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Intelligent Control and Automation, 2010, 1, 59-67 doi:10.4236/ica.2010.12007 Published Online November 2010 (http://www.SciRP.org/journal/ica) Copyright © 2010 SciRes. 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 E-mail: kn22@yandex.ru 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 Copyright © 2010 SciRes. ICA 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 SSF 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 Copyright © 2010 SciRes. ICA 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 fUnF 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 Copyright © 2010 SciRes. ICA 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 uF 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 Copyright © 2010 SciRes. ICA 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 fF. 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 fUnF 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 fUnF 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 Copyright © 2010 SciRes. ICA 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 Copyright © 2010 SciRes. ICA 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 fUnF (,,)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 Copyright © 2010 SciRes. ICA 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 SSS to which be- long secants , ij , and 1,1,ij aa are constructed. Then ijSS, 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 fUnF. 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. 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