Linear Homogeneous System Research Articles

К проблеме устойчивости периодического решения в условиях бифуркации Хопфа

Данная работа посвящена проблеме устойчивости малого периодического решения нормальной автономной системы обыкновенных дифференциальных уравнений. При исследовании устойчивости периодического решения автономной системы естественно анализировать локальную динамику пересечений возмущенных траекторий с ортогональными сечениями соответствующего цикла. Путем введения специальной системы координат, в которой одна из осей направлена по касательной к траектории периодического решения, задача об орбитальной устойчивости периодического решения сводится к задаче об устойчивости по Ляпунову нулевого решения вспомогательной системы с периодической по t правой частью. Для вспомогательной системы, размерность которой на единицу меньше размерности исходной системы, в линейном приближении вопрос об устойчивости нулевого решения сводится к оценке мультипликаторов матрицы монодромии. Таким образом, по теореме Андронова — Витта реализуется классический подход к исследованию орбитальной устойчивости периодического решения. При этом имеет место некритический случай орбитальной устойчивости. Такой подход традиционно используется и в условиях бифуркации типа Хопфа для систем с параметром. В данной работе для автономной системы с параметром получены условия бифуркации малого решения, период которого близок к периоду решений соответствующей линейной однородной системы. Сформулировано определение свойства орбитальной устойчивости по параметру, согласно которому возмущенные правые полутраектории сколь угодно близки к исследуемому циклу не только за счет малости возмущений начальных значений, но и за счет малости параметра. При этом использована идея ослабления требований определения устойчивости ляпуновского типа, предложенная М.М. Хапаевым. Свойство орбитальной устойчивости по параметру может иметь место и при наличии орбитальной неустойчивости исследуемого цикла в классическом смысле. Для исследования орбитальной устойчивости малого периодического решения по параметру использовано нелинейное приближение упомянутой выше вспомогательной системы возмущенных движений.

Dynamic research of elementary differential gear with elastic links

Many research works are devoted for study of static and dynamic parameters of planetary gears of various designs. A number of dynamic problems can be solve accurately enough without taking into account the elasticity of loaded elements. In calculations of mining machines transmissions dynamic processes must be considered gear flexibility as is much more than flexibility of large diameter and small length shafts. In this article are considered possibilities and features of research of dynamic phenomena that take a place in planetary mechanisms of mining machine transmissions. For this purpose planetary mechanisms are presented as elastic mechanical systems. For research the dynamic model of elementary differential mechanism with taking into account the link elasticity is used. For this purpose the differential gear presented as equivalent planetary row with inertialess planetary carrier. Main links of such mechanism have inertia moments determined by known formulas. The equivalent dynamic model of elementary differential gear is presented as three-mass branched system. Moments of elasticity, occurring in mechanical constraints of system, are determined. Equations of movement for each of the masses give the linear homogeneous differential equation system or free oscillations equation system of mechanism. This system is representable as linear algebraic equation system. Particular solutions of inhomogeneous differential equation system are of practical interest, as they permit to make the analysis of the effect of external loads, acting on the mechanism. Operator solution method application will give particular solutions of linear inhomogeneous differential equations. Received systems will allow solving many dynamic tasks of elastic mechanical systems: character change and maximum values of mechanism links dynamic loads, periods and frequencies of oscillations, conditions of the resonant state of the mechanical system.

Статическая термоустойчивость пологой геометрически нерегулярной оболочки из ортотропного термочувствительного материала

Рассматривается пологая ортотропная геометрически нерегулярная оболочка (ГНО) постоянного кручения, термомеханические параметры которой линейно зависят от температуры. При достижении температуры определенного значения происходит скачкообразно смена формы равновесия, что вызывает изменение первоначальной геометрии оболочки. Эти значения температур называют критическими. Для практики значительный интерес представляют соотношения, связывающие критические температуры с геометрическими и термомеханическими параметрами ГНО. Решение задач статической термоустойчивости ГНО, как правило, начинается с анализа их исходного безмоментного состояния. Тангенциальные усилия, вызванные нагревом оболочки, определяются как решения системы сингулярных дифференциальных уравнений безмоментной термоупругости. Эти усилия содержатся в формах Брайена или Рейсснера в уравнениях статической термоустойчивости, и от их структуры существенно зависит успех в дальнейшем решении задачи. В работе решение сингулярной безмоментной термоупругости найдено в элементарных функциях. Уравнения моментной термоупругости, записанные в компонентах поля перемещений, методом функции перемещений сведены к одному сингулярному дифференциальному уравнению в частных производных восьмого порядка. Решение записывается в виде двойного тригонометрического ряда, коэффициенты которого на основании процедуры Галeркина определяются как решения линейной однородной алгебраической системы уравнений. Из равенства нулю определителя этой системы (в первом приближении) получено алгебраическое уравнение пятой степени для относительной критической температуры, наименьший положительный действительный корень которого и есть искомая температура. Проводится количественный анализ влияния геометрических и термомеханических параметров ГНО на величины критических температур.

Dynamic Quantized Consensus of General Linear Multi-agent Systems under Denial-of-Service Attacks

In this paper, we study a multi-agent consensus problem under Denial-of-Service (DoS) attacks with data rate constraints. We consider leaderless consensus under an undirected communication graph and assume that the graph is connected in the absence of DoS. The dynamics of the agents take general forms modeled as homogeneous linear time-invariant systems. In our analysis, we derive specific bounds on the data rate for the multi-agent system to achieve consensus even in the presence of DoS attacks. The main contribution of the paper is the characterization of the trade-off between the tolerable DoS attack level and the required data rates for the communication among the agents. To avoid quantizer saturation under DoS attacks, we employ dynamic quantization with zoom-in and zoom-out capabilities.

Optimal Prestress Investigation on Tensegrity Structures Using Artificial Fish Swarm Algorithm

To obtain the optimal uniform prestress of a tensegrity structure with geometric configuration given, a novel method is developed for prestress design of tensegrity structures by utilizing the artificial fish swarm algorithm (AFSA). In the beginning, the form‐finding process is implemented by solving a linear homogeneous system concerning the self‐equilibrium system. The issue is subsequently performed as a minimum problem by regulating the value of an objective function where the unilateral condition and the stress uniformity condition are entirely considered. The AFSA is adopted to search for the global minimum, leading to a set of initial prestresses that guarantee all the above conditions. Two illustrative examples have been fully studied to prove the accuracy and efficiency of the presented approach in prestress design of tensegrities according to the practical requirements. Furthermore, the numerical examples investigated in this paper confirm that the AFSA has explicit advantages of rapid convergence and overcoming the local minima.

On the Kronecker canonical form of a full row rank matrix pencil and applications

Abstract The Kronecker canonical form (KCF) of matrix pencils plays an important role in many fields such as systems control and differential–algebraic equations. In this article, we compute a finite and infinite Jordan chain and also a singular chain of vectors corresponding to a full row rank matrix pencil using an extended algorithm, first introduced by Jones (1999, Ph.D. Thesis, Department of Mathematics, Loughborough University of Technology, Loughborough, UK). The proposed method exploits these vectors forming the chains corresponding to the finite and infinite eigenvalues and to the right minimal indices of the pencil. This leads to the computation of two transformation matrices for obtaining under strict equivalence the KCF of the pencil. An application to the study of homogeneous linear rectangular descriptor systems is considered and closed form solutions are obtained in terms of these two transformation matrices. All the results are illustrated with an example.

Some Properties of Oscillation Indicators of Solutions to a Two-Dimensional System

It is proved that all strong oscillation indicators considered as functionals on the set of solu¬tions to linear homogeneous two-dimensional differential systems with continuous coefficients bounded on the semi-axis are not residual (i.e. can be changed under changing the solution on a finite interval). An example of two-dimensional system with a solution whose all strong oscillation indicators differ from the corresponding weak ones is provided.

Input-to-state Stability of Nonlinear Positive Systems

In this paper, input-to-state stability (ISS), as a useful tool for robust analysis, is first applied to continuous-time and discrete-time nonlinear positive systems. For continuous-time and discrete-time positive systems, some new definitions of ISS are introduced. Different from the usual ISS definitions for nonlinear systems, our ISS definitions can fully reflect the positiveness requirements of states and inputs of the positive systems. By introducing the max-separable ISS Lyapunov functions, some ISS criterions are given for general nonlinear positive systems. Based on that, the ISS criterions for linear positive systems and affine nonlinear homogeneous systems are given. Through them, the ISS properties can be judged directly from the differential and algebraic characteristics of the systems. Simulation examples verify the validity of our results.

A Novel Reduced-Order Protocol for Consensus Control of Linear Multiagent Systems

This paper deals with output-feedback-based state consensus of homogeneous linear multiagent systems. A novel reduced-order dynamic output-feedback protocol is proposed, which does not need the absolute output information of neighboring agents. Necessary and sufficient conditions for the solvability of the consensus problem under the proposed protocol are derived, and it is shown that the protocol state is convergent if consensus is reached. A design algorithm is then constructed for computing the protocol gains. Furthermore, the protocol specialized for leader–follower consensus is addressed. In addition, the solvability of a truncated version of the proposed protocol without controller interaction is discussed from a low-gain perspective. A numerical example is finally presented to demonstrate the effectiveness of the proposed protocol and design method.

Time-Varying Output Formation Containment of General Linear Homogeneous and Heterogeneous Multiagent Systems

This paper investigates the time-varying output formation-containment control of general linear homogeneous and heterogeneous multiagent systems (MAS). This control aims at making each follower's output reach an agreement on the time-varying formation reference, namely, the centroid of the multiple leaders’ outputs, and keep a predefined time-varying offset with respect to it. Distributed static output-feedback (OPFB) and dynamic OPFB control protocols are first constructed for homogeneous and heterogeneous MAS, respectively, using only neighboring relative output information and a time-varying formation compensation signal. It is proved that the time-varying output formation-containment problem is solved by making certain output formation-containment errors go to zero asymptotically. Then, local sufficient conditions and explicit local design procedures are provided for both homogeneous and heterogeneous MAS. It is shown that the state/output consensus problems, the state/output containment problems, the state/output formation control problems, the time-varying state/output formation-tracking problems with a single leader, and target enclosing problems of homogeneous/heterogeneous MAS can all be treated as special cases of this paper. Finally, numerical simulations validate the effectiveness of the proposed control protocols.

Formulas for the general solution of weakly delayed planar linear discrete systems with constant coefficients and their analysis

The paper is concerned with weakly delayed linear discrete homogeneous planar systems with constant coefficients. By the method of Z-transform, formulas for the general solutions, dependent on the Jordan forms of the matrix of non-delayed linear terms, are derived and the influence is analyzed of the delay on the form of the general solutions. It is shown that, after several steps, the general solutions depend only on two arbitrary parameters which are linear combinations of the initial values. This property is used to prove results on conditional stability. Linear discrete homogeneous planar systems without delay are found to have the same general solutions as the initial one. The results are illustrated by examples. Previous results are analyzed, commented and improved.

Reducibility of Linear Differential Systems to Linear Differential Equations

Lyapunov reducibility of any bounded and sometimes unbounded linear homogeneous differential system to some bounded linear homogeneous differential equation is established. The preservation of the additional property of periodicity of coefficients is guaranteed, and for two-dimensional or complex systems the constancy of their coefficients is preserved. The differences in feasibility of asymptotic and generalized Lyapunov reducibility from Lyapunov one are indicated.

A linear state feedback switching rule for global stabilization of switched nonlinear systems about a nonequilibrium point

A switched equilibrium of a switched system of two subsystems is a such a point where the vector fields of the two subsystems point strictly towards one another. Using the concept of stable convex combination that was developed by Wicks et al. (1998) for linear homogeneous systems, Bolzern and Spinelli (2004) offered a design of a state feedback switching rule that is capable to stabilize a linear nonhomogeneous switched system to any switched equilibrium. The state feedback switching rule of Bolzern–Spinelli gives a nonlinear (quadratic) switching threshold passing through the switched equilibrium. In this paper we prove that the switching threshold (i.e. the associated switching rule) can be chosen linear, if each of the subsystems of the switched system under consideration is globally asymptotically stable.

Oscillation, Rotatability, and Wandering Characteristic Indicators for Differential Systems Determining Rotations of Plane

The oscillation, rotatability, and wandering characteristic indicators of Lyapunov type are studied for two-dimensional linear homogeneous differential systems determining rotations of the phase plane. A complete set of order relations between them is obtained. For each of those indicators it is established whether it is continuous or discontinuous as a function of the coefficients of the system.

Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

This work concerns with free vibration analysis of cracked nanobeam problems. Based on Eringen's nonlocal elasticity theory, the governing equation of Euler–Bernoulli and Timoshenko nanobeams, are derived. It is assumed that strain at a certain point is a function of the strains at all points within the influence domain. The cracked beam is modeled as multi-segments connected by a rotational spring located at the cracked sections. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. Polynomial based differential quadrature method is employed to solve the problem. Derivatives of the field quantities are approximated as a weighted linear sum of the nodal values. For different supporting cases, the boundary conditions are directly substituted in the equation of motion, such that the problem is reduced to that of linear homogeneous algebraic system. This suggested numerical scheme accurately determined angular frequencies of the problem. A comparative study is tabulated to compare the obtained results with the previous ones. Further, a parametric study is introduced to investigate the influence of crack locations, crack severity and the nonlocal scale parameter on the obtained results. The obtained results recorded that frequency values decrease with the increasing of both of crack severity and the nonlocal scale parameter. The results of the proposed scheme may be applied for structural health monitoring.

Buckling analysis of sandwich orthotropic cylindrical shells by considering the geometrical imperfection in face sheets

A three dimensional analysis for the buckling behavior of a sandwich orthotropic thick cylindrical shell under uniform lateral pressure has been presented. It is assumed that both ends of the shell have the simply supported conditions and the face sheets have an axisymmetric initial geometrical imperfection in the axial direction. The governing differential equations are derived based on the second Piola–Kirchhoff stress tensor and are reduced to a homogenous linear system of equations using differential quadrature method (DQM). The buckling pressures have been calculated for the shell with isotropic core and orthotropic face sheets with 0-degree orientation with respect to the hoop direction. Moreover, buckling pressure reduction parameter has been defined and computed for different imperfection parameters of face sheets and geometrical properties of sandwich shells. The results obtained in the present work are compared with finite element solutions and results reported in the literature and very good agreements have been observed. It is shown that the imperfections have higher effects on the buckling load of thick shells than thin ones. Likewise, it is found that the present method can capture the various geometrical imperfections observed during the manufacturing process or service life.

On the structure of the fundamental matrix of the linear homogeneous differenrial system of the special kind

For the linear homogeneous differenrial system, whose coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency, the kind of the fundamental matrix are established by the condidtion of the some resonance relations.

The existence of a two-dimensional bounded system with continual and coinciding spectra of frequencies and of wandering exponents

A two-dimensional linear homogeneous differential system with continuous bounded coefficients is constructed so that the frequencies and wandering exponents of each of its nonzero solutions coincide with each other and the set of their values at different solutions (the spectrum) is some interval of the number line. Bibliography: 12 titles.

The full separation of the countable linear homogeneous system of the differenial equations at the resonance case

For the countable linear homogeneous system of the differential equations, the coefficients of whose are represented by a absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency, the conditions of the existence of the linear transformation with the coefficients of the similar structure, which leads this system to a diagonal kind in resonance case, are obtained.

Buckling Analysis of Orthotropic Thick Cylindrical Shells Considering Geometrical Imperfection Using Differential Quadrature Method (DQM)

Abstract This paper presented a three dimensional analysis for the buckling behavior of an imperfect orthotropic thick cylindrical shells under pure axial or external pressure loading. Critical loads are computed for different imperfection parameter. Both ends of the shell have simply supported conditions. Governing differential equations are driven based on the second Piola–Kirchhoff stress tensor and are reduced to a homogenous linear system of equations using differential quadrature method. Buckling loads reduction factor is computed for different imperfection parameters and geometrical properties of orthotropic shells. The sensitivity is established through tables of buckling load reduction factors versus imperfection amplitude. It is shown that imperfections have higher effects on the buckling load of thin shells than thick ones. Results show that the presented method is very accurate and can capture the various geometrical imperfections observed during the manufacturing process or transportation.