ABOUT NON-PARAMETRIC IDENTIFICATION OF T-PROCESSES

Full Text

Introduction. In numerous multidimensional real processes output variables are available to measure not only at different time periods but also after a long time. This leads to the fact that dynamic processes have to be considered as non-inertial with delay. For example, while grinding the products time constant is 5-10 minutes, and control of output variable, for example fineness of grind- ing, is measured once in two hours. In this case investi- gated process can be presented as non-inertial with delay. If output variables of the object are somehow stochasti- cally dependent, then we call such processes T-processes. j = 1, n . The main feature of this task of modeling is that class of dependency F (×) is unknown. Parametric class of vector functions Fj (u, x, a), j = 1, n , where α is a vector of parameters, does not allow to use methods of paramet- ric identification [3; 4] because class of functions accurate to parameters cannot be defined in advance and well known methods of identifications are not suitable in this case [3; 4]. In this way the task of identification can be seen as solving of non-linear equations: Similar processes require special view on the problem of identification different from existing ones. The main thing Fj (u, x) = 0, j = 1, n (1) is that identification of such processes should be carried out differently from the existing theory of identification. We should pay special attention to the fact that the term “process” is considered below not as processes of prob- abilistic nature, such as stationary, Gaussian, Markov, martingales, etc. [1]. Below we will focus on T-processes actually occurring or developing over time. In particular technological process, industrial, economic, the process of person’s recovery (disease) and many others. Identification of multidimensional stochastical proc- esses is a topical issue for many technological industrial processes of discrete-continuous nature [2]. The main feature of these processes is that vector of output vari- ables x = ( x1, x2 , ..., xn ) , consisting of n component is relatively component vector x = ( x1, x2 , ..., xn ) at known values of u. In this case it is expediently to use methods of nonparametric statistics [5; 6]. T-processes. Nowadays the role of identification of non-inertial systems with delay is increasing [7; 8]. This is explained by the fact that measurement of some of the most important output variables of dynamic objects is carried out through long periods of time, that exceeds a constant of time of the object [9; 10]. The main feature of identification of multidimensional object is that investigating process is defined with the help of the system of implicit stochastic equations: Fj (u (t - t), x (t ), x(t )) = 0, j = 1, n , (2) such that the components of this vector are stochastically dependant unknown in advance way. We denote vector of input component - u = (u1, u2 , ..., um ) . This formulation of the problem leads to the fact that the mathematical description of the object is represented as some analogue of the implicit functions of the form Fj (u, x) = 0, where Fj (×) is unknown, τ is delay in different channels of multidimensional system. Further τ is omitted for sim- plicity. In general investigated multidimensional system implementing T-processes can be presented in fig. 1. Fig. 1. Multidimensional objects Рис. 1. Многомерный объект In fig. 1 the following designations are accepted: u = = (u1, ..., um ) - m-dimensional vector of input variables, x = ( x1, ..., xn ) - n-dimensional vector of output variables. Through various channels of investigated process depend- ence of j component of vector u can be presented as dewhere α is a vector of parameters. Then follows the evaluation of parameters according to the elements of training sample ui , xi , i = 1, s and solution of the system of nonlinear interrelated relations (5). Success in building a model will depend on qualitative parametrization of the pendence on components of vector u: j = 1, n . x< j > = f j (u< j > ), system (5). Further we will consider the problem of building T-models under nonparametric uncertainty, when the sys- tem (5) is unknown up to the parameters. Every j channel depends on several components of vector u, for example u<5> = (u1, u3, u6 ) , where u<5> is a compound vector. When building models of real techno- logical and industrial processes (complexes) often vectors x and u are used as compound vectors. Compound vector is a vector composed from several components of the vector, for example u< j> = ( x2 , x5 , x7 , x8 ) or another set of components. In this case, the system of equations will be Fˆj (u< j> , x< j> ) = 0, j = 1, n . T-models. The processes, which have output variables that have unknown stochastic relationships, were called T-processes, and their models were called T-models. Ana- lyzing the above information it is easy to see, that descrip- tion of the process in fig. 1 can be accepted as a system of Let the input of the object receive the input variables values, which, of course, are measured. Availability of training sample xi , ui , i = 1, s is necessary. In this case evaluation of vector components of output variables x at known values of u, as noted above, leads to the need to solve the system of equations (4). If dependence of output component from vector components of input variables is unknown, then it is natural to use the methods of non- parametric evaluation [5; 11]. At a given value of the vector of input variables u = u¢ , it is necessary to solve the system (4) with respect to the vector of output variables x. General scheme of solution of such a system: 1. First a discrepancy is calculated by the formula: ij j ( i , xs , us ), j implicit functions: Fj (u< j > , x< j > ) = 0, j = 1, n , (3) e = F u< j> , x< j> ( ) r r = 1, n , (6) where u< j> , x< j> are compound vectors. The main feawhere we take F u< j > , x< j > ( ) r r as nonparametric ( i , x , u )s s ture of modeling of such a process under nonparametric evaluation of regression of Nadaraya-Watson [10]: uncertainty is the fact that functions (3) e (i ) = F (u< j > , x (i )) = Fj (u< j > , x< j > ) = 0, j = 1, n are unknown. Obviously the j ej j system of models can be presented as following: s <n> æ uk¢ - uk [i] ö Fˆ (u< j> , x< j> r r ) = 0, j = 1, n , (4) å x j [i]ÕФ ç c ÷ j , xs , us = x (i ) - i=1 k =1 è suk ø , (7) where r r are temporary vectors (data received j s <n> æ u¢ - u [i] ö xs , us r åÕФç k k ÷ by s time moment), in particular xs = ( x1, ..., xs ) = i=1 k =1 ç csu ÷ k è ø = ( x11, x12 , ..., x1s , ..., x21, x22 , ..., x2s , ..., xn1, xn2 , ..., xns ) , but even in this case Fˆ (×), j = 1, n are unknown. In the where j = 1, n, , < m > is dimension of a compound vecj theory of identification such problems are not solved and tor uk , < m > £ m , further this designation is also used are not set. Usually parametric structure is chosen (3), unfortunately it is difficult to fulfill because of lack of apriori information. Long time is required to define for other variables. Bell-shaped functions Фæ uk¢ - uk [i] ö è ø k ç csu ÷ parametric structure, that is the model is represented as: and parameter of fuzziness csuk satisfy several conditions Fj (u< j> , x< j> , a) = 0, j = 1, n , (5) of convergence and have the following features: c Ф(×) < ¥; -1 Ф(c-1 (u - u ))du = 1; described, for example, by the following system of equations: s i ò s W(u ) ìFˆ ( x , x , u , u , u ) = 0; lims®¥ c-1Ф(c-1 (u - ui )) = d(u - ui ), lims®¥ cs = 0, x1 1 3 1 2 5 ï ˆ s s íF ( x , x , u , u ) = 0; (11) ï lims®¥ scs =¥ . 2. Next step is conditional expected value: x2 1 2 4 5 x3 1 2 3 2 3 5 ïîFˆ ( x , x , x , u , u , u ) = 0. x j = M {x | u< j> , e = 0}, j = 1, n . (8) We take nonparametric evaluation of regression of Nadaraya-Watson as an estimate (8) [10]: xˆ j = The system of equations (11) is a dependence, unlike the system (10), known from the available a priori information. Having got a sample of observation, we can proceed to a studied problem, which is finding the forecast values of output variables x at known input u. First, discrepans <n> æ uk - uk [i] ö <m> æ ek [i] ö cies are calculated (7) using the technique described ear- å x j [i]× ÕФç 1 1 ÷ ÕФç 2 ÷ lier. We introduce discrepancies as a system: = i=1 k1 =1 è csu ø k2 =1 è cse ø , j = 1, n, (9) s <n> æ uk - uk [i] ö <m> æ ek [i] ö ìe1 (i) = Fˆ (xi , xi , ul , ul , ul ); åÕФç 1 1 ÷ ÕФç 2 ÷ ï 1 1 3 1 2 5 i=1 k1 =1 è csu ø k2 =1 è cse ø ïe (i ) = Fˆ (xi , xi , ul , ul ); (12) where bell-shaped functions Ф(×) are taken as triangular ï í 2 2 1 2 4 5 ˆ i i i l l l core: ì uk - uk [i] uk - uk [i] ïîe3 (i) = F3 (x1 , x2 , x3 , u2 , u3, u5 ). where e j , j = 1, 3 are discrepancies, whose corresponding ï1- 1 1 , 1 1 < 1, ç ÷ í Фæ uk1 - uk1 [i] ö = ï c csu u - u [i] csu components of an output vector cannot can’t be derived from the parametric equations. è su ø ï ï0, î k1 k1 ³ 1. csu The forecast for the system (11) is carried out according to the formula (9) for each output component of the object. ì ek [i] ï1- 2 , ek [i] 2 < 1, First, we present the results of a computational experiment without interference. In this case, values of input ç ÷ í Фæ ek2 [i] ö = ï c cse e [i] cse variables of the newly generated input variables (not included in the training sample) go to the input of the è se ø ï ï0, î k2 ³ 1. cse object. A configurable parameter will be a parameter of fuzziness сs , which in this case, we take equal 0.4 (the Carrying out this procedure we obtain the value of output variables x under input influences on the object u = u¢ , this is the main purpose of a required model, which further can be used in different management sys- tems [8], including organizational one [12]. Computational experiment. For computational experiment a simple object with five input variables u = (u1, u2 , u3 , u4 , u5 ) taking random values in the interval u Î[0, 3] and with three output variables x = ( x1, x2 , x3 ) where x1 Î[-2; 11] , x2 Î[-1; 8] , x 3Î[-1; 8] was cho- sen. We will develop a sample of input and output vari- ables based on a system of equations: value was determined as a result of numerous experiments to reduce the quadratic error between model and object output [13; 14]) the parameter of fuzziness will be taken the same when calculating in the formulas (7) and (9), sample size is s = 500 . Let’s give graphs for object outputs by components x1, x2 and x3 . In fig. 2, 3 and 4 the output values of the variables are marked with a “point”, and the output value of the model are marked with a “cross”. The figures demonstrate the comparison of the true values of the test sample of the output vector components and their forecasted values ob- tained by using the algorithm (6)-(9). We will conduct the results of another computational ìx1 - 2u1 + 1.5 íx2 4 ï -1.5u - 0.3 ï ïîx3 - 2u2 + 0.9 - u2 - 0.3x = 0; 5 3 - 0.6 - 0.3x1 = 0; - 4u5 - 6.6 + 0.5x1 - 0.6x2 = 0. (10) experiment, in this case, interference x is imposed on values of the vector x components of the object output. The conditions of the experiment: sample size is s = 500 , interference acting on the output vector components of an object is x = 5 % , parameter of fuzziness is c = 0.4 As a result we get a sample of measurements r r r r us , xs s (fig. 5-7). where us , xs are temporary vectors. It should be noted that the process described by the system (10) is only nec- essary to obtain training samples, there is no other infor- mation about the process under investigation. Dealing with a real object, a training sample is formed as a result of measurements which are carried out with available control measures. In the case of stochastic dependence between output variables, the process is naturally The conducted computational experiments confirmed the effectiveness of the proposed T-models, which are presented not as generally accepted in the theory of model identification, but as some method of forecasting the output variables of the object at the known input u = u¢ . It should be noted that in this case we do not have a model in the sense generally accepted in the theory of identification [15]. Fig. 2. Forecast of the output variable x1 with no interference. Error d = 0.71 Рис. 2. Прогноз выходной переменной x1 при отсутствии помех. Ошибка d= 0,71 Fig. 3. Forecast of the output variable x2 with no interference. Error d = 0.71 Рис. 3. Прогноз выходной переменной x2 при отсутствии помех. Ошибка d= 0,71 Fig. 4. Forecast of the output variable x3 with no interference. Error d = 0.71 Рис. 4. Прогноз выходной переменной x3 при отсутствии помех. Ошибка d= 0,71 Fig. 5. Forecast of the output variable x1 with interference 5 %. Error d = 0.77 41 Рис. 5. Прогноз выходной переменной x1 с помехой 5 %. Ошибка d = 0, 77 Fig. 6. Forecast of the output variable x2 with interference 5 %. Error d = 0.77 Рис. 6. Прогноз выходной переменной x2 с помехой 5 %. Ошибка d = 0, 77 Fig. 7. Forecast of the output variable x3 with interference 5 %. Error d = 0.77 Рис. 7. Прогноз выходной переменной x3 с помехой 5 %. Ошибка d = 0, 77 Conclusion. The problem of identification of non- inertial multidimensional objects with delay in unknown stochastic relations of the output vector components is considered. Here a number of features arise, which mean that the identification problem is considered under condi- tions of nonparametric uncertainty and, as a consequence, cannot be represented up to a set of parameters. On the basis of available a priori hypotheses the system of equa- tions describing the process with the help of compound vectors x and u is formulated. Nevertheless functions F (×) remain unknown. The article describes the method of calculating the output variables of the object at the known input, which allows them to be used in computer systems for various purposes. Above some particular re- sults of computational studies are given. The conducted computational experiments showed a sufficiently high efficiency of T-modeling. At the same time, not only the issues related to the introduction of interference of different levels, different sizes of training samples, but also objects of different dimensions were studied.

About the authors

A. V. Medvedev

Reshetnev Siberian State University of Science and Technology

31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

D. I. Yareshchenko

Siberian Federal University

Email: YareshenkoDI@yandex.ru
26/1, Academician Kirensky St., Krasnoyarsk, 660074, Russian Federation

References

Supplementary files