The Dynamic Systems Adaptive Identification Algorithms on the Basis of the Regularity Principle

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 7, July 2015)

The Dynamic Systems Adaptive Identification Algorithms on

the Basis of the Regularity Principle

H. Z. Igamberdiyev

1

, J. U. Sevinov

2

1,2_{Department of Electronic and Automatic, Tashkent State Technical University, University st.2, 100095, Tashkent city,}

Republic of Uzbekistan

Abstract — The problems of adaptive algorithms synthesis of dynamic systems identification on the basis of a regularity principle are considered. An error estimation of a basic data task for the various modeling structures are received, allowing without doing the direct decision to estimate from above the error of the required decision and choosing the regularization parameter optimum value . On the basis of incorrectly put objectives decision methods the regular recurrent and iterative computing schemes of parametrical identification problem solutions are provided. Keywords — Object of control, control system, parametrical identification, dynamic filtration, regularization, regularization parameter.

I. INTRODUCTION

In the modern theory of idealization control based on the assumption that the mathematical model of object rather precisely describes its behavior and which is known in advance, practically leaves from consideration. At the solution of practical problems of technological processes control can appear that a number of characteristics of real object can be in advance unknown or change in the course of its functioning. In this regard the way of creation of adaptive control systems which are not demanding full aprioristic knowledge of controlling object and conditions of its functioning was very tempting. The systems named adaptive systems with adjusted models [1-5] treat systems of this kind. In such systems control algorithms should on the basis of the current information on input and output of object target signals to approach the model behavior to object behavior. Despite theoretical validity of adaptive optimum creation in classical sense of parametrical estimation algorithms and control, their practical realization in some cases is ineffective. The main lack of a method of adjusted model is complexity of receiving conditions of operability of system. For overcoming of arising difficulties two ways are used. The first – acceptance of special measures to enrichment of a range of an entrance signal with a view of ensuring convergence of estimates to true values. The first way is usually used for achievement of the difficult purposes of control, in particular, for synthesis of optimum systems. The second way consists in refusal of a strong solvency of parameters estimates, i.e. of convergence of estimates to true values.

Justification of operability of system is carried out under condition of weaker point achievement of adaptation – proximity of exits or conditions of model and object [2, 3]. Thus, as a rule, it is not possible to provide an optimality of system and it is necessary to be limited to weaker purposes of control, for example, system stabilization. It is caused by that many problems of identification and synthesis of operating influences belong to the class of the incorrect. In such situation the problem of synthesis of methods and algorithms of estimation is expedient for considering from the point of view of the theory of return tasks [6,7] when definition structures of algorithms and calculation of their parameters are carried out on the basis of a regularity principle. Such approach allows not only to provide proximity of exits or model and object conditions, and also strong convergence of estimates to true values that allows to keep operability of a synthesized control system, at least, at small influences of unrecorded factors at initial synthesis i.e. to possess roughness in relation to them. In this regard development of synthesis algorithms of adaptive control systems with adjusted models on the basis of a principle of regularity is represented actual.

II. TASK STATEMENT -I

Let's assume that the object of control in the entrance exit variables is described by the linear differential equation

t t

t B u

y

A()  ()  , (1) Where yt - l- vector of exits of object of control; ut - m-

vector of operating influences; А(), В() - polinomial matrixes respectively dimensions of ll and lm,

r r

n A A

I

A()  ₁ , r _r

B B

B() ₁ ,

I

_n - an individual ll- matrix;  - operator of an individual delay;



_t - the stationary vector hindrance formed by the

filter L()_tE()_t, where to p _p

n L L

I

L() ₁ , t

p p

n L

I

E()  , - vector random variables with the independent values, satisfying to conditions

0 ] [ _t 

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i.e. polynoms detA(



),detE(



) have no roots in the closed individual circle of the complex plane.

Controls

u

_t are formed by means of linear feedback

of a look [2, 3]:

t

t D y

u

C()  () , (2)

Where C()I_mC₁qC_q,

q q

D D

D

D() 0 1 , Сi - mm- matrixes, Di - ml- matrixes.

The functional of quality looks like



( , )



lim _{t t}

t MQ y u

J

 

 . (3)

Here Q - a positive square-law form of variables у, u, i.e. Q(y,u) > 0 at | у | + | u |  0 and

u Q u u Q y y Q y u y

Q( , ) T ₁₁ 2 T ₁₂  T ₂₂ , (4) 22

12 11,Q ,Q

Q - material matrixes of the corresponding dimensions.

It is required to synthesize a regulator (2) minimizing a functional of quality (3) - (4) in a class of stabilizing regulators. The different order of a regulator (2) usually isn't fixed in advance, i.e. minimization of a functional (3) is made in a class of all linear stabilizing feedback.

The problem of minimization of a functional of quality of adaptive control, as a rule, leads to tasks of the solution of the matrix linear algebraic equations of identification and control of a regulator of a look [3,4,8]

) ( θ ) ( ) (

yt T t  t , (5) F

Z A f z

A  , :  , (6)

Where





T

q t t

p t t

t) y ( 1),...,y ( ),u ( 1),...,u ( )

(  T  T  T  T 

 ;





T

q p,B, ,B

A , , A , A

θ _{1 2}  ₁  ; (t) - revolting influence of dimension of l1 with a covariational matrix of R; AA

 

 and f f

 

 are defined on the basis of a characteristic polynom of optimum system

         

 a  b   

g   and plenums

   

  

 , , , regulator 

 

, u_t

 

, y_t,



p p



col

z ₁,..., ,₁,..., .

The equations (5) and (6) from the computing point of view belong to the class incorrectly of objectives [6,7], i.e. small errors in basic data lead to big errors of the decision. It is connected with that stability conditions of solutions of the equations (5) and (6) in most cases are, as a rule, broken.

Therefore naturally there is an aspiration to analyze feature of problems of synthesis of systems of adaptive control with adjusted models from positions of methods of regular estimation.

Algorithms Of The Decision

For the purpose of unification and the analysis of possible versions of the mathematical description, the characterizing of structure and a set of models of dynamic objects of control is made in work. It is established that in the theory and practice of identification in the terms "entrance exit" the greatest distribution was received by modeling structures of a look:

, ..., , 2 , 1

), ( ) ( u B ) ( y A ) ( y

1 1

N t

t j t j

t t

q j

j p

j j



  









 



(7)

,..., 2 , 1 ), ( ) ( Bu ) (

yt  t  t t (8) Where u(t) - an entrance signal of dimension of m1, у(t)- a target signal of dimension of l1, (t) - revolting influence of dimension of l1 with a covariational matrix of R, A_i(ll)- matrixes, B_i(lm)- matrixes; В - lm matrix.

Modeling structures (7) and (8) at the solution of a problem of identification will usually be transformed to modeling representations of a look:

)

(

θ

)

(

)

(

y

t





T

t





t

. (9)









X

y

, (10)

Where (t)



yT(t1),...,yT(tp),uT(t1),...,uT(tq)



T,





T

q p,B, ,B

A , , A , A

θ _{1 2}  ₁ ; X – lnlm - matrix



X(1) X(2)  X(n)



T, X(t)I_lu(n)T;  - dimension vector

m

l

, made of

l

under columns

) (i

 (i1,2,...,l) matrixes of B, Il - individual (l×l) -

matrix,



– a symbol kronnekeras works, n-volume of supervision.

The matrix operator of T(t), a vector of parameters



and the right part y(t) of modeling structure (9) is representable in a look:





, ) ( | | ) 2 ( | ) 1 ( | ) ( | ...

| ) 2 ( | ) 1 ( ) (

l l

l l

l l

T

I q t u I t u I t u I p t y

I t y I t y t

  

    

    

 

 ₍₁₁₎

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



, ..., , 2 , 1 , ) ( ) ( ) ( )

( 1_{1 2}2

p i i t y i t y i t y I i t

y _{l l}l

      

  ₍₁₂₎

, ) ( 0 0 0 ) ( 0 0 0 ) ( ) ( , , ) ( 0 0 0 ) ( 0 0 0 ) ( ) ( 1 1 1 1 1                                   i t y i t y i t y i t y i t y i t y i t y i t y l l l l l                (13)





, ..., , 2 , 1 , ) ( ) ( ) ( )

( 11 22

q j j t u j t u j t u I j t

u l ml

      

  ₍₁₄₎

; ) ( 0 0 0 ) ( 0 0 0 ) ( ) ( , , ) ( 0 0 0 ) ( 0 0 0 ) ( ) ( 1 1 1 1 1 1                                   j t u j t u j t u j t u j t u j t u j t u j t u m m m l                (15)





T

b a 

 , (16)

Where





; ... ... ... ... ... ... ... ... ... ... a 1 2 12 1 11 1 1 1 1 2 1 12 1 1 1 11 p ll p l p l p p l p ll l l l a a a a a a a a a a a a  (17)





; ... ... ... ... ... ... ... ... ... ... b 1 2 12 1 11 1 1 1 1 2 1 12 1 1 1 11 q lm q m q l q q l q lm m l l b b b b b b b b b b b b  (18)





T

l t y t y t y t

y() ₁() ₂()  () . (19) Let's believe that components of u_j(t), j1,2,,m

and y_i(t),i1,2,,l of a vector of entrance and target variables are measured with a margin error with the

corresponding dispersions of 2 (t) i

y

 and 2 (t)

j u

 . Then

considering structure of the matrix operator of T(t) and the right part y(t) of modeling structure (9) taking into account representations (11) - (16), (19) in the assumption of a statsionarnost of variables u(t) and y(t)

it is possible to write down the following expressions for

an error of a task of a matrix of T(t) and vector y(t):

. ) ( , ) ( ) ( 2 1 1 2 2 1 1 2 1 2 2                           







   l i y m j u l i y t l t t l h i j i     (20)

The received expressions, namely estimates (20), allow without making to estimate the direct decision from above an error of the solution of the equation (9). On expressions (20) it is possible to receive also aprioristic information on an order of an error of the decision for receiving qualitative conclusions about with what accuracy it is reasonable to solve system further. Information received thus can be used for a choice of optimum value of parameter of regularization. Obviously, it is not meaningful to aspire to receive the system decision with a margin error, essentially smaller, than value of a tochnostny vector of  

 

,h .

Making the transformations similar (12) - (19), for modeling structure (10) it is possible to receive the following expressions for errors of h and δ the matrix operator of the X-th and a vector of the right part y:

.

)

(

,

)

(

)

(

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































































     n i l j y n i m k X l j y

i

j

l

i

h

j i j









(21)

Estimates (21) can be naturally used for a choice of optimum parameter of regularization α at the solution of the equation (10).

For the solution of the equation (9) the regular recurrent scheme of a look is offered:



() () ()| 1)



) ( ) 1 | ( ) |

(t t  t t K t yt T t  t t

   , (22)

) 1 | 1 ( ) 1 |

(t t  t t

  ,



()



) ( ) 1 | ( )

(t Pt t t G Dt

K    _ ,









, ) ( ) 1 | ( ) ( ) ( , ) ( ) ( ) ( 1 t t t P t t D I t R t D t D G T _         



() ( )



( 1) ( 1) )

1 ( 1 |

(t t K t Mvt v t K t Pt

P T T ,

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Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 5, Issue 7, July 2015) Where G_



D(t)



– generating system of functions for

a method of regularization,  – regularization parameter,

) ( ) ( )] ( ) (

[ t j Rt tj

M  T   , v(t)y(t)T(t)(t). At the solution of the equation (10) there is expedient the following regular iterative scheme [9]:





_,





, 1

1 ,

, b  U U b U Y w r

b_{ir ir n}T _{n h ir n}T _nT _{ r} , (23)

Where

w

_r – the random variables possessing

properties M w_kMw_k 2 2,k1,...,r;

m i

b_i, 1,..., – lines of a matrix of B.

Let's assume that



U_nTU_n



_h , U_nTU_n 1 and are satisfied the following conditions of approximation:



U_nTU_n



_hUT_nU_n h1,



U_nTY_nT



_U_nTY_nT 1. For the iterative scheme (23) various rules останова iterative process were considered. Let's give one of them:

П0: a1a2,



  



inf r 1:b_i,r b_i,r1

m , (24)

Where m – the moment of stop, a₁,a₂0.

On the basis of the theory of iterative methods of the solution of incorrect tasks it is possible to show [9,10] that regularization for a rule останова (24) is established at the following ratios between small parameters:

П0:  с



h



2, c0. (25)

Thus for a rule останова (25) the following limiting

ratio of lim ( ) ~ _* 0

0 ,

 

 I um u

h 

 D , where by





 n k

k

n w

W

1

, D_ 



w: W_n n,n1,2,...,



, 



 , u_*- the quasisolution of the equation is carried out (8), from where follows that to

0 ) )( ( lim

0 ,

 

 I hm

h





 D .

Estimates (20) and (21) allow to use a method of uncertain multipliers of Lagranzha, i.e. to find the vector

of _k minimizing smoothing functional

2 2

] , ,

[ _k d_k G hG _k d_{k k}

M     [6], and

parameter



to define from

  

  

  



 _



 _{k k}

k d h

G 2 , infTy

condition.

III. TASK STATEMENT -II

Let's consider now a class of models with noise in object and the actuation device, having a wide circulation in practical tasks. Such class of control systems with noise in object and the actuation device can be described the look equations:

, ,

2 2 1

1

1 1 1

1

n i

i n i i

i n i n

n q

i i n i p

i

i n i n

v u

d y

c u

v u

b y

a y

 

 

 



 











  

 

 



(26)

Where {un}, {уn} – observable sequences on an

entrance and on an object exit respectively; {v1n}, {v2n} –

gaussian sequences with a zero population mean and individual dispersion, joint distribution gaussian. Thus

. 0 ] [

] [

, 0 ] [

] [

, ] [ ] [

2 1

2 1

2 2 1

1

 

 

 

 

 

j n n j

n n

j n n j

n n

kj j k j

k

u v M u

v M

y v M y

v M

v v M v v

M 

The task consists in an assessment of parameters of object {ai}, {bi} and actuation device parameters {сi},

{di} by results of supervision {un}, {уn}, where

N

n0,1,..., . Under certain conditions parameters of object and the actuation device can be calculated independently [4]. These conditions are easily feasible at the solution of problems of identification and control of real objects.

Algorithms Of The Decision

Then estimates of parameters of object and the actuation device can independently be calculated on the basis of the equations:









N T T

T _Y

S

S_{1 2} ₁ ₂  , (27)









N T T

T _S

S

S_{1 3} ₃ ₄  , Where

) , , , , ,

( _{1 1}

1T  a ak b  bI

 ,

) , , , , ,

( _{1 1}

2T  ak ap bI  bq

 ,

) , , , , ,

( _{1 1}

3T  c ck d  dI

 ,

) , , , , ,

( _{1 1}

4  

T  c_k c d_I  d ;

N

S S S

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we will enter the following designations:



1 2



 0,

 

12 0

T T T

S S

S ;



S₁S₃



S_R,

 

₃T₄T T _R. The algorithm of estimation of a vector of parameters of object of ₀ on the basis of the equation (27) below is given. The same algorithm can be used and for estimation of a vector of parameters of a regulator of _R.

According to terminology of dynamic filtration algorithms for estimation of object parameters vector of

0



in the designations accepted above we will write

down the following equations: ₀(t1)₀(t)w₀(t), )

( ) ( ) ( )

(t S₀ t ₀ t v₀ t

Y_N    , where



w₀(t)w₀(t)



Q₀(t)

M T  , M



v₀(t)vT₀(t)



R₀(t). On the basis of methods of an optimum filtration theory and incorrect tasks solution [1,7,11] for a required

assessment of ₀,(k) it is possible to write down the following system of the recurrent equations:



~ ( 1) ~ ( 1) ( 1) ( 1) ( )



,

) 1 ( )

1 ( )

1 (

) 0 ( ,

, 0 0

, ,

0 ,

0

k Y k k

k S k Y

k K k k

N

N      



  

 



  



  

 



  

, ) ( ~ ) ( ~ ) ( ~

) ( ~ ) ( ~ ) 1 ( ~ ) 1 (

) 0 ( 1 )

0 ( 0

1 )

0 ( 0 1 )

0 ( 0 ,

0

k Y k k S

I k S k k

S k

N T

T



 

 

    



 _{  }



 









( 1)



,

) 1 ( )

1 ( ~ ) 1 ( )

1 (

1 ,

, 0

, ,



  

  

 

 

k

k N k S k M k

K T

 

  

 



1 )

0 ( 0 1 )

0 ( 0

, _{( 1)} ~ _{( )}~ _{( )}~ _{( ) }

  



 _{ }



 S k k S k I

k

M T  ,

) ( ~ ) ( ~ ) ( ~ ) 1 ( )

1

( , ₀(0) ₀1

,

k K k k S k M k

N     T  ,



( 1)~ ( 1) ( 1)



, ) 1 ( )

1 ( )

1 (

, 0

,

, ,

,

T T

k N k S k M

k K k M k

p

 

 



  

 



  



  

 



,

) 1 ( )

1 ( ~ ) 1

( , ,

, _{    }

 k  k  k ,



~ ( 1) ( 1)



, )

1 ( ) 1 ( ~ ) 1 ( ~

) 1 ( ~ ) 1 ( )

1 ( ~ ) 1 ( ~

, 0

, 0

0 ,

0 ,

T T

k N k S k N k S k R

k S k M k S k

 

  

  

  

    

  



  



). ( ~ ) ( ~ ) ( ~ ) ( ~ ) 1 ( ) ( ~ ) ( ~

) ( ~ ) ( ~ ) 1 ( ) 1 (

1 1

) 0 ( 0 ,

) 0 ( 0 1

, ,

k K k k

k S k M k S k

k K k K k k

T

T

   



_{   }



  

    

 

 

  



Then ₀,(k) can be set ₀,(k1)



S~₀T(0)(k)W~1(k)S~₀(0)(k) I



1S~₀T(0)(k)W~1(k)Y~_N(0)(k)

 

formula, and at an appropriate choice of regularization

parameter of  the limiting ratio of lim ₀ , ₀( )

0  k

  





 ,

where by 0(k)



– optimum in square mean 0(k)

assessment will be carried out at exact knowledge of model (26).

IV. CONCLUSION

The given algorithms allows to solve effectively a problem of identification of the multidimensional stochastic systems described by parameters matrix when components of a vector of supervision have unknown covariance’s. Estimates received at these possess properties of uniform optimality at all possible values of unknown parameters.

REFERENCES

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[3] Derevitskiy D.P., Fradkov A.A. 1981. Applied the theory of discrete adaptive control systems. Moscow: Science.

[4] Steinberg Sh.E. 1987. Identification in control systems. Moscow: Energoatomizdat.

[5] Eykhoff P. 1983. Modern methods of systems identification: Translation from English. Moscow: World.

[6] Tikhonov A.N., Arsenin V.Y. 1979. Incorrect problems decision methods. Moscow: Science.

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