Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems

The purpose of research is to establish and study conditions, that define when system identification problems are well-posed and when solutions become unstable and therefore practically unfit for parametric-structural identification on the base of description in the form of infinite expansions. Resu...

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Veröffentlicht in:	Кибернетика и вычислительная техника
Datum:	2014
Hauptverfasser:	Gubarev, V.F., Gummel, A.V., Melnichuk, S.V.
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Veröffentlicht:	Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України 2014
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spelling	Gubarev, V.F. Gummel, A.V. Melnichuk, S.V. 2015-07-09T19:31:07Z 2015-07-09T19:31:07Z 2014 Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems / V.F. Gubarev, A.V. Gummel, S.V. Melnichuk // Кибернетика и вычислительная техника. — 2014. — Вип. 177. — С. 5-15. — Бібліогр.: 10 назв. — англ. 0452-9910 https://nasplib.isofts.kiev.ua/handle/123456789/84521 681.5 The purpose of research is to establish and study conditions, that define when system identification problems are well-posed and when solutions become unstable and therefore practically unfit for parametric-structural identification on the base of description in the form of infinite expansions. Results: It was shown that solution high sensitivity is associated with illconditioned matrices that are used to estimate the coefficients of the model. For finite-frequency and subspace identification methods it was demonstrated that depending on the ratio of the input data error and the condition number of matrix the solution of identification problem can be both stable and unstable. Рассмотрены проблемы моделирования и идентификации сложных динамических систем. Для задач структурно-параметрической идентификации на основе описания в виде бесконечных разложений установлены и исследованы условия, при которых эти задачи являются корректно поставленными, а когда их решения становятся неустойчивыми и, следовательно, практически непригодными. Полученные результаты являются фундаментальными и дают более глубокое понимание процесса идентификации. Розглянуто проблеми моделювання та ідентифікації складних динамічних систем. Для задач структурно-параметричної ідентифікації на основі опису у вигляді нескінченних розкладів встановлено та досліджено умови, за яких ці задачі є коректно поставленими, а коли їх розв’язки стають нестійкими і, отже, практично непридатними. Отримані результати є фундаментальними і дають більш глибоке розуміння процесу ідентифікації. en Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України Кибернетика и вычислительная техника Системы и интеллектуальное управление Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems Корректная идентификация бесконечномерных систем ядерного типа и многомерных систем Коректна ідентифікація нескінченновимірних та багатовимірних систем ядерного типу Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems
spellingShingle	Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems Gubarev, V.F. Gummel, A.V. Melnichuk, S.V. Системы и интеллектуальное управление
title_short	Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems
title_full	Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems
title_fullStr	Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems
title_full_unstemmed	Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems
title_sort	well-posed identification of nuclear type infinite and multidimentional systems
author	Gubarev, V.F. Gummel, A.V. Melnichuk, S.V.
author_facet	Gubarev, V.F. Gummel, A.V. Melnichuk, S.V.
topic	Системы и интеллектуальное управление
topic_facet	Системы и интеллектуальное управление
publishDate	2014
language	English
container_title	Кибернетика и вычислительная техника
publisher	Міжнародний науково-навчальний центр інформаційних технологій і систем НАН України та МОН України
format	Article
title_alt	Корректная идентификация бесконечномерных систем ядерного типа и многомерных систем Коректна ідентифікація нескінченновимірних та багатовимірних систем ядерного типу
description	The purpose of research is to establish and study conditions, that define when system identification problems are well-posed and when solutions become unstable and therefore practically unfit for parametric-structural identification on the base of description in the form of infinite expansions. Results: It was shown that solution high sensitivity is associated with illconditioned matrices that are used to estimate the coefficients of the model. For finite-frequency and subspace identification methods it was demonstrated that depending on the ratio of the input data error and the condition number of matrix the solution of identification problem can be both stable and unstable. Рассмотрены проблемы моделирования и идентификации сложных динамических систем. Для задач структурно-параметрической идентификации на основе описания в виде бесконечных разложений установлены и исследованы условия, при которых эти задачи являются корректно поставленными, а когда их решения становятся неустойчивыми и, следовательно, практически непригодными. Полученные результаты являются фундаментальными и дают более глубокое понимание процесса идентификации. Розглянуто проблеми моделювання та ідентифікації складних динамічних систем. Для задач структурно-параметричної ідентифікації на основі опису у вигляді нескінченних розкладів встановлено та досліджено умови, за яких ці задачі є коректно поставленими, а коли їх розв’язки стають нестійкими і, отже, практично непридатними. Отримані результати є фундаментальними і дають більш глибоке розуміння процесу ідентифікації.
issn	0452-9910
url	https://nasplib.isofts.kiev.ua/handle/123456789/84521
citation_txt	Well-Posed Identification of Nuclear Type Infinite and Multidimentional Systems / V.F. Gubarev, A.V. Gummel, S.V. Melnichuk // Кибернетика и вычислительная техника. — 2014. — Вип. 177. — С. 5-15. — Бібліогр.: 10 назв. — англ.
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fulltext	5 Системы и интеллектуальное управление УДК 681.5 WELL-POSED IDENTIFICATION OF NUCLEAR TYPE INFINITE AND MULTIDIMENSIONAL SYSTEMS V.F. Gubarev, A.V. Gummel, S.V. Melnychuk Space Research Institute of the National Academy of Sciences of Ukraine and State Space Agency of Ukraine, Kiev, Ukraine Рассмотрены проблемы моделирования и идентификации сложных динамических систем. Для задач структурно-параметрической идентификации на основе описания в виде бесконечных разложений установлены и исследованы условия, при которых эти задачи являются корректно поставленными, а когда их решения становятся неустойчивыми и, следовательно, практически непригодными. Полученные результаты являются фундаментальными и дают более глубокое понимание процесса идентификации. Ключевые слова: системная идентификация, некорректно- поставленная задача, конечно-частотная идентификация, идентификация на основе выделения ортогонального подпространства (4SID), регуляризация. Розглянуто проблеми моделювання та ідентифікації складних динамічних систем. Для задач структурно-параметричної ідентифікації на основі опису у вигляді нескінченних розкладів встановлено та досліджено умови, за яких ці задачі є коректно поставленими, а коли їх розв’язки стають нестійкими і, отже, практично непридатними. Отримані результати є фундаментальними і дають більш глибоке розуміння процесу ідентифікації. Ключові слова: системна ідентифікація, некоректно- визначена задача, кінцево-частотна ідентифікація, ідентифікація на основі виділення ортогонального підпростору (4SID), регуляризація. INTRODUCTION System identification problems belong to a class of inverse problems which have the feature that under certain conditions its solutions become unstable i.e. sensitive to the errors in input data. Especially it takes place in multidimensional cases when it is necessary to assume a model set which adequately represents the original system, then select a model order and find other parameters using only general information about unknown plant. Processes in such system may be so complex that on the base of fundamental laws and theoretical results it is often impossible to define even the model set which includes a model that completely describes the system that generates data. Also it is supposed that there is a causation between finite input and output of a plant. The only possible way in such cases is to use different infinite-dimensional expansion as model set [1, 2].  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 6 Computational mathematic methods serve as a foundation of such approach (e.g. Galerkin method and other iterational methods, methods based on Green function or impulse response for system with disturbed or lumped parameters [3–5]). Example of such complicated system is a heating processes in power station. Fuel supply that affects vapor parameters is considered as an input. Processes of burning, heating and vaporizing are so complex that it is impossible to describe them in any other way besides approach proposed above, that is using infinite- dimensional model which links input (fuel supply) and output (vapor parameters). In this case input-output map is an impulse response that induces nuclear Hankel operator or equivalent non-rational transfer function. The approximation of infinite-dimensional linear system by finite-dimensional ones is a subject of interest in a mathematical theory. In [2] using output normal realization the convergence of the finite-dimensional approximation to precise model was proven and error bounds on the truncated realization were given. These results are important for model reduction problem, i.e. conversion of the complex infinite-dimensional description into simple low-dimensional model [6]. In identification problem the reduced model order selection is defined primarily by the well-posedness, for which it is necessary to satisfy conditions providing the stability of the solution with respect to errors in the initial data. In this sense truncated model order will be the main regularization parameter which means that the dimension of rational approximation should be in agreement with the errors in available data. Hence regularization procedure should be incorporated in the existing methods of identification allowing to find the stable solution and to extend model order when errors tend to zero. This paper shows how this problem may be solved correctly with selection of highest model order admissible by stability conditions using two popular identification methods. Besides numerous computational experiments that were conducted allowed us to clarify a cause of ill-posedness which is an essential to identification problem. Fundamental properties of the regularized solutions are established irrespectively to identification method used for plant that generated output data. Systems with single input and single output (SISO) are considered because even in this case nontrivial results were revealed. PROBLEM SETTING For infinite-dimensional nuclear type linear SISO systems model set can be written as η+=ξ++= czybuAxx ,& , (1) where A is a linear operator mapping between infinite-dimensional linear vector spaces, b is a linear operator mapping one-dimensional space into infinite- dimensional one, c is a projecting operator. Under suitable assumptions they lead to existence of a non-rational transfer function bAsIcsG )()( −= . Values ξ and η denote unknown perturbation in input and noise in output that may be present in real system. Description (1) be also appropriate for multi-dimensional case when system  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 7 has finite but large and unknown dimension. In this case A is a matrix, cb, are vectors and transfer function is a rational. Such system is also a nuclear type if it eigenvalues belong to left half-plane. Let description (1) be proper for unknown real system, i.e. is a nuclear type and (1) includes true model. It is required to find the finite-dimensional matrix A and vectors cb, using experimental data ))(),(( tuty that would provide a rational approximation of the real system described by (1) so that difference between system output and model output would be either in accordance with errors ηξ, or defined by stability condition of the solution. In other words it means to solve identification problem correctly. For finite multidimensional case it means to construct the model with significantly less dimension than dimension of the real system. Here is a full analogy with so-called model reduction problem [6]. In this case rational approximation (1) can be represented as ,, xcyubxAx T nnn =+=& (2) where nA is a matrix nn × , nb is a column-vector, Tc is a row-vector (T is a transposition). FINITE-FREQUENCY IDENTIFICATION WITH REGULARIZATION Model reconstruction using Frequency method of identification is realized on frequency domain parameters, extracted from experimentally measured output when input is excited by the harmonic test signal. The classical frequency approach has a long history and here we don’t give the review of these methods because readers can find it for example in the paper [7]. We note that in recent years the methods were significantly developed by Alexandrov and Orlov, see [7–11]. If (2) will be written in the form of Jordan realization it is easy to get the following input-output relation ( ) [ ] .)()(sin)(cos )()( 10 )( 1 0 ∑ ∫ ∑ ∫ = θ−α− = θ−λ− θθθ−β⋅+θ−β⋅+ +θθ= P p t p s pp c p t Q q t t q dutftfe duegty p q (3) For simplicity sake here it is accepted that initial state under 0t is zero: 00 =t ; eigenvalue multiplicity is missing; Q is a number of real and P is a number of complex-valued eigenvalues with 0,0 ≥α≥λ pq . Besides ,Pp bcbcf bcbcf Qqbcg c np s np s np c np s p s np s np c np c np c p nqnqq 1 , , ,1 , = −= += == , (4) nqc and nqb are components of the vectors nb and nc for real eigenvalues and  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 8 c npc , s npc , c npb , s npb are components of these vectors corresponding Jordan block       α−β− βα− pp pp . From equations (4) it is possible to select different canonical form. For example, assuming 1=nqb , 1=c npb , 0=s npb we receive the control canonical realization and for 1=nqc , 1=c npc , 0=s npc we have observable canonical realization. By analogy balanced realization may also be obtained. In frequency method the system and model inputs are usually exited by polyharmonic signal with K sinusoids of different frequencies kω and amplitudes ku : 0,sin)( 1 ≥ω= ∑ = ttutu K k kk . (5) For sample data the regression equation is derived if substitute (5) in (3) and integrate. As result we obtain ( ) ( )[ ] ( ) ( ) ,cos sin cossin )( 1 1 3222 1 1 4122 1 3223 1 1 22             ωγ−γ+ ω+λ ω + +ω         γ+γ+ ω+λ λ + +βγ−γ+βγ+γ+     + ω+λ ω = ∑ ∑ ∑ ∑ ∑ ∑ ∑ = = = = = λ− = = λ− tff g tff g etfftff e g uty k Q q P p s p c p kq qk k Q q P p s p c p kq qq P p t p s p c pp s p c p K k Q q t kq qk k p q (6) where nPQ =+ 2 , ( ) ( )222211 ),,( pkp p pkp p kpp β−ω+α α + β+ω+α α =ωβαγ=γ , ( ) ( )222222 ),,( pkp pk pkp pk kpp β−ω+α β−ω + β+ω+α β+ω =ωβαγ=γ , ( ) ( )222233 ),,( pkp p pkp p kpp β−ω+α α − β+ω+α α =ωβαγ=γ , ( ) ( )222244 ),,( pkp pk pkp pk kpp β−ω+α β−ω − β+ω+α β+ω =ωβαγ=γ . When ppqPQ βαλ ,,,, are known formula (6) becomes linear regression which can be used for qg , c pf , s pf determination. In this regard three stages of identification  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 9 are proposed. At first maximal admissible by stability condition Q and P are defined. In other words a solution of the identification problem for Q and P should be weakly sensitive to errors in available data and further increase of order n will lead to scattering of the estimated parameters. Then Q and P may be considered as a regularization parameters. Here we have analogy with function recovery problem which is given in the form of approximately defined coefficients of Fourier expansion. At the second stage eigenvalues (i.e. ppq βαλ ,, ) are calculated for known model order. At the final stage parameters qg , c pf , s pf are estimated from equations (4) using linear regression and then coefficients c np s npnq c np s npnq cccbbb ,,,,, that correspond to Jordan realization in suitable canonical form are founded. For finding solution on the first and second stages it is proposed to use the well known equations of the frequency identification method [7] ,,1,0)()()( KkjWjjV kkkk ==ωΨ+Φ−ω (7) where ,1−=j )( )()( k k k jW jVjG ω ω =ω and 012 2 1 1 )()()( vvjvjvjjV kn n kn n kk +ω++ω+ω=ω − − − − K , 011 1)()()( wwjwjwjjW kn n kn n kk +ω++ω+ω=ω − − K are analytical expressions for numerator and denominator of transfer function represented as frequency response, kk jΨ+Φ is a value of transfer function on imaginary axis which may be evaluated from measured in experiment output )(ty according to [7, 9] ∫ τ ω=τΦ 0 sin)(2)( tdtty u k k k , ∫ τ ω=τΨ 0 cos)(2)( tdtty u k k k , Kk ,1= . (8) For large but finite τ kΦ and kΨ are determined with errors. If ∞→τ kΦ and kΨ tend to precise values when certain requirements to the choice of kω are realized [7, 9]. Besides correct identification on the base (7) is achieved when nK ≥ . Then using corresponding part equations of (7) parameters iv )1,0( −= ni may be expressed through iw ),0( ni = and after that we eliminate iv from remaining equations. This procedure is correct because the relevant linear system equations has a Vandermonde determinant, composed from different kω only, i.e. it is a nonsingular and may be computed exactly. Details of this procedure one can find in paper [12]. Here we write the final result namely the linear equation of the system with respect to vector ),,,( 10 nwwww K= : gHw = , (9)  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 10 where matrix H and g on the right side have a complicated dependence on kΦ , kΨ , kω that is ),,( kkkHH ωΨΦ= , ),,( kkkgg ωΨΦ= . Their expressions are given in [12]. Large finite τ provides approximate kΦ , kΨ . As a result system (9) has given approximate matrix H and right hand side of equation. Nuclear type systems generate the data which lead to ill-conditionality of the matrix H even for not large n , i.e. the problem (9) becomes ill-posed for multidimensional model. It was established in [12] that solution of (9) will be stable if ,11)( <      τ ε+ +εcHk (10) where )(Hk is a matrix H condition number; ε is a boundary (or variance) for noise and perturbation; cε defines level of computational error. Detailed researches of the solution stability problem were realized by means of numerical experiments. Wide class of systems with different structure parameters and noise have generated data for identification which thereupon were utilized in algorithms that estimates Φ̂ , Ψ̂ and matrix H . Model order n and therefore a dimension of a system equation (9) was significant parameter. So matrix H and vector g were computed for different n , i.e. we formed set of nH and ng for different n . Condition number of the matrices obtained in numerical experiments is evaluated by SVD decomposition ,T nnnn VUH Σ= (11) where nU , nV are orthogonal matrices and nΣ is a diagonal matrix with the singular values in non-increasing order on the diagonal. Then matrix nH condition number in spectral norm 2⋅ is defined by expression , )( )()(κ 1 2 n n n n H HH σ σ = (12) where 1σ is the first and nσ is the last singular values. Typical for nuclear type system dependence )(κ2 nH from approximate model order n illustrates Fig. 1. Increase of the model dimension on unit leads approximately to two order )(κ2 nH enlargement. Now we address the stability problem that is how to determine the regularized solution of the identification problem. As it was said before model order n is regarded as regularization parameter. However it is more convenient to use condition number for numerical modeling of identification process instead n since condition number is roughly linked with n for each concrete system. In this case condition number may represent the class of system even with different n . Estimations of Φ̂ and Ψ̂ depend on observation interval τ . Obviously this dependence become tangible when τ is changed exponentially. So it is appropriate  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 11 to choose discrete time kτ as 010 τ=τ k k where 0τ is initial proper observation interval. Fig. 1. Condition number values for approximate models of different order We exploit small interval [ ]kkk ∆+ττ , near each kτ for data collection and estimations of Φ̂ and Ψ̂ . We use set of matrices H obtained from these data for stability analysis. We can judge about appearing unstable solution by the scattering of eigenvalues obtained from equation 0)( =sW with coefficients iw ( ni ,1= ) calculated according to system (9). Fig. 2 summarizes the results of multiple numerical experiments for data generating systems with different structures and parameters. On plane of two parameters )(κ2 H and τ are shown the stability domain A , range C where all systems were unstable and transition domain B where part of the systems provide the stable solution and another are unstable. Further τ increase leads to curves merging and after that joint curve tend to saturation that is explained by dominant influence of computational errors. As a result even for precise Φ , Ψ models with order 112 >+ PQ lie in unstable domain. For models from transition domain the checking test on stability is required. Fig. 2. Solution stability domains for frequency identification  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 12 REGULARIZED SUBSPACE (4SID) IDENTIFICATION METHOD To build a finite-dimensional model of a plant with a modification of the 4SID method difference approximation of system (1) was selected as a model set. This model set may be written as ,,1 t T tttt xcybuAxx =+=+ (13) where A is a matrix α×α , b is a vector 1×α , Tc is a vector α×1 , α is a model dimension, discretization t∆ is fitted so that 0lim =∆ ∞→α t , i.e. description (13) should in limit lead to result coinciding with continuous approximation. Subspace identification is a well developed method and description of it may be found for example in [13, 14]. Here we consider simplified version. Let input tu be exited by square wave-form oscillator alternating with relaxation intervals    ∈ ∈ = +12 2 ,0 , i ii t St Stu u , ,,,2,1,0 Ni K= (14) where K,, 20 SS are exiting intervals, K,, 31 SS are relaxation intervals. All 12 +iS ( Ni ,,1,0 K= ) have equal length M and duration of iS2 ( Ni ,,1,0 K= ) may be varied. Hankel matrix Y is formed from M measurements Miii yyy ,122,121,12 ,,, +++ K on each relaxation interval               = + + + MNMM N N yyy yyy yyy Y ,1231 2,123112 1,123111 K MMMM K K . By analogy we form matrix X from vectors 12 +ix relevant to origin of each relaxation interval , ,1231 2,123112 1,123111               = + + + MNMM N N xxx xxx xxx X K MMMM K K where α=+ ,1,,12 kx ki is a component of vector 12 +ix . Due to equations (13) and specifics of the matrix X , Y the following matrix equation may be written [13, 14] ,1 XY M ⋅Γ= + (15) where               =Γ + MT T T M Ac Ac c M 1 is an observability matrix.  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 13 According to realization theory if 21 YYY ⋅= is any given full-rank factorization of the Hankel matrix, then 1Y is the observability matrix and 2Y is the matrix of initial state on the relaxation intervals for some state-space realization. Optimal way for realization of such factorization is to use SVD decomposition. Let the SVD of Y be given by ,TVQY Σ= (16) where Q and V are orthogonal matrices and Σ is a diagonal matrix with singular values in non-increasing order on the diagonal. If the data generating system is n order, then in the ideal case (all 12 +i my are precisely given) Y is of rank n , so that only the n first singular values are non-zero provided that nNnMn >>≥α ,, . But in reality instead of precise Y we would have a matrix ZYY += ~ , where Y is a true matrix and Z is a matrix produced by perturbation of input and output noise. Then all singular values of matrix Σ will be positive. For multidimensional or infinite-dimensional systems condition number is evaluated as a ratio of singular values that grows exponentially when α incremented by one. So SVD decomposition (16) we write as follows ,T rrr T nnn VQVQY Σ+Σ= (17) where )( rn QQQ = , ),( rn VVV = ,       Σ Σ =Σ r n 0 0 . In the framework of statistical approach for finite-dimensional system the first term corresponds to a “signal” and the second is a “noise”. It is clear that 0=Σr in the absence of errors. So the dimension of nΣ defines first of all the system order n . In our case due to ill-conditioning of the matrix nY (first term) for large (or even for not large) n we consider partition (17) as regularization procedure. First term with maximal n defines the maximum order of approximate model when identification still gives the stable solution. Unlike to statistical approach the model order estimation depends on value YY − ~ . When 0~ →− YY , then following tendency holds ,nn → where n is an order of a model that corresponds to deterministic case. Value *n is entirely defined by inevitable computational errors. If partition in (17) is made correctly, which means the order of approximate model satisfy the stability condition, the next step is to find the model parameters. By setting nn Q=Γ −1 the matrix A for some realization can be obtained from the matrix equation ,1:2:1 +Γ=⋅Γ MM A (18) where 1:2 +Γ M is a submatrix derived from 1+ΓM crossing out the first row and M:1Γ is a submatrix derived from 1+ΓM crossing out the last row.  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 14 In non-singular case system in (18) is overdetermined and a least-squares or a total least-squares methods to solve the system with respect to A can be used. After obtaining matrix A from (18) we find its eigenvalues that allows to write matrix A in Jordan realization. For canonical controllable realization vector b is given and only vector c should be found. When canonical observable realization is preferable, then vector b should be found with given vector c . With given A and b from first equation in (13) and (14) it is easy to calculate tx in each point t of the observation interval. Then the second equation in (13) allows to find vector c from overdetermined linear system of equations with ty measured in experiment for the same input. Here we remark that problem estimation of c is more stable in comparison with determination of the matrix A eigenvalues. It means that well- posedness of the identification problem under consideration is completely defined by the splitting procedure (17) that is choise of term T nnn VQ Σ . Multiple numerical experiments show that condition number depends on n in just the same way as it reports in Fig. 1. Stability property in dependence on n was studied by numerical modeling. Two main parameters define the solution stability, namely, YY −=ε ~ and condition number 2κ for spectral norm of the matrix T nnn VQ Σ , i.e. n T nnn VQ σ σ1 2 )(κ =Σ where 1σ is the first and nσ is the last singular values of the matrix nΣ . Instead of regularization parameter n we choose 2κ on the same reason that was pointed out for frequency method. So the plane of these two parameters ε and 2κ was used for demonstration of stability result obtained in multiple experiments with different system structures and parameters generating the data. Fig. 3. Solution stability domains for subspace identification method Domain A in Fig. 3 conforms to stability and C is unstable region. Between them transition zone B is located where part of the identified models are stable and the other are unstable. Here as in Fig. 2 the additional stability test is required  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177 15 to use, for example set of data obtained by randomization. Fig. 2 and Fig. 3 can be used for model construction of reduced order or maximal order assumed by stability condition. CONCLUSIONS Identification problems of infinite or multidimensional systems may be fully treated as essentially ill-posed. Because of using only measured in experiment inaccurate data in many cases we not only look for a solution of the operator equation but also an operator itself. Hence the problem arises how to write the model set that includes true model of a plant. In such situation we propose in capacity of model set to represent it in the form of infinite expansion and to find finite approximation consistent with errors in available data using for this regularization. In other words the identification problem in such setting is always ill-posed. It is shown that this property is fundamental regardless what kind of method is applied. It should be pointed out that approximate model obtained in identification is similar to real system with respect to output only. Moreover model parameters (eigenvalues and others) can essentially differ from the same parameters of the real system. For example, in most experiments each model eigenvalue represent whole cluster of system eigenvalues, i.e. is determined as averaged estimation. It is also worse mentioning also about ill-posed identification of the finite- dimentional systems. All systems generating data with order corresponding to domain C in Fig, 2 and Fig. 3 lead to ill-posed identification. For such systems only approximate model of less order than at original system may be reconstructed. 1. Funrmann P.A. Linear Systems and Operators in Hilbert Space. McGraw-Hill, New York. 1981. 476 p. 2. Glower K., Curtain R.F., Partington J.R. Realization and Approximation of Linear Infinite- Dimensional Systems with Error Bounds. SIAM Journal Control and Optimization, 1988, vol. 26, no. 4, pp. 863–897. 3. Gubarev V.F. Problem-oriented identification and control in the system with distributed parameters. Problems of Control and Informatic, 2000, no. 3, pp. 26–37 (in Russian). 4. Gubarev V.F. Rational approximation of the system with distributed parameters. Cybernetic and system analysis, 2008, no. 2, pp. 99–116 (in Russian). 5. Gubarev V., Zhukov O. Rational Approximation and Identification of Distributed Parameter Systems. Proceedings of the 17-th World Congress IFAC. July 6–11 2008. Seoul, Korea, pp. 6452–6457 (in Russian). 6. Antoulas A.C., Sorensen D.C., Gugercin S. A survey of model reduction methods for large- scale systems. Contemporary mathematic, 2001, 280, pp. 193–219. 7. Alexsandrov A.G. Method of frequency parameters. Automatica and Telemechanick. 1988, 50, no. 12, pp. 3–5 (in Russian). 8. Alexsandrov A.G. Finite-frequency identification: multidimensional object. International conference on control problem. Proc. of the conf. Moscow: Institute of control problem RAS, 1999, no. 1, pp. 15–28 (in Russian). 9. Orlov Yu.F. Frequency parameters identification under parallel tests. Automatica and Telemechanick, 2007, 68, no. 1, pp. 20–40 (in Russian). 10. Gubarev V.F., Melnichuk S.V. Identification of multidimensional system using steady-state parameters. Problems of Control and Informatic, 2012, no. 5, pp. 26–43 (in Russian). Получено 26.06.2014  V.F. Gubarev, A.V. Gummel, S.V. Melnychuk, 2014 ISSN 0452-9910. Кибернетика и вычисл. техника. 2014. Вып. 177

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