Methods of using mobile Internet devices in the formation of the general professional component of bachelor in electromechanics competency in modeling of technical objects

. The article describes the components of methods of using mobile Internet devices in the formation of the general professional component of bachelor in electromechanics competency in modeling of technical objects: using various methods of representing models; solving professional problems using ICT; competence in electric machines and critical thinking. On the content of learning academic disciplines “Higher mathematics”, “Automatic control theory”, “Modeling of electromechanical systems”, “Electrical machines” features of use are disclosed for Scilab, SageCell, Google Sheets, Xcos on Cloud in the formation of the general professional component of bachelor in electromechanics competency in modeling of technical objects. It is concluded that it is advisable to use the following software for mobile Internet devices: a cloud-based spreadsheets as modeling tools (including neural networks), a visual modeling systems as a means of structural modeling of technical objects; a mobile computer mathematical system used at all stages of modeling; a mobile communication tools for organizing joint modeling activities.


Introduction
Modernization of professional training of specialists in mechatronics at Ukrainian technological universities [21] based on a balance between the fundamental [19] and technological component of the training process necessitates the search for ICT training tools [20], which not only provide the opportunity for active experimentation anytime, anywhere, but also support the development of professionally important qualities of the future electrical engineer, among which the main place is taken by competence in the modeling of technical objects.Today, such universal teaching tools are mobile Internet devices (MID) [12; 26; 35].This work is a further development of the research begun in articles [20] and [19], the purpose of which is to develop methods of using MID in the formation of bachelor in electromechanics competency in modeling of technical objects.The purpose of the article is the selection and justification of the MID-based software tools, the use of which contributes to the formation of the general professional component of this competency.

Use of mobile Internet devices in the formation of competence in using various methods of representing models
The formation of such a general professional component of the competence of a bachelor of electromechanics in the modeling of technical objects, as a competence in the application of various methods of representing models, involves the acquisition of knowledge and skills in the construction of computer mathematical and simulation models, their algorithmic and structural description, and the selection of adequate ways of representing computer modeling tools.Based on the content of the competence, its formation and development takes place throughout the entire training of the bachelor of electromechanics, therefore, it is inexpedient to single out the leading disciplines for this process.So, in the teaching the "Computing Engineering and Programming", it is possible to solve the problem of numerical integration considered in module 9 "Definite and improper integrals" of the discipline "Higher Mathematics" in a different formulation [19]: instead of using the table representation of the function, use analytic, and instead of interpolating formulas, use stochastic Monte Carlo method.In this case, the mean value theorem for integrals is used, according to which for a curved trapezoid whose area under the graph of a continuous function on a closed, bounded interval is equal to the area of a rectangle whose base is the length of the interval and height is the mean value of the integrand in the interval.
Ilia M. Sobol in [34] proposes an algorithm for the approximate calculation of a definite integral by the Monte Carlo method, noting that "in practice [one-time] integrals ... are not calculated by the Monte Carlo method: for this there are more exact methods -quadrature formulas.However, the transition to multiple integrals changes the situation: quadrature formulas become very complex, and the Monte Carlo method remains almost unchanged" [34, p. 52].
So, to determine the volume under the surface of a function, the following stochastic algorithm can be applied: 1. Limit the surface of a rectangular box, the volume of which V par calculated as the product of length (determined by the integration limits [a; b] by Ox axis) by width (determined by the integration limits [c; d] by Oy axes) by height (determined by the maximum value of the integrand f(x, y) on integration D). 2. Place in a certain parallelepiped a certain number of points N, the coordinates of which we will randomly choose.3. Determine the number of points K that will be located below the surface of the function.4. The volume V, limited by the function and coordinate axes, is given by the expression V = V par K/N.
In order to implement interdisciplinary integration, it is advisable to jointly use this algorithm and the mean value theorem for integrals of a multiple integral: 1. We limit the integration plane to the corresponding limits [a; b] by Ox axes and [c; d] by Oy axes.2. Place on a certain plane a certain number of points N, the coordinates of N we will choose at random.3.At each point we measure the value of the integrand f(x, y) and find the arithmetic mean M for all N points.4. The value of the double integral is given by the expression The first implementation is performed in the same SageCell modeling environment [17; 25], with which the value of the integral was calculated by deterministic methods (fig.1): x,y=var('x,y') # symbol integration variables @interact # interactive model with controls def _(f=input_box(default=x*y-x^2+y^2, label="$f(x,y)=$", width=20), a=input_box(default=0, label="$Lower\ integration\ limit\ by\ Ox\ " "axis\ a=$", width=3), b=input_box(default=2, label="$Upper\ integration\ limit\ by\ Ox\ " "axis\ b=$", width=3), c=input_box(default=1.2,label="$Lower\integration\ limit\ by\ Oy\ " "axis\ c=$", width=3), d=input_box(default=7, label="$Upper\ integration\ limit\ by\ Oy\ " "axis\ d=$", width=3), N=slider(vmin=1, vmax=100000, step_size=1, default=1000, label="$Grid\ points\ quantity$"), eps=input_box(default=0.1, label="$Accuracy $", width=6)): Another implementation will be directed by students to conduct as many experiments as possible in order to clarify the fact that with the same number of experiments, the simulation results can differ significantly.To do this, we use the mobile version of the spreadsheets, and enter the following values corresponding to the previous code.At first, the limits of integration: 7 The next step is to enter formulas for random values of x and y and calculate the function of them: The last step is to calculate the average value of the function and the integral: In order to increase the number of points, you must copy the range F2:H2 in any number of other lines.The simulation results are presented in fig. 2.

Fig. 2. Usage of the Monte Carlo method to calculate the value of a definite integral in mobile spreadsheets
A discussion of two implementations of the same model allows us to draw conclusions both on the correct implementation and on the advisability of using the selected modeling tool.So, when using spreadsheets, an increase in the number of experiments is possible only by adding new lines, however, any changes on the worksheet lead to the generation of new random numbers -instead of clarifying the results of the previous experiment, the student conducts a new experiment.At the same time, when using the model in a computer mathematics system, you can verify the erroneousness of the condition for completing the cycle -instead of abs(sumnext-sum)<eps must use condition abs(sumnext/N-sum/(N-1))<eps for all steps except the first: applying the precondition can lead to both early completion of the cycle (if the function value is close to zero in the first steps) and an overestimation of the number of iterations (if the function value becomes close to zero after reaching the required accuracy), and even to the "eternal" cycle (if the value of the function on the selected area will not be less accurate).An important component of competence in the application of various methods of representing models is the formation of the ability to select an adequate way of representing models of computer modeling tools.One of the traditional ICT training tools for modeling is computer mathematics systems.In [23] it was pointed out that it should be used together with support systems for teaching bachelors in electromechanics for modeling technical objects, such as the Moodle LMS, supplemented by the developed SageCell filter [18].The specified filter is able to implement a numerical solution of systems of differential equations describing mathematical models directly in the learning management system.
However, the use of only computer mathematics systems (even such powerful ones as SageMath [27]) for teaching modeling of technical objects of bachelors in electromechanics is not enough, since the synthesis and calculation of models of control systems, electric drive elements, etc. primarily use visual modeling tools that provide the opportunity to build dynamic models (discrete, continuous and models of systems with discontinuities elements).This determines the necessity and expediency of combining traditional computer mathematics systems with specialized libraries for modeling technical objects in environment for the visual construction of models.At the same time, the choice of environment for modeling should take into account the specifics of future professional activity, which for bachelors of electromechanics is the synthesis of the corresponding technical objects -electromechanical systems.
Mastering the modeling of technical objects provides theoretical and practical filling of the fundamental, general and specialized professional training for a bachelor's of electromechanics.In this regard, it is desirable that the environment for their modeling gives the user access not only to traditional libraries for modeling continuous and discrete dynamic systems, but also to libraries for electric machines and power converters.In addition, to achieve the goal of learning mobility, the modeling environment must have a high level of cross-platform access (in particular, access via a Web interface) and be freely distributed.
In order to make a reasonable choice of the environment for technical objects modeling for bachelors of electromechanics, an expert assessment of the most common systems of visual modeling was carried out, the results of which are presented in the table 1.
Currently, the Scilab has the highest expert assessment, the advantages of which were recognized in 2011 by the French Ministry of National Education, Higher Education and Science, giving Scilab a recognition of its pedagogical significance for teaching mathematics "Reconnu d'Intérêt Pédagogique" [29].
According to [28], Scilab is a package of scientific programs for numerical calculations that provides a powerful open environment for calculations, similar to the Matlab language and a set of functions for mathematical, engineering and scientific calculations.The package is suitable for professional use and use in the universities, providing tools for various calculations from visualization, modeling and interpolation to differential equations and mathematical statistics.Execution of scripts written for Matlab is supported.
Scilab was created in 1990 by INRIA scientists (Institut national de recherche en informatique et en automatique -State Institute for Computer Science and Automation Research) [9] and ENPC (École nationale des ponts et chaussées -National School of Bridges and Roads).At first it was called Ψlab (Psilab).The Scilab Consortium was created in May 2003.To promote the use of Scilab as open source software in the academic and industrial fields.In July 2008, Scilab joined the Digital Foundation to improve technology transfer [10].
In June 2010, the Scilab Consortium announced the creation of Scilab Enterprises [4].Scilab Enterprises develops and sales, directly or through an international network of affiliate service providers, a comprehensive suite of services for Scilab users.Scilab Enterprises also develops and maintains Scilab software.Scilab Enterprises' ultimate goal is to help make using Scilab more efficient and easier.Since February 2017, Scilab has been developed and published by ESI Group, an industrial virtual reality development company [5].
Scilab contains hundreds of mathematical functions with the ability to add new ones written in various languages (C, C++, Fortran, etc.).Various data structures are supported (lists, polynomials, rational functions, linear systems), an interpreter and a high-level language.
Scilab was designed as open system in which users can add their data types and operations on this data by overloading.
Many tools are available in the system: Scilab has a programming language similar to MATLAB; the system includes a utility that allows to convert Matlab documents to Scilab.Scilab allows to work with elementary and a large number of special functions (Bessel, Neumann, integral functions), has powerful tools for working with matrices, polynomials (including symbolic), performing numerical calculations (for example, numerical integration) and solving linear algebra problems, optimization and simulation, powerful statistical functions, as well as a tool for building and working with charts.For numerical calculations, the libraries Lapack, LINPACK, ODEPACK, Atlas and others are used.
The package also includes Scicos, a tool for editing block diagrams and simulations (the Xcos add-on is an analogue of the Simulink package in MATLAB).There is the possibility of collaboration between Scilab and LabVIEW.
Distinctive features of Scilab: ─ fee free; ─ small size (distribution takes less than 150 Mb); ─ the ability to run in the console without using a graphical interface.This allows automated calculations in batch mode.
Starting with version 6, the program is distributed under the GPL compatible license CeCILL license [1].Despite the lack of usability of the text interface, it is easier to adapt to MID.Among the well-known Scilab implementations for the Android OS, we distinguish two -Scilab on Aakash and Scilab Console Free.
Scilab on Aakash [30] is an Indian development that is supported in both the Android and GNU/Linux Aakash versions.In Android, Scilab is part of the APL library (Aakash Programming Lab [7], developed by Indian Institute of Technology Bombay), which provides Scilab 5.4.All Scilab functions cannot be runnging on Android, so the developers provide the user interface with only 2 windows (fig.3): enter commands and view the results of their execution.When executing graphic commands with the Plot option selected, the results of execution will be displayed in a separate window (fig.4).This version of Scilab may be admitted as mobile, but not fully functional.
Scilab Console Free (fig.5) -is a mobile version of Scilab for Android and iOS.Like Scilab on Aakash, this is not a fully functional version of Scilab 5.4.1: the graphic functions of Scilab and Xcos are not activated.The developers of this product went the easy way -to achieve mobility on the device you first have to install one of the Linux options (for example, using GNURoot Debian [2]).
The mobility of this version of Scilab is achieved by transferring the Linux mobile operating system and its software environment to Android and iOS.Therefore, Scilab Console Free is quite demanding on the internal memory of the mobile device and the version of the operating system.The standard Web interface to Scilab, which is proposed by Cloud Scilab [36], does not provide an interface similar to the full-featured version -it is only possible to host usercreated models with a Web interface (application deployment -"deploying" programs to the cloud server [15]) by creation and publication of interactive documents, the software part of which is performed on the server side, saves the resources of a MIDs (fig.6).Programs deployed on the Scilab Cloud are described by the Scilab language (both algorithms and the user interface).This allows you to create programs with a visual interface both for deployment in the cloud (work in Scilab Cloud with display via a web browser) and traditional execution (launch in Scilab on the user's computer).
Despite the high level of mobility, the use of Cloud Scilab is accompanied by a number of problems: ─ deployment of programs is only available with administrative privileges (requires an additional fee to the cloud service provider); ─ the interface is provided only for user models, and not to the entire Scilab interface.
In terms of functionality, this service is similar to the Wolfram Demonstrations Project, however, unlike the latter, all users can publish demonstrations, and not only service administrators comply with the document distribution model in CoCalc [14].However, Scilab Cloud does not provide access to the Xcos module.In addition, in fact, the only source about Scilab Cloud is the materials of webinars conducted by Scilab Enterprises [3], which does not contribute to the widespread use of this Web-interface.
W3 Scilab is significantly more open [13].W3 Scilab is the Indian Web-based interface to Scilab, which is the basis of Scilab on Aakash.The interface allows users to send short fragments of Scilab code to a remote server, receive Scilab code on this server, return and display execution results in a browser.
Xcos is an addition to Scilab, which allows the synthesis of mathematical models in the fields of mechanics, hydraulics, electronics and electromechanics.This visual modeling environment is designed to solve the problems of dynamic modeling of systems, processes, devices, as well as testing and analysis of these systems.In this case, the object is modeled (system, device, process), supplied graphically in the form of a block diagram that includes blocks of system elements and the connections between them (fig.7).The Xcos on Cloud service provides the ability to build simulation models of technical objects (in particular, electromechanical systems) in a mobile Web browser.The current goal of the Xcos on Cloud project (formerly Xcos on Web [22]) is to recreate a fully functional version of Scilab Xcos with access via a mobile Web browser [6].
The main components of the Xcos on Cloud main window are presented in fig.8.

Fig. 8. Xcos on Cloud main window components
The main part of the Xcos on Cloud main window is occupied by the model building area.On the left side is the so-called palette of blocks -a library of elements from which the model is built.To use any block, just drag it from the palette to the model building area.Blocks are interconnected by communication lines.In fig. 9 shows a DC motor model built in Xcos on Cloud.Blocks of the model have various parameters, user-configurable by double-clicking on the selected block.Unfortunately, the project is still at the preliminary development stage, so some settings are not available: for example, the first-order aperiodic unit (CLR) and amplifier (GAIN) blocks in the constructed model are unstable, as evidenced by the format specifier '%s'.This makes it impossible to conduct experiments on the model by the click of a button.Simulate.A temporary workaround for this problem is the ability of Xcos on Cloud to exchange data with the traditional version of Scilab Xcos through data export tools.<?xml version="1.0"encoding="UTF-8"?> <XcosDiagram background="-1" title="MavXcos"> <mxGraphModel as="model"> <root> <mxCell id="0"/> <mxCell id="1" parent="0"/> <BasicBlock blockType="c" id="2" interfaceFunctionName="STEP_FUNCTION" parent="1" simulationFunctionName="csuper" simulationFunctionType="DEFAULT" style="STEP_FUNCTION"> ... </root> </mxGraphModel> <mxCell id="1" parent="0" as="defaultParent"/> </XcosDiagram> The implementation of the full functionality of Scilab Xcos in Xcos on Cloud creates the conditions for a using MID in the training of bachelors in electromechanics, modeling of technical objects.Similar features are provided by the full virtualization of Scilab on rollApp [31].

2.2
Use of mobile Internet devices in the formation of competence in solving professional problems using ICT The formation of a general professional component of the competence of a bachelor of electromechanics in the modeling of technical objects, such as competence in solving professional problems using ICT, provides for the acquisition of knowledge in the field of computer engineering and programming, the ability to create application software, and the skills to work with ICT tools to solve problems in field of electromechanics.The formation of this competence begins in the academic discipline "Computing Engineering and Programming" and occurs along with the formation of the general scientific component of the competence of the bachelor of electromechanics in the modeling of technical objects, in particular competencies in information and communication technologies and in applied mathematics.Its further development takes place in the process of developing models; they are considered in the academic disciplines "Theory of Automatic Control" and "Modeling of Electromechanical Systems".The purpose of studying the discipline "Theory of Automatic Control" is to master the methods of setting tasks, the principles of building automatic control systems, methods of analysis and synthesis of linear, nonlinear, impulse, digital, adaptive and optimal systems using modern software systems for modeling dynamic systems.
As a result of studying the discipline, students, in particular, should receive skills: ─ compose differential equations for elements of automatic control systems and the system as a whole; ─ draw up and transform structural diagrams of automatic control systems; ─ determine the time functions and time characteristics of the automatic control system and its elements under the conditions of various types of signals; ─ determine the frequency functions and characteristics of automatic control systems; ─ analyze the stability of linear, impulse and nonlinear automatic control systems; ─ calculate control devices (controllers) that provide the necessary quality indicators, including using the state space method; ─ synthesize automatic control systems in the presence of random signals; ─ apply adaptive methods to control non-stationary objects.
At the lectures of the second module "Properties and characteristics of closed control systems.Synthesis of linear continuous control systems", the issues of influence on the system and requirements for the control process, stability of control systems, stationary (stable) modes of linear control systems, quality assessment of control systems with step and arbitrary actions, frequency methods for assessing the quality of control systems, approximate methods for choosing the control method and parameters are considered regulators, synthesis of control systems according to the logarithmic frequency characteristics.To consolidate this material, we propose laboratory works "Analysis of stability and quality of control systems" and "Synthesis and research of control systems for objects with a delay".Consider the use of ICT tools to solve the second of them.
Objective of the work: to study the influence of the delay link on the stability and quality of the automatic control system.The content of the work 1.The study of the influence of the delay link on the characteristics of the automatic control system.2. Experimental obtaining the transient and frequency characteristics of the system with delay.

Theoretical information
The automatic control systems may include time delay, the equations of which are of the form: where  -delay time.
The transfer function of a time delay in accordance with the delay theorem (properties of the Laplace transform): (2) Automatic control systems, which include a time delay, are called time delay systems.
It can be connected in the direct circuit of the system or to the feedback circuit.Moreover, regardless of the location of the delay link, the characteristic equation of a closed system with delay has the form: where Q(s) and R(s) -polynomials in the denominator and numerator of the transfer function of the open system without delay.This characteristic equation is not a polynomial and has an infinite number of roots.Therefore, to study the stability of delayed systems, it is necessary to use frequency stability criteria, such as the Nyquist stability criterion.The conclusion about the stability of the system can be made on the basis of the analysis of the amplitude-phase frequency response of an open system with delay.
It can be shown that the presence of a delay link does not change the module of A(ω) but introduces only an additional negative phase shift of -ωτ.By varying the delay time τ over a wide range, one can find its value at which the closed-loop system will be at the stability boundary.In this case, amplitude-phase frequency characteristic of the open-loop phase response of the delayed system will pass through the point (-1; 0) = -1+j0.
The latency  cr and the corresponding value of the frequency ω cr , at which the AFC passes through the point (-1; 0), are called critical.For a critical case, the following conditions are true: The automatic control system will be stable if the delay time  is less than critical:  <  cr .

Work order
The subject of the study is an automatic control system, with a delay link.
1.The transfer function of the open-loop automatic control system without delay has the form: The value of the transmission coefficient K and the time constants Using the Scilab package, construct the transition characteristic h(t) of the closed-loop system with unit negative feedback (fig.10, 11).Make a conclusion about the stability of the system.2. Insert a time delay function with the delay time  = 0,01 s (fig.12), into the direct chain of the system, construct the transition characteristic of the system (fig.13) and make a conclusion how the time delay affects the quality of the transition process.3. Increasing the time of delay, to trace how the transition characteristic of the system changes (fig.14) to make the system lose stability (the transition process diverges).4. To determine experimentally the value of the critical delay time  cr , when the system will be at the stability boundary (the transition process will be undamped).5. Repeat steps 2-4 for the case when the time delay function included in the feedback circuit.Make some conclusions.
Further development of competence in solving professional problems by means of ICTs occurs in the process of completing term paper in the discipline "Theory of automatic control".In order to develop skills for conducting professionally directed educational research, students can be invited to familiarize themselves with new results in the subject area of scientific publications in professional publications and start the coursework on their reproduction.So, in the work "Optimization of the servocontrolled automatic control system" [11] an example of servo-controlled control system of a working body of a paver with hydraulic drive is considered.When moving the stacker along the base, which is ready for laying pavement on it, its running equipment (tracked or wheeled) makes uncontrolled random movements in the vertical direction under the influence of roughnesses in the microrelief of the base of the road.These movements are transmitted through the stacker frame and suspension of the screed to the working equipment, causing in turn uncontrolled movements of the screed, which entail a random change in the thickness and angle of the transverse slope of the stacked layer, thereby deteriorating the quality of the coating.
A simulation model of the hydraulic drive tracking system for the screed plate can be implemented in MATLAB Online, as suggested by the authors of [11].The structure of the simulation model circuit (fig.15) includes the following elements: a bidirectional hydraulic cylinder; three-position valve; hydraulic pump; controlled hydraulic lock; ideal hydraulic pressure sensor; the "smooths the paver plate" element; cylinder rod movement and speed sensor (feedback sensor) the ideal force sensor; the element "hydraulic fluid" (Oil-30W oil) is proportional to the servo valve of the hydraulic actuator (electro-hydraulic distributor that converts the electrical signal to movement) element "viscous friction"; ideal source of strength; disturbing influence of "microrelief"; PS-converter; capacity for working fluid; disturbing effects due to the influence of the work of other elements.

Use of mobile Internet devices in the formation of competencies in electric machines and critical thinking
In the discipline "Electrical Machines", the formation of both the general scientific component of the competence of the bachelor of electromechanics in the modeling of technical objects (the leading means of competencies in basic sciences formation were mobile augmented reality tools), and such a general professional component as competencies in electrical machines, including knowledge of the structure and functioning principles of electrical machines, in particular: energy conversion processes (electromagnetic and electromechanical), characteristics of certain types of electrical machines, structure of asynchronous machines, synchronous machines, DC machines, transformers; the ability to calculate the parameters and characteristics of electrical machines.To form the latter, the leading tools are mobile computer mathematics systems.The last general professional component of a bachelor of electromechanics competence in modeling technical objects is competence in critical thinking -the knowledge and skills of setting a problem with an insufficient amount of input data, analyzing the availability of methods and means of solving a problem, assessing your own readiness to solve a problem, independently searching for missing data and ways to solve the problem; the ability to control their own activities -both mental and practical; the ability to control the logic of deploying your own thoughts; the ability to determine the sequence and hierarchy of the stages of activity and the like.As well as for competence in the application of various ways of representing models, leading academic disciplines cannot be distinguished to this competence -its formation takes place throughout the entire training of a bachelor of electromechanics.
Neural network modeling is an effective means of modeling technical objects with a hidden or fuzzy structure [32; 33].As the editors of the book "Neuro-Control and its Applications" note, the relevance of its application is due to the need to develop control methods for complex nonlinear systems: "The first examples of the development of control methods for nonlinear systems ... are mainly associated with methods for solving nonlinear differential equations that are adequate to a single-processor background of Neumann computational to cars.... The development of computers with mass parallelism ... has led to the creation of fundamentally new algorithms and methods for controlling non-linear dynamic systems.They are associated with neural network algorithms for solving ordinary nonlinear differential equations and, as a result, with the inclusion of a neurocomputer in the control loop of a nonlinear dynamic system.... The sufficiently wide development and spread of such algorithms has led ... to the creation of a whole branch of science called "neurocontrol" [24, pp. 9-10].
In an engineering context, intelligent management should have the following properties: learning ability and adaptability; survivability; simple control algorithm and user-friendly man-machine interface; the ability to incorporate new components that provide the best solutions in the face of limitations imposed by technical tools [24, p. 15].
The deep machine learning is a class of intelligent control algorithms that use multilayer neural networks with non-linear nodes.Let us consider the construction of a neural network model for approximating data described in Chapter 23 of "Neural Network Design" [8], and obtained using an intelligent sensor -one or more standard sensors connected to a neural network to obtain a calibrated measurement of one of the parameters.
The intelligent position sensor uses the voltage value from two photocells to estimate the location of the object.In fig.16, an object located between a light source and two photocells is shown around.The object, moving along the y axis, casts a shadow on the photocells, which leads to a change in the voltages v 1 and v 2 .When the position of the object y increases, the voltage v 1 first decreases, then the voltage v 2 decreases, then v 1 increases and finally v 2 increases (fig.17).To implement the deep learning model, we apply cloud-oriented Google Sheets with the Solver addition according to the technique developed in [16].
In order to determine the coordinate of the object, we will build a four-layer neural network with the architecture shown in fig.19: ─ input layer is a two-dimensional arithmetic vector (x 1 , x 2 ), with the components are the corresponding measured voltages v 1 , v 2 , normalized according to the network activation function; ─ the first hidden layer will have dimension 5 and is described by the vector (h 1 (1) , h 2 (1) , h 3 (1) , h 4 (1) , h 5 (1) ); ─ the second hidden layer will have dimension 3 and is described by the vector (h 1 (2) , h 2 (2) , h 3 (2) ); ─ output layer is y norm value normalized according to the network activation function.For convenience of processing, we transpose the obtained data and put them in cells A10:A76 for v 1 , B10:B76 for v 2 , C10:C76 for y.To do this, we add the following values to the table cells: =TRANSPOSE(B1:BP3) Based on the fact that the constructed neural network will have a polar activation function, all values at the network input should be normalized (reduced to the range [0; 1]).At the output of the network, you must perform the reverse operationdenormalization.
Normalizing is performed for each column separately.To do this, we find the minimum and maximum values for them by entering the following formulas in the cells: E5 =min(A10:A76) Further, the range E6:E7 is copied to F6:G7.The essence of normalizing is easy to understand by the expression (5): normalized value = input value-minimum value maximum value-minimum value . ( With this approach, the minimum value is normalized to 0, and the maximum up to 1.
The normalized voltage values must be feed to the input layer of the neural network: E8 input layer E9 x 1 F9 x 2 E10 =(A10-E$7)/(E$6-E$7) Cell E10 is distributed in the range E10:G76.In accordance with the selected architecture of the neural network, add to the 2 neurons of the input layer a bias neuron.To do this, insert its name (x 3 ), in cell G9, and its value (1) in the range G10: G76.At this stage, the input layer is formed in the form of a signal vector (x 1 , x 2 , x 3 ).The next step is to transmit the signal from the input layer of the neural network to the first hidden one.To determine the signal power, it is necessary to have weights of the neural network.Denote by: ─ w ij xh (1) the weight coefficient of the synapse, which connects the neuron x i (i = 1, 2, 3) of the input layer with the neuron h j (1) (j = 1, 2, ..., 5) of the first hidden layer; 0 To create matrices, it is necessary to copy K9 cell to the range L9:N9, I11 -to I12, J10 -to J10:N12, K15 -to L15, I17 -to I18:I21, J16 -to J16:L21, I26 -to I27:I28, J25 -to J26:J28 (fig.21).To calculate the scalar product of a row vector of input layer values by a column vector of the matrix of weight coefficients w xh (1) , it is advisable to use the matrix multiplication function: =1/(1+exp(-mmult($E10:$G10,J$10:J$12))) U10 1 Next, copy cell P10 to the range P10: T76, and U10 to U11: U76.Given that all elements of the matrix of weight coefficients w xh (1) are initially equal to zero, after copying the formulas, all measured elements of the hidden layer are equal to 0.5.
Similarly, we perform calculations of the elements of the second hidden and output layers: W8 second hidden layer W9 h 1 (2) X9 h 2 (2) Y9 h 3 (2) Z9 h 4 (2) W10 =1/(1+exp(-mmult($P10:$U10,J$16:J$21))) Z10 1 AB8 output layer AB9 y norm AB10 =1/(1+exp(-mmult($W10:$Z10,J$25:J$28))) Next, copy cell W10 to the range W10:Y76, Z10 -to Z11:Z76, AB10 -to AB11:AB76 (fig.22).Within this formulation of training, the neural network can be considered as an optimization problem in which the objective function (the sum of the squared deviations in the cell AH10) should be minimized by varying the weights of the matrices w xh (1)  (range J10:N12), w h(1)h(2) (range J16:L21) and w h(2)y (range J25:J28).To solve this problem, standard Google Sheets tools are not enough, so you need to install the Solver add-on by choosing Add-ons → Get add-ons ... (fig.23).To reduce the sum of squared deviations, Solver can be called up many times: it is advisable to experiment with the combined use of various optimization methods, changing the limits of variation of the weight coefficients.In this case, it is not necessary to try to prove the value of the sum of the distances to zero -this may be more than a (rather small) value.
For clarity of the simulation results, it is advisable to construct a graph of the ratio of experimentally obtained (measured) y values and calculated (approximated) y calc (fig.25).The quality of approximation is determined by the degree of deviation of the graph points from the beam, divides the first quadrant in half.From the figure and the sum of the squares of deviations obtained after optimization, we can conclude that the constructed neural network model is adequate.
To test the model limitations, we propose the next task: using given in the table 2 values of voltages v 1 , v 2 , calculate the y coordinate of the object, and explain the choice of voltages and obtained result.

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analytic integration of the double integral I0=integrate(integrate(f,x,a,b),y,c,d) v=html("The analytic value of the double integral %1.4lf"%I0) # Monte Carlo numerical integration # with a known number of experiments sum=0 # sum of function values f(x,y) for i in range(N): # grid points quantity cycle xi=random()*(b-a)+a # random value generation for x and y yi=random()*(d-c)+c #in range [a; b] and [c; d] sum=sum+f(xi,yi) # sums accumulation I=(sum/N)*(b-a)*(d-c) # integral calculations v=v+html("\nThe numerical value of the double integral " "calculated by the Monte Carlo method for %s grid" " points,\nis %1.4lf"%(N,I)) # Monte Carlo numerical integration # with predetermined accuracy sum=0 # sum of function values f(x,y) N=1 # minimum quantity of experiments while true: # cycle without quantity of experiments limit xi=random()*(b-a)+a # random value generation for x and y yi=random()*(d-c)+c # in range [a; b] and [c; d] sumnext=sum+f(xi,yi) # calculating the next sum # break condition for the cycle -the module of the # difference between the next and previous # sum becomes less than the specified accuracy if(abs(sumnext-sum)<eps): break else: sum=sumnext N=N+1 # move on to the next experiment I1=(sum/N)*(b-a)*(d-c) # integral calculations v=v+html("\nThe numerical value of the double integral " "calculated by the Monte Carlo method with accuracy %1.4lf,\n" "equals %1.4lf and demanded %s grid lines"%(eps, I1, N)) # function surface visualization show(plot3d(f, (x,a,b), (y,c,d))) show(v)

Fig. 1 .
Fig. 1.The use of various models to calculate the value of a definite integral in a computer mathematics system

Fig. 14 .
Fig. 14.Transient response of a closed-loop system with a negative feedback with delay  = 0,1 s

Fig. 15 .
Fig. 15.Structural model of the hydraulic drive tracking system for the screed slab

Fig. 18 .
Fig. 18.The dependence of the voltages v1, v2 on the y coordinate of the object

Fig. 19 .
Fig. 19.Neural network architecture for solving the problem of determining the position of an object

Fig. 21 .
Fig. 21.Fragment of a spreadsheet after normalizing input data and creating matrices of weighting coefficients

Fig. 22 .
Fig. 22. Fragment of the spreadsheet after calculating the initial values of the weighting coefficients of the hidden and output layer

Table 1 .
Comparison of environments for technical objects modeling 1and T 2 are shown in the table

Table 2 .
Model testing Measurement only with the second sensor.The result is adequate because the pair v1, v2 has a match in the test data set 2 0.000 1.000 1.366 Measurement only with the second sensor.The result is inadequate