Pade Approximants Research Articles

Couple New Iterative Method with Pade Approximation to Solve the Nonlinear Wave-Like Equations with Variable Coefficients

In this paper, some cases of wave-like equation have been solved by modified new iterative method as an infinite power series but it is impossible to compute the infinite series, therefore, the truncated power series was approximated by Pade approximation. Pade approximation approximates a truncated power series as ratio of two polynomials to reduce calculations and shorten the power series while maintaining accuracy. These cases are shown the successful use of solution approximation by Pade approximation compared to power series expansion.

Prescribed performance based data-driven adaptive sliding mode control for discrete-time nonlinear systems

This research explores the control problem of discrete-time nonlinear systems with prescribed performance utilizing the data-driven method. Firstly, the equivalent data model of the system is constructed employing the full-form dynamic linearization technique. Secondly, a novel error transformation function is proposed to transform the constrained tracking error into the unconstrained equivalent error. Subsequently, the new sliding function is established to achieve the original tracking error converges to the predetermined asymmetric domain. In contrast to existing methods, the proposed approach has a more straightforward controller structure. Additionally, the introduction of pade approximation simplifies the theoretical analysis. Finally, the superiority of the proposed scheme is validated through simulation results on the single-link manipulator system.

High-fidelity approximations for large holograms

Future holographic displays will require high-resolution holograms ranging from several hundred gigabytes to several terabytes. Hologram computations using the Fresnel approximation have been proposed; however, they are effective only in paraxial regions and result in significant errors for high-resolution holograms. This study proposes a high-resolution hologram calculation method using the Pade approximations, which can significantly improve the accuracy of the phase calculation despite a slight increase in computational complexity compared with the Fresnel approximation. In this study, image holograms with resolutions ranging from 10K to 30K pixels were generated using the Fresnel and Pade approximations, and the computation time and image quality of the reproduced images were evaluated. The Pade Approximation served as an effective solution for obtaining large holograms of high quality and reasonable computational complexity.

Evidence of dynamical dark energy in a non-flat universe: current and future observations

We investigate the dark energy phenomenology in an extended parameter space where we allow the curvature density of our universe as a free-to-vary parameter. The inclusion of the curvature density parameter is motivated from the recently released observational evidences indicating the closed universe model at many standard deviations. Here we assume that the dark energy equation-of-state follows the PADE approximation, a generalized parametrization that may recover a variety of existing dark energy models. Considering three distinct PADE parametrizations, labeled as PADE-I, SPADE-I and PADE-II, we first constrain the cosmological scenarios driven by them using the joint analyses of a series of recently available cosmological probes, namely, Pantheon sample of Supernovae Type Ia, baryon acoustic oscillations, big bang nucleosynthesis, Hubble parameter measurements from cosmic chronometers, cosmic microwave background distance priors from Planck 2018 and then we include the future Gravitational Waves standard sirens (GWSS) data from the Einstein telescope with the combined analyses of these current cosmological probes. We find that the current cosmological probes indicate a very strong evidence of a dynamical dark energy at more than 99% C.L. in both PADE-I, and PADE-II, but no significant evidence for the non-flat universe is found in any of these parametrizations. Interestingly, when the future GWSS data from the Einstein telescope are included with the standard cosmological probes an evidence of a non-flat universe is found in all three parametrizations together with a very strong preference of a dynamical dark energy at more than 99% C.L. in both PADE-I, and PADE-II. Although from the information criteria analysis, namely, AIC, BIC, DIC, the non-flat Λ-Cold Dark Matter model remains the best choice, however, in the light of DIC, PADE parametrizations are still appealing.

Neural networks-based adaptive fault-tolerant control for a class of nonstrict-feedback nonlinear systems with actuator faults and input delay

<abstract><p>This paper addresses the challenge of adaptive control for nonstrict-feedback nonlinear systems that involve input delay, actuator faults, and external disturbance. To deal with the complexities arising from input delay and unknown functions, we have incorporated Pade approximation and radial basis function neural networks, respectively. An adaptive controller has been developed by utilizing the Lyapunov stability theorem and the backstepping approach. The suggested method guarantees that the tracking error converges to a compact neighborhood that contains the origin and that every signal in the closed-loop system is semi-globally uniformly ultimately bounded. To demonstrate the efficacy of the proposed method, an electromechanical system application example, and a numerical example are provided. Additionally, comparative analysis was conducted between the Pade approximation proposed in this paper and the auxiliary systems in the existing method. Furthermore, error assessment criteria have been employed to substantiate the effectiveness of the proposed method by comparing it with existing results.</p></abstract>

A Method for Source Location of Harmonic Instability in Renewable Power Plants Considering the Feeder Distributed Parameters

The current analysis of harmonic instability in existing renewable energy substations mostly relies on the lumped parameter model of the power system, which fails to adequately describe the port characteristics of the system in a wide frequency range. However, considering the distributed parameter model of the transmission lines, the frequency domain network matrix becomes a transcendental equation, making it difficult to directly solve for the system’s inherent modes and leading to challenges in harmonic instability analysis and the accurate identification of unstable sources. Therefore, this paper proposes a new modal analysis method of harmonic stability in the frequency domain based on Pade approximation and line distribution parameters. The proposed method can not only accurately assess the stability of the system, but also locate the dominant instability source of the system.

A Current Differential Protection Scheme for Distribution Networks with Inverter-Interfaced Distributed Generators Considering Delay Behaviors of Sequence Component Extractors

The high-level proliferation of inverter-interfaced distributed generators (IIDGs) in modern distribution networks (DNs) has changed system topologies and fault current signatures, which compromises the protective relays in DNs. Investigating IIDG fault behaviors-based protection scheme will benefit the grid’s safety and stability. This paper proposes a novel current differential protection (CDP) scheme that considers the delay behaviors of positive- and negative-sequence component extractors for IIDGs in DNs. A frequency-domain analytical model of the fault current for a grid-connected IIDG with the PQ control strategy and a low-voltage ride-through (LVRT) capability is investigated. The dynamic behavior of the IIDGs considering the sequence-component extractor based on the Pade approximation is presented, where the T/4 delay extractor of the IIDGs causes a two-stage behavior in the fault transient process. It is found that a 5 ms error between the measured and actual values after the fault will affect the transient characteristics of the IIDGs. The transient current generated by the IIDGs during grid faults contains a large number of low-order harmonic components within the range of 0–200 Hz, which is significantly different to the current provided by the power grid. Therefore, the proposed CDP scheme uses protective relays at both terminals to obtain the required transient electric quantity using the Prony method. By constructing the frequency-characteristics ratio (FCR) and the exchanging FCR between two terminal relays, the developed protection criteria are implemented. The accuracy of the fault analysis method, whose maximum computational error is below 0.1%, and the feasibility of the proposed protection scheme are demonstrated by using a 10 kV DN in a PSCAD/EMTDC simulation, which can be applied to various fault conditions and traditional DNs without IIDGs.

Optimization of linearity of piezoresistive pressure sensor based on pade approximation

To optimize nonlinearity and temperature drift of silicon piezoresistive pressure sensor, novel calibration algorithm based on Pade approximation is adopted in the paper. The output data of the pressure sensor is processed by the novel algorithm to remove the influences of the cross-sensitivity and the unexpected jump points. High accuracy up to 0.01% FS can be achieved based on the algorithm. Compared with the other published algorithms, the proposed algorithm shows the merits of high accuracy, less memory consumption, and low cost. The algorithm can have potential application in the field of the intelligent sensor with the requirement of high output accuracy.

Confluent Padé approximant and their applications

New variants of the Padé approximants (PA) are defined aiming to reproduce the correct hierarchy of the renormalon behavior. These types of PAs are motivated by the appearance of branch cuts instead of poles in the Borel plane of the perturbative series. The first variant of PAs wants to highlight dominant cuts in the Borel plane and it is called D-Log Padé approximant. This method converts the original branch cut into poles through derivatives, where we implement the PA to finally return to the original function. The second method is named Confluent Padé approximant and takes the approximants to the subdominant confluent singularities in the Borel plane which must be considered before moving to the other singular values. We apply the method to the Bjorken Sum Rule, giving a prediction of the next coefficient in the perturbation series.

Регулярна та складна поведінка маятникової системи у магнітному полі

This paper considers the dynamics of an oscillatory dissipative system of two coupled pendulums in a magnetic field. The pendulums are coupled via an elastic element. The inertial components of the pendulums vary over a wide range, and in the analytical study the mass ratio is chosen as a small parameter. The magnetic forces are calculated using the Pade approximation, which best agrees with the experiment. This approximation describes the magnetic excitation to good accuracy. The presence of external inputs in the form of magnetic forces and various types of loads that exist in many engineering systems significantly complicates the mode shape analysis of nonlinear system. Nonlinear normal modes of this system are studied, one mode being coupled and the other being local. The modes are constructed by the multiple-scale method. Both regular and compound behavior is studied as a function of the system parameters: the pendulum mass ratio, the coupling coefficient, the magnetic intensity coefficient, and the distance between the axis of rotation and the center of gravity. The effect of these parameters is studied both at small and at sizeable initial pendulum inclination angles. The analytical solution is compared with the results of a numerical simulation based on the fourth-order Runge?Kutta method where the modes are calculated using the initial values of the variables found in the analytical solution. The numerical simulation, which includes the construction of phase diagrams and trajectories in the configuration space, allows one to assess the dynamics of the system, which may be both regular and compound. The stability of the coupled mode is studied using a numerical-analytical test, which is an implementation of the Lyapunov stability criterion. In doing so, the stability of a mode is determined by assessing the vertical off-trajectory deviation of the mode in the configuration space.

Advanced numerical techniques for predicting frequency response functions

Numerical methods have been widely used to predict Frequency Response Functions (FRF) of structures. The FRF can be used to characterize structural dynamics and support sound and vibration control. Generally, frequency sweep is conducted in the FRF calculations to identify damping and resonant frequencies of structures under excitations. However, as numerical models are getting complex and larger, the demands for computational resources (e.g., CPU time, memory, disk space) are greatly increased. Lately, high performance computing (HPC) at the DoD HPC center has been used with advanced numerical techniques to reduce the overall computational time. Those advanced numerical techniques include Krylov subspace and Galerkin Projection (KGP) and Pade Approximation. Significantly CPU time reduction has been demonstrated for relatively smaller FE models. This paper will further discuss adaptive KGP by automating the frequency sweep process via calculated relative residues, then combine with Finite Element Tearing and Interconnecting (FETI) to solve for FRF of a large FEM model in the order of 60M DOF. A flat plate with viscoelastic layer will serve as an example to demonstrate accuracy and efficiency of AKGP with FETI. The benefits of using HPC with advanced numerical techniques to solve for large FEM models are clearly displayed.

Adaptive finite-time anti-disturbance fuzzy control for uncertain time-delay nonlinear systems and its application to SVC

The adaptive finite-time anti-disturbance fuzzy control method for n-dimensional uncertain nonlinear systems with input delay is proposed in this paper. For unknown nonlinear terms of the controlled system, the fuzzy logic systems (FLSs) are utilized to approximate them. To further improve the robustness of the controlled system, the disturbance observer is established to estimate external disturbances, facilitating the designed controller to compensate for the influence of external disturbances. Moreover, the Pade approximation method is used to transform the time-delay system model, which is convenient for controller design. With the aid of the finite-time command filtered backstepping technique, the adaptive finite-time disturbance rejection controller with low computational complexity is designed, which can compensate for the errors caused by the filters and assure that system state variables are stable and bounded in a finite time. In the end, the simulation for the single-machine infinite bus (SMIB) system with static var compensator (SVC) is presented to illustrate the effectiveness of the proposed control approach.

Semi-Analytical Methods in the Problem of Deformation of a Fluid Strip

A problem from a class of unsteady plane potential flows with a free boundary is considered. The entire boundary occupied by the liquid is free, and a zero pressure is maintained. There are neither external nor capillary forces. The motion is driven by inertia. The parameters prescribed at the initial time are the velocity field and the domain occupied by the fluid. The task is to determine these parameters at subsequent time instants. The solution is sought in the form of power series, which are then summed up with the use of the Pade approximation.

An improved cubic approximation for Kepler’s equation

ABSTRACT A novel cubic Pade approximation of sin(e sin E) is used to solve Kepler’s equation and compute the eccentric anomaly with high accuracy without requiring iteration. It requires computation of sin, cos, atan, sqrt, and a cube root. A refinement of the higher order difference methods is described that is faster and gives improved numerical accuracy.

ANALYTICAL APPROXIMATIONS WITH EXACT NON-INTEGRAL PART FOR VOLTERRA’S POPULATION MODEL

The present paper is strongly motivated by the brilliant work ofWazwaz [9, Sections 4 and 5] computing analytical approximation in the form of a series truncated at t8 and applying [4/4] Pade approximant to the series. In this paper, we make an attempt to workout analytical approximations in the form of the series truncated at t4 and apply suitable Pade approximations as well as asymptotic approximations with the following features: i) The solution contains exact non integral part. ii) The solution exhibits the population rapid rise along logistic curve followed by decay to zero in the long run. iii) The solution is reasonably comparable with that of Wazwaz [9] using the information from the series with terms only upto t4.

Artificial boundary condition for Klein-Gordon equation by constructing mechanics structure

An innovative local artificial boundary condition is proposed to numerically solve the Cauchy problem of the Klein-Gordon equation in an unbounded domain. Initially, the equation is considered as the axial wave propagation in a bar supported on a spring foundation. The numerical model is then truncated by replacing the half-infinitely long bar with an equivalent mechanical structure. The effective frequency-dependent stiffness of the half-infinitely long bar is expressed as the sum of rational terms using Pade approximation. For each term, a corresponding substructure composed of dampers and masses is constructed. Finally, the equivalent mechanical structure is obtained by parallelly connecting these substructures. The proposed approach can be easily implemented within a standard finite element framework by incorporating additional mass points and damper elements. Numerical examples show that with just a few extra degrees of freedom, the proposed approach effectively suppresses artificial reflections at the truncation boundary and exhibits first-order convergence.

A note on homotopy perturbation approach for nonlinear coagulation equation to improve series solutions for longer times

In this work, the approach presented in a recent publication by Kaur et al. (2019) is improved. The truncated series solution derived from the existing Homotopy Perturbation method (HPM) behaves peculiarly for a longer time domain and provides accurate results only for a shorter time. The Homotopy perturbation approach and the Pade approximation are coupled to estimate the nonlinear coagulation equation to tackle this problem. This significantly improves solution quality over a longer time frame by consuming fewer terms of the truncated series. The effectiveness of the new approach is tested by deriving the new analytical solutions of the number density function for a bilinear kernel with exponential initial distributions. In addition, the new solutions for physical relevant shear, Ruckenstein/Pulvermacher and Brownian kernels corresponding to exponential and gamma initial distributions are also derived. Due to the non-availability of the analytical solutions, the verification of the new results is done against the mass conserving finite volume scheme (Singh et al., 2015) for shear stress, bilinear and Brownian kernels.

An effective controller design strategy for higher order systems

The recent methods that guarantee the stability of the reduced-order models are Mihailov stability, improved Pade approximation and truncation, improved generalized pole clustering, moment matching and salp swarm optimization. Further, these methods could overcome the limitations such as non-uniqueness, pole clustering, gain adjustment and difficulty to maintain the dominant roots in the lower order system for non-minimum higher order plants. This research emphasizes the design of the proposed compensating algorithm by using an additional open loop zero for the stable reduced-order models of large-scale single-input single-output linear time-invariant continuous time systems. The results of the proposed algorithm are compared with the existing compensating methods and the design is validated and illustrated numerically.

Iterative Solutions for the Nonlinear Heat Transfer Equation of a Convective-Radiative Annular Fin with Power Law Temperature-Dependent Thermal Properties

The temperature distribution in a conductive-radiative rectangular profiled annular fin with internal heat generation is scrutinized in the present investigation. The nonlinear variation of thermal conductivity and heat transfer coefficient governed by the power law is considered. The analytical approximation for the non-dimensional temperature profile is obtained using the differential transform method (DTM)-Pade approximant. The nondimensionalization of the governing energy equation using dimensionless terms yields a nonlinear ordinary differential equation (ODE) with corresponding boundary conditions. The resulting ODE is analytically solved with the assistance of the DTM-Pade approximant procedure. Furthermore, the impact of thermal parameters on the temperature field and thermal stress is elaborated with graphs. The important results of the report divulge that temperature distribution greatly enhances with an augmentation of the heat generation parameter, but it gradually reduces with an increment in the magnitude of the thermogeometric and radiative-conductive parameter.

Observer‐based finite‐time adaptive fault‐tolerant control for nonlinear system with input delay and prescribed performance

SummaryThis paper focuses on the issue of the adaptive fuzzy fault‐tolerant control for a type of nonlinear system with input delay and disturbances. Fuzzy logic systems (FLSs) are used to estimate the unknown nonlinear quantities of the system. The Pade approximation is applied to counteract the negative effects of input delay on the system. The prescribed performance control based on error transformation helps to guarantee the transient performance of system errors. A fuzzy state observer is established to estimate the unmeasurable states of the system. Then, in presence of disturbances and actuator faults, a finite‐time control scheme is designed using the Lyapunov function and the backstepping technique. Based on Lyapunov stability, it is proved that the tracking error can converge to a specified range in a finite time, and all signals in the system are bounded in the controller design. In the end, the simulations of a second‐order system and a multi‐area interconnected power system certify the feasibility and effectiveness of the control strategy.