Nonlinear Nonstationary Systems Research Articles

МЕТОД ТА ДОСЛІДЖЕННЯ ТОЧНОСТІ ОБЧИСЛЮВАННЯ МАТЕМАТИЧНОЇ МОДЕЛІ ЕЛЕКТРОГІДРАВЛІЧНОГО ЕФЕКТУ

The mathematical model of the electrohydraulic effect (EHE) is a nonlinear non-stationary system of partial differential equations (PDE), which reflects the motion and contact interaction of two media, gaseous and liquid, under the action of pulsed thermal excitation. Such PDE systems generally do not have analytical solutions and are solved only by numerical methods, which are approximate in nature, i.e. they provide a solution with some error. The peculiarity of the motion of media in the case of EHE is large deformations in the form of flows and vortices. This feature imposes significant restrictions on the choice of the method for solving the PDE model. The widely known Lagrangian finite element method used to solve contact problems, in the case of EHE modeling leads to a pathological change in the shape of the finite elements, and therefore to instability of the calculation process and an unlimited increase in error. In calculation practice, a variant of the finite element method, FEM-ALE, has become widespread. It uses both Lagrangian and Eulerian grids of nodes and a special advection procedure, i.e., transfer by interpolation of the solution from the Lagrangian grid to the Eulerian one, and then vice versa from the Eulerian one to another new Lagrangian grid. Such an additional interpolation procedure, which is used repeatedly in the calculation, leads to additional (in comparison with the Lagrangian FEM) errors. If the causes and kinetics, as well as the methods of error control in the case of the Lagrangian FEM, have been sufficiently studied in the literature, then the kinetics of error development in the simulation of the EGE using the FEM-ALE method has been practically not studied. The article is devoted to the study of the development of errors in the numerical solution in this particular case. An analytical solution of the problem in the asymptotic case for a technological system with a rigid chamber is obtained. This solution is analyzed in comparison with the numerical solution using the FEM-ALE method. Conclusions are made regarding the errors in calculating pressure as the main factor of mechanical impact of EGE on the technological object and technological equipment, as well as errors in calculating deformations of gaseous and liquid media.The scientific novelty consists in the development of a method for determining the accuracy of EHE parameter calculations using the MSE-ALE method. The practical value lies in determining the accuracy of calculating the parameters of the EHE mathematical model, which makes it possible to justify the use of numerical modeling as a method of researching technological processes and systems using EHE

Synthesis of adaptive observer for nonlinear nonstationary systems

Synthesis of adaptive observer for nonlinear nonstationary systems

Determination of the Speed and Surface Temperature of Aircraft Using the Second Approximation of the System of Moment Equations

This study focuses on solving the initial-boundary value problem for a one-dimensional non-stationary nonlinear system of moment equations in the second approximation. We prove the existence of a unique local in-time solution of the problem under consideration in the space of functions that are continuous in time and square integrable over the spatial variable x. To solve the direct and inverse problems for this system, we develop an iterative numerical method and reduce the system of moment equations to canonical form. Furthermore, we present the software implementation of the algorithms and their application for solving inverse problems related to determining the speed and surface temperature of an aircraft, as well as atmospheric parameters. We provide the results of numerical experiments conducted to validate our approach.

Decomposition of Equations of Nonlinear Affine Control Systems and its Application to the Synthesis of Regulators

The article deals with the decomposition of nonlinear differential equations based on the group-theoretic approach. At the beginning, the decomposition of differential equations of linear systems using a transition matrix of state is presented, and then, based on the theory of continuous groups (Lie groups), the process of decomposition of differential equations of nonlinear systems is shown. The decomposition approach is based on the isomorphism theorem of the space of vector fields and Lie derivatives, which allows us to consider vector fields as differential operators of smooth functions. A formula is derived about the adjoin representation of a Lie group in its Lie algebra, which actually determines the finding of a vector field that characterizes the interaction of two or more vector fields. The Lie algebra of derivatives makes it possible to determine the infinitesimal action of the Lie group, i.e. the linearization of this action is carried out (transformation of the points of the trajectory space of the original system in a small neighborhood). Decomposition allows, as in the linear case, to separate the finding of an action (only locally) of a group of transformations from the transformed points themselves. For linear systems, this separation is global. It is also shown that the decomposition of linear equations is a particular case of the decomposition of nonlinear equations. An algorithm of the method of model predictive control with Gramian weighting using this decomposition is presented. A practical example of decomposition and application of the model predictive control for stabilization of a nonstationary nonlinear system is considered.

Численное решение системы уравнений Навье–Стокса в случае сжимаемой среды с использованием нейронных сетей

The use of the Physics Informed Neural Networks (PINN) method for the numerical solution of a nonstationary nonlinear system of partial differential equations describing the process of motion of a one-dimensional heat-conducting gas is considered. The approach is based on the fact that a neural network approximates the solution of a system of differential equations, while taking into account the physics of the simulated process. The neural network is trained by minimizing a quadratic functional built on the difference between the predicted values and residuals of differential equations, boundary and initial conditions. Different types of approximation of the original equations are discussed in the case when the operator is continuous or discrete in time. An analysis of the presented methods is carried out. Their advantages and disadvantages are indicated.

A Deep Learning Neural Network Method Using Linear Eigenvalue Statistics for Schizophrenic EEG Data Classification

Electroencephalography (EEG) signals can be used as a neuroimaging indicator to analyze brain-related diseases and mental states, such as schizophrenia, which is a common and serious mental disorder. However, the main limiting factor of using EEG data to support clinical schizophrenia diagnosis lies in the inadequacy of both objective characteristics and effective data analysis techniques. Random matrix theory (RMT) and its linear eigenvalue statistics (LES) can provide an effective mathematical modeling method for exploring the statistical properties of non-stationary nonlinear systems, such as EEG signals. To obtain an accurate classification and diagnosis of schizophrenia, this paper proposes a LES-based deep learning network scheme in which a series of random matrixes, consisting of EEG data sliding window sampling and their eigenvalues, are employed as features for deep learning. Due to the fact that the performance of the LES-based scheme is sensitive to the LES’s test function, the proposed LES-based deep learning network is embedded with two ways of combining LES’s test functions with learning techniques: the first is to have the LES’s test function assigned, while, using the second way, the optimal LES’s test function should be solved in a functional optimization problem. In this paper, various test functions and different optimal learning methods were coupled in experiments. Our results revealed a binary classification accuracy of nearly 90% in distinguishing between healthy controls (HC) and patients experiencing the first episode of schizophrenia (FES). Additionally, we achieved a ternary classification accuracy of approximately 70% by including clinical high risk for psychosis (CHR). The LES-embedded approach yielded notably higher classification accuracy compared to conventional machine learning methods and standard convolutional neural networks. As the performance of schizophrenia classification is strongly influenced by test functions, a functional optimization problem was proposed to identify an optimized test function, and an approximated parameter optimization problem was introduced to limit the search area of suitable basis functions. Furthermore, the parameterization test function optimization problem and the deep learning network were coupled to be synchronously optimized during the training process. The proposal approach achieved higher classification accuracy rates of 96.87% between HC and FES, with an additional 89.06% accuracy when CHR was included. The experimental studies demonstrated that the proposed LES-based method was significantly effective for schizophrenic EEG data classification.

Adaptive observer of state variables of a nonlinear nonstationary system with unknown constant parameters

Adaptive observer of state variables of a nonlinear nonstationary system with unknown constant parameters

Algorithm of adaptive estimation of parameters for a class of nonlinear non-stationary systems

Algorithm of adaptive estimation of parameters for a class of nonlinear non-stationary systems

Asymptotic Stability Investigation of the Zero Solution for a Class of Nonlinear Nonstationary Systems by the Averaging Method

A system of nonlinear differential equations is considered that describes the interaction of two coupled subsystems, one of these subsystems is linear, and the other is nonlinear and homogeneous with an order of homogeneity greater than one. It is assumed that this system is affected by nonstationary perturbations with zero mean values. Using the averaging method, sufficient conditions are determined under which perturbations do not disturb the asymptotic stability of the zero solution. The derivation of these conditions is based on the use of a special construction of the nonstationary Lyapunov function which takes into account the structure of the acting perturbations. In addition, we consider the case where there is a constant delay in the right-hand sides of the system. An original approach to the construction of the Lyapunov-Krasovskii functional for such a system is proposed. Using this functional, conditions are found that guarantee the preservation of the asymptotic stability for any positive delay.

Finite difference method for numerical solution of the initial and boundary value problem for boltzmann’s sixmoment system of equations

Boltzmann’s one-dimensional non-linear non-stationary moment system of equations in the third approximation is presented, in which the first, third and fourth equations corresponds to the laws of conservation of mass, momentum and energy, respectively. This system contains six equations and represents a nonlinear system of hyperbolic type equations. For the Boltzmann’s six- moment system of equations an initial and boundary value problem is formulated. The macroscopic boundary condition contains the moments of the incident particles distribution function on the boundary and moments of the reflected particles distribution function from the boundary. The boundary condition depends on the temperature of the wall (boundary). In this work, using the finite-difference method, an approximate solution of the mixed problem for the Boltzmann system of moment equations is constructed in the third approximation under the boundary conditions obtained by approximating the Maxwell boundary condition. For given values of the coefficients included in the moments of the nonlinear collision integral and the parameter depending on the wall temperature, as well as for fixed values of the initial conditions, a numerical experiment was carried out. As a result, the approximate values of the particle distribution function incident on the boundary and reflected from the boundary, as well as the density, temperature and average velocity of gas particles, as moments of the particle distribution function, are obtained.

The Solvability of Mixed Value Problem for the First and Second Approximations of One-Dimensional Nonlinear System of Moment Equations with Microscopic Boundary Conditions

The paper gives a derivation of a new one-dimensional non-stationary nonlinear system of moment equations, that depend on the flight velocity and the surface temperature of an aircraft. Maxwell microscopic condition is approximated for the distribution function on moving boundary, when one fraction of molecules reflected from the surface specular and another fraction diffusely with Maxwell distribution. Moreover, macroscopic boundary conditions for the moment system of equations depend on evenness or oddness of approximation {f}_{k}(t,x,c), where {f}_{k}(t,x,c) is partial expansion sum of the molecules distribution function over eigenfunctions of linearized collision operator around local Maxwell distribution. The formulation of initial and boundary value problem for the system of moment equations in the first and second approximations is described. Existence and uniqueness of the solution for the above-mentioned problem using macroscopic boundary conditions in the space of functions Cleft(left[0,Tright];{L}^{2}left[-a,aright]right) are proved.

Solution of a local boundary problem for a non-linear non-stationary system in the class of discrete controls

This article proposes an algorithm of construction for the discrete controlling function which is restricted by a norm and provides transition for the wide class of the systems of nonstationary nonlinear ordinary differential equations from the initial state to the setting final state. A constructive sufficient condition that provides this transition is obtained. Efficiency of the method is demonstrated by the solution of the robot-manipulator control problem and its numerical modeling.

Применение уравнения Больцмана для определения аэродинамических характеристик летательных аппаратов

In this paper we present the derivation of new one-dimensional non-stationary nonlinear system of moment equations and an approximation of the microscopic boundary condition when part of molecules reflect from the surface specularly and part diffusive with Maxwell’s distribution. Macroscopic boundary conditions for the system of moment equations depend on the evenness and oddness of the approximation , where is the partial sum of expansion of the distribution function of molecules  into eigenfunctions of the linearized collision operator. The formulation of the initial and boundary value problem for the system of moment equations in the third approximation under the Maxwell-Auzhan macroscopic boundary conditions is given. To analyze aerodynamic characteristics of aircraft in transient regime was used complete integro-differential Boltzmann equation, which contains a term depending on the moving speed of aircraft, and under Maxwell's microscopic boundary conditions, depending on the surface temperature.

Experience and features of innovative training of specialists in the field of automation at the proving ground of power plant APCS

Power units and power plants as automation and control objects are characterized by high dimension and redefinition, the presence of the uncertainty factor of most characteristics of a generally non-linear non-stationary system, as well as the action of random, as a rule, uncontrolled disturbances and a significant number of parameters inaccessible for direct measurement. The complexity of the technical system determines the appropriate level of requirements for the training of engineering and operational personnel in the field of modern automation and control systems, and at the same time requires the creation of adequate training tools and methodology by integrating fundamental engineering education and modern achievements in the field of information management technologies. The article summarizes the experience of 20 years of use and development of the educational and research complex «The proving ground of power plants APCS» for training specialists on the problems of modern technology of automation of energy facilities, advanced training of personnel of power plants, energy associations and engineering companies.

On totally global solvability of controlled second kind operator equation

We consider the nonlinear evolutionary operator equation of the second kind as follows $\varphi=\mathcal{F}\bigl[f[u]\varphi\bigr]$, $\varphi\in W[0;T]\subset L_q\bigl([0;T];X\bigr)$, with Volterra type operators $\mathcal{F}\colon L_p\bigl([0;\tau];Y\bigr)\to W[0;T]$, $f[u]$: $W[0;T]\to L_p\bigl([0;T];Y\bigr)$ of the general form, a control $u\in\mathcal{D}$ and arbitrary Banach spaces $X$, $Y$. For this equation we prove theorems on solution uniqueness and sufficient conditions for totally (with respect to set $\mathcal{D}$) global solvability. Under natural hypotheses associated with pointwise in $t\in[0;T]$ estimates the conclusion on univalent totally global solvability is made provided global solvability for a comparison system which is some system of functional integral equations (it could be replaced by a system of equations of analogous type, and in some cases, of ordinary differential equations) with respect to unknown functions $[0;T]\to\mathbb{R}$. As an example we establish sufficient conditions of univalent totally global solvability for a controlled nonlinear nonstationary Navier-Stokes system.

On Partial Stability and Detectability of Functional Differential Systems with Aftereffect

We consider a general class of the nonlinear non-stationary system of functional differential equations with aftereffect that admits a “partial” (with respect to a part of the variables) zero equilibrium position. We obtain conditions under which stability (asymptotic stability) with respect to a part of the variables of the “partial” equilibrium position implies its stability (asymptotic stability) in all variables. We analyze these conditions from the standpoint of the problem of partial detectability of the system in question and introduce the concept of its partial zero dynamics. We also study an application to the problem of partial stabilization of controllable systems.

Justification of Macroscopic Boundary Conditions for One-Dimensional Nonlinear Non-stationary Moment System of Equations of Boltzmann

Justification of Macroscopic Boundary Conditions for One-Dimensional Nonlinear Non-stationary Moment System of Equations of Boltzmann

Macroscopic Boundary Conditions on a Solid Surface in Rarefied Gas Flow for a One-Dimensional Nonlinear Nonstationary 12-Moment System of Boltzmann Equations

Boundary conditions for a one-dimensional nonlinear nonstationary system of Boltzmann equations are formulated in the fifth approximation. The Maxwell microscopic boundary conditions are approximated in the case of the one-dimensional Boltzmann equation when some of the molecules reflect specularly from the surface, while the others reflect diffusely with Maxwell’s distribution. An initial-boundary value problem for the 12-moment system of Boltzmann equations with Maxwell–Auzhani boundary conditions is stated. For the 12-moment system of Boltzmann equations, six boundary conditions are set at the left and right endpoints of the interval ( $$ - a$$ , $$a$$ ).

Synthesis of Guaranteed Stability Regions of a Nonstationary Nonlinear System with a Fuzzy Controller

The paper proposes a method for constructing guaranteed regions of stability of nonstationary nonlinear systems on the plane of parameters of a fuzzy PID controller. It is shown that this method allows to determine the full stability areas, which are significantly larger than the areas determined by classical methods (frequency circle criterion, quadratic Lyapunov functions). This improvement is achieved by using the algorithm for constructing spline Lyapunov functions. This type of Lyapunov functions is based on the necessary and sufficient conditions of stability, while the classical methods are only sufficient conditions of stability. In this regard, on the basis of the proposed method, it is possible to calculate the maximum sizes of the sectors in which the nonlinear characteristics in the channels of the fuzzy PID controller should be located. Examples of the synthesis of fuzzy P, PI, PID controllers for a nonstationary control object of the third order are given. Numerical experiments show that the expansion of the boundaries of nonlinear characteristics allows to improve the accuracy in the steady state, and also to almost double the speed of the automatic control system with a nonstationary object. The advantages over linear controllers are demonstrated. The proposed method guarantees the stability inside the calculated stability regions for any character of the change in the nonstationary parameter, as well as for any character of the change in the nonlinear characteristics in the corresponding sectors.

Соотношения взаимности для нелинейных систем

The quantum proof of the reciprocal relations for nonlinear non-stationary systems in a magnetic field is obtain in the approach of Markov’s relaxation.