Infinite System Research Articles

Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper.

Plume Spreading Due to Floor Conditions of a Plunging Liquid Jet Using Stereographic Backlit Imaging

Abstract Plunging liquid jets are a multiphase flow studied to understand how gas is entrained in a liquid and the resulting mixing capabilities. From existing literature, it has been hypothesized that rising bubbles play a noticeable role in the multiphase hydrodynamics of the plunging liquid jet bubble plume and that separating the rising bubbles from the incoming liquid jet can result in a significant increase in the depth of the bubble plume. This study explores the effects of separating the incoming liquid jet from the rising bubble plume through floor interactions and compression effects due to a finite tank depth. This configuration is found in many natural and industrial systems, but not within published literature. Using existing theoretical models of infinite depth plunging liquid jet systems, which align reasonably well with captured baseline data, two models are developed for when floor interactions are present, one theoretical and one empirical. The models show a correlation between plume spread and floor interaction with the incoming plunging liquid jet bubble plume. Data acquired through stereographic backlit imaging over a range of flow rates show a reasonable agreement with the proposed models.

An optimal membership function based Fuzzy-[formula omitted] stabilizer design to suppress low frequency oscillation in synchronous generator

An optimal membership function-based fractional order fuzzy proportional integral (Fuzzy-PIλ) power system stabilizer (PSS) is designed to suppress low frequency oscillation (LFO) in a synchronous generator connected to an infinite bus power system. Speed deviation and the change in speed deviation of synchronous generator due to LFO are taken as inputs of the fuzzy controller, and the parameters of a fractional-order PI controller (proportional gain, integral gain, and the fractional term in integral control action) are taken as fuzzy output variables. The range, scaling, and shape of the membership functions of the fuzzy controller are optimized such that it can tune the fractional order proportional integral (PIλ) stabilizer under any disturbance or parametric uncertainty in the synchronous generator. The advantage of the proposed stabilizer is that it can handle a wide range of disturbances and adjust the gains of the control parameters for the system. A nonlinear dynamic acceleration coefficient-based class-topper optimization is applied to optimize the fuzzy settings. The robustness and sensitivity of the proposed optimal Fuzzy-PIλ stabilizer are verified under different scenarios for the synchronous generator and compared with available results.

On a Linear Differential Game of Pursuit with Integral Constraints in ℓ2

In this paper, we study the stability, controllability, and differential game of pursuit for an infinite system of linear ODEs in ℓ2. The system we consider has a special right-hand side, which is not diagonal and serves as a toy model for controllable system of infinitely many interacting points. We impose integral constraints on the control parameters. We obtain criteria for stability and null controllability of the system. Further, we construct a strategy for the pursuer that guarantees completion of the pursuit problem for the differential game. To prove controllability we use the so called Gramian operators.

MFCDFT and impedance characteristic-based adaptive technique for fault and power swing discrimination

ABSTRACT In today’s fast-changing power systems, it is vital to have a protection system that can distinguish between power swing and fault conditions. With advanced protective devices (Digital Relays), an appropriate protection strategy can be established to identify power swing and fault conditions individually. The proposed algorithm uses the sliding window concept, which requires very little computational data unless any abnormal conditions exist in the power system. Modified Full Cycle Discrete Fourier Transform (MFCDFT) is more effective in the extraction of the fundamental phasors from the original complex signal by effectively removing the harmonics and DC components. An adaptive impedance characteristic-based MFCDFT is utilized for the discrimination of power swing and fault in this work. Several fault scenarios and power swings are generated on the test system concurrently and independently from the relay station at varied distances. PSCAD/EMTDC is utilized on a single-machine infinite bus system to analyze various test conditions. The simulation is utilized in this case to collect data from various test circumstances. After obtaining the data, the MFCDFT algorithm along with adaptive impedance estimation is performed using MATLAB code. After examining various test results, it is discovered that the suggested technique correctly recognizes power swings and various fault scenarios. It can also identify the difference between stable and unstable power swings. According to the results, the proposed technique is proficient at spotting fault and power swing conditions. In addition, the algorithm ensures dependable operation with quick detection and avoids maloperation under a typical swing condition.

Reduced order in domain control of distributed parameter port-Hamiltonian systems via energy shaping

An in-domain finite dimensional controller for a class of distributed parameter systems on a one-dimensional spatial domain formulated under the port-Hamiltonian framework is presented. Based on (Trenchant et al. 2017) where positive feedback and a late lumping approach is used, we extend the Control by Interconnection method and propose a new energy shaping methodology with an early lumping approach on the distributed spatial domain of the system. Our two main control objectives are to stabilize the closed-loop system, as well as to improve the closed-loop dynamic performances. With the early lumping approach, we investigate two cases of the controller design, the ideal case where each distributed controller acts independently on the spatial domain (fully-actuated), and the more realistic case where the control action is piecewise constant over certain intervals (under-actuated). We then analyze the asymptotic stability of the closed-loop system when the infinite dimensional plant system is connected with the finite dimensional controller. Furthermore we provide simulation results comparing the performance of the fully-actuated case and the under-actuated case with an example of an elastic vibrating string.

Account of additional factors for damping torsional oscillations

Some parameters are influenced by the turbine unit's torsional oscillations. The fundamental comes from damping these oscillations, which are brought on by a departure in the turbine blades' speed from the device's prediction of the steam volume, and attenuation of fluctuations due to the distribution of energy in the turbine's productive components. The usual single-machine infinite bus system is used for the analysis. For various turbine-generator shafts and various generator operating situations, rotating mass mechanical system evaluations for small-signal stability and large disturbance are conducted. It is demonstrated that the shaft's “structural' damping (H) and “steam' damping (Kn) coefficients have a considerable impact on the damping of torsional modes. The goal of this work is to determine the effect of changing the damping factors in the mathematical model of the steam turbine shaft on the system's static stability, as well as the extent to which these variables' limits on damping rotational oscillations on the maximum torsional torques generated in the shaft masses. The mathematical model of the steam turbine shaft with a single machine and transmission line to an infinite bus system was simulated using Dymola software, and the static and dynamic effects of damping factors (H) and (Kn) on system stability were demonstrated. By evaluating the best case for parameters with the least influence on the system's stability, the results were obtained by changing the factors (Kn) from 0.005 to 0.5 and (H) from 0.005 to 0.2 and the extent of its effect on the maximum torque of the steam turbine masses and reducing it by 8.4 %, as well as by reducing the settling time of the system after disturbances occur and reaching to Steady state by about 90 %.

Research on robust adaptive control of strong nonlinear complex large power grids.

In this paper, a Multi-Index Nonlinear Robust Adaptive Control (MINRAC) method was proposed for ultra-complex multi-input multi-output power systems with uncertain parameter factors and external disturbances. The controller designed by this method has excellent static and dynamic characteristics. Under the condition of the uncertainty of system parameters and external disturbance at the same time, the MINRAC method can ensure that the multiple indexes concerned by ultra-complex power systems can be controlled at their expected values. The simulation results showed that the control mechanism of MINRAC method was consistent in both single-machine infinite bus system and multi-machine interconnected coupling system. The output function chooses power angle, angular frequency and terminal voltage as constraints. When the system has parameter uncertainty and external interference, the uncertain parameter values are adjusted by adaptive control to force these indicators to tend to the given expected value. For three-phase short circuit, which is the most serious fault in power system, the use of multi index nonlinear robust controller can ensure that the system is stable in a wide range and has better dynamic performance.

Conditions of Existence and Uniqueness of Limit Cycle for Grid-Connected VSC With PLL

This paper investigates the large disturbance stability of single voltage source converter (VSC) connected to the infinite bus system through the transmission line. One conservative stable region is obtained via the direct method of Lyapunov. Based on the Poincaré Theorem, it is interesting to find there might exist limit cycle between the obtained stable boundary and the angle of unstable equilibrium point (UEP). Accordingly, the nonlinear damping effect on the system stability is defined, and one limit cycle exists when the positive and negative damping effect over one period is mutually balanced. A conservative existence condition of limit cycle is further derived by constructing a proper piece-wise curve to approximate one specified unstable system trajectory. The derived analytical existence condition can directly predict the limit cycle without numerical simulations. Finally, the paper also proves that the studied dynamic system has at most one limit cycle, and if it exists, it is unstable.

Moment Closure for a Birth-Death Model of Antimicrobial Heteroresistance

Developing predictive Antimicrobial Resistance (AMR) models supporting optimal treatment design to combat ”superbugs” poses a significant challenge for mathematical biology. Birth-Death (BD) processes constitute an intuitive and flexible approach for modelling biological systems’ stochastic dynamics that is regaining attention due to the advances in computational techniques. This work presents a multivariate BD model of antimicrobial heteroresistance, a phenotype in which a bacterial isolate contains many subpopulations with heterogeneous antimicrobial responses. The model includes Lotka-Volterra competition between subpopulations, leading to an infinite coupled system of equations for the moment dynamics of the BD process. Then, a moment closure is proposed by assuming a log-normal distribution for a univariate BD process approximating the total population behaviour. The results are compared with stochastic simulations of the multivariate BD process during a typical time-kill assay.

From the binomial reshuffling model to Poisson distribution of money

<abstract><p>We present a novel reshuffling exchange model and investigate its long time behavior. In this model, two individuals are picked randomly, and their wealth $ X_i $ and $ X_j $ are redistributed by flipping a sequence of fair coins leading to a binomial distribution denoted $ B\circ (X_i+X_j) $. This dynamics can be considered as a natural variant of the so-called uniform reshuffling model in econophysics. May refer to Cao, Jabin and Motsch (2023), Dragulescu and Yakovenko (2000). As the number of individuals goes to infinity, we derive its mean-field limit, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. Our aim of this work is then to prove (using a coupling argument) that the distribution of wealth converges to the Poisson distribution in the $ 2 $-Wasserstein metric. Numerical simulations illustrate the main result and suggest that the polynomial convergence decay might be further improved.</p></abstract>

ON THE EXISTENCE OF SOLUTIONS OF INFINITE SYSTEM OF HADAMARD-TYPE FRACTIONAL FUNCTIONAL INTEGRAL EQUATION IN THE BANACH SEQUENCE SPACE LP, P >1 AND COUPLED SEMI-ANALYTICAL METHOD TO SOLVE IT.

This article presents a study where the tool measure of noncompactness (MNC) and Meir-Keeler condensing operator (MKCO) are employed to establish the existence of solutions to an infinite system of Hadamard-type fractional functional integral equation in the Banach sequence space lp, p > 1. Additionally, an illustrative example is provided to validate the obtained results. Furthermore, we employ a coupled semianalytical method based on the modified homotopy perturbation method (mHPM) and Adomian’s decomposition method (ADM), to approximate the solution for the given example.

Слойная конечность в некоторых группах

Infinite groups with finiteness conditions for an infinite system of subgroups are studied. Groups with a condition: the normalizer of any non-trivial finite subgroup is a layer-finite group or the normalizer of any non-trivial finite subgroup has a layer-finite periodic part are studied for beginning in the locally finite class of group, then in the class of periodic groups of Shunkov and finally in the class of Shunkov groups which are contain a strongly embedded subgroup with an almost layer-finite periodic part. The group $G$ is called the Shunkov group if for any prime $p$ and for every finite subgroups $H$ from $G$ any two conjugate elements of order $p$ from the factor-group $N_G(H)/H$ generate a finite subgroup. Results for almost layer-finite groups and groups with almost layer-finite periodic part are transferred to layer-finite groups and groups with layer-finite periodic part. New characterizations of layer-finite groups and groups with layer-finite periodic part are obtained.

Regulated functions space R(R+,R∞) and its application to some infinite systems of fractional differential equations via family of measures of noncompactness

Regulated functions space R(R+,R∞) and its application to some infinite systems of fractional differential equations via family of measures of noncompactness

On constraint qualifications for an infinite system of quasiconvex inequalities

On constraint qualifications for an infinite system of quasiconvex inequalities

Solution of infinite system of third-order three-point nonhomogeneous boundary value problem in weighted sequence space $${{\varvec{bv}}}_{{{\varvec{p}}}}^{\omega }$$

Solution of infinite system of third-order three-point nonhomogeneous boundary value problem in weighted sequence space $${{\varvec{bv}}}_{{{\varvec{p}}}}^{\omega }$$

Study on phase transition critical phenomena in group movement based on eigen microstate theory

Group motion exists widely in nature. Physicists use abstract models, such as the Vicsek model, to simplify group motion to obtain universal laws. However, the order parameters and phase transition types that describe the characteristics of the Vicsek model system are still controversial. For example, how to characterize the order parameters and types of phase transitions that characterize the system. This paper starts with the eigen microstate method and studies the Vicsek model. First, the continuous phase transition in the system is deeply studied through scaling analysis, and two critical exponents about the noise are obtained. Second, considering the density effect, similar to the noise study, three eigen microstates in the Vicsek model are revealed through scaling analysis, showing discontinuous and continuous phase transitions. Moreover, it is proved that in the actual infinite system, continuous phase transition and discontinuous phase transition still exist. Finally, the experimental data under different noises and densities are obtained through a large number of numerical simulations, and the phase diagram of the Vicsek model is depicted.

Periodic traveling waves in Fermi–Pasta–Ulam type systems with nonlocal interaction on 2d-lattice

The paper deals with the Fermi--Pasta--Ulam type systems that describe an infinite systems of nonlinearly coupled particles with nonlocal interaction on a two dimensional lattice. It is assumed that each particle interacts nonlinearly with several neighbors horizontally and vertically on both sides. The main result concerns the existence of traveling waves solutions with periodic relative displacement profiles. We obtain sufficient conditions for the existence of such solutions with the aid of critical point method and a suitable version of the Mountain Pass Theorem for functionals satisfying the Cerami condition instead of the Palais--Smale condition. We prove that under natural assumptions there exist monotone traveling waves.

SPECTRAL DYNAMIC STIFFNESS METHOD FOR THE FLUTTER PROBLEM OF COMBINED PLATES

At present time, the spectral dynamic stiffness method is being actively developed as an alternative to the finite element method for vibration and stability problems of composite structures from beams, rods, plates and shells. This approach, based on exact solutions of governing differential equations, makes it possible to more effectively study the problem in the medium and high frequency ranges, and gives analytical expressions for natural modes. It is proposed to use the advantages of this method to study the problems of dynamic stability and flutter of an orthotropic composite plate in a supersonic gas flow. Using the linear approximation of piston theory, solution of the problem is investigated according to the Galerkin method on the basis of the eigenforms of a composite plate in vacuum. According to this approach the boundary value problem is reduced to a homogeneous infinite linear algebraic system of equations with coefficients are depending from physical-mechanical and geometrical parameters of the problem. The frequency parameter is included in the system linearly, that allows us to reduce eigenproblem for infinite system to the problem of determining the eigenvalues and vectors of a matrix. The convergence of the Galerkin method depending on the number of basis functions is studied numerically. It is shown that the first 16 eigenforms provide the good convergence of the method. Examples of numerical implementation are given, obtained solution allow us to study the dependence of the critical velocity from the properties of the material and geometry of the combined plate.

Cauchy Problem for the Nonlinear Liouville Equation in the Class of Periodic Infinite-Gap Functions

The inverse spectral problem method is used to integrate the nonlinear Liouville equation in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution of the nonlinear Liouville equation is introduced. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of three times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the Dubrovin system of equations and using the first trace formula satisfies the Liouville equation.