Функциональный метод локализации и принцип инвариантности Ла-Салля

A functional method of localization has proved to be good in solving the qualitative analysis problems of dynamic systems. Proposed in the 90s, it was intensively used when studying a number of well-known systems of differential equations, both of autonomous and of non-autonomous discrete systems, including systems that involve control and / or disturbances.The method essence is to construct a set containing all invariant compact sets in the phase space of a dynamical system. A concept of the invariant compact set includes equilibrium positions, limit cycles, attractors, repellers, and other structures in the phase space of a system that play an important role in describing the behavior of a dynamical system. The constructed set is called localizing and represents an external assessment of the appropriate structures in the phase space.Relatively recently, it was found that the functional localization method allows one to analyze a behavior of the dynamical system trajectories. In particular, the localization method can be used to check the stability of the equilibrium positions.Here naturally emerges an issue of the relationship between the functional localization method and the well-known La Salle invariance principle, which can be regarded as a further development of the method of Lyapunov functions for establishing stability. The article discusses this issue.В задачах качественного анализа динамических систем хорошо зарекомендовал себя функциональный метод локализации. Предложенный в 90-хх гг., он активно использовался в исследовании ряда известных систем дифференциальных уравнений, как автономных, так и неавтономных, дискретных систем, в том числе систем включающих управление и/или возмущения.Суть метода состоит в построении такого множества в фазовом пространстве динамической системы, которое содержит все инвариантные компактные множества. Понятие инвариантного компактного множества включает положения равновесия, предельные циклы, аттракторы, репеллеры и другие структуры в фазовом пространстве системы, играющие важную роль в описании поведения динамической системы. Построенное множество называют локализирующим. Оно служит внешней оценкой соответствующих структур в фазовом пространстве.Относительно недавно было установлено, что функциональный метод локализации позволяет анализировать поведение траекторий динамической системы. В частности, с помощью метода локализации можно проверять устойчивость положений равновесия.Здесь естественным образом возникает вопрос о связи функционального метода локализации с известным принципом инвариантности Ла-Салля, который можно рассматривать как дальнейшее развитие метода функций Ляпунова для установления устойчивости. Настоящая статья посвящена обсуждению этого вопроса

Наблюдатель состояния для модели кардиостимулятора на основе уравнения Ван дер Поля

A Dutch physiologist and a founder of electrocardiography V. Einthoven [10] proposed the first known model of the cardiac electrical activity.  Later, van der Pol and van der Mark [11] developed a model of the heart, where the heartbeat is considered as a relaxation oscillation. From this point of view, to model the operation of pacemakers, the van der Pol equation [14,15,19] can be useful. The paper offers modeling of only one heart node that is the S-A (sinoatrial) node, which is the main heart pacemaker [20].Many control algorithms for dynamic systems are based on feedback, which involves the full state vector of a dynamic system. However, in practice, the full state vector is not always known. So, in the case of cardiac electrical activity, the potentials of the nodes rather than their changing rates are measured. To restore the full state vector from existing measurements, state observers are often used.In this paper, we solve the task of constructing an observer with linear error dynamics [22.25]. A necessary condition for the existence of such an observer is the system observability. The sufficient conditions can be formulated in the framework of the differential-geometric approach [25] using the ideas of duplicity [25,26]. Within this approach, an algorithm for observer construction can be developed. In the paper, a general problem to construct an observer for two-dimensional systems is solved and the results obtained are applied to the pacemaker model based on the Van der Pol oscillator. The numerical simulation enables us to illustrate operation of the observer developed.Первая известная модель электрической активности сердца была предложена нидерландским физиологом и основоположником электрокардиографии В. Эйнтховеном [10]. Позже Ван дер Поль и Ван дер Марк [11] построили модель сердца, где сердцебиение рассматривается как релаксационное колебание. С этой точки зрения для моделирования работы кардиостимуляторов можно использовать уравнение Ван дер Поля [14,15,19]. В данной работе моделируется работа только одного узла сердца — сино-атриального узла (САУ), являющегося основным кардиостимулятором сердца [20].Многие алгоритмы управления динамическими системами базируются на обратной связи, в которой задействуется полный вектор состояния динамической системы. Однако на практике полный вектор состояния известен не всегда. Так, в случае электрической активности сердца измеряют потенциалы узлов, а скорости их изменения не измеряются. Для восстановления полного вектора состояния по имеющимся измерениям часто применяют наблюдатели состояния.В данной работе решается задача построения наблюдателя с линейной динамикой ошибки [22,25]. Необходимым условием существования такого наблюдателя является наблюдаемость системы. Достаточные условия можно сформулировать в рамках дифференциально-геометрического подхода [25] с использованием идей двойственности [25,26]. В рамках этого подхода можно разработать алгоритм построения наблюдателя. В работе для двумерных систем решается общая задача построения наблюдателя, затем полученные результаты применяются к модели кардиостимулятора на основе осциллятора Ван дер Поля. Работа построенного наблюдателя иллюстрируется путем численного моделирования

The Dini Derivative and Generalization of the Direct Lyapunov Method

The contemporary theory of stability for systems of differential equations is based on the concept of Lyapunov stability, A.M. Lyapunov’s results and their certain generalizations. Analysis of the first approximation of a system is used as a main method to study a stability of the equilibrium points. In publications this method is known as the first Lyapunov method. But this method does not allow drawing conclusions in the critical case, and then the second Luapunov method can be used, which is also known as a direct Lyapunov method.The direct Lyapunov method is based on existing function with certain properties. The function has to be positive definite. If in the certain vicinity of the equilibrium point a function derivative by virtue of the system is not positive, then it is called Lyapunov function. The existence of Lyapunov function means that the equilibrium point is stable.A role of the Lyapunov function is not only to establish the fact of the equilibrium point stability (or stronger property of asymptotic or exponential stability). It gives a lower bound of the region of attraction of an equilibrium point, which can be important in the control theory. So, to construct the Lyapunov function is a problem of importance, even if the fact of equilibrium stability has been already established.Herewith there are no universal methods to construct the Lyapunov function for the autonomous system of differential equations. To solve the problem are used various numerical methods in which it is difficult to provide the important condition —differentiability of function under construction. At the same time, in the Lyapunov method the differentiability condition is non-essential and is related only to the mathematical technique. Therefore, generalizations of the Lyapunov method, which are directed to the abandonment of strong conditions of function smoothness, are important. One of such generalizations is the use of the Dini derivative. The use of the Dini derivative allows us to construct, for example, the piecewise linear Lyapunov functions.The results connected with the Dini derivative are rarely included in standard monographs, and an objective of the present article is to make these results more accessible

Realization of the Iteration Procedure in Localization Problems of Autonomous Systems

In the last 15 years one way for a qualitative analysis of dynamical systems was formed i.e. the localization of invariant compact sets of a dynamical system. Here the localization means creating a system of such sets, which contain all invariant compact sets of a dynamic system [1], in the phase space.Invariant compact sets are closely connected with bounded trajectories of the system, the structure of which in the phase space play key role in many applications of dynamical system theory. The problems of invariant compact sets localization abut upon other important problems, for instance, the problems of estimation of attractor basins, control problems, etc.Back investigations of localization problems was oriented both to development of solving methods [28] and to investigation of particular dynamical systems encountered in applications (see, for example, [9 { 16]).One of quite efficient methods of localization problem solving is based on smooth functions defined in the phase space. It is so called functional method [1 { 3]. Effectiveness of the method is enhanced when we use several functions. Thus, using the next function gives the restriction of the already constructed localizing set. An iteration procedure for sequential narrowing of the localizing set [1 { 2] arises.The paper presents analysis of the iteration procedure, which naturally occur in the autonomous systems of special type where the right side of each differential equation is resolvable relative to the corresponding phase variable. Such systems are encountered in applications [17].</p

The Qualitative Analysis of a Lorenz-Type System

In modern natural sciences, the term of a dynamic system plays an important role and is a common type of mathematical models. Dynamical systems are rarely come to simple functional dependencies. Therefore, qualitative analysis methods of dynamical systems are crucial. In the paper, we consider the simplest type of dynamic systems | continuous dynamical systems described by the systems of ordinary differential equations.Qualitative analysis of differential equations systems usually starts with a search for equilibrium points and a study of the behaviour of a dynamic system in the neighborhood of each equilibrium points. The main attention is paid to the stability of equilibrium, as well as their behaviour type classification. Effective qualitative analysis of differential equations systems is best approached through the bifurcation theory which explains modification of quality in the behaviour of a dynamic system if its parameters are changed.In the behavior of dynamic systems, in addition to the equilibrium points, other bounded trajectories (for example, boundary cycles or separatrix) and their certain conglomerates (such as attractors, invariant tori) play an important role. Investigation of bounded trajectories, in particular, attractors is a difficult task and a lot of scientific articles deal with this problem.In this paper, we study a continuous Lorenz-type system. For this system, all of the equilibrium points are defined and the analysis of equilibrium points types are performed in accordance with the system parameters. The analysis of some bifurcations of equilibrium points are carried out. In particular, the Andronov | Hopf bifurcation is determined and it is shown that it leads to a bifurcation of boundary cycles.DOI: 10.7463/mathm.0315.078949

The plane motion control of the quadrocopter

Among a large number of modern flying vehicles, the quadrocopter relates to unmanned aerial vehicles (UAV) which are relatively cheap and easy to design. Quadrocopters are able to fly in bad weather, hang in the air for quite a long time, observe the objects and perform many other tasks. They have been applied in rescue operations, in agriculture, in the military and many other fields.For quadrocopters, the problems of path planning and control are relevant. These problems have many variants in which limited resources of modern UAV, possible obstacles, for instance, for flying in a cross-country terrain or in a city environment and weather conditions (particularly, wind conditions) are taken into account. Many research studies are concerned with these problems and reflected in series of publications (note the interesting survey [1] and references therein). Various methods were used for the control synthesis for these vehicles: linear approximations [2], sliding mode control [3], the covering method [4] and so on.In the paper, a quadrocopter is considered as a rigid body. The kinematic and dynamic equations of the motion are analyzed. Two cases of motion are emphasized: a motion in a vertical plane and in a horizontal plane. The control is based on transferring of the affine system to the canonical form [5] and the nonlinear stabilization method [6].DOI: 10.7463/mathm.0215.078947

The Behavior of a Two-Component Population System in Vicinity of the Zero Equilibrium Point

A clinical behaviour of the cell-based therapy in recent decades has encouraged a great interest in the culture of cell populations in vitro. One of the directions of cell therapy is transplantation of stem cells. The cell material for transplantation is obtained by culturing the patient’s cells. However, there are often problems because of the genetic mutations of cells in the process of culture, namely, the degeneration of a portion of mutated cells into "immortal" (cancerous) cells, which makes the transplantation of such material unsafe for a patient. To study development dynamics of the cell populations in vitro is quite costly. Such studies are, usually, conducted at the beginning of culture, in the middle of the process, and when completing the process of culture. It is difficult to judge t the cell population development in detail by such data. Here mathematical modelling is of importance.The papers [8-11] propose a cell population system consisting of two types of cells, namely normal (healthy) and abnormal (aneuploid) cells. Interest in such a population system is due to the fact that, although the aneuploid cells have a life time less than the normal ones, a small portion of the aneuploid cells can degenerate into practically "immortal" cancer cells, whose population may, eventually, become dominant.In the qualitative analysis of the nonlinear dynamic systems, a standard component is information on the number of rest points, their nature and location. Earlier, [16] a detailed study of the rest points and their possible nature was performed depending on the biological parameters, such as the proportion of dead cells, the average time of the cell cycle, the proportions of normal cells becoming the population of abnormal ones, etc. However, there is no exhaustive answer, yet, concerning this issueThe paper continues to study the two-component population model considered earlier [9-11, 16]. The study focuses on the zero equilibrium point. The conditions of stability are specified taking into account the fact that the dynamic system, by virtue of its biological content, must be considered in the first quadrant of the plane. In addition, a study of the zero equilibrium point has been conducted in critical cases in which the method to investigate linear approximation stability the does not work