Quadratic Integrals Research Articles

Integrable time-dependent central potentials

The integrable time-dependent central potentials that admit linear and quadratic first integrals other than those constructed from the angular momentum are determined. It is shown explicitly that previous answers to this problem are incomplete. The results are applied in order to find the integrable time-dependent oscillators, the integrable time-dependent generalized Kepler potentials, a class of integrable binary systems with variable mass, and the integrable Yukawa and interatomic potentials with time-dependent parameters. Finally, a new class of integrable potentials is integrated and the corresponding wavefunction is determined.

A reliability analysis method of dynamic irregularity for track–viaduct system with low stiffness

To evaluate the reliability of the dynamic irregularity of a track structure in satisfying the safety requirements, a stochastic virtual track inspection (SVTI) system that can be used to predict the time-varying probability distribution of dynamic irregularity was developed by combining virtual track inspection (VTI) system and the generalized probability density evolution method (GPDEM) in this study. A train–track–bridge dynamic interaction model (TTBDIM) was built using the Universal Mechanism platform. The inertial reference method (IRM) was simulated by finding the frequency-domain quadratic integration of the axle box acceleration and filtering it, based on which the deterministic VTI system was established. For the sake of probability analysis, the track geometry irregularities (TGI) was considered as a stochastic variable. A number-theory-based uniform experimental design was used to determine the representative points in the high-dimensional probability space. SVTI was established by coupling the deterministic VTI system and the generalized probability density evolution equation. The probability density evolution function of the dynamic irregularity was solved for using the finite difference method with the total variation diminishing (TVD) format. In this study, floating slab tracks on a U-shaped beam viaduct were taken as an example to analyze the influence of track stiffness irregularity (TSI) on the basic signal characteristics of the dynamic irregularity of the tracks using the deterministic VTI system. Furthermore, the SVTI system was used to analyze the influence of TSI on the time-varying probability distribution and determine whether the dynamic inspection requirements of the dynamic irregularity were satisfied. The research findings suggested that the TSI of the track–viaduct system with a low stiffness had a significant influence on the time-varying characteristics of the dynamic irregularity probability distribution. As the stiffness of the track–viaduct system decreased, the mean probability distribution of the dynamic irregularity extrema shifted to the higher-level area with an increasing standard deviation. When rubber spring floating slab tracks (RSFSTs) and steel spring floating slab tracks (SSFSTs) were laid on the 35-m-span U-shaped beam, the probabilities of dynamic irregularity exceeding the limit were 16.13% and 30.02%, respectively. The deterministic VTI system might not be able to detect irregularities exceeding the limit within the limited simulation mileage, which could lead to misjudgments and track regularity overestimation. In conclusion, the proposed SVTI system will provide direction for the design of track–viaduct systems with low stiffnesses in urban rail transit construction.

Evaluation of Newly Developed Liquid Level Process with PD and PID Controller without Altering Material Characteristics

This article explains the design of fuzzy logic controllers (FLCs) for level processes which is generally used in numerous control operations. The main purpose of the proposed design is to maintain the liquid level in the tank at the desired level. In this paper, the fuzzy logic controller is chosen as the controller for the level process because of its fault tolerance, knowledge representation, expertise, non-linearity, uncertainty, and real-time manipulation. Fuzzy logic controllers have been developed and compared in the Mamdani version. Performance on proportional derivatives (PD) and proportional-integral-derivatives (PID) controllers. Whereas traditional PD and PID controllers are simple, dependable and eliminate steady-state errors, fuzzy logic controllers are rule-based systems that are a logical model of human behavior in processes of the proposed design. The response is provided as follows: The LabVIEW software has been validated. It is used to simulate the proposed system. Comparing error indicators such as PD controller, PID controller, fuzzy logic controller integral absolute error, integral quadratic error, time and absolute error integral, time and quadratic error integral, fuzzy logic controller is observed from the simulation results. increase. It offers better performance than other controllers.

An efficient and quadratic accurate linear-gradient smoothing integration scheme for meshfree Galerkin methods

To address domain integration of meshfree Galerkin methods with quadratic base, we propose an efficient and accurate linear-gradient smoothing integration (LGSI) scheme in this study. In our scheme, the smoothed gradient is expressed as the linear polynomial form with respect to the center of the smoothing domain by means of Taylor's expansion. The unknown coefficients can be uniquely determined in terms of the smoothed gradient technique, which is low-cost because it transforms the complex domain integration into its boundary integration. The LGSI is also simple because there is no need to correct test functions and introduce additional computational costs. The integration error of the LGSI scheme for the stiffness matrix in the 2D case is proved. Consequently, the LGSI is exact with respect to the quadratic meshfree Galerkin method. Numerical examples demonstrated the performance of the LGSI scheme for solving the 2D anisotropy potential and elasticity problems as well as the 3D potential problem.

Quantization of inhomogeneous spacetimes with cosmological constant term

We show that the Szekeres system with cosmological constant admits a sufficient number of conservation laws, which allow to claim the integrability of the system. The main novelty in this investigation is that we find that the unique attractor of the Szekeres system is the isotropic inhomogeneous de Sitter (-like) Universe, contrary to the original system in which the attractors describe Kantowski–Sachs (-like) spacetimes. We also study the existence of quantum corrections and the emergence of classicality by considering the linear and quadratic conserved quantities at the quantum level. We perform an analysis considering different approaches, involving the Bohmian quantum potential and a probability analysis. The result is that there are no quantum corrections for the quadratic integrals, while there exists a linear case for which we find quantum corrections.

Quadratic First Integrals of Time-Dependent Dynamical Systems of the Form q¨a=−Γbcaq˙bq˙c−ω(t)Qa(q)

We consider the time-dependent dynamical system q¨a=−Γbcaq˙bq˙c−ω(t)Qa(q) where ω(t) is a non-zero arbitrary function and the connection coefficients Γbca are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) I we assume that I=Kabq˙aq˙b+Kaq˙a+K where the unknown coefficients Kab,Ka,K are tensors depending on t,qa and impose the condition dIdt=0. This condition leads to a system of partial differential equations (PDEs) involving the quantities Kab,Ka,K,ω(t) and Qa(q). From these PDEs, it follows that Kab is a Killing tensor (KT) of the kinetic metric. We use the KT Kab in two ways: a. We assume a general polynomial form in t both for Kab and Ka; b. We express Kab in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of t. In both cases, this leads to a new system of PDEs whose solution requires that we specify either ω(t) or Qa(q). We consider first that ω(t) is a general polynomial in t and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities Qa(q) to be the generalized time-dependent Kepler potential V=−ω(t)rν and determine the functions ω(t) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x¨=−ω(t)xμ+ϕ(t)x˙(μ≠−1) and compute the relation between the coefficients ω(t),ϕ(t) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.

Three-Component Stäckel Model of the Galaxy Based on the Rotation Curve from Maser Data

A three-component Stackel model of the Galaxy, including the bulge, disk, and halo, is constructed. Parameter estimates of the potential are obtained as a result of fitting the model rotation curve to azimuthal velocities found from data on trigonometric parallaxes and spatial velocities of masers. The fitting method takes into account the measurement and natural dispersions of azimuthal velocities and uses an algorithm for excluding objects with excessive residuals. In order to obtain more uniform samples, the objects were divided into two groups: masers associated with high-mass star forming regions and masers of other types. A significant kinematic inhomogeneity of these groups was identified and taken into account: the azimuthal velocity dispersion is $\sigma_{0,1}=4.3\pm 0.4$~km\,s$^{-1}$, in the first group and $\sigma_{0,2}=15.2\pm1.3$~km\,s$^{-1}$ in the second. After constructing the model of the Galactic-plane potential, it was generalized to the entire space under the assumption of the existence of a third quadratic integral of motion. When reconstructing the Galactic rotation curve in detail, the used algorithm gives an analytical expression for the Stackel potential, which significantly simplifies the task of constructing the Galaxy's phase density model in the Stackel approximation. In order to make the Stackel model more realistic, one needs to develop methods of direct account of data on the vertical distribution of density in the Galaxy.

Optimal affine image normalization approach for optical character recognition

Optical character recognition (OCR) in images captured from arbitrary angles requires preliminary normalization, i.e. a geometric transformation resulting in an image as if it was captured at an angle suitable for OCR. In most cases, a surface containing characters can be considered flat, and a pinhole model can be adopted for a camera. Thus, in theory, the normalization should be projective. Usually, the camera optical axis is approximately perpendicular to the document surface, so the projective normalization can be replaced with an affine one without a significant loss of accuracy. An affine image transformation is performed significantly faster than a projective normalization, which is important for OCR on mobile devices. In this work, we propose a fast approach for image normalization. It utilizes an affine normalization instead of a projective one if there is no significant loss of accuracy. The approach is based on a proposed criterion for the normalization accuracy: root mean square (RMS) coordinate discrepancies over the region of interest (ROI). The problem of optimal affine normalization according to this criterion is considered. We have established that this unconstrained optimization is quadratic and can be reduced to a problem of fractional quadratic functions integration over the ROI. The latter was solved analytically in the case of OCR where the ROI consists of rectangles. The proposed approach is generalized for various cases when instead of the affine transform its special cases are used: scaling, translation, shearing, and their superposition, allowing the image normalization procedure to be further accelerated.

Quadratic double ramification integrals and the noncommutative KdV hierarchy

In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double ramification integrals are the main ingredient for the computation of the double ramification hierarchy associated to the infinite dimensional partial cohomological field theory given by $\exp(\mu^2 \Theta)$ where $\mu$ is a parameter and $\Theta$ is Hain's theta class, appearing in Hain's formula for the double ramification cycle on the moduli space of curves of compact type. This infinite rank double ramification hierarchy can be seen as a rank $1$ integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the KdV hierarchy on a noncommutative Moyal torus.

Characterizing Existence of Minimizers and Optimality to Nonconvex Quadratic Integrals

Quadratic functions play an important role in applied mathematics. In this paper, we consider the problem of minimizing the integral of a (not necessarily convex) quadratic function in a bounded subset of nonnegative integrable functions defined on a finite-dimensional space that is not compact with respect to any (locally convex) topology in the space of integrable functions. We establish a complete description about the existence or nonexistence of solution in terms of the (strict) copositivity of the matrix involved in the integrand. In addition, we characterize optimality via the Hamiltonian function.

새로운 구속된 이차함수 음 조건을 이용한 시간지연 선형 시스템의 개선된 안정성

This paper considers the delay-dependent stability of linear systems with a time-varying delay in the frame of the Lyapunov-Krasovskii functional(LKF) approach. First, we choose an LKF that is the form of a constrained quadratic function(CQF) for the delay in its time-derivative and a quadratic integral. Second, the Bessel-Legendre inequality and the reciprocally convex inequality are used to find the upper bound of the quadratic integral. Third, a new form of necessary and sufficient condition is adopted to equivalently transform the CQF into the form of Linear matrix inequality(LMI). Finally, two well-known numerical examples are given to show the usefulness of the proposed result.

Turbojet direct-thrust control scheme for full-envelope fuel consumption minimization

Purpose Reducing fuel consumption of unmanned aerial vehicles (UAVs) during transient operation is a cornerstone to achieve environment-friendly operations. The purpose of this paper is to develop a control scheme that improves the fuel economy of a turbojet in its full operating envelope. Design/methodology/approach A novel direct-thrust linear quadratic integral (LQI) approach, comprised by an optimal observer/controller satisfying specified performance parameters, is presented. The thrust estimator, based in a Wiener model, is validated with the experimental data of a micro-turbojet. Model uncertainty is characterized by analyzing variations between the identified model and measured data. The resulting uncertainty range is used to verify closed-loop stability with the circle criterion. The proposed controller provides stable responses with the specified performance in the whole operating range, even with after considering plant nonlinearities. Finally, the direct-thrust LQI is compared with a standard thrust controller to assess fuel economy and performance. Findings The direct-thrust LQI approach reduced the fuel consumption by 2.1090% in the most realistic scenario. The controllers were also evaluated using the environmental effect parameter (EEP) and transient-thrust-specific fuel consumption (T-TSFC). These novel metrics are proposed to evaluate the environmental impact during transient-thrust operations. The direct-thrust LQI approach has a more efficient fuel consumption according to these metrics. The results also show that isolating the thrust dynamics within the feedback loop has an important impact in fuel economy. Controllers were also evaluated using the EEP and T-TSFC. These novel metrics are proposed to evaluate the environmental impact during transient-thrust operations. The direct-thrust LQI approach has a more efficient fuel consumption according to these metrics. The results also show that isolating the thrust dynamics within the feedback loop has an important impact in fuel economy. Originality/value This study shows the design of an effective direct-thrust control approach that minimizes fuel consumption, ensures stable responses for the full operation range, allows isolating the thrust dynamics when designing the controller and is compatible with classical robustness and performance metrics. Finally, the study shows that a simple controller can reduce the fuel consumption of the turbojet during transient operation in scenarios that approximate realistic operating conditions.

Quadratic integrals of geodesic flow, webs, and integrable billiards

We present a geometric interpretation of integrability of geodesic flow by quadratic integrals in terms of the web theory and construct integrable billiards on surfaces admitting such integrals.

A preliminary assessment of an innovative air-launched wave measurement buoy

In this study, an air-launched buoy is designed for wave measurement. It has the advantages of both ordinary spherical wave measurement buoys and cylindrical air-launched buoys. Instead of a parachute, the buoy uses an air-brake flap that can expand to help it drop into the water safely. This also improves its wave-following characteristics compared with a simple cylindrical buoy. The buoy can also maintain the shape of a cylinder when stored and transported. In addition, an integration algorithm for wave measurement is proposed based on a micro-electro mechanical system accelerometer. Owing to the interference caused by the accelerometer, the proposed wave measurement algorithm aims to decrease the errors of existing quadratic integration, thereby controlling low-frequency noise components. The velocity of the buoy is obtained by low-frequency attenuation integration, while the displacement is obtained by polynomial fitting integration. The experimental results show that the measured wave data is similar to those measured by a wave height gauge in a multifunction wave basin. The accuracy is within the allowable error range compared with the data obtained by ordinary wave measurement buoys in the ocean.

Linear Quadratic Minimum-Effort, Specify-Performance Approach to Missile Guidance

A generic guidance design framework is proposed that is based on the idea of minimizing the guidance effort expressed as the quadratic integral of the guidance command and imposing performance constraints on various trajectory parameters at arbitrary points in time during the flight. This framework extends previous work on guidance algorithms for active aerial defense and for obstacle avoidance. Using some concrete examples it is shown that the proposed framework is flexible enough to tackle interesting guidance problems related to multiple targets scenarios. It is also shown that the same framework offers a practical solution for the optimal intercept guidance problem with constraints on the lateral missile acceleration.

Quadratic Element Integration of Approximated First Order Polarization Tensor for Sphere

Polarization tensor is an object-specific property in order to indicate its shape, size and also the material used. This information is beneficial to determine between used and unused material underground for example. In this paper, we describe an accurate and easy-implemented method based on numerical integration in order to compute the first order polarization tensor. We proposed an alternative method to deal with boundary integral equation of first order polarization tensor which is quadratic element numerical integration. This method uses standard three points Gaussian quadrature in order to generate the singular integral operator matrix of polarization tensor. Different values of object’s conductivity are used in order to study the behavior of the polarization tensor. The validation of the results was based on the exact solution provided for sphere and ellipsoid geometry by previous researcher. Moreover, numerical computation showed that the quadratic element integration generates more accurate numerical results for the approximated first order polarization tensor and better in its convergence. The numerical results are illustrated in graphical form in order to show the validity of the proposed scheme.

Fuzzy Fault Tolerant Control of a Bacterial Growth Process Using a Multiple Lyapunov Function

Abstract This paper presents the problem of actuator fault estimation and fault-tolerant control (FTC) of a biological process using Takagi–Sugeno fuzzy formulation. The goal is to ensure the desired outputs tracking even if the time-varying actuator faults occur. We propose to use a proportional multi-integral (PMI) observer to estimate both the time-varying actuator faults and the state of system. The reconstructed faults are used to reconfigure the nominal controller. As a nominal control, we use a fuzzy linear quadratic integral (LQI) law. To ensure the global asymptotic convergence of the PMI observer and to improve the compensation speed of faults, we propose to use the multiple Lyapunov function by introducing a convergence rate. Sufficient conditions in terms of linear matrix inequalities (LMIs) are developed. The obtained results show that, the proposed approach is successfully applied to the problem of actuator fault-tolerant control of a bacterial growth process.

Optimal Robust LQI Controller Design for Z-Source Inverters

This paper investigates the linear quadratic integral (LQI)-based control of Z-source inverters in the presence of uncertainties such as parameter perturbation, unmodeled dynamics, and load disturbances. These uncertainties, which are naturally available in any power system, have a profound impact on the performance of power inverters and may lead to a performance degradation or even an instability of the system. A novel robust LQI-based design procedure is presented to preserve the performance of the inverter against uncertainties while a proper level of disturbance rejection is satisfied. The stability robustness of the system is also studied on the basis of the maximum sensitivity specification. Moreover, the bat algorithm is adopted to optimize the weighting matrices. Simulation results confirm the effectiveness of the proposed controller in terms of performance and robustness.

Integrable and Superintegrable Potentials of 2d Autonomous Conservative Dynamical Systems

We consider the generic quadratic first integral (QFI) of the form I=Kab(t,q)q˙aq˙b+Ka(t,q)q˙a+K(t,q) and require the condition dI/dt=0. The latter results in a system of partial differential equations which involve the tensors Kab(t,q), Ka(t,q), K(t,q) and the dynamical quantities of the dynamical equations. These equations divide in two sets. The first set involves only geometric quantities of the configuration space and the second set contains the interaction of these quantities with the dynamical fields. A theorem is presented which provides a systematic solution of the system of equations in terms of the collineations of the kinetic metric in the configuration space. This solution being geometric and covariant, applies to higher dimensions and curved spaces. The results are applied to the simple but interesting case of two-dimensional (2d) autonomous conservative Newtonian potentials. It is found that there are two classes of 2d integrable potentials and that superintegrable potentials exist in both classes. We recover most main previous results, which have been obtained by various methods, in a single and systematic way.

Superintegrable geodesic flows versus Zoll metrics

Koenigs constructed a family of two dimensional superintegrable (SI) models with one linear and two quadratic integrals in the momenta, shortly (1,2). More recently Matveev and Shevchishin have shown that this construction does generalize to models with one linear and two cubic integrals i.e. (1,3), up to the solution of a non-linear ordinary differential equation. Our explicit solution of this equation allowed for the construction of these SI systems and led to the proof that the systems globally defined on S2 are Zoll. We will generalize these results to the case (1,n) for any n≥2. Our approach is again constructive and shows the existence, when n is odd, of metrics globally defined on S2 which are indeed Zoll (under appropriate restrictions on the parameters), while if n is even the metrics we found are never globally defined on S2, as it is already the case for the (1,2) model constructed by Koenigs.