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Binder jetting is a crucial process in additive manufacturing (AM) and is widely used in sand mold casting. This study explores the challenges of simulating binder droplets in ANSYS Fluent, including complexity and computational time. To overcome these challenges, we propose a geometric approach that models the binder droplet as a circular shape instead of an actual droplet. Additionally, the dynamic mesh feature is employed to transform the initial boundary condition into a wall condition at a specified time interval (Δt). This simplified approach eliminates the need to simulate actual droplets, leading to significant computational resource and time savings. By adopting this geometric approach, we can accurately predict the diffusion and penetration behavior of binder droplets with varying materials and volumes in porous media with different porosities. Through data analysis, it was found that the main variables affecting the diffusion diameter and penetration depth are binder volume and porosity. The successful implementation of this simplified model enables researchers and engineers to expedite the simulation of binder behavior, facilitating process optimization and enhancing the understanding of binder jetting technology in the field of additive manufacturing.

In quantum mechanics, one can express the evolution operator and other quantities in terms of functional integrals. The main goal of this paper is to prove corresponding results in the geometric approach to quantum theory. We apply these results to the formalism of L-functionals., 17 pages

This paper presents a model-free fault detection and isolation (FDI) method for nonlinear dynamical systems using Koopman operator theory and linear geometric technique. The key idea is to obtain a Koopman-based reduced-order model of a nonlinear dynamical system and apply the linear geometric FDI method to detect and isolate faults in the system. Koopman operator is an infinite-dimensional, linear operator which lifts the nonlinear dynamic data into an infinite-dimensional space where the correlations of dynamic data behave linearly. However, due to the infinite dimensionality of this operator, an approximation of the operator is needed for practical purposes. In this work, the Koopman framework is adopted toward nonlinear dynamical systems in combination with the linear geometric approach for fault detection and isolation. In order to demonstrate the efficacy of the proposed FDI solution, a mathematical nonlinear dynamical system, and an experimental three-tank setup are considered. Results show a remarkable performance of the proposed geometric Koopman-based fault detection and isolation (K-FDI) technique. Scopus

Using the geometric approach, we formulate a quantum theory in terms of Jordan algebras. We analyze the notion of a (quasi)particle (=elementary excitation of translation-invariant stationary state) and the scattering of (quasi)particles in this framework.

A geometric approach to the proof of Fermat's last theorem is proposed. Instead of integers a, b, c, Fermat's last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers satisfying the inequalities n>m, n

This paper presents a geometric approach for real-time forward kinematics of the general Stewart platform, which consists of two rigid bodies connected by six general serial manipulators. By describing the rigid-body motion as exponential of twist, and taking advantage of the product of exponentials formula, a step-by-step derivation of the proposed algorithm is presented. As the algorithm naturally solves all passive joint displacements, the correctness is then verified by comparing the forward-kinematic solutions from all chains. The convergence ability and robustness of the proposed algorithm are demonstrated with large amounts of numerical simulations. In all test cases, the proposed algorithm terminates within four iterations, converging with near-quadratic speed. Finally, the proposed algorithm is also implemented on a mainstream embedded motion controller. Compared with the incremental method, the proposed algorithm is more robust, with an average execution time of 0.48 ms, meeting the requirements of most applications, such as kinematic calibration, motion simulation, and real-time control.

This paper presents a comparison of fuel-optimal and shortest paths of an unmanned combat aerial vehicle (UCAV) with obstacle avoidance. A nonlinear constrained optimization algorithm is applied to obtain the optimal paths. An initial value problem (IVP) and an inverse-dynamics approach are used separately to determine optimal paths for various scenarios and in order to reduce computation time. While inputs of the optimization algorithm are discrete control variables in the IVP method, discrete state variables are used as inputs in the inverse-dynamics method. The minimized path segments of the geometrical model provide an initial estimation of the heading angle for the aircraft flight mechanics model. The number of variables used by the optimization algorithm has a direct effect upon the optimal accuracy; however, the computation time is inversely proportional to the number of the variables. Simulation results demonstrate that the proposed IVP method effectively converges to optimal solutions.

We study an SAIRS-type epidemic model with vaccination, where the role of asymptomatic and symptomatic infectious individuals are explicitly considered in the transmission patterns of the disease. We provide a global stability analysis for the model. We determine the value of the basic reproduction number $\mathcal{R}_0$ and prove that the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_01$, condition under which a positive endemic equilibrium exists. We investigate the global stability of the endemic equilibrium for some variations of the original model under study and answer to an open problem proposed in Ansumali et al. \cite{ansumali2020modelling}. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when $\mathcal{R}_0=1$. We provide a thorough numerical exploration of our model, to validate our analytical results., 29 page, 7 figures

In recent years, rabies virus transmission has affected the community widely. Therefore, the study of this deadly virus transmission acquires a significant place in the epidemic spreading. One way to know about the inner sight of such diseases is to consider them mathematically. In this research, we develop a mathematical model for rabies transmission under harmonic mean type incidence rate and consider its qualitative behavior. Using the Next Generation matrix technique, we have derived the threshold number R 0 for the given model. Local and Global stabilities for the disease-free equilibrium are discussed using the Castillo-Chavez method. We have further derived the conditions under which R 0 > 1 and have shown that the model is locally and globally stable at an endemic equilibrium point. For stability, a geometrical approach which is the generalization of Lyapunov theory is used by using a third additive compound matrix. The sensitivity analysis of the basic reproductive number R 0 is carried out and some important parameters are discussed in the last section.

In this paper, we discuss the Anthroponotic Cutaneous Leishmania transmission. In addition, we develop a mathematical model for the Anthroponotic Cutaneous Leishmania transmission and consider its qualitative behavior. We derive the threshold number $R_{0}$ R 0 of the model using the next generation method. In the disease-free case, we carry out the local and global stability under the condition $R_{0} R 0 < 1 . Moreover, we derive the global stability at the disease-free equilibrium point by utilizing the Castillo-Chavez method. On the other hand, at the endemic equilibrium point, we show the local and global stability to be held under specific conditions and $R_{0}>1$ R 0 > 1 . We also establish the global stability at the endemic equilibrium point with the help of a geometrical approach, which is a generalization of Lyapunov theory, by using a second additive compound matrix. Finally, we take into account the sensitivity analysis of the threshold number with other parameters. We also discuss several graphs of important parameters.

We define inclusive scattering matrix in the framework of geometric approach to quantum field theory . We review the definitions of scattering theory in the algebraic approach and relate them to the definitions in geometric approach., New results added

In this chapter, it will be shown how the tools and methodologies introduced in the previous chapters can be used to address classical control problems, underlying also the main difference in the results with respect to the delay-free case.

In this chapter, we introduce the main tools that will be used in the book to deal with nonlinear time-delay systems affected by constant commensurate delays. We will introduce basic notions such as the Extended Lie derivative and the Polynomial Lie Bracket (Califano et al. 2011a; Califano and Moog 2017) which generalize to the time-delay context the standard definitions of Lie derivative and Lie Bracket used to deal with nonlinear systems. We will finally show how changes of coordinates and feedback laws act on the class of systems considered.

International audience

This paper presents a new theoretical approach to the study of robotics manipulators dynamics. It is based on the well-known geometric approach to system dynamics, according to which some axiomatic definitions of geometric structures concerning invariant subspaces are used. In such a framework, certain typical problems in robotics are mathematically formalised and analysed in axiomatic form. The outcomes are sufficiently general that it is possible to discuss the structural properties of robotic manipulation. A generalized theoretical linear model is used, and a thorough analysis is made. The noninteracting nature of this model is also proven through a specific theorem.

Identification of microorganisms by MALDI-TOF mass spectrometry is a very efficient method with high throughput, speed, and accuracy. However, it is significantly limited by the absence of a universal database of reference mass spectra. This problem can be solved by creating an Internet platform for open databases of protein spectra of microorganisms. Choosing the optimal mathematical apparatus is the pivotal issue for this task. In our previous study we proposed the geometric approach for processing mass spectrometry data, which represented a mass spectrum as a vector in a multidimensional Euclidean space. This algorithm was implemented in a Jacob4 stand-alone package. We demonstrated its efficiency in delimiting two closely related species of the Bacillus pumilus group. In this study, the geometric approach was realized as R scripts which allowed us to design a Web-based application. We also studied the possibility of using full spectra analysis (FSA) without calculating mass peaks (PPA), which is the logical development of the method. We used 74 microbial strains from the collections of ICiG SB RAS, UNIQEM, IEGM, KMM, and VGM as the models. We demonstrated that the algorithms based on peak-picking and analysis of complete data have accuracy no less than that of Biotyper 3.1 software. We proposed a method for calculating cut-off thresholds based on averaged intraspecific distances. The resulting database, raw data, and the set of R scripts are available online at https://icg-test.mydisk.nsc.ru/s/qj6cfZg57g6qwzN.

Fault diagnosis techniques can be classified into passive and active types. Passive approaches only utilize the original input and output signals of the system. Because of the small amplitudes, the characteristics of incipient faults are not fully represented in the data of the system, so it is difficult to detect incipient faults by passive fault diagnosis techniques. In contrast, active methods can design auxiliary signals for specific faults and inject them into the system to improve fault diagnosis performance. Therefore, active fault diagnosis techniques are utilized in this article to detect and isolate incipient faults based on the fault structure. A new framework based on observer approach for active fault diagnosis is proposed and the geometric approach based fault diagnosis observer is introduced to active fault diagnosis for the first time. Based on the dynamic equations of residuals, auxiliary signals are designed to enhance the diagnosis performance for incipient faults that have specific structures. In addition, the requirements that auxiliary signals need to meet are discussed. The proposed method can realize the seamless combination of active fault diagnosis and passive fault diagnosis. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed approach, and it is indicated that the proposed method significantly improves the accuracy of the diagnosis for incipient faults.

This paper investigates an SIS epidemic model with variable population size including a vaccination program. Dynamics of the endemic equilibrium of the model are obtained, and it will be shown that this equilibrium exists and is locally asymptotically stable when $${\mathcal {R}}_0 > 1$$ . In this case, the disease uniformly persists, and moreover, using a geometric approach we conclude that the model is globally asymptotically stable under some conditions. Also, a numerical discussion is given to verify the theoretical results.

This paper provides a complete characterization of solvability of the problem of structural model matching by output feedback in linear impulsive systems with nonuniformly spaced state jumps. Namely, given a linear impulsive plant and a linear impulsive model, both subject to sequences of state jumps which are assumed to be simultaneous and measurable, the problem consists in finding a linear impulsive compensator that achieves exact matching between the respective forced responses of the linear impulsive plant and of the linear impulsive model, by means of a dynamic feedback of the plant output, for all the admissible input functions and for all the admissible sequences of jump times. The solution of the stated problem is achieved by reducing it to an equivalent problem of structural disturbance decoupling by dynamic feedforward. Indeed, this latter problem is formulated for the so-called extended linear impulsive system, which consists of a suitable connection between the given plant and a modified model. A necessary and sufficient condition for the solution of the structural disturbance decoupling problem is first shown. The proof of sufficiency is constructive, since it is based on the synthesis of the compensator that solves the problem. The proof of necessity is based on the definition and the geometric properties of the unobservable subspace of a linear impulsive system subject to unequally spaced state jumps. Finally, the equivalence between the two structural problems is formally established and proven.

In this paper, we study a new SEIRS epidemic model describing nonlinear incidence with a more general form and the transmission of influenza virus with disease resistance. The basic reproductive number $\Re_{0}$ is obtained by using the method of next generating matrix. If $\Re_{0}1$ , by using the geometric method, we obtain some sufficient conditions for global stability of the unique endemic equilibrium. Finally, numerical simulations are provided to support our theoretical results.

This paper deals with developing tests for checking whether an unknown system has certain structural properties. The tests that we are aiming at are in terms of noisy input–state–output data obtained from the unknown system. Since, in general, the data do not determine the unknown system uniquely, many systems are compatible with the same set of data. Therefore we cannot apply system identification and apply existing, model based, tests. Instead, we will use the concept of informativity, and establish tests for informativity of the given noisy data. We will do this for a range of system properties, among which strong observability and detectability and strong controllability and stabilizability. These informativity tests will be in terms of rank tests on polynomial matrices that can be constructed from the noisy data. We will also set up a geometric framework for informativity analysis. Within that framework we will give geometric tests for informativity for strong observability, observability, and left-invertibility.

An HIV/AIDS epidemic model with general nonlinear incidence rate and treatment is formulated. The basic reproductive number $\Re_{0}$ is obtained by use of the method of the next generating matrix. By carrying out an analysis of the model, we study the stability of the disease-free equilibrium and the unique endemic equilibrium by using the geometric approach for ordinary differential equations. Numerical simulations are given to show the effectiveness of the main results.

In this paper we study the question under which conditions a linear multi-modal system is disturbance decoupled. We establish necessary and sufficient geometric conditions from which the existing results on switched linear systems and conewise linear systems can be recovered as special cases. Also, we apply these conditions to a class of linear complementarity systems in order to obtain a more crisp characterization.

International audience; A full characterization of accessibility is provided for nonlinear time–delay systems. It generalizes the rank conditionwhich is known for weak controllability of linear time-delay systems, as well as the celebrated geometric approachfor delay free nonlinear systems and the characterization of their accessibility. Besides, fundamental results are derived onintegrability and basis completion which are of major importance for a number of general control problems for nonlinear time–delay systems. They are shown to impact preconceived ideas about canonical forms for nonlinear time–delay systems.

В магістерській дисертації описано керування матричним перетворювачем частоти (МПЧ) із застосуванням геометричного підходу для опису змінних − керування за допомогою вхідного параметру, в якості якого обрано плечові напруги, двома вихідними параметрами, за які виступають вихідні фазні напруги та синусоїдальні струми фаз. Розглянуто побудову вихідних параметрів керування на двовимірній площині з тривимірного простору вхідних величин. По обраній схемі МПЧ проаналізовано його роботу на підвищеній частоті (150 Гц) та пониженій частоті (25 Гц), і показано відповідні часові діаграми змінної напруги. Побудовано таблицю що описує струми та напруги в МПЧ при відповідних станах вентилів. За приведеною таблицею виведені матриці які описують перехід від тривимірного простору вхідних величин до двовимірних просторів вихідних величин. Розраховано силові елементи схеми та обрано відповідні модулі. Обрано просторово-векторну систему керування для даного МПЧ. За допомогою програмного забезпечення Matlab промодельована обрана система. Показано відповідні вхідні та вихідні характеристики. Master's thesis describes the control of the matrix frequency converter (MPC) with the use of a geometric approach to describe the variables - control using an input parameter, which is selected shoulder stresses, two output parameters, for which the output phase voltages and sinusoidal phase currents. The construction of control output parameters on a two-dimensional plane from a three-dimensional space of input quantities is considered. According to the chosen scheme, the MFC analyzes its operation at high frequency (150 Hz) and low frequency (25 Hz) and shows corresponding timing diagrams of alternating voltage. A table is created describing the currents and voltages in the MFP at the respective states of the valves. The following table shows the matrices that describe the transition from three-dimensional space of input quantities to two-dimensional spaces of initial values. The power elements of the circuit are calculated and the corresponding modules are selected. The spatial-vector control system for this MFC is selected. The selected system is modeled with the help of Matlab software. The corresponding input and output characteristics are shown. В магистерской диссертации описано управления матричным преобразователем частоты (МПЧ) с применением геометрического подхода для описания переменных − управление с помощью входящего параметра, в качестве которого выбраны плечевые напряжения, двумя выходными параметрами, за которые выступают выходные фазные напряжения и синусоидальные токи фаз. Рассмотрено построение выходных параметров управления на двумерной плоскости из трехмерного пространства входных величин. По выбранной схеме МПЧ проанализированы его работу на повышенной частоте (150 Гц) и пониженной частоте (25 Гц), и показано соответствующие временные диаграммы переменного напряжения. Построено таблицу описывающую токи и напряжения в МПЧ при соответствующих состояниях вентилей. По приведенной таблицей выведены матрицы описывающие переход от трехмерного пространства входных величин в двумерных пространств выходных величин. Рассчитано силовые элементы схемы и избран соответствующие модули. Выбрано пространственно-векторную систему управления для данного МПЧ. С помощью программного обеспечения Matlab промоделирована выбранная система. Показано соответствующие входные и выходные характеристики.

In this survey we present recent results on switching linear systems. In particular, we recall structural-geometric notions of invariance, controlled invariance and conditioned invariance for switching linear systems and we show how they can be used to provide solutions to a number of control and application problems.

Ovaj rad opisuje glavne sastavnice linearnog programiranja, funkciju cilja i dopustivo područje, uključujući njihove moguće oblike i metode rješavanja problema linearnog programiranja – geometrijski pristup rješavanju problema linearnog programiranja i opis koraka simpleks metode, najpoznatijeg algoritma za rješavanje problema linearnog programiranja. Posebno pažnja je usmjerena na geometrijski pristup rješavanju problema linearnog programiranja i njegovu primjenu u srednjoškolskoj nastavi matematike. Nadalje, u ovom radu možemo pronaći područja i načine primjene linearnog programiranja u ravnini i u višedimenzionalnim prostorima. Također, opisuje se i korištenje informatičke tehnologije u svrhu rješavanja i vizualizacije problema linearnog programiranja koja su primjerena učenicima srednjih škola. Zaključno, ovaj rad je opis procesa linearnog programiranja i njegove primjene u nastavi matematike u srednjim školama. S obzirom na to, nadam se da će mnogima biti pomoć u radu te poticaj i ohrabrenje za poučavanje matematike. This thesis describes the main ideas and steps in linear programming, objective function and feasible area, including their possible forms and methods of solving linear programming problems - a geometric approach to solving the problem of linear programming and a description of the steps of the simplex method, well known algorithm for solving problems of linear programming. Particular attention is focused on the geometric approach to solving the problem of linear programming and its application in secondary school mathematics. Furthermore, in this thesis we can find areas and ways of applying linear programming in the plane and in multidimensional spaces. We also describe the use of informatic technology for the purpose of solving and visualizing linear programming problems that are appropriate for high school students. To conclude, this thesis describes the process of linear programming and its application in teaching mathematics in secondary schools. With this in mind, Ihope that this thesis will be helpful in work, thrust and encouragement for many to teach mathematics.

In this paper, the control of a matrix frequency converter (MFC) is considered using a geometric approach. In the previous work, methods of forming the output voltages according to the law of change of input voltages were considered, but the issue of simultaneous formation of both the output voltage and the input current was not considered. To solve this problem, it is advisable to consider the simplest transforming system using a matrix frequency converter. The purpose of this work is to obtain, along with matrices for the transition from shoulder voltages, as independent variables, to the output values of a two-dimensional space when providing a sinusoidal input current and a voltage of a given frequency.The construction of controlling output parameters on a two-dimensional plane from a three-dimensional space of input quantities is considered. Shows the three-dimensional appearance of the independent variables in the coordinates of the indicated vectors Ual1, Ual2, Ual3 and the projection of the vectors of the independent variables on the two-dimensional planes V and W. To obtain the vectors of currents on the two-dimensional plane V, it is necessary to make a projection of the input vectors of voltages with a gain on the coefficient of translation voltages in currents. For the input values the shoulder voltages Ual1, Ual2 and Ual3 are selected. For the output values, the voltage on the loading phases Ua, Ub, Uc and the currents of the iA, iB, iC phases. Accordingly, this will enable the MFR to calculate the valve control law to generate phase voltages and sinusoidal current consumption.The work of the MFC will be considered at an elevated frequency (150 Hz) and at a lower frequency (25 Hz). The base vectors have chosen those shoulder voltages that are formed by only one input vector (one of the three shoulder voltages was equal to one, and the other two to zero). The tables describing the currents and voltages in the MFC at the appropriate valve states at each frequency are constructed. The following tables show matrices that describe the transition from the three-dimensional space of input quantities to two-dimensional spaces of the output quantities. At 25 Hz, there are two switches when the input shoulder voltage is formed by one vector (line 5 and 6 of Table 2), so we have two matrices for currents. Also, at these frequencies, the output voltages are constructed on each of the phases and the input voltages with separated intervals under each switching of valve states.The resulting matrices for the voltages and the resulting matrix for currents at different frequencies were compared with each other. With the help of elementary transformations of the matrices, their equivalence with each other is proved. Thus, the received matrices allow to calculate independent vectors of three-dimensional space, source vectors in two-dimensional space that will be dependent on each other. That extends the ability to operate with shoulder voltages and key states in the MFC.Ref. 10, fig. 4., В данной статье рассмотрено управление матричным преобразователем частоты (МПЧ) с помощью геометрического подхода. Рассмотрено построение управляющих выходных параметров на двумерную плоскость из трехмерного пространства входных величин. Составлена таблица, описывающая токи и напряжения в МПЧ при соответствующих состояниях вентилей. По приведенной таблице выведены матрицы, описывающие переход от трехмерного пространства входных величин к двумерным пространствам выходных величин. Проанализирована работа МПЧ по выбранной схеме на повышенной частоте (150 Гц) и пониженной частоте (25 Гц) и показаны соответствующие временные диаграммы переменного напряжения.Библ. 10, рис. 4., В даній статті розглянуто керування матричним перетворювачем частоти (МПЧ) за допомогою геометричного підходу. Розглянуто побудову керуючих вихідних параметрів на двовимірну площину з тривимірного простору вхідних величин. Побудовано таблицю, що описує струми та напруги в МПЧ при відповідних станах вентилів. За приведеною таблицею виведені матриці, які описують перехід від тривимірного простору вхідних величин до двовимірних просторів вихідних величин. По обраній схемі МПЧ проаналізовано його роботу на підвищеній частоті (150 Гц) та пониженій частоті (25 Гц) і показано відповідні часові діаграми змінної напруги.Бібл. 10, рис. 4.

—In this paper the elementary cell of the inverter is considered without taking into account losses in the converter circle and internal resistance. The use of a geometric approach for the analysis of a single-phase inverter voltage is considered. The vectors of input and output voltages on the load and current consumed from the source are analyzed. It can be seen that during the transition from the space of inputs to the space of output quantities one degree of freedom is lost. Using the cell states indicated in the table, the formation of a sinusoidal load voltage with the use of a pause is shown. In forming the output voltage in this way, the current consumed will be in the form of pulses and such a current may not satisfy the conditions that are pushed to the inverter from the source side. The given method of forming a sinusoidal load voltage, in which on the positive half-period is formed an interval with a negative voltage, and on the negative one - with a positive one. The expressions for calculating the average values of the load voltage and the supply current during the switching period of the transistor are given. Proposed the method of forming the necessary value of the output voltage with the provision of constant consumed current. In this case, the control with ensuring the flow of direct current requires the construction of a system of vectors with the projection of input values on the plane of the output quantities for both the voltage and the currents. It is shown that using the method of providing the necessary voltage using the change in polarity provides a constant current consumption, however, the use of power source is reduced. Vector diagrams are presented, and a timeline describing the work of the inverter cell. Constructed a table in which are formed the voltages and currents that correspond to the particular condition keys of the inverter. The matrices of the transition from the input values in a two-dimensional space to the initial quantities in a one-dimensional space are constructed. It is shown that, unlike voltage vectors, current vectors coincide, so it is expedient to consider them as one on both half-cycles. Thus the vector is obtained that corresponds to the value of the current in the positive and negative half-cycle of the voltage, provided that the half-waves of the voltage is symmetric. Using the data obtained in the article, based on the analysis of the scheme with the help of a geometric approach, it is possible to develop the rules of control of the cell of the inverter.Ref. 10, fig. 3, tabl. 1., Рассмотрено использование геометрического подхода для анализа однофазного инвертора напряжения. Проанализированы векторы входных и выходных величин по напряжению на нагрузке и по току, потребляемом из источника. Предложен способ формирования требуемого значения выходного напряжения с обеспечением постоянного тока потребления. Построена таблица, в которой приведены сформированные напряжения и токи, соответствующие конкретным состояниям ключей инвертора. Выведены матрицы перехода от входных величин в двумерном пространстве до выходных величин в одномерном пространстве.Библ. 10, рис. 3, табл. 1., Розглянуто використання геометричного підходу для аналізу однофазного інвертору напруги. Проаналізовані вектори вхідних і вихідних величин за напругою на навантаженні та струмом, що споживається з джерела. Запропонований спосіб формування необхідного значення вихідної напруги з забезпеченням постійного струму споживання. Сформована таблиця, що описує стани ключів інвертора та напруги і струми, що формуються таким чином. Побудовані матриці переходу від вхідних величин в двовимірному просторі до вихідних величин в одновимірному просторі.Бібл. 10, рис. 3, табл. 1.

In the conditions of increasing use of dynamic and nonlinear loads, it is important to construct systems with frequency converters that will be invariant to the maximum number of external perturbations. For an asynchronous electric drive, it is advisable to use a matrix frequency converter (MFC), among which the advantages are: free power transfer and power supply; lack of energy accumulation; the formation of curves input and output currents and voltages with a minimum contribution of higher harmonics.This article presents the use of a geometric approach to control the MFC. The geometric approach method for describing variables is promising in relation to the choice of a system of variable transducers, which determines the maximum number of control channels that is the greatest problem in developing invariant systems. According to the scheme of the selected MFC, the formation of a sinusoidal input current and output voltage using a geometric approach is described for the description of variables. In [1, 2, 3, 5] we present a geometric approach for describing the input and output parameters of various systems. In this case, [1, 3, 5] only the formation of the output voltage according to known shoulder voltages is considered. At the same time, the simultaneous formation of the voltage of the given frequency and the sinusoidal current of consumption is appropriate for the construction of the power supply of the asynchronous electric drive. To construct a control system by the geometric approach, the input values form a three-dimensional space in which the vectors of these quantities are located, and their projections form a two-dimensional space of output quantities, while the dimensions of the input and output voltage voltages of the converter depend on the number of the arms of the converter.For the formation of sinusoidal input current and output voltage, a geometric approach is used to describe the variables in the three- shoulders MFC (Fig.1), where each key represents a set of two transistors and two diodes that are engaged in counter-parallel (Fig.2). For all possible logical switches of the keys a table is created, where the input voltage is selected for the output voltage, the voltage at the load phases and the phase currents are selected for the output values. The MFC works both on the high and on the lower frequency (Fig.4). In the three-dimensional space of input values, vectors of shoulder voltages are constructed. The vectors of the shoulder voltages of the MFR form a three-dimensional space of input quantities (shoulder voltages O2O1, O2K, O2L, O1O2, O1H, O1F), shown in Fig.3. Projections of these input values into two-dimensional quantities: stresses on loading phases (Ua, Ub, Uc) and input phase currents (ia, ib, ic). Thus, the system of vectors in Fig.3 illustrates that at the same time forming the voltage in the loading and syncope phase phases, in the independent system of vectors in a three-dimensional space, at least one vector appears, depending on both output variables.Ref. 8, Fig. 4, Table. 1., В данной статье представлено использования геометрического подхода для управления матричным преобразователем частоты. По схеме выбранного матричного преобразователя частоты (МПЧ) рассмотрено формирование синусоидального входного тока и выходного напряжения с использованием геометрического подхода для описания переменных. Создана таблица со всеми возможными логическими включениями ключей, где за входную величину избраны плечевые напряжения, за выходные величины избраны напряжения на фазах нагрузки и токи фаз. Проанализирована работа преобразователя как на повышенной, так и на пониженной частоте. В трехмерном пространстве входных величин построены векторы плечевых напряжений. Проекции этих входных величин на двумерное пространство сформируют векторы выходных величин.Библ. 8, рис. 4, табл. 1., В даній статті представлено використання геометричного підходу для керування матричним перетворювачем частоти. По схемі обраного матричного перетворювача частоти (МПЧ) розглянуто формування синусоїдального вхідного струму та вихідної напруги з використанням геометричного підходу для опису змінних. Створено таблицю з усіма можливими логічними включеннями ключів, де за вхідну величину обрано плечові напруги, за вихідні величини обрано напруги на фазах навантаження та струми фаз. Проаналізовано роботу перетворювача як на підвищеній, так і на пониженій частоті. В тривимірному просторі вхідних величин побудовані вектори плечових напруг. Проекції цих вхідних величин на двовимірний простір сформують вектори вихідних величин.Бібл. 8, рис. 4, табл. 1.

В даній статті представлено використання геометричного підходу для керування матричним перетворювачем частоти. По схемі обраного матричного перетворювача частоти (МПЧ) розглянуто формування синусоїдального вхідного струму та вихідної напруги з використанням геометричного підходу для опису змінних. Створено таблицю з усіма можливими логічними включеннями ключів, де за вхідну величину обрано плечові напруги, за вихідні величини обрано напруги на фазах навантаження та струми фаз. Проаналізовано роботу перетворювача як на підвищеній, так і на пониженій частоті. В тривимірному просторі вхідних величин побудовані вектори плечових напруг. Проекції цих вхідних величин на двовимірний простір сформують вектори вихідних величин. In the conditions of increasing use of dynamic and nonlinear loads, it is important to construct systems with frequency converters that will be invariant to the maximum number of external perturbations. For an asynchronous electric drive, it is advisable to use a matrix frequency converter (MFC), among which the advantages are: free power transfer and power supply; lack of energy accumulation; the formation of curves input and output currents and voltages with a minimum contribution of higher harmonics. This article presents the use of a geometric approach to control the MFC. The geometric approach method for describing variables is promising in relation to the choice of a system of variable transducers, which determines the maximum number of control channels that is the greatest problem in developing invariant systems. According to the scheme of the selected MFC, the formation of a sinusoidal input current and output voltage using a geometric approach is described for the description of variables. In [1, 2, 3, 5] we present a geometric approach for describing the input and output parameters of various systems. In this case, [1, 3, 5] only the formation of the output voltage according to known shoulder voltages is considered. At the same time, the simultaneous formation of the voltage of the given frequency and the sinusoidal current of consumption is appropriate for the construction of the power supply of the asynchronous electric drive. To construct a control system by the geometric approach, the input values form a three-dimensional space in which the vectors of these quantities are located, and their projections form a two-dimensional space of output quantities, while the dimensions of the input and output voltage voltages of the converter depend on the number of the arms of the converter. For the formation of sinusoidal input current and output voltage, a geometric approach is used to describe the variables in the three- shoulders MFC (Fig.1), where each key represents a set of two transistors and two diodes that are engaged in counter-parallel (Fig.2). For all possible logical switches of the keys a table is created, where the input voltage is selected for the output voltage, the voltage at the load phases and the phase currents are selected for the output values. The MFC works both on the high and on the lower frequency (Fig.4). In the three-dimensional space of input values, vectors of shoulder voltages are constructed. The vectors of the shoulder voltages of the MFR form a three-dimensional space of input quantities (shoulder voltages O2O1, O2K, O2L, O1O2, O1H, O1F), shown in Fig.3. Projections of these input values into two-dimensional quantities: stresses on loading phases (Ua, Ub, Uc) and input phase currents (ia, ib, ic). Thus, the system of vectors in Fig.3 illustrates that at the same time forming the voltage in the loading and syncope phase phases, in the independent system of vectors in a three-dimensional space, at least one vector appears, depending on both output variables. В данной статье представлено использования геометрического подхода для управления матричным преобразователем частоты. По схеме выбранного матричного преобразователя частоты (МПЧ) рассмотрено формирование синусоидального входного тока и выходного напряжения с использованием геометрического подхода для описания переменных. Создана таблица со всеми возможными логическими включениями ключей, где за входную величину избраны плечевые напряжения, за выходные величины избраны напряжения на фазах нагрузки и токи фаз. Проанализирована работа преобразователя как на повышенной, так и на пониженной частоте. В трехмерном пространстве входных величин построены векторы плечевых напряжений. Проекции этих входных величин на двумерное пространство сформируют векторы выходных величин.

В даній статті розглянуто керування матричним перетворювачем частоти (МПЧ) за допомогою геометричного підходу. Розглянуто побудову керуючих вихідних параметрів на двовимірну площину з тривимірного простору вхідних величин. Побудовано таблицю, що описує струми та напруги в МПЧ при відповідних станах вентилів. За приведеною таблицею виведені матриці, які описують перехід від тривимірного простору вхідних величин до двовимірних просторів вихідних величин. По обраній схемі МПЧ проаналізовано його роботу на підвищеній частоті (150 Гц) та пониженій частоті (25 Гц) і показано відповідні часові діаграми змінної напруги. In this paper, the control of a matrix frequency converter (MFC) is considered using a geometric approach. In the previous work, methods of forming the output voltages according to the law of change of input voltages were considered, but the issue of simultaneous formation of both the output voltage and the input current was not considered. To solve this problem, it is advisable to consider the simplest transforming system using a matrix frequency converter. The purpose of this work is to obtain, along with matrices for the transition from shoulder voltages, as independent variables, to the output values of a two-dimensional space when providing a sinusoidal input current and a voltage of a given frequency. The construction of controlling output parameters on a two-dimensional plane from a three-dimensional space of input quantities is considered. Shows the three-dimensional appearance of the independent variables in the coordinates of the indicated vectors Ual1, Ual2, Ual3 and the projection of the vectors of the independent variables on the two-dimensional planes V and W. To obtain the vectors of currents on the two-dimensional plane V, it is necessary to make a projection of the input vectors of voltages with a gain on the coefficient of translation voltages in currents. For the input values the shoulder voltages Ual1, Ual2 and Ual3 are selected. For the output values, the voltage on the loading phases Ua, Ub, Uc and the currents of the iA, iB, iC phases. Accordingly, this will enable the MFR to calculate the valve control law to generate phase voltages and sinusoidal current consumption. The work of the MFC will be considered at an elevated frequency (150 Hz) and at a lower frequency (25 Hz). The base vectors have chosen those shoulder voltages that are formed by only one input vector (one of the three shoulder voltages was equal to one, and the other two to zero). The tables describing the currents and voltages in the MFC at the appropriate valve states at each frequency are constructed. The following tables show matrices that describe the transition from the three-dimensional space of input quantities to two-dimensional spaces of the output quantities. At 25 Hz, there are two switches when the input shoulder voltage is formed by one vector (line 5 and 6 of Table 2), so we have two matrices for currents. Also, at these frequencies, the output voltages are constructed on each of the phases and the input voltages with separated intervals under each switching of valve states. The resulting matrices for the voltages and the resulting matrix for currents at different frequencies were compared with each other. With the help of elementary transformations of the matrices, their equivalence with each other is proved. Thus, the received matrices allow to calculate independent vectors of three-dimensional space, source vectors in two-dimensional space that will be dependent on each other. That extends the ability to operate with shoulder voltages and key states in the MFC. В данной статье рассмотрено управление матричным преобразователем частоты (МПЧ) с помощью геометрического подхода. Рассмотрено построение управляющих выходных параметров на двумерную плоскость из трехмерного пространства входных величин. Составлена таблица, описывающая токи и напряжения в МПЧ при соответствующих состояниях вентилей. По приведенной таблице выведены матрицы, описывающие переход от трехмерного пространства входных величин к двумерным пространствам выходных величин. Проанализирована работа МПЧ по выбранной схеме на повышенной частоте (150 Гц) и пониженной частоте (25 Гц) и показаны соответствующие временные диаграммы переменного напряжения.

Розглянуто використання геометричного підходу для аналізу однофазного інвертору напруги. Проаналізовані вектори вхідних і вихідних величин за напругою на навантаженні та струмом, що споживається з джерела. Запропонований спосіб формування необхідного значення вихідної напруги з забезпеченням постійного струму споживання. Сформована таблиця, що описує стани ключів інвертора та напруги і струми, що формуються таким чином. Побудовані матриці переходу від вхідних величин в двовимірному просторі до вихідних величин в одновимірному просторі. In this paper the elementary cell of the inverter is considered without taking into account losses in the converter circle and internal resistance. The use of a geometric approach for the analysis of a single-phase inverter voltage is considered. The vectors of input and output voltages on the load and current consumed from the source are analyzed. It can be seen that during the transition from the space of inputs to the space of output quantities one degree of freedom is lost. Using the cell states indicated in the table, the formation of a sinusoidal load voltage with the use of a pause is shown. In forming the output voltage in this way, the current consumed will be in the form of pulses and such a current may not satisfy the conditions that are pushed to the inverter from the source side. The given method of forming a sinusoidal load voltage, in which on the positive half-period is formed an interval with a negative voltage, and on the negative one - with a positive one. The expressions for calculating the average values of the load voltage and the supply current during the switching period of the transistor are given. Proposed the method of forming the necessary value of the output voltage with the provision of constant consumed current. In this case, the control with ensuring the flow of direct current requires the construction of a system of vectors with the projection of input values on the plane of the output quantities for both the voltage and the currents. It is shown that using the method of providing the necessary voltage using the change in polarity provides a constant current consumption, however, the use of power source is reduced. Vector diagrams are presented, and a timeline describing the work of the inverter cell. Constructed a table in which are formed the voltages and currents that correspond to the particular condition keys of the inverter. The matrices of the transition from the input values in a two-dimensional space to the initial quantities in a one-dimensional space are constructed. It is shown that, unlike voltage vectors, current vectors coincide, so it is expedient to consider them as one on both half-cycles. Thus the vector is obtained that corresponds to the value of the current in the positive and negative half-cycle of the voltage, provided that the half-waves of the voltage is symmetric. Using the data obtained in the article, based on the analysis of the scheme with the help of a geometric approach, it is possible to develop the rules of control of the cell of the inverter. Рассмотрено использование геометрического подхода для анализа однофазного инвертора напряжения. Проанализированы векторы входных и выходных величин по напряжению на нагрузке и по току, потребляемом из источника. Предложен способ формирования требуемого значения выходного напряжения с обеспечением постоянного тока потребления. Построена таблица, в которой приведены сформированные напряжения и токи, соответствующие конкретным состояниям ключей инвертора. Выведены матрицы перехода от входных величин в двумерном пространстве до выходных величин в одномерном пространстве.

Planning accurate manipulation for deformable objects requires prediction of their state. The prediction is often complicated by a loss of stability that may result in collapse of the deformable object. In this work, stability of a fabric strip folding performed by a robot is studied. We show that there is a static instability in the folding process. This instability is detected in a physics-based simulation and the position of the instability is verified experimentally by real robotic manipulation. Three state-of-the-art methods for folding are assessed in the presence of static instability. It is shown that one of the existing folding paths is suitable for folding of materials with internal friction such as fabrics. Another folding path that utilizes dynamic motion exists for ideal elastic materials without internal friction. Our results show that instability needs to be considered in planning to obtain accurate manipulation of deformable objects., Comment: In review for IEEE/ASME Transactions On Mechatronics

International audience; In the present paper, the problem of attacks on cyber-physical systems via networked control system (NCS) subject to unmeasured disturbances is considered. The geometric approach is used to evaluate the security and vulnerability level of the controlled system. The presented work deals with the so-called false data injection attacks and shows how imperfectly known disturbances can be used to perform undetectable, or at least stealthy, attacks that can make the NCS vulnerable to attacks from malicious outsiders. A numerical example is given to illustrate the approach.

Observability and detectability on one hand as well as accessibility and controllability on the other, are related properties and have not been closely investigated for nonlinear-time delay systems until these recent years. Currently accessibility for nonlinear time delay systems is defined as the absence of autonomous functions on the extended state space. Such a property is expressed in terms of the Accessibility matrix rank, extending to the nonlinear context the well known Kalman Criterion. A weaker notion is introduced in this paper which is motivated by the fact that even if autonomous functions exist for the given system, it may be still possible to move from a fixed point to an arbitrary one in some open subset of $I\!\!R^{n}$ . The study is performed on the class of driftless single input systems

In this paper, a susceptible-vaccinated-exposed-infectious-recovered (SVEIR) epidemic model for an infectious disease that spreads in the host population through horizontal transmission is investigated, assuming that the horizontal transmission is governed by an unspecified function f ( S , I ) $f(S,I)$ . The role that temporary immunity (vaccinated-induced) and treatment of infected people play in the spread of disease, is incorporated in the model. The basic reproduction number R 0 $\mathcal{R}_{0}$ is found, under certain conditions on the incidence rate and treatment function. It is shown that the model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. By constructing a suitable Lyapunov function, it is observed that the global asymptotic stability of the disease-free equilibrium depends on R 0 $\mathcal{R}_{0}$ as well as on the treatment rate. If R 0 > 1 $\mathcal{R}_{0}>1$ , then the endemic equilibrium is globally asymptotically stable with the help of the Li and Muldowney geometric approach applied to four dimensional systems. Numerical simulations are also presented to illustrate our main results.

This work deals with the problem of noninteraction (and the dual problem of fault detection) for discrete-time linear time-invariant finite-dimensional state space systems. In particular, the standard, infinite horizon, noninteracting control requirement is replaced by a less demanding one, where noninteracting control is only sought on finite time horizons, suitably defined in connection with some structural properties of the system subblocks. A necessary and sufficient condition for solvability of the finite horizon problem thus stated is derived in terms of the almost controllability subspaces associated with the block structure of the system. The proof of the condition is constructive in the sense that it leads to a design procedure for the feedforward compensators that guarantee noninteracting control over the given time horizons. The underlying idea of the proof, i.e. the exploitation of the almost controllability subspace algorithm in the case of finite horizon noninteraction, is also compatible with a modified procedure for designing the compensators achieving infinite horizon noninteraction, which may be admissible for some specific subblocks of the system. In fact, in this latter case, it is the effective use of the controllability subspace algorithm which plays a key role. The dual counterpart in the context of fault detection introduces a structural means to identify and treat the cases where, due to the structural properties of the monitored system, some of the residuals which can be generated are only significant in a limited time. These concepts are also illustrated with a detailed numerical example.

This paper presents the investigation of direct and inverse kinematics for piezo robot with spherical actuators. Two different actions of spherical piezo actuators are intro-duced. Several methods to solve inverse kinematics were analyzed. A geometric approach is chosen for solving the inverse kinematics, because it has a lot of advantages in ac-cordance to other methods. A theory and a concise algo-rithm to find the solutions of the inverse kinematics using the global projections of robot links is presented.

International audience; This paper proposes a novel approach to obtain the maximal singularity-free re-gions of planar parallel mechanisms, which is based on a constructive geometric reasoning. The proposed approach consists of two algorithms. First, the bor-ders of the singularity-free region associated with an arbitrary starting point of the moving platform is obtained. Then, the second algorithm finds the center of the maximal singularity-free circle, which is obtained using the so-called offset curve algorithm. The procedure is applied to a 3-PRR planar parallel mechanism as an example and the obtained results illustrate graphically the effectiveness of the proposed algorithm. The proposed approach can be directly applied to obtain the maximal singularity-free circle of similar parallel mechanisms, which is not the case for other approaches proposed in the literature that are limited to a given parallel mechanism, namely, the 3-RPR planar parallel mechanism. Moreover, as the main feature of the proposed approach, it can be implemented both in a CAD system and in a computer algebra system where non-convex and reentrant curves can be considered.

Este trabalho propõe investigar o significado da abordagem geométrica para o estudo das funções, e certificar-se de como ela pode contribuir para a aquisição de competências em outras áreas do conhecimento. Para tanto foram realizadas entrevistas com professores de outras áreas do conhecimento, além da aplicação de atividades a uma turma do primeiro ano do Ensino Médio. Dos professores foram ouvidos relatos de como os gráficos de funções se apresentam em suas disciplinas, a habilidade dos alunos em lidar com esses gráficos e de como seria possível sanar possíveis dificuldades. Aos alunos foram aplicadas atividades que constam nos livros didáticos de outras disciplinas, a saber, Física, Química, Geografia, Biologia e Sociologia, com as devidas adaptações aos estudo do conceito de função. O que nos permitiu avaliar a compreensão e assimilação dos conteúdos. Os resultados nos leva a ressaltar a importância da abordagem geométrica das funções, não só no estudo da Matemática, como também, na relação dessa com outras disciplinas. This work proposes to investigate the meaning of the geometric approach for the study of the functions, and to make sure how it can contribute to the acquisition of competences in Other areas of knowledge. For that purpose, an interview was conducted with teachers from other areas of knowledge, besides the application of activities to a class of the first year of High School. From the teachers were heard reports of how the function graphs present themselves in their disciplines,the students’ ability to give you these graphs, and how you could remedy possible difficulties. Activities were applied to students Which are included in textbooks of other disciplines, namely Physics, Chemistry, Geography, Biology and Sociology, with appropriate adaptations to the study of the concepts of functions.The results lead us to emphasize the importance of the geometric approach of functions, not only in the study of Mathematics, but also in the relation of this with other disciplines.

The set of m x n singular matrix pencils with normal rank at most r is an algebraic set with r + 1 irreducible components. These components are the closure of the orbits (under strict equivalence) of r 1 matrix pencils which are in Kronecker canonical form. In this paper, we provide a new explicit description of each of these irreducible components which is a parametrization of each component. Therefore one can explicitly construct any pencil in each of these components. The new description of each of these irreducible components consists of the sum of r rank-1 matrix pencils, namely, a column polynomial vector of degree at most 1 times a row polynomial vector of degree at most 1, where we impose one of these two vectors to have degree zero. The number of row vectors with zero degree determines each irreducible component. (C) 2017 Elsevier Inc. All rights reserved.

This work deals with output regulation in multivariable hybrid systems featuring a continuous-time linear dynamics periodically affected by instantaneous changes of the state. More precisely, given a hybrid linear plant and a hybrid linear exogenous system, with periodic state jumps, the problem consists in finding a hybrid feedback regulator, with the same characteristics, achieving global asymptotic stability of the closed-loop dynamics and asymptotic tracking of the reference generated by the exogenous system for all the initial states. Starting from a general, necessary and sufficient condition for the existence of a solution, the discussion leads to a more specific, sufficient condition which outlines the computational framework for a straightforward synthesis of the compensator. The internal model principle is shown to hold in a more general formulation than the original one, adapted to the hybrid systems considered. A numerical example is worked out with the aim of illustrating how to implement the devised technique. The geometric approach is the key methodology in attaining these results.

In this work, we develop a novel fault detection and isolation (FDI) scheme for discrete-time multi-dimensional (n-D) systems for the first time in the literature. These systems represent as generalization of the Fornasini-Marchesini model II two- and three-dimensional (2-D and 3-D) systems. This is accomplished by extending the geometric FDI approach of one-dimensional (1-D) systems to n-D systems. The basic invariant subspaces including unobservable, conditioned invariant and unobservability subspaces of 1-D systems are generalized to n-D models. These extensions have been achieved and facilitated by representing an n-D model as an infinite dimensional system, and by particularly constructing algorithms that compute these subspaces in a finite and known number of steps. By utilizing the introduced subspaces the FDI problem is formulated and necessary and sufficient conditions for its solvability are provided. Sufficient conditions for solvability of the FDI problem for n-D systems using LMI filters are also developed. Moreover, the capabilities and advantages of our proposed approach are demonstrated by performing an analytical comparison with the only currently available 3-D geometric methods in the literature. 1 2016, Springer Science+Business Media New York. This publication was made possible by NPRP Grant No. 4-195-2-065 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. Scopus

International audience; Linearization by output injection has played a key role in the observer design for nonlinear control systems for almost three decades. In this paper, following some recent works, ge- ometric necessary and sufficient conditions are derived for the existence of a dynamic compensator solving the problem under regular output transformation. An algorithm which computes a compensator of minimal order is given.