Pade Approximation Method Research Articles

Adaptive backstepping control for strict‐feedback non‐linear systems with input delay and disturbances

The problem of adaptive neural networks (NNs) control for a class of uncertain non-linear systems with input delay and disturbances is studied. By using Pade approximation method, an auxiliary system is constructed to compensate the input delay based on the introduced variable. NNs are used to approximate the unknown non-linear functions. With the aid of backstepping technique, adaptive NNs controllers are designed which can guarantee all the signals in the closed-loop systems are semi-globally uniformly ultimately bounded and the tracking error can be adjusted around the origin with a small neighbourhood. The stability of the closed-loop systems is proved by using the Lyapunov stability theorem and two simulation examples are given to illustrate the effectiveness of the proposed methods.

Adaptive Control Design with Assigned Tracking Accuracy for a Class of Nonlinearly Parameterized Input-Delayed Systems

This paper addresses the adaptive control problem of a class of nonlinear systems with unknown parameters and input delay, and the tracking accuracy of the controlled system is assigned a priori. The Pade approximation method is introduced to deal with the problem from the input delay. By creating a group of nonnegative functions, an appropriate controller is designed with the backstepping technology. It is shown that under the obtained controller, the boundedness of all the closed-loop signals is guaranteed, and the tracking error especially can converge to the accuracy assigned a priori. Finally, a simulation example is given to verify the effectiveness of the proposed scheme.

BER analysis of EGC receiver over correlated Nakagami-m fading channels with phase error and asynchronous CCI

Performance analysis of an L-branch Equal Gain Combining (EGC) receiver with phase estimation error and asynchronous Co-Channel Interference (CCI) over correlated Nakagami-m fading channel is presented. Both the desired and interfering signals are assumed to be Nakagami-m distributed. The output Signal to Noise Ratio (SINR) of the EGC combiner is derived. Closed form expression for the n-th moment of the output SINR is derived. Using the n-th moment expression, Average Bit Error Probability (ABEP) for Binary Phase Shift Keying (BPSK) modulation is derived using Moment Generating Function (MGF) and Pade approximation method. Numerical and simulation results of ABEP for various desired and interfering signal parameters are obtained. ABEP results for various degrees of phase errors are also obtained. The results for special cases are verified with the available results.

Qualitative Analysis of Stable Reduced Order Models for Interval Systems Using Mixed Methods

ABSTRACT A large number of problems are available for complex higher order systems. These systems are difficult to analyze and synthesize. Therefore it is needful to approximate higher order models to lower order models. Interval systems have constant coefficients which are uncertain within a finite range. In this paper, the frequency domain reduction methods are applied for reducing the order of interval systems. The three mixed methods are considered for reducing the linear dynamic interval systems. In these mixed methods, denominators of the reduced models are obtained by using differentiation method while the numerators of the reduced models are obtained by using differentiation, factor division, and pade approximation methods. The novel feature of the proposed methods is that it guarantees the stability of reduced order models if the original interval systems are stable. In addition, the mixed methods are compared qualitatively in terms of integral square error and integral absolute error to know about the best method among three mixed methods. Numerical examples are given to show the effectiveness of the mixed methods.

Online Voltage Stability Assessment for Load Areas Based on the Holomorphic Embedding Method

This paper proposes an online steady-state voltage stability assessment scheme to evaluate the proximity to voltage collapse at each bus of a load area. Using a non-iterative holomorphic embedding method (HEM) with a proposed physical germ solution, an accurate loading limit at each load bus can be calculated based on online state estimation on the entire load area and a measurement-based equivalent for the external system. The HEM employs a power series to calculate an accurate P – V curve at each load bus and accordingly evaluates the voltage stability margin considering load variations in the next period. An adaptive two-stage Pade approximant method is proposed to improve the convergence of the power series for accurate determination of the nose point on the P – V curve with moderate computational burden. The proposed method is first illustrated in detail on a four-bus test system and then demonstrated on a load area of the Northeast Power Coordinating Council 48-geneartor 140-bus power system.

Improved Reduced-Order Modeling Using Clustering Method with Dominant Pole Retention

This paper provides a quantitative measure criterion for selection of poles from a higher order model. The selection of poles is important because it determines both the transient and steady-state information of the dynamical system. The new indices specify which poles are dominant even when they are not the slowest. On the basis of important poles contribution to the system poles, they are selected to form cluster center. Pade approximation method is used to estimate the coefficients of numerator polynomial. Besides, this technique gives a better approximation in both the transient and the steady-state responses of the large-scale system. The effectiveness of the proposed method is demonstrated through numerical test examples and compared with other well-known published methods.

Observer-Based Adaptive Fuzzy Control of Nonlinear Non-strict Feedback System with Input Delay

This paper investigates the adaptive fuzzy output feedback control problem for nonlinear non-strict feedback system with input delay. In the control design, the Pade approximation method and fuzzy logic system are used to handle the input delay and unknown functions, and a state observer is designed to approximate the unmeasured states of the system. By backstepping technique, an observer-based adaptive fuzzy control design method is given. The stability of the controlled system and the tracking performance of the system are demonstrated by utilizing the Lyapunov function theory. Finally, a simulation example of electromechanical system is provided to confirm the effectiveness of the proposed control method.

Fractional radiative transport in the diffusion approximation

The fractional radiative transport equation describes the propagation of particles, whose path length distribution $$p(\ell )=-\partial _\ell E_\alpha (-\sigma _t\ell ^\alpha )$$ satisfies the generalized Lambert–Beer law $$\begin{aligned} \partial _\ell ^\alpha p(\ell )=-\sigma _t p(\ell ),\quad \ell >0, \end{aligned}$$ for $$\alpha \in (0,1]$$ with $$E_\alpha (x)$$ being the Mittag-Leffler function and $$\sigma _t$$ is the total attenuation coefficient. Within the classical radiative transport theory the diffusion equation is known to be as the most often considered approximation. In this paper, we derive a generalized diffusion model on the basis of the fractional radiative transport equation using the Fourier transform in combination with the Pade approximation method. Moreover, the associated diffusive flux vector is given in form of a generalized Fick’s law containing the fractional gradient operator. It is shown via comparisons to the Monte Carlo method and an exact analytical solution that the derived diffusion approximations agree quite well with the exact solution of the fractional radiative transport equation even for highly absorbing media.

An A-stable Explicit Numerical Integration Method Based on Padé Approximation

Numerical integration method is the basic method of power system transient stability analysis and calculation. Looking for a fast and reliable numerical method can effectively improve the calculation speed and accuracy of power system transient stability. This paper puts forward a numerical integration method of power system transient stability which is based on Pade approximation, analyzes the numerical stability of this method and uses transient IEEE145-node calculation system for simulation calculation. By comparing the results of Pade approximation method with the results of Taylor series expansion method, it proves that the Numerical integration method based on Pade approximation has high accuracy. Therefore the new method could be more suitable to numerical analysis of transient stability and other like-wise problems.

A transfer function type of simplified electrochemical model with modified boundary conditions and Padé approximation for Li-ion battery: Part 2. Modeling and parameter estimation

Abstract The electrochemistry-based battery model can provide physics-meaningful knowledge about the lithium-ion battery system with extensive computation burdens. To motivate the development of reduced order battery model, three major contributions have been made throughout this paper: (1) the transfer function type of simplified electrochemical model is proposed to address the current-voltage relationship with Pade approximation method and modified boundary conditions for electrolyte diffusion equations. The model performance has been verified under pulse charge/discharge and dynamic stress test (DST) profiles with the standard derivation less than 0.021 V and the runtime 50 times faster. (2) the parametric relationship between the equivalent circuit model and simplified electrochemical model has been established, which will enhance the comprehension level of two models with more in-depth physical significance and provide new methods for electrochemical model parameter estimation. (3) four simplified electrochemical model parameters: equivalent resistance R eq , effective diffusion coefficient in electrolyte phase D e eff , electrolyte phase volume fraction e and open circuit voltage (OCV), have been identified by the recursive least square (RLS) algorithm with the modified DST profiles under 45, 25 and 0 °C. The simulation results indicate that the proposed model coupled with RLS algorithm can achieve high accuracy for electrochemical parameter identification in dynamic scenarios.

A transfer function type of simplified electrochemical model with modified boundary conditions and Padé approximation for Li-ion battery: Part 1. lithium concentration estimation

Abstract To guarantee the safety, high efficiency and long lifetime for lithium-ion battery, an advanced battery management system requires a physics-meaningful yet computationally efficient battery model. The pseudo-two dimensional (P2D) electrochemical model can provide physical information about the lithium concentration and potential distributions across the cell dimension. However, the extensive computation burden caused by the temporal and spatial discretization limits its real-time application. In this research, we propose a new simplified electrochemical model (SEM) by modifying the boundary conditions for electrolyte diffusion equations, which significantly facilitates the analytical solving process. Then to obtain a reduced order transfer function, the Pade approximation method is adopted to simplify the derived transcendental impedance solution. The proposed model with the reduced order transfer function can be briefly computable and preserve physical meanings through the presence of parameters such as the solid/electrolyte diffusion coefficients (Ds&De) and particle radius. The simulation illustrates that the proposed simplified model maintains high accuracy for electrolyte phase concentration (Ce) predictions, saying 0.8% and 0.24% modeling error respectively, when compared to the rigorous model under 1C-rate pulse charge/discharge and urban dynamometer driving schedule (UDDS) profiles. Meanwhile, this simplified model yields significantly reduced computational burden, which benefits its real-time application.

Padé approximant related to inequalities for Gauss lemniscate functions

Based on the Pade approximation method, we present new inequalities for Gauss lemniscate functions. We also solve a conjecture on inequalities for Gauss lemniscate functions proposed by Sun and Chen.

Investigation of Electronic and Magnetic Properties of Iron(II)-Bromide Compound by First Principle, Mean Field, Series Expansion Calculations and Monte Carlo Simulation

The self-consistent abinitio calculations, based on DFT (density functional theory) approach and using FLAPW (full-potential linear augmented plane wave) method, are performed to investigate both electronic and magnetic properties of the FeBr 2 compound. Polarized spin and spin-orbit coupling are included in calculations within the framework of the ferromagnetic state between two adjacent Fe atoms. Magnetic moments considered to lie along (001) axes are computed. The antiferromagnetic and ferromagnetic energies of FeBr 2 are estimated. Obtained data from abinitio calculations are used as input for the high-temperature series expansion (HTSE) calculations to compute other magnetic parameters. The exchange interactions between the magnetic atoms Fe–Fe in FeBr 2 are established by using the mean field theory. The critical temperature TC(K) is obtained by HTSEs combined with the Pade approximant method. The critical exponent γ associated with the magnetic susceptibility is established. The critical temperature and magnetic hysteresis cycle are obtained by Monte Carlo simulation.

Padé approximant related to remarkable inequalities involving trigonometric functions

In this paper we, respectively, give simple proofs of some remarkable trigonometric inequalities, based on the Pade approximation method. We also obtain rational refinements of these inequalities. We are convinced that the Pade approximation method offers a general framework for solving many other similar inequalities.

Effects of PLL and frequency measurements on LFC problem in multi-area HVDC interconnected systems

Recent advancements in power electronics have made HVDC links and renewable based generation more popular in power systems application with better grid support functionalities like frequency control and inertia emulation tasks. Conventional operation and control strategies are undergoing of different changes and all the infrastructure of future modern power system should efficiently support the delivery of ancillary services in complex scenarios of AC/DC multi-area interconnected system. The AGC system of tomorrow must be able to handle complex interactions between control areas with HVDC links and distributed generation equipment. In such scenario, the effects of wide-area interconnections, PLL (Phase Locked Loop) and frequency measurements cannot be ignored. The dynamics effects of PLL and frequency measurements are very important for HVDC operation. For obtaining an acceptable performance of AC/DC system, the dynamic models of PLL and measurements need to be taken into account. This paper focused on the effects of PLL and frequency measurements in frequency supports of HVDC interconnected system. A novel approach for analyzing the dynamic effects of HVDC links considering PLL effects during coordination with AC system is presented and discussed. The effects of PLL are considered by introducing a second-order function. A Pade approximation method is also introduced for adding the effects of communication delays on AGC operation and the state space models are presented. The proposed model is analyzed for different multi-area test systems which contains parallel AC/DC transmission links.

A generalized Padé–Lindstedt–Poincaré method for predicting homoclinic and heteroclinic bifurcations of strongly nonlinear autonomous oscillators

By combining the generalized Pade approximation method and the well-known Lindstedt–Poincare method, a novel technique, referred to as the generalized Pade–Lindstedt–Poincare method, is proposed for determining homo-/heteroclinic orbits of nonlinear autonomous oscillators. First, the classical Pade approximation method is generalized. According to this generalization, the numerator and denominator of the Pade approximant are extended from polynomial functions to a series composed of any kind of continuous function, which means that the generalized Pade approximant is not limited to some certain forms, but can be constructed variously in solving different matters. Next, the generalized Pade approximation method is introduced into the Lindstedt–Poincare method’s procedure for solving the perturbation equations. Via the proposed generalized Pade–Lindstedt–Poincare method, the homo-/heteroclinic bifurcations of the generalized Helmholtz–Duffing–Van der Pol oscillator and \(\Phi ^{6}\)-Van der Pol oscillator are predicted. Meanwhile, the analytical solutions to these oscillators are also calculated. To illustrate the accuracy of the present method, the solutions obtained in this paper are compared with those of the Runge–Kutta method, which shows the method proposed in this paper is both effective and feasible. Furthermore, the proposed method can be also utilized to solve many other oscillators.

Electronic Correlations Effects and Magnetic Properties in Manganese–Bismuth Compound

Self-consistent ab initio calculations, based on the density functional theory approach and using the full potential linear augmented plane wave method, are performed to investigate both electronic and magnetic properties of the MnBi compounds. Polarized spin and spin–orbit coupling are included in calculations within the framework of the ferromagnetic state between two adjacent Mn atoms. The magnetic moment considered to lie along (001) axes are computed. The obtained data from ab initio calculations are used as input for the high-temperature series expansions calculations to compute other magnetic parameters. The exchange interactions between the magnetic atoms Mn–Mn in MnBi are given using the mean field theory. The high-temperature series expansions of the Ising model magnetic susceptibility is given up to the tenth order series in x = J1(Mn–Mn) / kBT. The Curie temperature, TC, is obtained using high-temperature series expansions of the magnetic susceptibility series combined with the Pade approximant method. The critical exponent γ associated with the magnetic susceptibility is deduced as well.

Ab Initio, Mean Field and High-Temperature Series Expansion Calculation Study of Structural Stability and Magnetism of MnHg

The self-consistent ab initio calculations, based on density functional theory (DFT) approach and using a full-potential linear augmented plane wave (FLAPW) method, are performed to investigate both electronic and magnetic properties of the MnHg compound. Polarized spin and spin–orbit coupling are included in calculations within the framework of the antiferromagnetic state between two adjacent Mn atoms. Magnetic moment considered to lie along (001) axis is computed. The antiferromagnetic and ferromagnetic energies of MnHg systems are obtained. Obtained data from ab initio calculations are used as input for the high-temperature series expansion (HTSE) calculations to compute other magnetic parameters. The exchange interactions between the magnetic atoms Mn–Mn in MnHg are given using the mean field theory. The HTSEs of the magnetic susceptibility of the magnetic moments in MnHg (mMn) through the Ising model are given up to tenth order series in (x=J(Mn–Mn) / kBT). The Neel temperature (TN (K)) is obtained by HTSEs applied to the magnetic susceptibility series combined with the Pade approximant method. The critical exponent (γ) associated with the magnetic susceptibility is deduced as well.

THE NUMERICAL SOLUTION OF HELMHOLTZ EQUATION VIA MULTIVARIATE PADÉ APPROXIMATION

In this study, we consider the numerical solution of the Helmholtz equation, arising from numerous physical phenomena and engineering applications including energy systems. We make use of a Pade Approximation method for the solution of the Helmholtz equation. Firstly, Helmholtz partial differential equation had been converted to power series by two-dimensional differential transformation, then the numerical solution of the equation was put into Pade series form for accelerating convergence and decreasing computational time. Hereby, we obtained numerical solution of Helmholtz type partial differential equation.

An effective method to suppress numerical dispersion in 2D acoustic and elastic modelling using a high-order Padé approximation

In this paper, we propose a method that can effectively reduce the numerical dispersion for solving the acoustic and elastic-wave equations. The method is a fourth-order Pade approximation scheme, in which the time difference operator is a rational function and a block tridiagonal system needs to be solved at each step. On the one hand, to efficiently solve a large linear system of equations we propose an explicit method for this implicit algrithom. On the other hand, to approximate the high-order spatial derivatives we use an eighth-order stereo-modelling method using wavefield displacements and their gradients simultaneously. For this new method, we investigate some mathematical properties including the stability, errors and the numerical dispersion relationship for 1D and 2D cases. We also present numerical results computed by the Pade approximation and compare them with the eighth-order Lax–Wendroff correction method and the eighth-order staggered-grid method. Numerical results show that the high-order Pade approximation scheme can effectively suppress the numerical dispersion caused by discretizing the wave equations when coarse spatial grids are used or models have strong velocity contrasts between adjacent grids. In contrast to other high-order finite-difference methods, the new method takes substantially less computational time and requires less memory because large spatial and time increments can be used. Thus the high-order Pade approximation method can potentially be used to solve large-scale wave propagation problems and seismic tomography based on the wave equations.