Automation and Remote Control, Vol. 63, No. 6, 2002, pp. 946–959. Translated from Avtomatika i Telemekhanika, No. 6, 2002, pp. 85–98. c 2002 by Ginsberg. Original Russian Text Copyright

MODELING OF BEHAVIOR AND INTELLIGENCE

A New Approach to the Problem of Structural Identification. II K. S. Ginsberg Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia Received August 6, 2001

Abstract—The subject under discussion is a new approach to the problem of structural identification, which relies on the recognition of a decisive role of the human factor in the process of structural identification. Potential possibilities of the suggested approach are illustrated by the statement of a new mathematical problem of structural identification.

1. INTRODUCTION We will illustrate the content of the problem of structural identification by the following example. Let us prescribe a family of mathematical equations, which is parameterized by a scalar or vector parameter. It is necessary (the practical demand in the framework of the applied problem solved) to select a value of the parameter on the basis of the available collection of measurements. The conventional procedure (in the context of the theory of parametric identification) of the solution of this problem includes the following: (1) the process of the formation (with due regard for the properties of the applied problem) of the mathematical statement in which the problem of the choice is interpreted as the problem of estimating an unknown nonrandom parameter; (2) the process of the solution of the formed statement on the basis of the body of the theory; (3) the calculation of a value of the parameter by means of the developed estimation algorithm using the available collection of measurements as initial data. In the practical use of the described procedure, a task arises of the search for an adequate applied problem of the initial family of mathematical equations. In the classical theory of identification, this search is commonly called structural identification. More exactly, the structural identification is called the process of the formation of the family of alternatives that is adequate to the applied problem (mathematical equations) for the parametric and the nonparametric identification. At present, the generally recognized methodology of structural identification is unavailable. The cause appears to lie in the fact that in the medium of specialists there exist two different disciplinary patterns of structural identification. At the conceptual (explanatory) level, all specialists, for example, agree that the intuition and experience of the decision-maker (DM) play an appreciable role in the process of structural identification. But at the level of a specific theoretical investigation, the basic intellectual efforts are aimed at the structurization and absolute formalization of a given process. In the best case, it is admitted that the DM plays the role of a designer who defines the type of algorithm of processing until the beginning of structural identification. In the framework of the mathematical disciplinary pattern, the most important theoretical investigations are thought to be the ones on the development of algorithms of generation and survey of structures (the structure is a prescribed family of mathematical equations), the choice and estimation of the quality of the “best” structure. A constant internal conflict between various images of structural identification is an important factor of the statement and the solution of new theoretical problems. The preceding discussion should not be taken up as a statement of the scientific inefficiency of the classical (mathematical) direction of the theory of structural identification in modern conditions. c 2002 MAIK “Nauka/Interperiodica” 0005-1179/02/6306-0946$27.00

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The developments of this direction were and will be of great importance for the solution of urgent practical problems. It is only owing to a vast mathematical and methodological reserve created in the framework of the classical direction that the potential possibility appeared for the development of the nonclassical conception of structural identification, which is set forth in this work and in [1–5]. 2. NEW APPROACH The solution of applied problems on the basis of the use of the notions, mathematical problems, and methods of the theory of automatic control is commonly carried out by a group of specialists, among which the leading ones are the technologist and the subject analyst (briefly, the analyst). We will settle to call the technologist a specialist professionally knowing the technology of production, within which a practical problem subject to the solution arises. The analyst is a specialist of another skill, who, in contrast to the technologist, is a master of the ways of the solution of practical problems on the basis of the use of the tools of automatic control theory. Therefore, the analyst must know how to form the mathematical description of a practical problem, to solve mathematical problems, and to handle the procedures of interpretation and scientific investigation. In the course of the solution, the analyst or the technolgist or both take the part of DMs. Here and later on, by an applied problem is meant a problem of design of the mathematical description of an automatic system (control, monitoring, estimation, measurement, and diagnostication). The solution of an applied problem is taken to be the mathematical description of an automatic system displaying the properties desirable for the user. In this case, the user is a development engineer of a specific and hardware support for an automatic system. The theory of identification is the means of information support of the DM at definite stages of the solution of an applied problem. The conventional approach to the definition of the notion of the “theory of identification” stems from the fact that the final result of identification is the mathematical model of a physical object. Therefore, in the classical interpretation, the identification is the construction of an adequate mathematical model of a physical object on the basis of measurements; the theory of identification is the system of mathematical methods of designing mathematical models of a physical object. The adequacy of a model means its compliance with an applied problem, i.e., the acceptability of the model for the DM as an element of the process of the solution of the applied problem. Various refinements of the notions “method,” “model,” and “object” do not change the essence of the approach. The principle remains invariable: the aim of identification involves finding the mathematical description of the cause-and-effect links of input and output variables of a physical object; the theory of identification deals only with the development of mathematical methods for designing mathematical models. In the framework of the conventional approach, the term “identification” is commonly used together with the terms “system,” “object,” and “process,” denoting an object of identification. In [6], the most common definition is given of the notion “identification,” suitable for most of the scientific disciplines: “identification” (Latin: idem—the same, facere—to make)—likening, establishment of the equivalence, identity of some objects on the basis of certain signs, i.e., the content of the universe notion “identification” is revealed. The classical theory of identification defines concretely the given definition in accordance with its comprehension of the main problems of the process of the solution of a practical problem. Historically, at the time of the origin of the theory, the most substantial issue of the applied use of problems and methods of the theory of automatic control was considered to be the absence of the theoretically justified methods of constructing the models of controllable objects on the basis of measurements. Therefore, in what follows, the basic attention is given to the development of the mathematical methods of the choice of an optimal or suboptimal model from the specified family of mathematical equations. Owing to the given efforts, the theory of parametric and nonparametric identification and the bases of the AUTOMATION AND REMOTE CONTROL

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mathematical theory of structural identification were worked out. As a result, the circle of original problems decreased markedly and the problem of the search for the initial family of mathematical equations, which is adequate to an applied problem, was put in the forefront. Next, we assume the following: The unit mathematical problem is a mathematical problem not containing prescribed parameters; here and later on, the notion “parameter” is treated in a wide sense, including not only scalar and vector parameters, but also functional parameters (possible values of which are functions), and also other types of letter constants; in the mathematical interpretation of the notion “prescribed parameter,” we conceptually hold the viewpoint outlined in [7]: “Let us note that parameters are not necessarily scalars, but they can be vectors and even elements of a functional space; roughly speaking, parameters are all that is preset in the statement of a problem. For example, for the differential equation dx/dt = f (x, t), the entire right side can be thought of as a functional parameter, although this does not fully comply with the traditional terminology.” The mathematical problem with specified parameters is a family of unit mathematical problems parametrized by a finite sequence of prescribed parameters. The adequate statement of an applied problem (briefly, the adequate statement) is a unit mathematical problem the approximate solution of which, being implemented, ensures the acceptable for the DM the fulfillment of desirable requirements for the quality of the solution of the applied problem; the description of an identification object, which exists in the adequate statement, is the final aim of structural identification in the conventional sense. The ideal statement of an applied problem (briefly, the ideal statement) is a unit mathematical problem, the approximate solution of which, being implemented, ensures an exact fulfillment of the desired requirements for the quality of the solution of the applied problem. The identification in a narrow sense is the process of constructing an adequate mathematical model of a physical object (the identification object) on the basis of the adequate statement and measurements: the adequacy of a model implies its correspondence to the applied problem, i.e., the acceptability of the model for the DM as an element of the process of the solution of the applied problem; the identification in a narrow sense consists of two stages. (1) For the first approximation to an adequate model, the description of an identification object is chosen, which is available in the adequate statement; in the given description, the DM singles out definite unknown nonrandom parameters that are replaced by numerical estimates obtained by means of algorithms of parametric or nonparametric identification; (2) The second found approximation to an adequate model is defined more exactly by the DM directly on the basis of the intuition and the personal skill; (1) the statement of an applied problem (briefly, the statement) is a unit mathematical problem that is examined by the DM as a theoretically possible claimant on the role of the adequate statement; (2) the set of alternatives of the preliminary choice is the set of statements from which a trial statement is chosen; (3) the trial statement is the “best” alternative of the preliminary choice. We will assume that the solution of an applied problem consists of two steps: (1) the development of an adequate statement of the applied problem; (2) the solution of the applied problem with the known adequate statement containing, if necessary, the phase of identification in a narrow sense. The first step is likely to be implemented best at the stage of the investigation of the possibilities of design of an automatic system (i.e., before the beginning of the development stage or at the modernization stage (redesign) of the acting automatic station. The development of an adequate statement contains as a component the structural identification in the traditional sense. The twostep procedure of the solution of an applied problem is the most desirable one, but not a uniquely admissible rational normative model of the process of the solution. A more complete procedure is AUTOMATION AND REMOTE CONTROL

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the one that accounts for the possibility of a substantial change at the second step of the perceptions of the DM relative to the properties that the adequate statement must possess. In the framework of the new procedure, in the case of an appreciable change in the image of the adequate statement, the implementation of the second step discontinues and the third step begins, which is brought about by the need of the repetition of the process of developing an adequate statement. It is suggested to consider the following: (1) The development of an adequate statement as the structural identification of an applied problem (briefly, structural identification); (2) The identification in a narrow sense as the identification of an applied problem with the known adequate statement (briefly, generalized parametric identification; here, the notion “parameter” is treated in a wide sense); (3) The structural identification and the generalized parametric identification as the identification of an applied problem (briefly, identification). The use of the term “identification” in the outlined context, first, does not contradict the universal definition of the word “identification” [6]; second, it does not contradict the traditional definition because it can be regarded as its conceptual generalization related to the nonclassical understanding of the word “construction” (“construction” as the development initially of an adequate statement and then the adequate model of a physical object). The new approach markedly extends the boundaries of the possible application of the methods of identification theory, including an applied problem into the composition of objects. In particular, the feature of the applied problem consists in that it is a theoretical object of the perception, which is mainly built up by the joint intellectual activity of the owner of the applied problem and the DM. In the context of the new approach, the notion “theory of identification” acquires another content [1–5]. Here, the role of the theory of identification appears at the first plan as a section that must set out the methodology of the practical use of the mathematical bases of the theory of automatic control. According to the given approach, the theory of automatic control consists of the following three sections: (1) the conceptual bases containing the conceptual model of reality; (2) the mathematical bases containing interpreted in the language of the conceptual model, mathematical problems with specified parameters and methods of their solution (here, each problem is interpreted as a theoretical model of a wide class of applied problems of the theory of automatic control); (3) the theory of identification that helps the DM select in the framework of mathematical bases the adequate statement for a specific applied problem and work out the adequate mathematical model of a physical object on the basis of the found adequate statement. It is assumed that the association of the mathematical bases with practice cannot be carried out without a complex intellectual activity of the DM. This DM must possess the professional ability (the intuitive gift, inexplicable from the viewpoint of the formal logic) to develop an adequate statement for each applied problem. According to the new approach in the context of the theory of automatic control: (1) The identification is the entire cognitive activity of the DM, which produces adequate models necessary for the practical use of the mathematical bases of the theory of automatic control in solving a specific applied problem (the primary definition from which, as a consequence, it is possible to obtain the above-stated operational definition of identification). (2) The theory of identification is the system of methods for constructing normative models of identification; ideally, the theory includes methods by the use of which the DM can independently produce normative specimens of his own identification activity.

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(3) The structural identification is the entire perceptive activity of the DM, related to the search in the mathematical bases of the theory of automatic control for an adequate statement of the applied problem solved; the theory of identification supports this activity, producing the methods of the development—with the aid of the DM—normative specimens of structural identification; the structural identification of an applied problem contains as a component the structural identification in the traditional sense. The term “cognitive activity” here denotes the sequence of formal and informal cognitive actions of the DM. The cognitive action is said to be formal if the model of an action (a plan, diagram, project, and mechanism) is built up in the framework of a definite theoretical or formal system; here, the case in point is the model that the DM will practically implement to reach the aim of the action. The type of model of the cognitive action (humanitarian, mathematical, and humanitarianmathematical) is likely to be determined (it is only possible to guess about it) by the requirements for the quality of achieving the aim of the action and by the individual characteristics of the DM, i.e., a priori, without the analysis of real properties of the applied problem, it is impossible to indicate the adequate language of the description of the cognitive action. As regards the methods of constructing normative models, here we are evidently at the stage of the development of block diagrams (graphic logic models) of structural identification of a different degree of detailing. Ideally, it is desirable to have a block diagram consisting of blocks (geometric figures) denoting the process of the solution of a mathematical or practical problem, which the DM must carry out independently, having, in the worst case, the verbal information support on the side of the theory of identification. We will clarify the introduced notions on the basis of the following model representations. Construction of the statement. In the text of the statement, we will single out three fragments: 0 is the description of the actual model of a physical system; M is the description of the actual model of measurements; and A is the description of the aim of the solution of the statement. The fragment A contains the requirement of finding at least one solution of the extremal problem f0 (x) → inf,

x ∈ C,

C = {x ∈ X | f (x) ∈ Q},

where f (x) = hf1 (x), . . . , fm (x)i is the finite sequence of functionals; fi : X → R (i = 0, . . . , m) are functionals determined on the set X; R = R1 ∪ {−∞, +∞}; R1 is the set of all real numbers; C is the constraint on the variable x; and Q ⊆ Rm is the prescribed set. As regards the functional fi (i = 0, . . . , m), we will assume that they are implicitly prescribed by the text of the fragments O, M , and A. The latter means that the DM can uniquely find these functionals on the basis of O, M , and A. We will display this property by the notation fi = Li (O, M, A)

(i = 0, . . . , m),

where Li (i = 0, . . . , m) are known transformations. Symbolically, the above-stated construction of the statement, we will write in the form def

S = hO, M, Ai, def

A = “f0 (x) → inf, x ∈ C”, def

(1)

def

C = {x ∈ X | hf1 (x), . . . , fm (x)i ∈ Q},

fi = Li (O, M, A)

(i = 0, . . . , m),

where hO, M, Ai is the finite sequence of descriptions, which we will call the “statement” and denote by the letter S; fi (i = 0, . . . , m) are prescribed functionals. In the context of the applied problem under study, the letter S identifies the statement of the problem of designing the mathematical description of an automatic system. The statement S is AUTOMATION AND REMOTE CONTROL

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commonly expressed in the language of the mathematical analysis, which is understood in the widest sense of this term. Therefore, the use of statements in the process of the solution of an applied problem is most often possible only on the basis of the application of numerical methods. Estimation of the quality factors of the trial solution of an applied problem. Let I be a family of statements that the DM can invent theoretically. F : I → X is the method of an approximate solution of any statement S ∈ I, which places in correspondence an approximate solution h = F (S) ∈ X with the statement S. Each h is regarded by the DM as a theoretically possible claimant on the role of an ideal or an adequate algorithm of synthesis of a trial solution of the applied problem. As an object of the practical use, h displays a collection of real properties. Particularly important properties are normed by the DM, i.e., a system of measuring-computational procedures is developed, which enables us to use specially planned real (natural) experiments to measure values of normed quality factors of a trial solution of the applied problem on the basis of h. We assume that gei (h) = gi (h) + ϕi (h)

(i = 1, . . . , p),

(2)

hϕ1 (h), . . . , ϕp (h)i ∈ Φ(h), where gei (h) (i = 1, . . . , p) is an estimate of the normed quality factor gi (x) at x = h; ϕi (h) (i = 1, . . . , p) is an error of the estimate; Φ(h) is the set known to DM for each solution h; and gei , gi , ϕi (i = 1, . . . , p) are functionals of the type fi : X → R. We also assume that the properties of the functional gei gi , ϕi (i = 1, . . . , p) are not known to the DM, i.e., he cannot define acceptable numerical estimates of the functionals gi for various algorithms h only on the basis of computational experiments with mathematical models. In the framework of the applied problem under study, the symbol h identifies a synthesis algorithm of the mathematical description of an automatic system, which contains the procedure of estimating the prescribed set (selected by the synthesis method F ) of unknown nonrandom parameters of the statement S. The prescribed set consists of parameters the knowledge of which is necessary to obtain for the mathematical description of an automatic system. If the automatic system contains an operative identifier, its description includes the algorithm of the current identification, i.e., the automatic procedure of the periodic overestimation in prescribed or calculated instants of time of the chosen set of unknown nonrandom functions of time. The estimation of gi (h) is performed in three steps: (1) the synthesis of the mathematical description of an automatic system on the basis of the statement S, the method F , and the initial data for the synthesis, which are not contained in the statement S; (2) the development of a prototype of the automatic system; (3) the use of the prototype for determining the quality of the trial solution of the applied problem; the estimation of values of gi (h) (i = 1, . . . , p) of the normed quality factors of the trial solution of the applied problem. Ideal statement. The approximate solution h of the statement S ∈ I is considered to be an ideal algorithm of the synthesis of the trial solution of an applied problem if there exists g(h) ∈ G,

(3)

where g(x) = hg1 (x), . . . , gp (x)i; G is the prescribed set adopted as a norm for the quality factors gi ; and g(x) ∈ G are technical requirements for the quality of the trial solution of the applied problem. The statement S for which g(h) ∈ G is said to be an ideal statement. For the given Φ(h), G, ge(h) = hge1 (h), . . . , gep (h)i, the indicator of the ideal statement is the set ψ(h) = {ϕ ∈ Φ(h) | (ge(h) − ϕ) ∈ G} , AUTOMATION AND REMOTE CONTROL

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where ϕ = (ϕ1 , . . . , ϕp ) ∈ RP is the vector of numerical variables and ge(h) is the p-dimensional numerical vector. The proposition “S is ideal statement” is true if ψ(h) = Φ(h), and false if ψ(h) = φ. If ψ(h) 6= Φ(h) and ψ(h) 6= φ, from the given Φ(h), G, and ge(h), we cannot establish the true meaning of the given proposition. Adequate statement. The statement (1), the measurement model (2) and condition (3) can be considered in turn as elements of the theoretical problem of structural identification, which is given by the set of four quantities hI, F, M, Ai,

(4)

where I is the family of statements (1); F is the method of an approximate solution of any statement S ∈ I; M is the measurement model (2); and A is the requirement for developing the model of the search for the ideal statement, which is postulated as a formulation of the aim of the solution of the problem (4). In the framework of the problem (4), the choice of various statements of the family I and the check of the conditions (a)

ψ(F (S)) = Φ(F (S)),

(b) ψ(F (S)) = ∅

(5)

against the results of real experiments form the content of the process of structural identification. If we account for the fact that the mapping ψ is unknown, this process will evidently be shaped up by the DM on the basis of the trial-and-error method. The model (4) does not take into account the fact that DM makes complex decisions not only on the basis of factors external with respect to him (prior and experimental information), but also under the effect of internal motives (experience, common sense, intuition, etc.). For example, the instant of the end of structural identification is most often defined (on account of the appreciable uncertainty in the knowledge of errors ϕi (h)) not by the condition (a) of formula (5), but by the system of values and experimental estimates of the DM. Therefore, the purpose of the solution of any realistically stated theoretical problem of structural identification must undoubtedly be laid down in the form of the requirement for the development of the method of the search for an adequate statement. We will clarify the notion “adequate statement” on the basis of the following definitions. The approximate solution h of the statement S ∈ I is said to be adequate to the applied problem in the system of values and expert estimates of the DM if the DM selects the alternative “h is an ideal synthesis algorithm” in the situation where he is not able to prove formally logically its truth. The statement the approximate solution of which is adequate to the applied problem is said to be adequate. Parametric structural identification. The text of a traditional problem of the mathematical bases of the theory of automatic control commonly contains parameters. The parameter is considered to be a letter with or without indices, which in the text of the problem and in the procedures of deriving the solution, denotes one and the same mathematical object (a number, numerical vector, set, function, functional, operator, and relation), in which case (1) by the form of a letter it is impossible to establish what object it designates; (2) the object denoted belongs to the set prescribed in the text of the problem. The mathematical object denoted by a letter will be called a true value of the parameter, and the elements of the prescribed set by values of the parameter. In the text of the problem, each parameter is necessarily interpreted as a prescribed or as an unknown one. The terms “prescribed” and “unknown” are not mathematical objects. Their availability is due to two causes. First, they facilitate the understanding of a problem. Second, they simplify the formulation of the aim of the AUTOMATION AND REMOTE CONTROL

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Search for adequate values of prescribed parameters of a mathematical problem

Solution of the trial

Synthesis

Synthesis of mathematical

statement of an

algorithm

description of an automatic system

applied problem

Trial statement

Mathematical description

of an applied problem

of an automatic system

Preliminary choice Design of the prototype

of an adequate

of an automatic system

statement

Prototype of an automatic system

Substantive statement

Testing of the prototype of Correction of the

an automatic system for

Adequate

substantive statement

agreement with requirements

statement

of an applied problem

desired for the user

Mathematical problem Choice of a mathematical problem with specified parameters

Substantive statement Development of the substantive statement of an applied problem

Idea of design of an automatic system, proof of the idea, and aim of search for an adequate statement

Block diagram of structural identification.

solution of the problem. Using the term “parameter,” it is possible to give the definition of the parametric version of structural identification. The parametric structural identification is a cyclic iterative process, each cycle of which consists of the following three steps: (1) The development of a substantive statement of the applied problem; (2) The choice of a mathematical problem I(βn ) with specified parameters in the mathematical bases of the theory of automatic control (n is the number of a cycle and βn is the finite sequence of prescribed parameters); (3) The search for adequate values of the prescribed parameters of the problem I(βn ), ending in the solution: (1) the “end of structural identification” if the DM will find a value βn0 for which the unit mathematical problem I(βn0 ) is an adequate statement; (2) the “transition to the first step of the (n + 1)th cycle” if DM stops the search in the nth cycle, not finding an adequate statement. AUTOMATION AND REMOTE CONTROL

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In turn, the search for adequate values of the prescribed parameters consists in the sequence of single-type macrosteps (iterations) of the cognitive activity of the DM. Each iteration of the nth cycle contains the following six steps: (1) the correction of the substantive statement of the applied problem; (2) the preliminary choice of the adequate statement: (1) the assignment of a family of statements of the preliminary choice of the adequate statement, which is parametrized by the finite sequence of prescribed parameters (briefly, the mathematical problem of a preliminary choice); (2) the choice of values of the prescribed parameters of the mathematical problem of the preliminary choice; (3) the solution of the trial statement of the preliminary choice (the result of the solution is the algorithm of a preliminary choice); (4) the definition of the trial statement of the applied problem on the basis of the developed algorithm of a preliminary choice; (3) the solution of the trial statement of the applied problem (the result of the solution is the algorithm of synthesis of the mathematical description of an automatic system; the solution relies on the solution method developed in the theory of automatic control); (4) the synthesis of the mathematical description of an automatic system; (5) the design of the prototype of an automatic system; (6) the testing of the prototype of the automatic system for the correspondence of the requirements desired for the user. Figure displays a block diagram of structural identification, in which each rectangle (block) denotes a stage or a step of the cognitive activity of the DM. Stages are denoted by natural numbers 1, 2, 3, and steps by symbols 3.1, 3.2, . . . , 3.6. The direction of an arrow beginning at one block and ending at the other, points out only the sequence of the transition from one block to another. If an arrow goes out of a block, the signature near the arrow denotes the main informational result of the cognitive activity. According to the earlier stated definitions, the sequence of steps 3.1, 3.2, . . . , 3.6 is called a macrostep or an iteration, and the sequence of stages 1, 2, 3 is called a cycle. If a few arrows extend from a block, this means the presence of alternative versions of the cognitive activity. At the current instant of time, only one version is implemented. The basic results of structural identification include an adequate statement of the applied problem, a demonstration version of the mathematical description of an automatic system, and a demonstration prototype of the automatic system. We note that the difficulties encountered in the search for values arise not for all prescribed parameters. In the main, this concerns the prescribed parameters that lack the technological interpretation. 3. NEW MATHEMATICAL MEANS OF THE INFORMATION SUPPORT OF THE DM The new approach to a problem of structural identification involves the statement and the solution of complex mathematical problems. As an example illustrating this thesis, we will consider a problem of designing the mathematical description of the system of the estimation of energy liberation in the active section of a nuclear reactor of the water-cooled power reactor (WCPR) type [5]. It is conventional to try to solve this problem on the basis of special programs of the threedimensional neutron-physical calculation, using additionally as testing and normalizing signals, the readings of intrareactor sensors of energy liberation. We will consider the case where the dependence of systematic errors of the physical calculation of energy liberation on time displays a quasistationary character. The latter means that in a rather large time interval, the magnitude of systematic errors practically depends only on spatial coordinates of the active section of the reactor. Our interest will be only Step 3.2 of the block diagram of structural identification. The key role here is played by two substeps (the first and the third), which are impossible to implement AUTOMATION AND REMOTE CONTROL

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without carrying out theoretical investigations on the development of the principle of design and the methods of the solution of mathematical problems of the preliminary choice. Next, we will site a substantive formulation of the mathematical problem of the preliminary choice, which consists of seven fragments. (1) Let the reliable description of the proper model of systematic errors of the physical calculation have the form w = ψ(u)ω T ,

ω ∈ Ω ⊆ R k1 ,

u ∈ U = {uj ∈ R3 | j = 1, 2, . . . , L},

where w is an unknown systematic error, uj is a vector row of the known spatial coordinate of the jth calculated cell of the active section of the nuclear reactor; ω is the vector row of unknown nonrandom parameters; ψ(u) = [ψ1 (u), . . . , ψk1 (u)]; ψi (u) (i = 1, 2, . . . , k1 ) is the system of the prescribed linearly independent functions of three variables; U , Ω are prescribed sets; Rki is the ki -dimensional Euclidean space; and L is the number of calculated cells. (2) Let the reliable description of the proper model of results of the physical calculation have the form υjn = yjn + wjn + εjn ,

wjn = ψ(uj )ω T

(ω ∈ Ω; j = 1, 2, . . . , L; n = 1, 2, . . . , n2 ), where n is number of the instant of time tn ; j is the number of a calculated cell; υjn = υj (tn ) is the result of the physical calculation at the instant of time tn in the jth calculated cell; yjn = yj (tn ) is an unknown mean linear energy liberation; yjn (j = 1, 2, . . . , L; n = 1, 2, . . . , n2 ) are unknown nonrandom parameters; wjn = wj (tn ) is an unknown value of the systematic error at the instant of time tn in the jth calculates cell; εjn = εj (tn ) is a random error of the result of the physical calculation; εn = [ε1n , . . . , εLn ] (n = 1, 2, . . . , n2 ); [ε1 , . . . εn2 ] is an unobservable random vector with the distribution function F (x; λ), where x ∈ RLn2 is the vector of variables; λ ∈ Λ ⊆ Rk3 is the vector of unknown nonrandom parameters; Λ is a prescribed set; F : RLn2 × Λ → R1 is a 2 (j = 1, 2, . . . , L; n = 1, 2, . . . , n ) are prescribed prescribed function; E{εjn } = 0, E{ε2jn } = σ1j 2 quantities; E{•} is the symbol of the mean value; n2 is the number of the known instant of time of the beginning of the next structural identification. (3) Let the reliable description of the proper model of estimates of systematic errors, which are taken from the readings of the intrareactor sensors of energy liberation have the form ∆jn = υjn − sjn ,

sjn = yjn + ϕjn

(j = j1 , . . . , jr ; n = 1, . . . , N ; tN ≤ t < tn1 ), where ∆jn = ∆j (tn ) is the estimate of a systematic error; sjn = sj (tn ) is the estimate of the quantity yjn from the readings of the intrareactor sensors of energy liberation; ϕjn = ϕj (tn ) is a random error; ϕn = [ϕj1 n , . . . , ϕjr n ] (n = 1, 2, . . . , N ); [ϕ1 , . . . , ϕN ] is an unobservable random vector 2 (n = 1, 2, . . . , N ); [ε , . . . , ε ], with the known distribution function; E{ϕjn } = 0, E{ϕ2jn } = σ2j 1 n2 [ϕ1 , . . . , ϕN ] are statistically independent random vectors; r(r > k1 ) is the number of calculated cells in which it is possible to determine sjn by the reading of the intrareactor sensors; tN is the prescribed instant of time; t is the current instant of time, which divides the past and the future; n1 (n1 > N ) is the number of the known instant of time of the beginning of operation of the projectable system of estimation. (4) Let S(p) = hO(p), M (p), A(p)i be the mathematical problem of estimation of the energy liberation with the prescribed vector parameter p ∈ P ⊆ Rk2 in which: (a) O(p) is the family of descriptions of the proper model of systematic errors: w = ψ(u)ω T , AUTOMATION AND REMOTE CONTROL

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where q : P → H(Rk1 ) is the prescribed inverse function with the range of the definition of P and the range of values in H(Rk1 ); H(Rk1 ) is the set of all subsets of the space Rk1 , i.e., sets are values of the function q; P is the prescribed set; q(p0 ) = Ω, p0 ∈ P ; (b) M (p) = hM1 (p), M2 (p)i, where M1 (p) is the family of descriptions of the proper model of the results of the physical calculation: ω ∈ q(p) ⊆ Ω

wjn = ψ(uj )ω T ,

υjn = yjn + wjn + εjn ,

(j = 1, 2, . . . , L, n = 1, 2, . . . , n2 ); M2 (p) is the family of descriptions of the proper model of the estimates of systematic errors: ∆jn = υjn − sjn ,

sjn = yjn + ϕjn

(j = j1 , . . . , jr , n = 1, . . . , N, tN ≤ t < tn1 ); (c) A(p) is the requirement for finding the family of optimal algorithms h0p : R4rN → Rk1 for estimating the unknown vector ω by observations ZN : h0p = arg inf ∗

sup

h ∈H ω∈q(p),λ∈Λ

E{J(h∗ (ZN ))},

(6)

L 2 X  2 X 1 e T − yjn , ρj υjn − ψ(uj )ω n2 − n1 + 1 n=n1 j=1 n

e = J(ω)



e ω J(h∗ (ZN )) = J(ω) e =h∗ (Z ) , N

e is an arbitrary estimate of the vector ω by the observations where H is the prescribed set; ω e an unobservable ZN ; ZN = [x1 , . . . , xN ]; xn = [∆j1 n , uj1 , . . . , ∆jr n , ujr ] (n = 1, 2, . . . , N ); J(ω) quality factor of the projectable mathematical description of the system of the estimation of energy liberation; ρj ≥ 0 (j = 1, 2, . . . , L) are prescribed coefficients; E{J(h∗ (ZN ))} is the risk of the estimation of energy liberation with the use of h∗ (ZN ) as an estimate of the vector ω. (5) Let there be approximations hp of the optimal algorithm h0p (see (6)) for all values of the parameter p ∈ P . Then, the function h : R4rN ×P → Rk1 (interpreted as the method of calculation of an estimate h(ZN ; p) of the parameters by the observations ZN for any value of the vector p ∈ P ) is given by the formula

h(ZN , p) = hp (ZN )∀ZN ∈ R4rN ,

∀p ∈ P.

(6) Let the desirable magnitude of the risk of estimation of the energy liberation R0 (ω, λ) be defined in the form R0 (ω, λ) = E{J(h(ZN ; m))}, m = arg inf E{J(h(ZN ; p))} = g0 (ω, λ), p∈P0

P0 = {p ∈ P | η(p) ≤ η(p0 )},

sup ω∈q(p),λ∈Λ



e ω J(h(ZN ; m)) = J(ω) e =h(Z

η(p) =

N ;m)

,

E{J(h(ZN ; p))},

e ω J(h(ZN ; p)) = J(ω) e =h(Z

N ;p)

,

where g0 : Ω × Λ → Rk2 is a physically unrealizable algorithm (from the viewpoint of the risk of the estimation of E{J(h(ZN ; p))}) of the choice of the problem S(g0 (ω, λ)) from the family AUTOMATION AND REMOTE CONTROL

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of unit mathematical problems S(p) = hO(p), M (p), A(p)i; h(ZN ; m) is a physically unrealizable estimate of the vector ω by the observations ZN , the calculation algorithm hm (ZN ) of which is an approximate solution of the problem S(g0 (ω, λ)). (7) It is necessary to find an optimal algorithm (within the limits of the step of the preliminary choice) of the choice of the problem of the estimation of energy liberation by the observation Z` : g∗ = arg inf δ(g), g∈G

R(ω, λ, g) − R0 (ω, λ) δ(g) = sup , R0 (ω, λ) ω∈Ω,λ∈Λ

(7)

f ))} , R(ω, λ, g) = E {J (h (ZN ; m

f = g(Z` ), m f)) = J(ω) e ω J (h (ZN ; m e , e =h(ZN ;m)

where g : R4r` → Rk2 is the algorithm of the choice of the problem S(g(Z` )) of estimation of energy liberation from the family of unit mathematical problems S(p) = hO(p), M (p), A(p)i by the observations Z` = [x1 , . . . , x` ]; xn = [∆j1 n , uj1 , . . . , ∆jr n , ujr ] (n = 1, 2, . . . , `; ` ≤ N ); g∗ is an optimal algorithm of the choice (within the limits of the step of the preliminary choice); δ(g) is the quality factor of the algorithm g at the step of the preliminary choice; G is the specified f) is an estimate of the vector by the observations ZN in the choice of the problem of set; h(ZN ; m the estimation of energy liberation with the aid of the algorithm g; R(ω, λ, g) is the risk of the estimation of energy liberation in the choice of the problem of the estimation of energy liberation with the aid of the algorithm g. The ratio [R(ω, λ, g) − R0 (ω, λ)]/R0 (ω, λ) represents a relative deviation of a possible risk R(ω, λ, g) of the estimation of energy liberation from the desirable estimation risk R0 (ω, λ). This ratio is interpreted as risk of the use of the algorithm g as a procedure of the final choice. Similarly, the quantity δ(g) is interpreted as a risk of the use of the algorithm g as a procedure of the preliminary choice. A similar understanding of δ(g) relies on the definite model of the cognitive activity of the “rational” DM. It is assumed that in the course of structural identification, the DM performs the most complex control functions in the intellectual respect; more exactly, his cognitive processes and consciousness implement these functions. Here and later on, the procedure of the choice is treated as a control device, and the result of the choice is interpreted as a control action. At Step 3.2 (see figure), the DM need select from the specified family of the statements of the applied problem, a trial statement that (in the context of the adopted terminology) is called a control action. The complexity of the choice of a trial statement results from the dual character of a control action. On the one hand, the trial statement is a tool of studying of an applied problem, in the course of which (Step 3.6) the DM can establish new facts, in particular, to determine what properties are displayed by the prototype of the automatic system produced with the aid of the trial statement. Under the action of these facts, the DM forms at Step 3.1 (see figure) a new image of the applied problem, on the basis of which he determines (Step 3.2) new values of the prescribed parameters of the mathematical problem of a preliminary choice. The cognitive (trial, studying) aspect of the trial statement is an obligatory element of the preliminary choice because it is the shortage of prior information and the need for its receiving that are the basic reasons of the beginning of structural identification. On the other hand, the trial statement is a necessary element of the process of design of the mathematical description with the properties desirable for the user. Ideally, the control action at Step 3.2 must be chosen so (must so combine the “cognitive” and “working” aspects) as to achieve at the next iteration the finite aim of structural identification. The trade-off between various aspects of the aim of the preliminary choice, in our opinion, can be reached with AUTOMATION AND REMOTE CONTROL

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the aid of various quality factors of the algorithm g, in particular, on the basis of the functional δ(·) (7) used in this work. Replacing in all seven fragments of the stated mathematical problems all prescribed parameters, apart from the vector p, by values, we will obtain the trial statement of the preliminary choice. In the actual cycle of structural identification, it is advisable to change only values of the constants Ω, q, P , `, N . The value q 0 of the parameter q changes when the DM understands that the selected function does not allow us to obtain the required estimation accuracy; similarly, the value Ω0 of the parameter Ω changes when the DM comes to a new perception of possible values of the unknown parameters of the proper model of systematic errors. A value of the parameter N is mainly defined from the perception of the DM as to the desired accuracy of the estimation of energy liberation. The transition to the next cycle must occur with a change in the value of the finite sequence of the parameters hψ, G, Hi. As regards other specified parameters, it is assumed that their values remain invariable in the course of identification. Here, the prime above the symbol means that this symbol is replaced in the text by the value of this symbol. To carry out effectively the structural identification, it is desirable to know how to state analogous mathematical problems of the preliminary choice for other applied problems, too. For this, in turn, it is necessary to develop the principles of their construction and the methods of the solution. It seems that the investigations in this direction are now the most actual theme of the theory of structural identification. Of no less importance are the following problems: (1) structurization and partial formalization of the cognitive activity of the DM, related to the choice of values of the prescribed parameters of the mathematical problem of a preliminary choice; (2) construction of the mathematical problem of a preliminary choice on the basis of the selected family of regressive statements of the applied problem; (3) formulations and studies of the mathematical problems of a preliminary choice for the classical problems of the theory of optimal control; (4) analysis and development of the modern theory of structural identification in the context of the outlined nonclassical conception of structural identification; (5) investigations of “mechanisms” of the generation of a new knowledge in the course of structural identification. 4. CONCLUSIONS A new approach to the problem of structural identification is suggested, which is aimed at the development of necessary initial data for the effective use of mathematical methods in the solution of urgent applied problems. The outlined approach is rendered concrete as to applications and problems of the theory of automatic control. Basic conceptual notions are introduced and cleared up; a new procedure is put forth for the transition from the substantive statement of an applied problem to its adequate mathematical analog on the basis of the statement and solution of mathematical problems of the preliminary choice. The potential possibilities of the suggested approach are illustrated by an example of the problem of design of the mathematical description of the system of the estimation of energy liberation in the active section of a nuclear reactor of the WCPR type. There are convincing reasons to believe that the development of all directions of the theory of identification will make it possible to implement technically in full measure the conception of adaptive systems with identifiers [8–12]. REFERENCES 1. Ginsberg, K.S., The Theory of Identifications: Stimuli, Prerequisites, and Prospects of Development, Prib. Sist. Upravlen., 1996, no. 12, pp. 27–30. 2. Ginsberg, K.S., Basics of System Modeling of a Real Process of Structural Identification: Key Notions, Avtom. Telemekh., 1998, no. 8, pp. 97–108. AUTOMATION AND REMOTE CONTROL

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3. Ginsberg, K.S., Nonclassical Problems of the Theory of Structural Identification. Part 1, Proc. Internat. Conf. “Identification of Systems and Problems of Control,” (SICPRO’2000), Moscow: Inst. Probl. Upravlen., 2000, pp. 992–1005. 4. Prangishvili, I.V., Lototskii, V.A., and Ginsberg, K.S., International Conference “Identification of Systems and Problems of Control,” Vest. RFFI, 2001, no. 3, pp. 44–57. 5. Ginsberg, K.S., System Regularities and the Theory of Identification, Avtom. Telemekh., 2002, no. 5. pp. 156–170. 6. Kondakov, N.I., Logicheskii slovar’-spravochnik (Logic Dictionary—Handbook), Moscow: Nauka, 1975. 7. Blekhman, N.N., Myshkis, A.D., and Panovko, Ya.G., Mekhanika i prikladnaya matematika: logika i osobennosti prilozhenii matematiki (Mechanics and Applied Mathematics: Logic and Features of Applications of Mathematics), Moscow: Nauka, 1983. 8. Raibman, N.S., The Application of Identification Methods in the USSR. A Survey, Automatica, 1976, vol. 12, no. 1, pp. 73–95. 9. Osnovy upravleniya tekhnologicheskimi protsessami (Basics of Control of Technological Processes), Raibman, N.S., Ed., Moscow: Nauka, 1978. 10. Raibman, N.S., Identification of Objects of Control (A Survey), Avtom. Telemekh., 1979, no. 6, pp. 60–93. 11. Raibman, N.S. and Chadeev, V.M., On the Conception of Adaptive Control Systems with an Identifier, Avtom. Telemekh., 1982, no. 3, pp. 54–60. 12. Lototskii, V.A., Identification of Structures and Parameters of Control Systems, Izmeren., Kontrol’, Avtomatiz., 1991, no. 3–4, pp. 30–39.

This paper was recommended for publication by V.A. Lototskii, a member of the Editorial Board

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