ) 22 (

| 1

~

_Q

j i j i T i

i

k

p

(4)

International Journal of Emerging Technology and Advanced Engineering

800

To satisfy the conditions (16) the amount of δ in (17) can be selected based on the ratio

min 1

0

(

)

Q







l

n

l



. This is because

.

]

)

(

[

)

(

_min



₀

M

l

n

l

1 _min

n



2

M

Q









Q



This method does not require any additional a priori information on an approximate equation system. Following [11,14] we can show that if the condition

lim

1 (_22₁_|)*





0 ,



G

T i i n

n

P

is rated

k

~

_i(_opt₁_,_j) and

k

ˆ

_i*_₁_,_j will be consistent and asymptotically efficient.

In order to select the optimal regularization parameter instead of a class with a non-biased estimates of square of length

K

we consider the class of regularized estimates

}

{

~

( )

, 1



j i

k

_



K

.

This class is determined from the Euler equation (18), and the regularization parameter



is determined from the condition of the residual:

,

~

2

min 2 ) 1 (

, 1 ) (

, 1 * ) 22 (

|

1







_

_

_

 



k

p

m

P

_i _i T _i _j _i _j

Q

(20) It can be shown [11,15,16] that the regularized estimates

) (

, 1



j i

k

_ , determined from the equations (18) and (20) satisfy

the condition

2 *

, 1 2 ) (

,

1j i j

i

k

M

_



_ . At the same time for optimal regularized estimates

k

~

_i(_opt₁_,_j) equality holds

0 )

(



p

i(1)1,j



M

. This fact leads to a shift of

* , 1 ) (

, 1

~

j i opt

j

i

k

M

_



_ and

M

(

p

~

_i(_1₁)_,_j



P

_i_(22₁_|_i)*T

k

~

_i(_opt₁_,_j)

)



0

, which ultimately negatively affect the value of the mean square error estimates of solutions

2 *

, 1 ) (

, 1

~

j i opt

j i

k

M

_



_ . To ensure conditions

2 *

, 1 ) (

, 1 2

* , 1 ) (

, 1

~

j i opt

j i j

i j

i

k

M

k

M

_



_



_



_ parameter γ in (20) can be found from the condition:

,

min

1 ) (

 

_





n

i i

e

M

(21)

where

e

()



(

e

₁()

,...,

e

_n()

)

T



~

p

_i(_1₁)_,_j



P

_i(_22₁_|_i)*T

k

_i(_₁)_,_j. (21) that the parameter



is expedient to choose the form:



  

 



n

i i m

p

e

j

i 1

) (

] ~

, 0 [ *

2 min 2 ) 1 (

, 1

min

arg



 



Q

.

III. TASK STATEMENT -II

We now consider the adaptive estimation algorithms with mutual correlation noise object and interference measurements. System model and the measurements will be written as:

,

1 |

1

1  





i i i



i i

i

A

x

Г

w

x

(22)

1 1

1

1   





i i



i i

i

H

x

G

v

z

, (23) where



1



1



1



0



0 0



1

~

N

0 ,

Q

,

v

~

N

0 ,

R

,

x

~

N

x

,

P

w

_i _i _i _i ,

ij i T i

i i

j j

i i

R

C

Q

v

w

v

w





















































,

cov

,

ij



– Kronecker delta,

Q

_i



0

,

C

_i



0

,

R

_i



0

,



,



0 ;

cov



,



0 cov

x

0

w

i



x

0

v

i



.

Algorithms Of The Decision

Based on the theory of dynamic evaluation methods [1-4] the estimation algorithm in this case can be written as:



i i i i



i i i i

i

x

K

z

H

x

_₁_|_₁



_₁_|



_₁ _₁



_₁ _₁_| , (24)



i i ii



p i i i i i i

i

A

x

K

z

H

x

_₁_|



_₁_| _|



_₁



_| , (25)





1

1 1 1 1 | 1 1 1 | 1 1

        







i i iT

T i i i i T i i i

i

P

H

P

H

G

R

G

K

, (26)



 



,

| 1 | |

1 |

1

p i T i i i p i T i i i

i p i i i i i i p i i i i i

K

G

R

G

K

Г

Q

Г

H

K

A

P

H

K

A

P









_ _



(27)



i i



i i

i

I

K

H

P

_₁_|_₁





_₁ _₁ _₁_| . (28) For the solution of this problem we use the method of "de correlating" noise model of the system and noise measurements. Accordingly, we shall choose

K

_ip of the expressions (25) and (27) so that

w

_i* and

v

_i were not correlated with each other:

,

0 ]

[

}

)

)(

(

)

(

{

)

,

(

cov

* *











i i p i i i

T i i i i i p i

i i i i

i

R

G

K

C

Г

v

G

K

w

Г

M

v

w

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International Journal of Emerging Technology and Advanced Engineering

801

where

w

*_i



Г

_i

w

_i



K

_ip

G

_i

v

_i.

From (29) it follows that the gain matrix

K

_ip should be determined on the basis of the expression:

i i i i p

i

G

R

Г

C

K

[

]



. (30) Thus, the accuracy of the solution during the filtration of correlated noise and interference measurement object substantially depends on the accuracy of

K

_ip gain calculation based on equation (30). This system of equations is often ill-conditioned and leads to the application of regularization methods [7,9,10].

For convenience of presentation and brevity we write the equation (30) as follows:

j i j i j i j i

i

k

d

L

_,



_,



_,





_, , (31)

Where

L

_i



(

R

_iT

G

_iT

)

- linear operator in a Hilbert space of H to H;

k

_i_,_j - i-th row of the matrix

K

_ipT,

n

j



1 ,

2 ,...,

;

d

_i_,_j - i-th column of the matrix

T i T i i

C

Г

D



approximation prerequisite form

2

, 2

E j i

d





, _max2

2

, 2

min





E j i

d

,

j



1 ,

2 ,...,

n

;

j i

d

_, - the exact value of the right-hand side of equation (31).

Consider the case where the operator

L

_i is self-adjoint and positive. Then M.M.Lavrentev method leads to an equation of the form:





k

i,j



L

i

k

i,j



d

i,j, (32)

Where



- the regularization parameter. 

R

family of operators defined by the formula:

1

)

(

)

(









L

_i

L

_i

R

_



_



, (33) generates a bounded approximation to the problem (31), if the relation

lim

, , ,1

0







R



L

i

k

i j

k

i j

 ,

k

i,j,1



ker

L

i,

subject to the conditions of the form:

 

1

,

)

(

sup

vrai



_





_





_









K

E

,

)

1 (

)

(



2





_



c



,



_

(



)





c

(

1 



)

[9,10], where

E

_ - spectral family of

L

_i,

с

- constant independent of



.

Equation (32) can be expressed in the form of   



i i j



i i j

j

i

I

L

d

L

d

k

, ,

1 ,

,



(



)



(

)



,

wherein



_

(



)



(







)

1 - generating system functions

0 







- spectral parameter.

Closer to

k

_i _j

 

L

_i

d

_i,_j *

,





pseudo solution equation (30) will be constructed in the form of:

 

 

 i i j

j

i

g

L

d

k

, ,



, , (34)

 

   

 i i i j i i j

j

i

g

L

k

g

L

d

k

, ,



, ,



, ,

 



_

 





g

2

g



. (35) Approximation (34) and (35) can also be written as

 

 i j

j

i

B

d

k

, ,



, ,

 

 i j

j

i

B

d

k

, ,



, ,

Where the operators:

 

L

i

g

B

_



_ ,

B

_



B

_

L

_i

B

_, (36)

approximate

L

_i . Then the approximation (34), (35) and (36) take the form:







 i



ij

j

i

L

I

d

k

,

1 ,

,







,







, ,

1 ,

,j i i i j

i

L

I

L

k





 ,





1





L

I

B

 i



,



 

1



1





L

I

L

I

B

 i



i i



.

When implementing the above algorithms regularization parameter



is advisable to determine method based on the discrepancy [7].

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International Journal of Emerging Technology and Advanced Engineering

802

1 ,

0 ,

, ,

,



















j i j i i j

i

L

k

d

k

. (37)

Instead of (37) can also use the expression of type





 



i i j

j

i

v

i

I

L

d

k

_, _,



Re

,





1 _, ,

or a special computational scheme Tikhonov method: 

 



k

i j

L

i

k

i,j,

L

i

d

i,j

2 , ,

2

_

_

.

On the basis of the results of [18] we can show that for much better conditioning of the system (31) it is advisable to consider a system of the form:

 

 ij



i j

i

k

d

L

_, _, _,



_, , (38) where

)]

)(

1 (

)

[(

1

, L L i L

i

L

_



D





D











D

,

L

D

–

L

_i diagonal matrix,















(

_,

,

~

_, _,

)

~

_, _, 2

E j i i j i i j

i

L

k

L

k

d

 _ _





,





1/2 2 2

2 1

1 

_





_









_





_





 

E L i E L E L

L

D

L

D





,



, ,

~

j i

k

- solution of system

L

_i_,_

k

~

_i_,_j_,_



d

_i_,_j.

Determining the vector

k

ˆ

_i,_j is reduced to the definition

of the vector _, _,_

~

j i

k

, satisfying the inequality

2 min 2

, , ,

~

_





_

_

E j i i j i

L

k

d

. (39)

Finding _, _,_

~

j i

k

vector associated with the consideration of the set of values of



:



₁

,



_l



q

_l



_l_₁

,

l



2 ,

3 ,...,

which

q

_l are the values that should be in the process of counting. These advantageous to determine the magnitude of the formula [18]:



₂



1/2

1 2 min 



l

q



v

,

wherein

2

, , ,

2

1 i j i i j ₁ _E l



d



L

k

_l



v

.

The value of



₁ can be selected on the basis of the approximate equality of the form

2 max 2 , , ,

2

1 ₁



_

_



E j i i j i

L

k

d

v

. Subsequent values



_l should provide a sufficiently rapid determination of the vector

k

~

_i_,_j, for which _min2

2

, 2 ,

~

_





_

_

E j i i j

i

L

k

d

, where

2



is based on the solution of the equation

2 min 2

, , 2 ,

~

_



_



_

_

E j i i j

i

L

k

d

. Defining vector

k

_i,_j

~

on the

basis of (39), then you can find along the vector

k

_i,_j

~

vector

j i

k

ˆ

_, , satisfying the following two conditions:

,

0 )

ˆ

,

ˆ

(

,

ˆ

, ,

, 2 2

,







i ij i j



i i j



E

j i i j

i

L

k

L

k

d

L

k

d

 

Where the



2



(



_min2





_max2

)

/

2

. It should be noted here that the values



and



are based on the selected parameter value



. In this case,







_ is determined by the principle of conservation of the Euclidean norm of the matrix with its regularization.

IV. CONCLUSION

Thus, the above expression synthesize algorithms for adaptive state estimation of dynamic systems under correlated noise and interference measurements of the object on the basis of approximate methods for solving ill-conditioned or singular stochastic systems of linear algebraic equations, and thus ensure the convergence of the estimates and to improve the accuracy of their evaluation.

REFERENCES

[1] Ogarkov, M.A. 1990. Methods of statistical estimation of random process quality. Moscow: Energoatomizdat.

[2] Kuzovkov, N.T., Salychev O.S. 1982. Inertial and optimal filtering. Moscow: Mechanical Engineering.

[3] Leondes, K.T. 1980. Filtration and stochastic management in dynamic systems. Moscow: World.

[4] Sinitsyn, I.N. 2006. Filter of Kalman and Pugachev. Publishing house: Lagos.

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803

[6] Igamberdiyev, H.Z., Yusupbekov, A.N., Zaripov, O.O. 2012. Regular methods of estimation and control of dynamic objects in the conditions of uncertainty. Tashkent: Tashkent State Technical University.

[7] Tikhonov, A.N., Goncharsky, A.V. 1987. Ill-posed problems of natural science. Moscow: Moscow University.

[8] Kabanikhin, S.I. 2008. Inverse and ill-posed problems. - Novosibirsk: Siberian Scientific Publishing House.

[9] Vajnikko, G.M., Veretennikov, A.Ju. 1986. Iterative procedures in ill-conditioned problems. Moscow: Science.

[10] Bakushinskij, A.B., Goncharsky, A.V. 1989. Iterative approach of ill-conditioned problems solution. Moscow: Science.

[11] Zhdanov, A.I. 1988. Solution of ill-posed stochastic linear algebraic equations regularized maximum likelihood method. Volume 28, №9, 1988.

[12] Albert, A. 1977. Regression pseudoinverse and recurrent otsenivanie. Moscow: Science.

[13] Demidenko, E.Z. 1981. Linear and nonlinear regression. Moscow: Finance and statistics.

[14] Draper, N., Smith H. 2007. Applied Regression Analysis. Moscow: Dialectics.

[15] Russell, D. 2013. Regression analysis. Moscow: Science.

[16] Malov, S. 2013. Regression analysis. Theoretical basis and practical recommendations. - Publishing House: Publisher of St. Petersburg University.

[17] Bakushinskii, A.B., Sizikov, V.S. 1982. Some non-standard regularization algorithms and their numerical implementation. Zh. - 1982 - T.22, №3.