Intelligent Control and Automation, 2010, 1, 59-67
doi:10.4236/ica.2010.12007 Published Online November 2010 (http://www.SciRP.org/journal/ica)
Structures, Fields and Methods of Identification of
Nonlinear Static Systems in the Conditions of Uncertainty
Nikolay N. Karabutov
Moscow State Institute of Radio Engineering, Electronics and Automation, Russia
Received March 8, 2010; revised September 2, 2010; accepted September 13, 2010
Abstract
The field of structures on set of secants is offered and methods of its construction for various classes of
one-valued nonlinearities of static systems are considered. The analysis of structural properties of system is
fulfilled on specially generated set of data. Representation on which modification it is possible to judge to
nonlinear structure of static systems is introduced. It is shown, that structures of nonlinear static systems
have a special V-point. The adaptive algorithm of an estimation of structure of nonlinearity on a class poly-
nomial function is offered.
Keywords: Structure, Identification, Algorithm, Field of Structures, Nonlinear System
1. Introduction
Statistical methods are applied to a solution of a problem
of structural identification basically, based on correlative
and an analysis of variance [1,2]. Various approaches
and methods are applied to identification of nonlinear
systems. In [1-3] the statistical methods based on the
correlation and dispersive analysis are used. In identifi-
cation systems genetic algorithms [4-6] are widely ap-
plied. For synthesis of models neural networks [7-11]
and their combination with various methods of approxi-
mation [12,13] also are used. Genetic and neural network
algorithms allow to find structure of model of nonlinear
systems on the set of approximating functions. In a
number of works the aprioristic information [14-16] and
the subsequent approximation on the set class of func-
tions [17-19] is used. Thus the choice of a class of func-
tions is not always proved. The carried out analysis
shows, that despite set of the publications, any formal-
ized procedures of an estimation of structure it was of-
fered not as the considered subject domain is very diffi-
cult for describing in existing language of mathematical
concepts, objects and decision-making. The problem of
structural identification even more becomes complicated
for static systems as for them it is complicated to open
the existing correlations, expressing through the inte-
grated indicator - a system exit. Here a priori data can
render the big help. In the conditions of a priori uncer-
tainty informational synthesis of system can give the
additional information only.
Attempt to estimate area to which should belong re-
quired nonlinear structure is natural. For nonlinear dy-
namic systems the concept of a sector condition [20], to
which should belong investigated nonlinearity has been
introduced. This condition is applied only at a stage of
synthesis of dynamic systems. In identification systems
to formalize such condition till now it was not possible.
For the first time such attempt has been undertaken in
[21,22] for dynamic systems where on the basis of
analysis of results of measurements the set which con-
tains the information on nonlinearity has been generated,
and also the method of structural identification is offered.
Properties of nonlinear dynamic systems can be esti-
mated on a phase or observable informational portrait
(OIP). For static systems of such universal graphic rep-
resentation does not exist. The problem even more be-
comes complicated that in the majority of the real sys-
tems described by static models, the informational set
generated on the basis of results of measurements, has
irregular character.
More low the approach to construction of a field of
structures on a class of secants for nonlinear static sys-
tems which is development and generalization of the
outcomes received in [21,22] is offered. The concept of a
sector condition is introduced and algorithms of a deci-
sion making about structure of nonlinear system are of-
fered. The structure on which it is possible to make a
solution on nonlinearity of system is offered.

N. N. KARABUTOV
60
2. Field of Structures on Class Fm
The plant is considered


,
T
nn n
yAUfUn

  (1)
 

,,
,,
,|: ,
,,
,1 ,
in inN
p m
mnijinjn
ij
uRuUfRJ R
fUnfU uu
ijmp k


 


 



 


F,
where k
n
UR, n yR is an input and a output,
k
A
A
R  is a vector of parameters, belong limited,
but a priori unknown area A
, ij

is some numbers,
N
nJ is discrete time, nR

 is a disturbance on an
output.
For (1) the informational set is known
 


II ,, ,0,
oonn N
yUy UnJN 
and mapping :
oUy
N
nJ corresponding to it.
Mapping o  describes an observable informational
portrait. Restriction OIP i
i
u
oo
uU
 1,ik is de-
fined and for everyone i
u
o
 secant
0,1, ,
(, )
iiiin
yuaa u

 , where 0, ,
i
a1, i a is some real
numbers is constructed.
Secants (, )
lj
yuu

for lj m uu F are constructed.
Let introduce the set of secants
   

,,,,,,,,,1
ilj
Uyyuyuuiiklj   
set on Io.
Definition 1. A field of structures S S of system (1)
on class m
F we will name a collection of mappings
(,)1, ,
ii
y
uuyik   (, )
lj lj
yuuuu y


(, )1lj on Euclidean plane 
 

 

,1,,,,,,1
Si lj
yuikyuuljl j   
m
SF .
It is necessary on the basis of analysis o
, Io to
construct a field of structures S S for system (1) on set
(,)Uy.
As mapping o  at 2k does not give in to obvi-
ous interpretation, that, we will be limited to its restric-
tions i
i
u
oo
uU
 1,ik constructed on plane
(,)
i
uy. As a result we will receive some final set of
mappings i
u
o
, lj
uu
o
, which represents a field of struc-
tures S S of system (1) on plane .
As secants with various ranges of definition the area of
definition of field ()
Sm
SF on plane  will be equal
dom( )
i
i
u are considered, and area of values ()
Sm
SF
will coincide with a area of values of mapping o

  
rng rngrng
So
y
S.
Let construct mappings i
u
o
, lj
uu
o
 on plane

. We
will add to them elements of set (,)Uy. Further we
will be limited only to the analysis of properties (, )
i
yu

,
(, )
lj
yuu

.
The field of structures allows to simplify the analysis
of structure of system (1) and to estimate as degree of
linearity (nonlinearity) of system through available set of
secants, and influence of elements of vector k UR on
shaping of a nonlinear part of system (,)
f
Un. Corre-
sponding results are more low give. The problem of lo-
calization of function (,)
f
Un on the basis of available
set of experimental data Io is not less important. In
conditions uncertainty the solution of the given problem
is connected with overcoming of some problems. In
work it is shown, that one of the approaches, allowing to
localize required area, it is based on sector construction
on set of secants. The quantity indicator (coherence coef-
ficient), allowing on the basis of properties of secants to
decision making about belong of nonlinearity to sector is
introduced.
Theorem 1 [21]. The system (1) with nonlinearity
(,) m fUn

F has a linear field of structures ()
Sm
SF .
Let consider sector ()
ijS m
SSF limited to secants
(, )
i
yu

, (, )
j
yu

. These secants have angular coeffi-
cients 1, 1, ,
ij
aa. We consider, that 1, 1,ij aa. If it will
appear, that in this sector almost secant (, ,)
lj
yu u

probably lays, i.e. Its angular coefficients belongs to in-
terval 1, 1, (, )
ij
aa component ij uu F can be included
in structure of model for system (1). The substantiation
of this statement will be more low given. The analysis of
all candidates a priori entering into function (,)
f
Un in
(1) similarly realize.
With each element (,)
qUy
  , 1, #(,)qUy ,
where #( ,)Uy is a cardinal number (,)Uy, we will
associate local structures of static system. We will des-
ignate through q
y
r

coefficient of mutual correlation
between q

and y.
Definition 2. We will say, that variable lj uu is co-
herent with y or has relationship with y, if secant
(, ,)()
ljij Sm
yu uS


SF almost sure. In this case sec-
tor ij S we will name area of a coherence of variable
lj
uu .
For an estimation of closeness of relation lj uu with
y we will introduce coherence coefficient
ij
iju yuy
Qrr

. (2)
Let notice, that 1
ij
Q

and, basically, should be
close to ij
uuy
r. If the given condition is not fulfilled, it
speaks about incoherence to considered component
ij
uu .
Theorem 2 [21]. If the coefficient of coherence ij Q
is equal ,
ij ij
Q


where min( ,),
ij
ijuyuy
rr

 and
ij
iju uy
r


secant (,)
iji j
yuu


belongs to sector ij S.
The stated approach remains fair also for a class of

N. N. KARABUTOV
61
nonlinearities d F



,,
,, |
:,
,,
,1 ,
j
l
k
ii
N
p md
dd
nljlnjn
lj
uRuUUR
fRJ R
fUn fU uu
ljmpk













 


F,
where
j
d, l d its are some numbers.
So, the method of secants allows to carry out the
analysis of structural properties of static system on the
basis of construction of a field of structures. The basic
virtue of the given approach is the possibility of repre-
sentation of structure of system on set of linear functions
(secants). Analysis S S allows to select those elements
from class m F, for which
 

min ,&
ij
u yuyijijij
rrQuu S.
The field of structures can be constructed for a wide
class of static systems. On the basis of ()
Sm
SF it is pos-
sible to make an inference about structure of static sys-
tem in the conditions of uncertainty and by that essen-
tially to narrow a class of research models.
The field of structures can be under construction on set
Ie [21-23], i.e. the set, the received ambassador of
elimination linear making of n y. It is easy to prove cor-
respondence of fields of structures on sets Io and Ie
only for those (,)m fUnF for which for factor of co-
herence ij Q the Theorem 2 is fulfilled.
The example of construction of a field of structures is
reduced in [22,23].
3. Field of Structures on Class s F
Let's spread the approach offered above to a class of
nonlinearities




,,,|:,
,,,.
k
ii N
d
iii i
f
UnuRuUURf R JR
fudu d 
  

s
s
F
Let show, how for class s F to receive coherence area,
i.e. sector to which belong nonlinearity , (,) i
d
in
f
Un u,
i
d. The further account is based on a straightening
method [21,22]. The field of secants is under construc-
tion on set ,, {,}
i
eu nn
ke , where I
ne
e is an error of
forecasting of an output signal of system n y by means
of model ˆˆ
nn
yas, where T
nn
s
IU, k
I
R is an
unit vector, i
ue
k, is coefficient structural properties [21]

,, ,
,,
i
eu nsinin
kkeuneu.
Let consider restriction ,, {}{}
i
eeunn
ke and its
contractions ,eu
i
k
e
 1,ik. For everyone ,eu
i
k
e
 we
will construct a secant


,,0,1,
,ii
ii
euieeeu
ekaa k   .
Secants ,,
(, )
j
j
eu d
ek

for mapping ,,eu d
j
j
k
e
j
d
j
s
u

F
(1)j are set


,,,,0,,1, ,,
,iiiiiii
eu didededeu d
ekaak   , (3)
where ,0, ,1, ,
ii
ed ed
aa
is some numbers,


,,, ,
,,, ,
j
jj
d
eu d nsjjnjnj
kkeudneud

. (4)
Let’s introduce set of secants





,
,,,,I
eedne
KeK eKee

  ,
where , (,){(, ), 1,}
i
eeu
K
eekik   ,
1
,,
[,, ]
k
T
eeu eu
Kk k, ,,,
ii
eu ded
kK, ,
q
ed
K
R, qk

,
,,,
(,){(, ),1}
jj
edeud
Ke ekj 

.
Statement 1. Any secant , (,)(, )
i
eu e
ekKe   at
ji

limits element ,, , (,)(, )
jj
eu ded
ekK e   from
below.
Upper bound of sector i
d
S to which should belong
secant ,, (, )
ii
eu d
ek

, on the basis of set (,)
e
K
e to
define it is not possible. Therefore to the received field of
structures we will add secant ,, (, )
ii
es d
ek

, where
i
s
R

, (/)
T
ii
s
IUu. As ,es k represents a transmis-
sion coefficient between n
s
R and n e, and local co-
efficient structural properties ,i
eu
k are stationary the
same property will possess and ,i
es
k. Hence, the coeffi-
cient of determination ,, (, )
ii
es d
ek

will be not less 2
ui
ke
r
and ,,
2
eu d
ii
ke
r at least for 01
i
d

. Thus, ,,
(, )
ii
es d
ek

it
is possible to use as an upper bound of area of coherence
i
d
S. It is fair, as transformation ,
i
d
in
u conducts to a
modification only the statistical moments of signal
j
u,
but not its structures. In the supposition, that 01
i
d

,
we receive reduction of the two first statistical moments
that leads to a raise of factor of determination
,,
(, )
ii
es d
ek

.
The field of structures for system (1) on class s F
looks like
 
 

 ,,
,,, ii
Ss esd
Ke ek  SF ,
where (0; 1),
i
d

1ik

.
In structure of model of system (1) we will include those
functions i
d
is
u

F for which condition ,,
22
ieud
ii
keke
rr is
satisfied. The evaluation of coefficient of a coherence (2)
in this case leads to an inadequate estimation of connec-
tion between e and i
d
i
u.
Let’s reduce algorithm of decision-making relative to
structure of model of system (1) on class
s
F.
1) We build secants , (, )
i
eu
ek  on plane , (,)
i
eu
ke for
,eu
i
k
e
 .
2) We define coefficient of determination ,
2
eu
i
ke
r for

N. N. KARABUTOV
62
,
(, )
i
eu
ek

.
3) If ,
2
eu
i
ker
r

, where 0
r

 is some set quantity
the system is linear on variable i
uU.
4) If condition ,
2
eu
i
ker
r

 is not fulfilled, is find
,,
(, )
ii
es d
ek

and defined ,,
2
eu d
ii
ke
r. At ,,
2
eu d
ii
ke r
r

 ele-
ment ,
i
d
in s
uF is included in model structure.
Remark. Condition ,
2
eu
i
ker
r

 is an indication of
linearity of system on variable i uU.
e
 for class
s
F it is possible to present mapping
also in space (,||)
en
K
e. Thus result of synthesis will
not.
On Figure 1 the field of structures ()
Ss
SF for plant
(1) with vector [0.8 1.42]T
A, function
0,3
2,
(,)0,5 n fUn u and limited noise n

is reduced ary.
For secants e  following values of coefficients of de-
termination are received: ,1
20.97
eu
ke
r, ,2
20.91
eu
ke
r,
,3
20.98
eu
ke
r. 0.94
r

. Comparison of received values
2
r with r

has allowed to exclude
s
F elements 1
1,
d
u
3
3
d
u from set. For 2
2
d
u on the basis of a straightening
method value 20.3d, and ,,0,3
1
20.99
eu
ke
r is received.
On Figure 1 the area of coherence 2
d
S for variable
2
2
d
u is shown.
So, the method of construction of a field of structures
S
S on sets m F and
s
F for nonlinear system (1) is of-
fered. It is based on the analysis of an informational por-
trait of system and in many respects depends on an kind
of class m
F. Very often (class
s
F) portrait e  is under
construction in the space received by transformation of
initial variables of system of identification. It speaks, in
particular, complexity of a problem of structural identi-
fication for which solution nontrivial approaches and
methods should be attracted. In particular, on class
s
F
for decision-making on structure of model of system (1)
field of structures ()
Ss
SF is under construction on set
of coefficients structural properties systems as it is a
measure of linearity of system and allows to establish
degree of a deviation from this indicator of performances
of system.
4. Field of Structures on Class Ff
Let consider plant (1) with nonlinearity




,,
,,|:,
,,1,
fi iN
in in
fUnuRuUf R JR
fu cui




  


F
(5)
where c

is some numbers.
Let on the basis of Io set I{,, }
ennN
Uen J,
where ˆ
nnn
eyy, ˆˆ
nn
yas is generated.
It is necessary for class
f
F on the basis of handling
of set Ie to construct a field of structures for system (1).
For a problem solution we will take advantage of the
approach stated in Section 2. We take variable i
u for
-0,50-0,250,00 0,25 0,50
-1,0
-0,5
0,0
0,5
1,0
1
,ue
k
e

2
d
S
),( 3.0
2,, se
ke 
1
,ue
k
e

),( 3.0
2,,ue
ke 
),( 3.0
2,, se
ke 
),( 3
,ue
ke 
),(,,i
ue
kee 
i
ue
k,
),( 2
,ue
ke 
),( 1
,ue
ke 
Figure 1. A field of structures of system (1) with
(,) s fUn

F ary.
which condition i
ue e
r

, where 0
e

 is some set
magnitude is satisfied. We will construct for i u map-
ping ,, {}{}
i
eeunn
ke and a secant


,,0,1,
,ii
ii
euieeeu
ekaa k   .
Let consider function i
u

and for everyone
[1; ]J




 we will define coefficient structural
properties ,,
i
eu
k

, where
J

is some segment of the
real numbers which choice will be given more low.
For everyone
J



we will construct secant
,,
(, )
i
eu
ek


(3) for mapping ,,eu
i
k
e

.

we will change
until for coefficient of determination of secant
,,
(, )
i
eu
ek

 the condition will be satisfied
,,
2
eu
i
ke e
r


, (6)
where ()0
eee



 .
Let introduce set of secants






,,
,, ,,
2
,,
such, that
ii
eu
i
eu eu
ke e
ke ekJ
r





 
.
Upper bound

of interval
J

is defined from a
condition of violation (6).
Definition 3. We will name set ,, (,)
i
eu
ke

 a field of
structures ,, (,( ,))
i
Sf eu
ke

SF of the system (1), cover-
ing nonlinearity (,)fUn on class
f
F.
The field of structures for system (1) on class
f
F
looks like



 


,, ,,
,, ,,,1
ii
S feueu
keke Ji



SF .
Let consider secant ,, (, )
i
eu
ek

 with indicator



and sector ,,1 ,, ((, ),(,))
ii
eu eu
Sek ek

 . We
will define coefficient of coherence Q

on class
f
F
,, ,,1eu eu
ii
keke
Qr r



.
On Q

we will judge membership , ()
in
fu to sector

N. N. KARABUTOV
63
S

.
Let consider a choice of magnitude e

 in (6), and,
therefore, and

. For everyone ,in u

we will calculate
coefficient of correlation
i
ue
r

. We will define parameter
J


 from a condition of violation of an inequality

max
ie
ue
r



 . (7)
On the basis of

we will set magnitude e

 in (6)
and we will find Q

.
Theorem 3. Let for system (1) the field of structures
()
Sf
SF is constructed. If condition (),
i
fu e
rQ


nonlinearity , ()
in f
fu F is coherent with signal n y is
satisfied and it can be included in model structure.
On set , {}
in
u

the field of structures be of the form
 

 
,,,
Sf ii
eu eu


 SF .
Unlike classes ,
s
FF, field ()
Sf
SF allows to estimate
structure of nonlinearity (,)fUn on final set of ap-
proximating functions i u

with index
J


. In some
cases by an amount of members of an approximating
polynomial it is possible to set structure of function
(,)fUn.
The example of a field of structures ,, (,( ,))
i
Sf eu
ke

SF
for system (1) with nonlinearity 11 () sin()fu u and a
vector of parameters [1.3 1.61.7]T A is shown on
Figure 2.
In Figure 2 following symbols are used: a straight line
with a rhomb is a secant with 1

; a straight line with
quadrate is a secant with 2


; a straight line with a
triangle is a secant with 2.5

; a straight line with a
circle is a secant with 3

. Here the example of modi-
fication n e for case 1

 is shown. Maximum value
in (7) is reached at 2.5

. For this value 0.81
e
 
.
0.81Q

. sin(),0.83
i
ue
r and, therefore, 1, ()
n
fu is
coherent with n y. Let’s notice, that at (0;2)


-0,4 -0,20,00,20,40,6
-1,0
-0,5
0,0
0,5
1,0

S
n
e
n
e
 ,, i ue
k
),( ,,


i
ue
ke
Figure 2. A field of structures of system (1) on class
f
F
with nonlinearity 11
( )sin( )
f
uu.
11
() cos()fu u

, and at (1;3)


11 ( )sin( )fu u, and
adequacy of model with 11 () cos()
c
fu fu on set Ie
above, than with 11 () sin()
s
fu fu. For deci-
sion-making concerning structure 1 ()fu in the field
,,
(,( ,))
i
Sf eu
ke

SF we fulfil check of candidates ,
cs
ff
on set Io. Results of an estimation of adequacy speak
about necessity for model to use element
s
f
fF.
Remark. If to build a field of structures ()
Sf
SF on
set Io owing to low coefficients of correlation for deci-
sion-making it is necessary to pass in space of coeffi-
cients structural properties. Here it is completely appli-
cable the approach stated above.
The offered approach allows to estimate nonlinearity
structure through variable n e. It is connected with that,
the problem of estimation (,)
f
fUnF on set Ie de-
mands performance double approximation, that in a con-
sidered case is not realized. Further the method of identi-
fication of function (,)
f
fUnF on a basis on the
adaptive approach is offered.
5. Adaptive Procedure of an Estimation of
Function


,
f
Un on Class f F
Let it is known variable i
uU and maximum degree



in (5). We will designate
2
,, ,12
,
TT
p
p
nininin p
X
uuuRC cccR




 
.
Desired dependence looks like


,
T
nin n
efu CX  
, (8)
where R


.
For estimation ,C

it is applicable adaptive model


1, 11
ˆˆ
ˆˆ ˆT
nninnnn
efu CX 

, (9)
where ˆn

, ˆn C is adjusted parameters.
Let designate an error of prediction n
e by means of
model (9) through ˆ
nnn
ee


. We will introduce mis-
alignment ,,
ˆ()()
nin in
fu fu  .
Algorithms of adaptation ˆn

, ˆn C we search from a
condition
22
ˆ
ˆ
ˆ
ˆ
min,min,
nn
oo
nn
C
C

   (10)
From (10) it is received
1
ˆˆ
nn Cnn
CC X 

 , (11)
11
ˆ
ˆˆ T
nn nnn
CX

 

 , (12)
where 0, 0
C 

 is the parameters ensuring con-
vergence of algorithms.
As function , ()
in
f
u is unknown, we will use estima-
tion ,1
ˆ
() /
inn n
fu e



.

N. N. KARABUTOV
64
Let note the Equations (11), (12) are rather misalignment
model parameters
1nnCnn
CC X





 , (13)
11
ˆT
nn nnn
CX



 , (14)
where ˆ
nnn
CCC   , ˆ
nnn
 
.
Theorem 4. Let ||


 , || ||
n
X, and also ex-
ists such 1
0





 , that 2
nnn 

 . Then algo-
rithms (13), (14) converge, if
2
2
0,
C
n
X
  (15)

021,




  (16)
where 1
ˆ
||
T
nn
CX

 0n , || || is Euclidean norm.
If n
e contains uncertainty n

limited on level
||
n

  Algorithms (13), (14) will converge if it is
fulfilled (15) and

1max
n
n


   ,

21
0








 ,
where 1

.
Theorem 5. Algorithms (13), (14) do not possess an
asymptotic stability.
So, Algorithms (13), (14) do not allow to receive as-
ymptotically an estimation of parameters of system (8).
Here uncertainty in which algorithms are applied affects.
The example of work of adaptive system of identifica-
tion of nonlinear system (1) with 11
() sin()
f
uu and
[1.3 1.61.7]T A is shown on Figures 3, 4. Vector
2
,,
[]
T
ninin
Xuu. The Figure 3 reflects process of tuning
of parameters of model (9). On Figure 4 the informa-
tional portrait reflecting results of adequacy of an esti-
mation of function 1
()
f
u is shown. The determination
coefficient was equaled 0.99, that follows from position
of secant ˆ
(,)
f
f  . The expectation and average quad-
ratic deviation for 1
()
f
u and 1
ˆ()
f
u are accordingly
equal:

10.78fu ,

1
ˆ0.767
fu ,


10.26fu
 ,


1
ˆ0.28
fu  
So, the mode of construction of a field of structures for
nonlinear static system (1) on the basis of the analysis of
set Ie is offered. The sector condition is introduced and
the mode of its construction for various classes of
nonlinearities is offered. The analysis of structural prop-
erties is fulfilled in space of secants of an observable
informational portrait of system or its virtual analogues.
0 204060
-0,6
0,0
0,6
1,2
1,4
1,6
n

ˆ
n
 ˆ
nncc ,2,1 ˆ
,
ˆ
n
c,2
ˆ
n
c,1 ˆ
n
Figure 3. Tuning of parameters of model (9).
-0,5 0,00,51,0
-1,0
-0,5
0,0
0,5
1,0
f
ˆ

1
uf
)(
ˆ1
uf


Figure 4. An estimation of adequacy of model (9) on plane
ˆ
(,)
ff
.
Algorithms and methods of identification of nonlinear
component system are developed for various classes of
nonlinearities.
6. Structures of Nonlinear Static Systems
The approaches stated above allow to make for a various
class of nonlinearities a solution on structure of model of
system. Field S S gives conception about system struc-
ture on a vector space of secants. Naturally there is a
problem on search of mapping for nonlinear static sys-
tem which would allow to make imprisonment before
trail about its nonlinear properties. Complexities of in-
troduction such mapping were considered above. Despite
it, the informational structures, allowing to receive con-
ception about nonlinearity of static system are more low
offered.
Let consider informational set

 



II,, ,0,
eenn N
eUe UnJN .
Appropriate Ie informational portrait :{} {}
en n
Ue
owing to an operation of perturbation nR

 has ir-
regular character and the solution does not allow to make
on nonlinear properties of system (1). The told confirms

N. N. KARABUTOV
65
Figure 5 on which the informational portrait for the plant
considered in Section 3 is shown.
Let consider set




II, ,,0,
kknn N
eke knJN ,
where n kR is factor structural properties systems (1)
on class r F (,,)rmsf for ,in n uU.
Let order values n k on increase on set
N
J
, that is
we will construct a variation number series. As a result
we will receive set {}
q
v
k, where [0, ]
v
N
qJ N . To
everyone v
q
k there corresponds value v
q
e. Hence
 


II,,,0,
kk
vvvv v
qq N
eke kqJN .
On Ik
v we will construct mapping :{} {} vv v
eq q
ke to
which on Euclidean plane (,)
vv
qq
ke there corresponds
some structure ,ke
v
S. We assume, that function v
q
e is
simple, univalent and intersects axis v
q
k in some point,
that is ,ke
v
S has a singular point. n e represents an error
of prediction of an exit of system (1) on the basis of lin-
ear static model with input T
nn
s
IU. v
q
k also is func-
tion n
e. Therefore mapping v
e
 will represent some
function, tending growth. It is fair for any class r
F. It is
known [21], that for linear static systems the coefficient
structural properties is an estimation of parameter of
model for ,in n
uU. For nonlinear systems it represents
the function varying in some limited range. Starting with
Ik
v it is concluded, that process of modification v
q
k at
magnification q has monotone character. The same
character will be carried also by function v
q
e but as v
q
e
depends from (,)
f
Un and n

its modification will
differ from the linear. For an estimation of degree of
nonlinearity we will construct secant (,)
vv
qq
ek  . Hence,
unlike e
 mapping v
e
 is structurally informative. So,
fairly
Statement 2. For system (1) with (,)r fUn

F
(,,)rmsf on set Ik
v there is mapping
:{} {}
vv v
eq q
ke to which on Euclidean plane (,)
vv
qq
ke
there corresponds the informational structure ,,
ke
v
S al-
lowing to analyze nonlinear properties of system.
Unlike Ie on set Ik
v structure ,ke v
S has more or-
dered character. It is necessary to notice, that ,ke
v
S it is
applicable also for mapping of linear static systems. For
(,) r fUnF (,,)rmsf structure ,ke
v
S contains a
singular point, who unlike dynamic systems is not char-
acteristic performance of a stability of system (1). She
serves as confirmation of a monotonicity of the curve
described by mapping v
e
. If to take advantage
Lyapunov’s

of characteristic indexes they, as one
would expect, state the estimation of index

close to
zero.
Let state a method, allowing to define type of a singu-
lar point for the given class of systems. We will intro-
duce functions
0,5 1,01,5 2,02,5
-1,0
-0,5
0,0
0,5
1,0
n
e
n
u,2
Figure 5. Projection of informational portrait

e to plane
2
(,)ue.
,,
ln ,ln
vv
kqqeqq
ke   (24)
Let find such value v
N
qJ, for which
,
min kno
qq

, , min en o
qq

.
Knowing o q, from (24) we define values ,
v
qo
k, ,
v
qo
e,
and, therefore, and singular point ,, (, )
vv
qo qo
Mk e. We
name ,, (, )
vv
qo qo
Mk e V-point. This title follows from the
following. Let's construct on plane ,, (, )
kq eq

 a portrait
of system (1) to which there corresponds structure ,ke
v
SL .
As show simulation data, for system (1) with
(,) r fUn

F it will look like, shown on Figure 6.
On Figure 6 the structure of nonlinear system (1) in
spaces ,, (, )
kq eq
YY , ,, (,)
kq eq
KE is presented. From
Figure 6 we see, that ,, (, )
kq eq
YYstructure ,ke
v
SL con-
tains V-point in space. In her the rate of motion of points
of a trajectory with magnification q comes nearer to
zero, reaching the minimum value at o qq. Then
o
qq

 the rate of motion of a point again increases.
Such behaviour of a trajectory speaks properties of the
logarithm. So, fairly
Statement 3. In space ,, (, )
kq eq
YY structure ,ke
v
SL
of system (1) has a special V-point.
Structure ,ke
v
S in space ,, (,)
kq eq
KE allows to esti-
mate nonlinear properties of system. From Figure 6 it is
see, that ,ke
v
S has nonlinear character and reflects a state
system (1) with ()f

s F, 0,3
2,
()0.5n
f
u. Representa-
tion ,ke
v
S is more informative, than a portrait on Figure
5. For decision-making on that, how much the change of
trajectory ,ke
v
S differs from linear, it is possible to take
advantage of methods of secants or straightening.
Representation ,ke
v
S for some class of nonlinearities
allows to make a solution on structure function
f
.
Theorem 6. Let the coefficient structural properties v
q
k
is defined on segment [, ]
v
v
vq
kq
J
kk, where v
q
k

,
v
q
k

, v
N
qJ, (,)fUn

s
F. If exists such *0d

,

N. N. KARABUTOV
66
-7 -6 -5 -4 -3 -2 -10
-6
-4
-2
0-1,5 -1,0 -0,50,00,5
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
qe,

qe,

v
q
e
qe,

v
SL ek,
v
Sek,
v
q
e
v
q
k
V-point
q
Figure 6. Informational structures ,
v
SLke ,
v
Ske of system
(1).
that

|1
v
v
q
q
kk O,
where (1)O is some neighborhood 1 * (, )
i
f
ud defines
structure of nonlinearity of system (1) on class s
F.
Remark. The Theorem 6 gives the new formulation of
a method of the straightening stated in Section 3. In her
the criterion of linearity directly is used.
The Theorem 6 is applicable only to so-called control-
lable models [21].
Let consider structure ,ke
v
S of system (1) on class
f
F.
In this case the theorem 6 is not applicable, so function
(,)
f
Un explicitly does not contain the controllable pa-
rameter. For a choice of an amount of members in an
approximating Polynomial (5) it is possible to take ad-
vantage of the approach stated in Section 4. As informa-
tional set it is necessary to use ,,, {,}
i
vv
euq q
ke

, v
N
qJ.
Let consider nonlinearity (,)
ijm
fuu F, (, )
ij
uuU

.
Let generate sets ,, , I(, ),I(,),I(,)
kikjkij
vv v
eu eueuu
ekekek to
which there correspond structures ,(I )
ke i
v
iu
v SS
,,
(I ),(I)
kejkei j
vv
juijuu

vv
SS SS. For each structure secant


is received (,,)ijij

, , (, )
vv
eu
ek


 .


it is
characterized by parameters 0, 1, ,
ii
aa.
Theorem 7. Let on Euclidean plane ,, (,)
i
vv
eu qq
ke
structures ,,
ijij
SS S, sector (,)
ij
SSS to which be-
long secants ,
ij

, and 1,1,ij aa are constructed.
Then ijSS, if for almost v
N
qJ
 
,,, ,, ,
vv v
iqi oiqij oijjqj oj
eae ae a  ,
where

,
,,
min min
max max
v
qqi
qq
iv
qqiq j
gg
eu
euu
 ,

,
,,
max max
min mi n
v
qqi
qq
jv
qqiq j
gg
eu
euu
 .
For ,ke
v
S it is possible to construct sector, using the
ideas explained in Section 2. For the nonlinearities be-
longing to class m F, it is easy to estimate influence
which on her render the elements of vector n U setting
structure of function (,) m fUnF. For this purpose it is
possible to take advantage of the approach offered in
[21].
7. Conclusion
The concept of a field of structures for nonlinear static
systems on set of secants is introduced. The mode of
construction of a sector condition for a nonlinear part of
system is offered. Algorithms and methods of deci-
sion-making in the form of nonlinearity are described.
For the nonlinearity described by a class polynomial of
functions, the adaptive approach for an estimation of her
structure in the conditions of uncertainty is offered. The
structure on which it is possible to make a solution on
nonlinearity of system is introduced.
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