Quadratic Integrals Research Articles

Feynman integral reductions by intersection theory with orthogonal bases and closed formulae

We present a prescription for choosing orthogonal bases of differential n-forms belonging to quadratic twisted period integrals, with respect to the intersection number inner product. To evaluate these inner products, we additionally propose a new closed formula for intersection numbers beyond d log forms. These findings allow us to systematically construct orthonormal bases between twisted period integrals of this type. In the context of Feynman integrals, this represents all diagrams at one-loop.

A Novel LQI Control Technique for Interleaved-Boost Converters

Hybrid electric vehicles (HEVs) and fuel cell electric vehicles (FCEVs) utilize boost converters to gain a higher voltage than the battery. Interleaved boost converters are suitable for low input voltage, large input current, miniaturization, and high-efficiency applications. This paper proposes a novel linear quadratic integral (LQI) control for the interleaved boost converters. First, the small-signal model of the interleaved-boost converter is derived. In the proposed method, an output voltage and a current signal error between two-phase input currents are selected to control not only the output voltage but also a balance between two-phase input currents. Furthermore, steady-state characteristics in terms of the output voltage and the input current are demonstrated by experiments and simulations using an experimental apparatus with a rated power of 700 W. The validity of the proposed method’s tracking performance and load response is demonstrated by comparing it with that of the conventional PI control. The tracking performance of the LQI control for the 40 V step response has a ten times faster response than that of the PI control. Also, the experimental results demonstrate that the proposed method maintains a constant output voltage for a 300 W load step while the PI control varies by 10 V during 70 ms. Additionally, the proposed method has an excellent disturbance rejection.

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

A mobile prototype-based localization approach using inertial navigation and acoustic tracking for underwater

During underwater operations, divers must determine their own trajectories using the Inertial Navigation System (INS) they carry to improve operational efficiency. However, the INS contains a sensor bias that is also incorporated into the quadratic integration process to obtain the displacement, resulting in trajectory drift of the divers during prolonged self-guidance. To overcome the above problem, other aids are needed to correct the accumulated error of the INS. The single-beacon Assisted Inertial Navigation (AIN) method can improve the flexibility of inertial error correction while simplifying the localization equipment, which is suitable for the INS cumulative error correction scenario of divers. However, most of the traditional single-beacon assisted correction methods do not consider the effect of acoustic line bending on hydroacoustic ranging, and at the same time, they do not consider the problem of singular or pathological coefficient matrices introduced by inertial navigation neighbor localization deviations. Based on the above two shortcomings, this paper uses the acoustic velocity profile for acoustic line tracking, combines the localization idea of Mobile Primitives (MP), and proposes an MP-based acoustic line tracking-Assisted Inertial Navigation Localization (AINL) method, which constructs a sliding time window (STW) by taking the historical positioning of divers as a virtual primitive, and combines the nonlinear optimization method for iterative optimization search as a means to improve the accuracy and stability of self-navigation of the divers.

On the stability of linear systems with a quadratic integral

The problem of stability of non-degenerate linear systems admitting a first integral in the form of a non-degenerate quadratic form is considered. New algebraic criteria for stability, as well as complete instability of such systems, have been established in the form of equality to zero of traces of products of matrices, which include an additional symmetric matrix. These conditions are closely related to the symplectic geometry of the phase space, which is determined by the matrix of the original linear system and the symmetric matrix defining the first integral. General results are applied to finding conditions for complete instability of linear gyroscopic systems.

Finite Difference Models of Dynamical Systems with Quadratic Right-Hand Side

Difference schemes that approximate dynamic systems are considered discrete models of the same phenomena that are described by continuous dynamic systems. Difference schemes with t-symmetry and midpoint and trapezoid schemes are considered. It is shown that these schemes are dual to each other, and, from this fact, we derive theorems on the inheritance of quadratic integrals by these schemes (Cooper’s theorem and its dual theorem on the trapezoidal scheme). Using examples of nonlinear oscillators, it is shown that these schemes poses challenges for theoretical research and practical application due to the problem of extra roots: these schemes do not allow one to unambiguously determine the final values from the initial values and vice versa. Therefore, we consider difference schemes in which the transitions from layer to layer in time are carried out using birational transformations (Cremona transformations). Such schemes are called reversible. It is shown that reversible schemes with t-symmetry can be easily constructed for any dynamical system with a quadratic right-hand side. As an example of such a dynamic system, a top fixed at its center of gravity is considered in detail. In this case, the discrete theory repeats the continuous theory completely: (1) the points of the approximate solution lie on some elliptic curve, which at Δt→0 turns into an integral curve; (2) the difference scheme can be represented using quadrature; and (3) the approximate solution can be represented using an elliptic function of a discrete argument. The last section considers the general case. The integral curves are replaced with closures of the orbits of the corresponding Cremona transformation as sets in the projective space over R. The problem of the dimension of this set is discussed.

Improvement of the linear quadratic regulator control applied to a DC-DC boost converter driving a permanent magnet direct current motor

This article discusses a new robust control technique that enables the DC-DC boost converter driving a permanent magnet direct current (PMDC) motor to operate in high static and dynamic performances. The new technique is based on the design of a both linear quadratic regulator (LQR) and linear quadratic regulator-proportional integral (LQR-PI) type controllers, which have the advantage of eliminating oscillations, overshoots and fluctuations on different characteristics in steady-state system operation. In order to increase the output voltage, the LQR regulator is combined with a first-order system represented in the form of a closed-loop transfer function, the latter raising the output voltage to 24 volts, this voltage is enough to drive the permanent magnet direct current motor. The contribution of this paper is the creation of a robust control system represented in the form of a hybrid corrector able to regulate steady-state and transient disturbances and oscillations as well as to increase DC-DC boost converter output voltage for the PMDC motor to operate at rated voltage. The results of the three control techniques are validated by MATLAB Simulink.

Quadratic Starlike Trees

We introduce the notion of a quadratic graph, which is a graph whose eigenvalues are integral or quadratic algebraic integral, and we determine nine infinite families of quadratic starlike trees, which are just all the quadratic starlike trees including integral starlike trees. Thus, the quadratic starlike trees are completely characterized. The expressions for characteristic polynomials of quadratic starlike trees are also given.

An approach for the estimation of lateral track irregularity using axle-box acceleration

PurposeThis article aims to predict the rapid track geometry change in the short term with a higher detection frequency, and realize the monitoring and maintenance of the railway state.Design/methodology/approachFirstly, the ABA data needs to be filtered to remove the DC component to reduce the drift due to integration. Secondly, the quadratic integration in frequency domain for concern components of the vertical and lateral ABA needs to be done. Thirdly, the displacement in lateral of the wheelset to rail needs to be calculated. Then the track alignment irregularity needs to be calculated by the integration of lateral ABA and the lateral displacement of the wheelset to rail.FindingsBy comparing with a commercial track geometry measurement system, the high-speed railway application results in different conditions, after removal of the influence of LDWR, identified that the proposed method can produce a satisfactory result.Originality/valueThis article helps realize detection of track irregularity on operating vehicle, reduce equipment production, installation and maintenance costs and improve detection density.

On Erdélyi–Kober Fractional Operator and Quadratic Integral Equations in Orlicz Spaces

We provide and prove some new fundamental properties of the Erdélyi–Kober (EK) fractional operator, including monotonicity, boundedness, acting, and continuity in both Lebesgue spaces (Lp) and Orlicz spaces (Lφ). We employ these properties with the concept of the measure of noncompactness (MNC) associated with the fixed-point hypothesis (FPT) in solving a quadratic integral equation of fractional order in Lp,p≥1 and Lφ. Finally, we provide a few examples to support our findings. Our suppositions can be successfully applied to various fractional problems.

PROGRAM FOR DETERMINING THE INFORMATIVE PARAMETERS OF SURFACE ELECTROMYOGRAPHIC SIGNALS

The article analyzes and calculates informative parameters of surface electromyographic signals (sEMS), which can be used to control biotechnical systems, as well as to diagnose the state of the musculoskeletal system. The analysis parameters in the time and frequency-time domains of the signal are considered. A program has been developed for calculating the informative indicators of the signal in the indicated areas. The program is implemented in the LabVIEW environment. To analyze the sEMG signal in the time domain, using the developed program, such indicators as Integral EMG, Average amplitude change, Wavelength, Simple quadratic integral, Absolute value of the 3rd time moment, and others were calculated; and to describe the signal spectrum by methods of time-frequency analysis, the average frequency of the spectrum (mean power frequency-MPF), the median frequency of the spectrum (median Frequency-MF), root mean square (RMS), power density spectrum (PDS), half width - the width of the spectrum at half maximum amplitude (HW). To test the program, files of real sEMQ signals were used. The calculated parameters of the sEMG analysis in the time and frequency-time domains make it possible to non-invasively and objectively assess the state of the musculoskeletal system. Keywords: Surface electromyographic signals, biotechnical systems, time-frequency analysis, Labvıew software.

Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces

Abstract Superintegrable systems in two-dimensional (2D) Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.

Generalization and optimization of switching controller to control free vibration of a multi-linear state space system

This paper studies the optimized quadratic integral of a generalized multi-linear state space system under free vibration. The general formulation of the optimization problem is presented and can be solved by the standard numerical procedure. For demonstration, two examples are examined thoroughly and analytically. First, the minimized integral of mechanical energy for a single-dof semi-active damping or stiffness controlled system are shown analytically and compared with the best active and passive ones. The Pontryagin Maximum Principle is used to prove the true optimality of the presented semi-active damping control. The Linear Quadratic Regulator on/off or continuous clipping controller is shown to be truly optimized in the first example. In the second example, the maximized dissipation energy of the single-dof on-off damping system is studied and verified by a sensorless experiment.

Second order Killing tensors related to symmetric spaces

We discuss the pairs of quadratic integrals of motion belonging to the n-dimensional space of independent integrals of motion in involution, that provide integrability of the corresponding Hamiltonian equations of motion by quadratures. In contrast to the Eisenhart theory, additional integrals of motion are polynomials of the fourth, sixth and other orders in momenta. The main focus is on the second-order Killing tensors corresponding to quadratic integrals of motion and relating to the special combinations of rotations and translations in Euclidean space.

A Linear Quadratic Integral Controller for PV-Module Voltage Regulation for the Purpose of Enhancing the Classical Incremental Conductance Algorithm

As a result of the exhaustion of fossil energy sources and the corresponding increase of their negative environmental impact, recent research has intensively focused on regions of alternative energy resources and, especially, on solar energy. Slow tracking of the maximum power point (MPP) and fluctuations around the MPP reduce the efficiency of photovoltaic power generation systems (PV). This study offers a novel design for the MPPT controller, which we refer to as the “hybrid IC-LQI approach”, which combines the incremental conductance (IC) technique and the linear quadratic integral (LQI) controller based on the boost converter’s small signal model. We conduct a comparative study of the proposed hybrid IC-LQI, and the classical one-stage IC technique in order to show the effectiveness of our proposal under three different scenarios of weather conditions and load. According to simulation findings, the proposed hybrid IC-LQI approach has a high tracking efficiency of up to 98.92%, owing to faster tracking of MPP with very large reduction of oscillations. On the other hand, the IC technique provides less efficiency, up to 96.1%, showing very slow tracking and high oscillations. The presented analysis of the results confirms the superior performance of the developed hybrid IC-LQI technique to the classical IC technique.

Dynamical large deviations of linear diffusions.

Linear diffusions are used to model a large number of stochastic processes in physics, including small mechanical and electrical systems perturbed by thermal noise, as well as Brownian particles controlled by electrical and optical forces. Here we use techniques from large deviation theory to study the statistics of time-integrated functionals of linear diffusions, considering three classes of functionals or observables relevant for nonequilibrium systems which involve linear or quadratic integrals of the state in time. For these, we derive exact results for the scaled cumulant generating function and the rate function, characterizing the fluctuations of observables in the long-time limit, and study in an exact way the set of paths or effective process that underlies these fluctuations. The results give a complete description of how fluctuations arise in linear diffusions in terms of effective forces that remain linear in the state or, alternatively, in terms of fluctuating densities and currents that solve Riccati-type equations. We illustrate these results using two common nonequilibrium models, namely, transverse diffusions in two dimensions involving a nonconservative rotating force, and two interacting particles in contact with heat baths at different temperatures.

A Prediction Method of Ship Motion Based on LSTM Neural Network with Variable Step-Variable Sampling Frequency Characteristics

In active heave compensation, in order to realize the smooth control of the heave compensation platform, it is necessary to use the ship motion measurement system to accurately obtain the ship displacement signal, invert the ship displacement signal, and then control the expansion and contraction of the electric cylinder so that the compensation platform remains horizontal. The ship displacement measurement system generally adopts the second integral of the acceleration sensor to obtain the ship displacement signal. During the acquisition process of the ship displacement signal, the quadratic integration process of the acceleration, and the communication process of the output control command, there is a processing lag which makes the error accumulate, resulting in a delay in the measurement of the ship motion. In order to collect the ship motion more accurately and control the heave compensation platform more precisely, this paper proposes a ship motion prediction method based on a variable step-variable sampling frequency characteristic LSTM (Long Short-Term Memory) neural network. First, we use the autocorrelation function algorithm to calculate the inherent delay of the lag in the process of signal acquisition by the measurement system. Secondly, the LSTM neural network is used to predict the inherent delay step of the lagging ship displacement signal. During the prediction process, it is found that the difference in the sampling frequency of the displacement signal will lead to a change in the step of the inherent delay—experiment in the laboratory to verify. By controlling the motion platform to simulate the motion of the ship and using the ship motion measurement system and the laser sensor system to measure the displacement signal of the motion platform synchronously, it is verified that the ship motion measurement system does have an inherent delay. Thirdly, on a sailing ship, ship displacement signals are collected by setting multiple sets of ship motion measurement systems. Finally, multiple sets of sampling frequency and multiple steps are set, and the ship motion is predicted based on the variable step-variable sampling frequency LSTM neural network. It is verified that the prediction accuracy is related to the sampling frequency of the signal collector and the prediction step of the LSTM neural network, which improves the prediction accuracy of the model and the timeliness of ship motion acquisition.

Linearizabiliy and Lax representations for cubic autonomous and non-autonomous nonlinear oscillators

In this work we consider a family of cubic, with respect to the first derivative, second-order ordinary differential equations. We study linearizability conditions for this family of equations via generalized nonlocal transformations. We construct linearizability criteria in the explicit form for some particular cases of these transformations. We show that each linearizable equation admits a quadratic rational first integral. Moreover, we demonstrate that generalized nonlocal transformations under certain conditions preserve Lax integrability for equations from the considered family. Consequently, we find that any linearizable equation from the considered family possesses a Lax representation. This provides a connection between two different approaches for studying integrability and demonstrates that the Lax technique can be effectively applied to finding and classifying dissipative integrable autonomous and non-autonomous nonlinear oscillators. We illustrate our results by several examples of linearizable equations, including a non-autonomous generalization of the Rayleigh–Duffing oscillator, a family of chemical oscillators and a generalized Duffing–Van der Pol oscillator.

New classes of quadratically integrable systems in magnetic fields: The generalized cylindrical and spherical cases

We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no longer be connected to separation of variables in the Hamilton–Jacobi equation and can have more general leading order terms. We focus on two cases extending the physically relevant cylindrical- and spherical-type integrals. We find three new integrable systems in the generalized cylindrical case but none in the spherical one. We conjecture that this is related to the presence, respectively absence, of maximal abelian Lie subalgebra of the three-dimensional Euclidean algebra generated by first order integrals in the limit of vanishing magnetic field. By investigating superintegrability, we find only one (minimally) superintegrable system among the integrable ones. It does not separate in any orthogonal coordinate system. This system provides a mathematical model of a helical undulator placed in an infinite solenoid.

Finite time stability of linear time varying delay systems using free matrix based integral inequalities

This paper discusses the problem of finite time stability (FTS) for linear delayed systems where the time-varying delay is bounded. First, based on a Lyapunov–Krasovskii Functional (LKF) which contains terms of triple integrals, delay-dependent FTS conditions are provided by introducing some free-weighting matrices. Then, new approximations of the single and multiple quadratic integrals that appear in the LKF derivative are established using integral inequalities, called free-matrix-based integral inequalities. The dimensions of these free matrices integral in our results are less than those obtained in the literature. This reduction in the number of variables does not mean that our method is a particular case but simply that our approach is completely different from the others and therefore more effective. Finally, less conservative design conditions in terms of Linear Matrix Inequalities (LMIs) are proposed and solved by the LMI Tools of MATLAB to show the advantage and effectiveness of the proposed approach.