^{1}

^{*}

^{2}

^{3}

^{3}

^{*}

^{3}

A universal regression-tensor approach is developed in the mathematical modeling of optimal parameters of chemical-technological process of complex mechanical products. The testing of developed algorithms was performed on the example of multi-factorial process of low-temperature sulfur-chromium plating of precision mechanical parts.

Originally regression analysis acquired theoretical and applied interest in problems of optimizing the parameters of linear stationary systems (type “black box”). In most cases, the studies were limited with analysis of finite- dimensional systems [

In this article regression analysis differs from the traditional presentation [

The approbation of theoretical apparatus of nonlinear vector regression in the article stands the problem of optimization (as a way of technological calculation) of characteristics of multi-factorial chemical-technological process. Payment of optimization of process of low-temperature sulfur-chromium plating of precised mechanical products is taken as example.

Let R-field of real numbers,

relative to the standard algebraic basis ( [

Let

process with fixed origin in

chemical-technological process. In this setting we select for consideration a nonlinear system of type “input-output” described by vector-tensor k-valent equation of multiple regression

where

The statement of the problem of optimization of chemical-technological process consists of three steps:

(i) for a fixed index k given

(ii) to build a posteriori estimates c, A,

parametric regression model (1):

here_{(l)}―“reaction” on “variation” v_{(l) }

with respect to “point” of reference mode of chemical-technological process

(iii) for

where

and variables of vector-function

In this section we will examine the analytic properties of nonlinear vector regressions of many variables that look like behavior of holomorphic functions (problem (i) from paragraph 1). In connection with this the presentation will be based on the concept of the Frechet derivative ( [

Proposition 1. Let W―open area in^{n} and w―point from W. If there is a Frechet derivative of order k, then the Frechet differential of k-th order ^{m} has a representation

Proof. Each derivative

Before we take a further step we note that formulation of Proposition 1 essentially imposes on the map w(×) one additional requirement, namely, position of analytical representation of vector-function w(×). In the case of a posteriori modeling w(×) this requirement is not feasible, so above we are limited with the analysis according to the problem (i) less realistic, but more logically verified task of analysis of the properties of mapping w(×).

In next assertion we establish an important property which a vector-function w(×) should have clarifying: when the mapping w(×) satisfies at least under some reasonable assumptions on it one of those special laws from which a concept of tensor regression (1) happened as a natural product of a continuous process of consolidation, abstraction and generalization.

Proposition 2. Let W―open domain in

Proof. By theorem 2 ( [^{(k)}(×) of mapping

where vector-function e(w,×) of class

Thus, the compilation of this proposal with Formula (4) leads to

where

Remark 1. Everywhere further we believe a priori that the simulated chemical-technological process satisfies the Proposition 2 for some index k ³ 2.

The Proposition 2 provides the most direct way of interpreting the concept of model complexity because it shows a direct link between the approximate model and the way the model should be evaluated from experimental data which in the strict sense refuse it; when we set the maximum allowable inconsistency

Let’s start with specification of tensor construction of Equation (1); this specification has a special character but its use in potential allows not to attract complex computational algorithms for calculating an optimal vector of variables of chemical-technological process. We consider (including Remark 1) the case k = 2. We also agree that the coordinates

where _{i}―an upper triangular matrix ( [

By Proposition 2 and Theorem 12 ( [

here

We will associate methodologically the parametric identification in multi-criteria vector-matrix-tensor formulation (2) for multiple stationary nonlinear model of type “black box” in class of regressions (5) with the concept of normal pseudo-solution for a finite-dimensional system of linear algebraic equations.

As usual ( [^{+ }we de-

note the generalized inverse (pseudo-inverse) matrix of Moore-Penrose ( [

Then (see Formula (50), [

We assume that during the operation of chemical-technological process there were conducted q-experiments of type “input-output”. For parameters of quadratic regression system (5) and q-data (general sample) of con-

ducted experiments we denote through

We call complete matrix of experimental data of input variables of chemical-technological process (6) q ´ m(m + 3)/2-matrix of type

respectively vector

we will call the complete vector of experimental data of i-th output variable.

Further considering that in system (5) each matrix B_{i} is upper triangular the structure of i-th equation

It is clear that in view of algebraic structure of Equation (7) the problem of parametric identification (2) should be solved on some basis of q-experiments with respect to the next group of vectors (of dimension

it is obvious that this group of vectors completely determines (sets) elements of matrix

Now we can give the solution to the problem of parametric identification of model of bilinear-tensor regression of chemical-technological process only by a posteriori information on the basis of preliminary passage of q- experiments.

Proposition 3. The problem of identification (2) in terms of parameters (8) of regression model (5) has a solution:

where U―full matrix of experimental data of input actions (6),

Proof. Below we will give a sketch of the proof. Following the standard arguments regression (5) for each l-th experiment according to the relations (6), (7) takes the following compact form:

Thus, if we reformulate according to the last system, optimization problem of parametric identification (2) applied to the equations of regression in tensor structure (5), then we arrive to the following multi-criteria formulation with respect to vectors

It is not difficult to establish that this multi-criteria formulation has (according to Formula (50), [

Corollary 1 ( [

sion model (5) which characterizes the behavior of chemical-technological process such that

(*)

(**)

Remark 2. Ratings (*), (**) primarily depend on the “volume” of a posteriori information in the formation of matrix U and vectors

An attractive idea to create engineering projects and algorithms that are adapted to changing conditions of studied (in the frameworks of these projects) chemical-technological processes, requires the use of nonlinear regression models of class (5) which are optimal flexible (tunable) during the variety of experimental data. Therefore the parametric identification of the functional model of chemical-technological process of class of regressions (5) studied in the previous section was necessary “technological” requirement in solving the problem of “synthesis control”

Proposition 4. Let

may have an inner extremum (at

where

-if^{*};

-if^{*};

-if^{*}.

Remark 3. In the first two cases of definite sign of quadratic form

Proof of Proposition 4. Since

then the necessary conditions of a local extremum have ( [

that is equivalent to the system of equations (below

which (as it has been easily seen) determine in the space

On the other hand, the definite sign of the second differential

determines sufficient conditions ( [

Coordinates of the stationary point (9) allow us to answer the question about the meaning of the functional

Corollary 2. If D_{i}―is a negative definite (similarly positive definite) matrix, then maximum (or minimum) value of the functional

where c_{i}―i-th coordinate of the vector

The proof is constructed by substituting (9) in (1).

Now we will turn to the study of more complex (task (iii)) variant of the problem of optimization of characteristics of chemical-technological process which plays the fundamental role in a more realistic and at the same time more difficult problems in calculating the optimal technological parameters of the mode of functioning of chemical-technological process. Its basis is the methodological position―each functional

Proposition 4. Let

while a sufficient condition that v^{*} provides the quality for a chemical-technological process

is the following requirement: stationary point v^{*} has an elliptic type which is equivalent to the position:

where_{p} of matrix Dcorrespond to inequalities

Proof. Main provisions of the proof repeat the conclusion of Proposition 4 that’s why we are restricted by the scheme of the proof. Necessary conditions of a local extremum have ( [

which is equivalent to the system of n equations:

the last system leads to the solution (10). □

If algebraic conditions (11) (equivalent to (12)) don’t meet then the critical point (10) of functional quality of chemical-technological process is either ( [

Speaking more formally we can quote: the presence of a saddle point warrants a change in at least one (but not all) inequality “<” from (11) (or (12)) on inequality “>”; while a similar change of relation “<” on “£” may causes the structure of parabolic point in the analytical solution of the problem of optimization.

The presented approach methodologically extends the standard procedure of planning experiment of chemical-technological process. Thus, if the calculated (predicted) coordinates of the stationary point (10) of any che- mical-technological parameters are outside the area of adequacy of the identified model (5), it is necessary to conduct an additional experiment while implement chemical-technological process with vector

The previous sections have been conceived as an attempt to bring “compactly” together under the same terms and notations large but diverse enough number of rigorous mathematical results that are dedicated to such a broad topic as multivariate regression analysis with emphasis on methods of covariant-tensor representation of functional derivatives (Frechet derivatives) involving the method of the least squares and their practical application to the optimization of complex multivariate processes. Next section is devoted to a detailed study of related concepts while the basic attention is focused on experimental testing of the theoretical results from Paragraphs 1 - 4 on the basis of experimental studies of process of low-temperature sulfur-chromium.

Numerical modeling was carried out in the environment of software package [

Without loss of generality as a reference mode of process of low-temperature sulfur-chromium we can take some point w of space R^{m} empirically selected from the overall composition of the experimental data; it is clear that in this case coordinates

Parameters of reference mode: w_{1} = 125˚C, w_{2} = 0.92 hour, w_{3} = 43% NaOH, w_{4} = 0.5% S, w_{5} = 1% Na_{2}S, w_{6} = 2% Na_{2}S_{2}O_{3}, w_{7} = 10% CrO_{3}.

Multivariate synthesis of sulfur-chromium layer in the series from field experiments q (due to m = 7 and p. (**) of Corollary 1 the number of experiments q £ 35) we will describe with the following chemical-technolo- gical variables:

Input data:

_{1}) of temperature of process 10^{−2} [˚C],

_{2}) of duration of process [hour],

_{3}) of concentration of hydroxide of sodium 10^{−2} NaOH [%],

_{4}) of concentration of sulfur S [%],

_{5}) of concentration of sulfide of sodium Na_{2}S [%],

_{6}) of concentration of hypophosphite of sodium 10^{−1} Na_{2}S_{2}O_{3} [%],

_{7}) of concentration of three oxide of chromium 10^{−1} CrO_{3} [%].

Output data:

The solution of the problem of parametric identification (2) for the regression Equation (5) of process of low- temperature sulfur-chromium presented in

Critical analysis of efficiency of model of the mathematical description of process of low-temperature sulfur- chromium expressed by Equations (13) gives the comparison of the last two columns of _{1}―expe- riment, ŵ_{1}―forecast according to (13). The graphic illustration of the index of quality

Number of experiment | Parameters of mode of coating of sulfur-chromium layer | Thickness of layer | |||||||
---|---|---|---|---|---|---|---|---|---|

№ | w_{1} + v_{1} | w_{2} + v_{2} | w_{3} + v_{3} | w_{4} + v_{4} | w_{5} + v_{5} | w_{6} + v_{6} | w_{7} + v_{7} | w_{1}(w + v) | ŵ_{1}(w + v) |

1 | 1 | 0.5 | 0.4 | 0.2 | 0.7 | 0.05 | 0.7 | 7.2 | 7.2 |

2 | 1.1 | 0.7 | 0.41 | 0.3 | 0.8 | 0.1 | 0.8 | 8.1 | 8.1 |

3 | 1.2 | 0.83 | 0.42 | 0.4 | 0.9 | 0.15 | 0.9 | 8.7 | 8.7 |

4-w | 1.25 | 0.92 | 0.43 | 0.5 | 1 | 0.2 | 1 | 9 | 9 |

5 | 1.3 | 1 | 0.44 | 0.6 | 1.1 | 0.25 | 1.1 | 9.5 | 9.5 |

6 | 1.35 | 1.8 | 0.45 | 0.7 | 1.2 | 0.3 | 1.2 | 9.5 | 9.5 |

7 | 1.4 | 1.17 | 0.46 | 0.8 | 1.3 | 0.35 | 1.3 | 9.5 | 9.5 |

Combining the results of Paragraphs 1 - 4 sulfur-chromium mode providing maximum thickness of physical structure of sulfur-chromium layer of machined surface of precision item we will contact with the solution of the optimization problem of the following form:

Development of new technological methods of processing of metals requires an adequate mathematical model allowing to predict interrelated effect of various factors of physical-chemical environment of the metalworking and mechanical-geometric characteristics of the treated surface of the item on the obtained results. Mathematical model of optimization (14) for a multivariate process of sulfur-chromium gives such an opportunity, namely, to identify the most critical parameters and set determined areas of improvement of used and developed technological plants for obtaining sulfur-chromium layer. Proposition 4 and Formula (9) allowing to calculate geometric coordinates of the stationary point for the problem of optimization determine (in terms of system (13)) the following highly effective technological parameters of the mode of sulfur-chromium: by virtue of system (5) (or that is equivalent to the Equations (13)) a stationary point (9) in the coordinate representation (of a row-vector) has the form:

or the same in the physical dimensions of given “reference” from the mode w:

Mathematical result obtained above (the coordinates of the stationary point of sulfur-chromium mode (9)) is in accordance with the logic of physical reasoning; illustration

Since own values of the matrix D_{1} respectively equal

then it speaks about the stationary saddle point of the functional F(v).

According to (12) and (16) in the obtained stationary point v^{*} functional F(v) reaches its “max” in the variables _{7}.

The foregoing discussion can be summarized in one sentence: if there is w_{7} = 9.917% CrO_{3}, then it is necessary to fulfill the conditions

if the position w_{7} = b ¹ 9.917% CrO is implemented, then it is necessary to decide the correction of the problem (3), (14) of the form in full capacity including the identification (2)

This rule is of course largely engineering (not mathematical); from a purely mathematical point of view it only specifies the behavior of chemical-technological process stating that in any case it is necessary to describe (to explain in a heuristic level) the original choice of the percentage in the solution of three chromium oxide CrO_{3}. In this connection we will mention another unexpected result: the first six parameters _{7}―content of three chromium oxide.

The problem of the analytical description of the a posteriori set of data occurs in many sections of science and technology associated with the modeling and/or identification of cognitive systems. In this context the article discusses theoretical issues of regression-tensor modeling of multivariate chemical-technological process in the class of systems (1) and on its basis rigorous analytical interpretations are given which were imposed as nonlinear constraints of theoretical nature as providing the optimal mode of chemical-technological process.

In Paragraph 2a detailed mathematical study of the question of existence of regression model while particular attention was paid to the role of differential calculation (in the constructions of strong Frechet derivatives) in finite-dimensional Euclidean spaces for receiving qualitative conditions (Proposition 2) in the solution of the task of “satisfactory” modeling. In this regard we will note that the description of chemical-technological process by regression system (1) and differential models [

The problem of identification of the method of least squares of coordinates of covariant tensors of both linear and bilinear is considered in Paragraph 3 of common positions formalized by criterion (2). In the large extent in this part of work the confirmation of algorithmic theory of nonlinear regression-tensor modeling of chemical- technological process in terms of designing rules for calculating parameters (8) by conditions suitable for the application of the optimal estimation (2) of operators of the regression model (1) was received in terms of Proposition 3.

In Paragraph 4 the importance of the theory of a posteriori mathematical modeling of chemical-technological process outlined in previous sections is confirmed by the fact that it is not only analytical (which is important itself) but the fact that it leads to efficient algorithms for synthesis of the optimal chemical-technological process. In this context the formula (9) for calculating geometric coordinates of the stationary point of the optimal mode of chemical-technological process was obtained according to the target criterion (3) as well as sufficient conditions are given to guarantee maximum quality of chemical-technological process in practice.

Paragraphs 5, 6 show the results of the numerical solution based on the experimental data of the problem of identification of bilinear tensors of nonlinear regression model of sulfur-chromium coating layer having the optimal thickness of sulfur-chromium layer. The stages of the numerical solution of the problem of parametric identification were considered while the detected deviations of calculated (predicted) values of synthesized sulfur-chromium layer and experimental data aren’t of fundamental nature in consequence of which effective mathematical method (finite chain of algebraic operations (9)) of calculating the optimal coating thickness providing the parameters of nonlinear multivariate mode of sulfur-chromium space of precision item was investigated and confirmed.

For a more complex chemical-technological process, a broader “dictionary of modeling” and the best knowledge of the theory of multivariate regression-tensor modeling are necessary to describe the structure of the functional (3) and use its properties due to additional research (in the spirit of [

on identification and algorithmization of procedure of selection of weighted coefficients

on the expansion of the linear-quadratic form of equations of regression (5) of “Taylor decomposition” of vector-function of regression of higher order;

on registration of additional parameters-coordinates of vector-function of regression model such physical- mathematical parameters of chemical-technological process as surface hardness, wear resistance, coefficient of dry friction of treated surface as well as fragility of the resulting metal plating;

on the development of nano-metric indicators of chemical-technological process and their qualitative account in the structure of nonlinear-tensor multidimensional regression model (1).

This work was supported by the Program “Leading Scientific Schools” (project no. NSh-5007.2014.09).