Parallel FETI‐DP algorithm for efficient simulation of large‐scale EM problems

An efficient parallelization of the dual‐primal finite‐element tearing and interconnecting (FETI‐DP) algorithm is presented for large‐scale electromagnetic simulations. As a nonoverlapping domain decomposition method, the FETI‐DP algorithm formulates a global interface problem, whose iterative solution is accelerated with a solution of a global corner problem. To achieve a good load balance for parallel computation, the original computational domain is decomposed into subdomains with similar sizes and shapes. The subdomains are then distributed to processors based on their close proximity to minimize inter‐processor communication. The parallel generalized minimal residual method, enhanced with the iterative classical Gram‐Schmidt orthogonalization scheme to reduce global communication, is adopted to solve the global interface problem with a fast convergence rate. The global corner‐related coarse problem is solved iteratively with a parallel communication‐avoiding biconjugate gradient stabilized method to minimize global communication, and its convergence is accelerated by a diagonal preconditioner constructed from the coarse system matrix. To alleviate neighboring communication overhead, the non‐blocking communication approach is employed in both generalized minimal residual and communication‐avoiding biconjugate gradient stabilized iterative solutions. Three numerical examples are presented to demonstrate the accuracy, scalability, and capability of the proposed parallel FETI‐DP algorithm for electromagnetic modeling of general objects and antenna arrays. Copyright © 2016 John Wiley & Sons, Ltd.     

The application of the generalized product‐type method based on Bi‐CG to accelerate the sparse‐matrix/canonical grid method

When the sparse‐matrix/canonical grid (SMCG) method is applied to analyse scattering of randomly positioned dielectric spheroids, the impedance matrix is decomposed into a strong interaction matrix and a weak interaction matrix. The strong interaction portion of the matrix–vector multiplication is computed directly as the moment method (MOM). The far‐interaction portion of matrix–vector multiplication is computed indirectly using fast Fourier transforms by a Taylor series expansion of impedance matrix elements about the canonical grid point. However, the condition number of the impedance matrix obtained from the SMCG method becomes large compared to the one from MOM. As a result, the conjugate gradient (CG) method converges slowly. To attack such a trouble, the generalized product‐type method based on Bi‐CG (GPBi‐CG) is used as an iterative solver in this paper. The numerical results show that the GPBi‐CG method can achieve good convergence improvement compared to the other iterative methods. Copyright © 2005 John Wiley & Sons, Ltd.     

An interface‐enriched generalized finite element analysis for electromagnetic problems with non‐conformal discretizations

An interface‐enriched generalized finite element method is presented for analyzing electromagnetic problems involving highly inhomogeneous materials. To avoid creating conformal meshes within a complex computational domain and preparing multiple meshes during optimization, enriched vector basis functions are introduced over the finite elements that intersect the material interfaces to capture the normal derivative discontinuity of the tangential field component. These enrichment functions are directly constructed from a linear combination of the vector basis functions of the sub‐elements. Several numerical examples are presented to verify the method with analytical solutions and demonstrate its h‐refinement convergence rate. The proposed interface‐enriched generalized finite element method is shown to achieve the same level of accuracy as the standard finite element method based on conformal meshes. Two examples, involving multiple microvascular channels and circular inclusions of different radii, are analyzed to illustrate the capability of the proposed approach in handling complicated inhomogeneous geometries. Copyright © 2015 John Wiley & Sons, Ltd.     

Online concurrent reinforcement learning algorithm to solve two‐player zero‐sum games for partially unknown nonlinear continuous‐time systems

Online adaptive optimal control methods based on reinforcement learning algorithms typically need to check for the persistence of excitation condition, which is necessary to be known a priori for convergence of the algorithm. However, this condition is often infeasible to implement or monitor online. This paper proposes an online concurrent reinforcement learning algorithm (CRLA) based on neural networks (NNs) to solve the H _∞ control problem of partially unknown continuous‐time systems, in which the need for persistence of excitation condition is relaxed by using the idea of concurrent learning. First, H _∞ control problem is formulated as a two‐player zero‐sum game, and then, online CRLA is employed to obtain the approximation of the optimal value and the Nash equilibrium of the game. The proposed algorithm is implemented on actor–critic–disturbance NN approximator structure to obtain the solution of the Hamilton–Jacobi–Isaacs equation online forward in time. During the implementation of the algorithm, the control input that acts as one player attempts to make the optimal control while the other player, that is, disturbance, tries to make the worst‐case possible disturbance. Novel update laws are derived for adaptation of the critic and actor NN weights. The stability of the closed‐loop system is guaranteed using Lyapunov technique, and the convergence to the Nash solution of the game is obtained. Simulation results show the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Predictor‐based iterative neural dynamic surface control for three‐phase voltage source PWM rectifier

This paper addresses the control problem of a three‐phase voltage source pulse width modulation rectifier in the presence of parametric uncertainties and external time‐varying disturbances. An adaptive controller is designed by combining a modified dynamic surface control method and a predictor‐based iterative neural network control algorithm. Especially, neural networks with iterative update laws based on prediction errors are employed to identify the lumped uncertainties. Besides, a finite‐time‐convergent differentiator, instead of a first‐order filter, is used to obtain the time derivative of the virtual control law. Using a Lyapunov–Krasovskii functional, it is proved that all signals in the closed‐loop system are ultimately uniformly bounded. Both simulation and experimental studies are provided to show the effectiveness of the proposed approach. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.     

A matrix‐exponential decomposition‐based time‐domain method for band structure calculation of one‐dimensional periodic structures

A time‐domain method for calculating the band structure of one‐dimensional periodic structures is proposed. During the time‐stepping of the method, the column vector containing the spatially sampled field data is updated by multiplying with an iteration matrix. The iteration matrix is first obtained by using the matrix‐exponential decomposition technique. Then, the small nonzero elements of the matrix are pruned to improve its sparse structure, so that the efficiency of the matrix–vector multiplication involved in each time‐step is enhanced. The numerical results show that the method is conditionally stable but is much more stable than the conventional finite‐difference time‐domain (FDTD) method. The time‐step with which the method runs stably can be much larger than the Courant–Friedrichs–Lewy (CFL) limit. And moreover, the method is found to be particularly efficient for the band structure calculation of large‐scale structures containing a defect with a very high wave speed, where the conventional FDTD method may generally lose its efficiency severely. For this kind of structures, not only the stability requirement can be significantly relaxed, but also the matrix‐pruning operation can be very effectively performed. In the numerical experiments for large‐scale quasi‐periodic phononic crystal structures containing a defect layer, significantly higher efficiency than the conventional FDTD method can be achieved by the proposed method without an evident accuracy deterioration if the wave speed of the defect layer is relatively high. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive neural network output feedback stabilization of nonlinear non‐minimum phase systems

This paper presents an adaptive output feedback stabilization method based on neural networks (NNs) for nonlinear non‐minimum phase systems. The proposed controller comprises a linear, a neuro‐adaptive, and an adaptive robustifying parts. The NN is designed to approximate the matched uncertainties of the system. The inputs of the NN are the tapped delays of the system input–output signals. In addition, an appropriate reference signal is proposed to compensate the unmatched uncertainties inherent in the internal system dynamics. The adaptation laws for the NN weights and adaptive gains are obtained using Lyapunov's direct method. These adaptation laws employ a linear observer of system dynamics that is realizable. The ultimate boundedness of the error signals are analytically shown using Lyapunov's method. The effectiveness of the proposed scheme is shown by applying to a translation oscillator rotational actuator model. Copyright © 2009 John Wiley & Sons, Ltd.     

Fast bias‐constrained optimal FIR filtering for time‐invariant state space models

This paper combines the finite impulse response filtering with the Kalman structure (predictor/corrector) and proposes a fast iterative bias‐constrained optimal finite impulse response filtering algorithm for linear discrete time‐invariant models. In order to provide filtering without any requirement of the initial state, the property of unbiasedness is employed. We first derive the optimal finite impulse response filter constrained by unbiasedness in the batch form and then find its fast iterative form for finite‐horizon and full‐horizon computations. The corresponding mean square error is also given in the batch and iterative forms. Extensive simulations are provided to investigate the trade‐off with the Kalman filter. We show that the proposed algorithm has much higher immunity against errors in the noise covariances and better robustness against temporary model uncertainties. The full‐horizon filter operates almost as fast as the Kalman filter, and its estimate converges with time to the Kalman estimate. Copyright © 2016 John Wiley & Sons, Ltd.     

Modeling of ultra‐wideband indoor channels with the modified leapfrog ADI‐FDTD method

Full‐wave time‐domain electromagnetic methods are usually effective in rigorously modeling and evaluating ultra‐wideband (UWB) wireless channels. However, their computational expenditures are expensive, when they are used to deal with electrically large‐size problems consisting of fine structures. In order to reduce computational time, the unconditionally stable leapfrog alternating‐direction implicit finite‐difference time‐domain (leapfrog ADI‐FDTD) method has been proposed recently. In this paper, the leapfrog ADI‐FDTD algorithm is developed for simulating lossy objects, such as office walls, floors, and ceilings, for UWB communication channel characterization. It leads to effective UWB channel characterization with power‐decay time constant, path loss exponent, and probability distribution of power gain. In comparison with the conventional FDTD, the proposed method can achieve 60% saving in computational time while retaining good accuracy. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of the short‐open calibration technique to vector finite element method for analysis of microwave circuits

A novel numerical de‐embedding scheme called the short‐open calibration (SOC) technique, in conjunction with the vector finite element method (FEM), has been developed to characterize two‐port network of arbitrarily shaped, three‐dimensional discontinuities in microwave circuits. This SOC technique is effectively implemented into the FEM for boundary truncation of the unbounded circuit structures. In such a manner, fast convergence of iterative solver for large‐sparse linear matrix equations from FEM is achieved. The SOC technique is used to remove parasitic effects brought by the approximation of the impressed voltage source and also the problem of resulting consistency between the two‐ and three‐dimensional simulations. Scattering parameters of discontinuous sections are constructed from the definition of port voltages and currents. Numerical solutions are well compared with those published in the available literatures. It is demonstrated that the features of the SOC technique are advantageous when combined with FEM for electromagnetic problems. Copyright © 2002 John Wiley & Sons, Ltd.     

Three‐dimensional finite element eddy current analysis by using high‐order vector elements

Sets of high‐order basis functions of a tetrahedral element are systematically constructed and applied to finite element analysis of eddy current problems. A polynomial space is divided into a lot of subspaces assigned on the edges, faces, and a volume of the tetrahedral element. Lagrange‐type vector basis functions of the subspaces are presented. The effect of the high‐order vector elements is investigated by a cubic conductor model located in AC steady‐state magnetic fields. In the calculations using the fundamental and second‐order elements, no convergent value of the eddy current power loss can be obtained in spite of fine meshes because the eddy current shifts to the surface of the conductor. The higher‐order vector elements give the convergent solutions in the coarse meshes. © 2004 Wiley Periodicals, Inc. Electr Eng Jpn, 147(4): 60–67, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.10306     

Modeling of doubly lossy and dispersive media with the time‐domain finite‐element dual‐field domain‐decomposition algorithm

A numerical scheme is presented for the time‐domain finite‐element modeling of an electrically and magnetically lossy and dispersive medium in the dual‐field domain‐decomposition method. Existing approaches for modeling doubly lossy and dispersive media are extended to the dual‐field case, yielding a general dual‐field domain‐decomposition scheme for modeling large‐scale electromagnetic problems involving such media. A quantitative analysis is performed to estimate the error induced by the modeling of medium dispersion. Copyright © 2012 John Wiley & Sons, Ltd.     

Modelling of lossy curved surfaces in the 3‐D frequency‐domain finite‐difference methods

A conformal first‐order or Leontovic surface‐impedance boundary condition (SIBC) for the modelling of fully three‐dimensional (3‐D) lossy curved surfaces in a Cartesian grid is presented for the frequency‐domain finite‐difference (FD) methods. The impedance boundary condition is applied to auxiliary tangential electric and magnetic field components defined at the curved surface. The auxiliary components are subsequently eliminated from the formulation resulting in a modification of the local permeability value at boundary cells, allowing the curved 3‐D surface to be described in terms of Cartesian grid components. The proposed formulation can be applied to model skin‐effect loss in time‐harmonic driven problems. In addition, the impedance matrix can be used as a post‐processor for the eigenmode solver to calculate the wall loss. The validity of the proposed model is evaluated by investigating the quality factors of cylindrical and spherical cavity resonators. The results are compared with analytic solutions and numerical reference data calculated with the commercial software package CST Microwave Studio™ (MWS). The convergence rate of the results is shown to be of second‐order for smooth curved metal surfaces. The overall accuracy of the approach is comparable to that of CST MWS™. Copyright © 2006 John Wiley & Sons, Ltd.     

Precise position control of an ultrasonic motor using the PID controller combined with NN

The input‐output characteristic of an ultrasonic motor (USM) has nonlinear elements and changes with a temperature rise or fluctuation of load‐mass. Therefore, it is difficult to accomplish satisfactory control performance by using conventional PID controllers. In this paper, we propose a PID controller combined with a neural network (NN‐PID controller). In this design method, the controller gains consist of constant gains of the PID controller and variable gains of the NN controller. The weights of the NN are adjusted by the backpropagation method so that the control error can be minimized. This method does not require long learning time of the NN. The effectiveness of the proposed design method is confirmed by experiments using an existing USM. © 2003 Wiley Periodicals, Inc. Electr Eng Jpn, 146(3): 46–54, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.10199     

Neural network adaptive robust control with application to precision motion control of linear motors

In this paper, neural networks (NNs) and adaptive robust control (ARC) design philosophy are integrated to design performance‐oriented control laws for a class of single‐input–single‐output (SISO) nth‐order non‐ linear systems. Both repeatable (or state dependent) unknown non‐linearities and non‐repeatable unknown non‐linearities such as external disturbances are considered. In addition, unknown non‐linearities can exist in the control input channel as well. All unknown but repeatable non‐linear functions are approximated by outputs of multi‐layer neural networks to achieve a better model compensation for an improved performance. All NN weights are tuned on‐line with no prior training needed. In order to avoid the possible divergence of the on‐line tuning of neural network, discontinuous projection method with fictitious bounds is used in the NN weight adjusting laws to make sure that all NN weights are tuned within a prescribed range. By doing so, even in the presence of approximation error and non‐repeatable non‐linearities such as disturbances, a controlled learning is achieved and the possible destabilizing effect of on‐line tuning of NN weights is avoided. Certain robust control terms are constructed to attenuate various model uncertainties effectively for a guaranteed output tracking transient performance and a guaranteed final tracking accuracy in general. In addition, if the unknown repeatable model uncertainties are in the functional range of the neural networks and the ideal weights fall within the prescribed range, asymptotic output tracking is also achieved to retain the perfect learning capability of neural networks in the ideal situation. The proposed neural network adaptive control (NNARC) strategy is then applied to the precision motion control of a linear motor drive system to help to realize the high‐performance potential of such a drive technology. NN is employed to compensate for the effects of the lumped unknown non‐linearities due to the position dependent friction and electro‐magnetic ripple forces. Comparative experiments verify the high‐performance nature of the proposed NNARC. With an encoder resolution of 1 µm, for a low‐speed back‐and‐forth movement, the position tracking error is kept within ±2 µm during the most execution time while the maximum tracking error during the entire run is kept within ±5.6 µm. Copyright © 2001 John Wiley & Sons, Ltd.     

Electromagnetic radiation from vertical dipole antennas near air–lossy soil interface—a finite‐difference time‐domain simulation

Radiation from vertical dipole antennas, which are located over or under the surface of lossy earth, is analysed by the finite‐difference time‐domain (FDTD) method in cylindrical coordinates. A novel generalized perfectly matched layer (PML) has been developed and used for the truncation of the lossy soil. In order to decrease the memory requirements and for having an accurate modelling, an efficient ‘non‐uniform’ mesh generation scheme is used. The excitation is considered in the form of sine carrier modulated by Gaussian pulse (SCMGP) and in each time step, computation is limited to that part of the mesh where the radiated pulse is passing (computational window). This could considerably reduce the required CPU time. In this manner, large‐scale problems can be solved and the values of radiated field at far distances (up to 500λ₀ in this work) can be obtained directly by the FDTD method. The frequency‐domain results are calculated from the obtained time‐domain results by taking the Fourier transform. The spatial distributions of the amplitude and phase of radiated field are shown in illustrations for different types of soil and different positions of antenna. The influence of the lossy soil on dipole's admittance is also shown. Copyright © 2005 John Wiley & Sons, Ltd.     

Indirect adaptive control of nonlinear system via dynamic multilayer neural networks with multi‐time scales

This paper deals with adaptive nonlinear identification and trajectory tracking problem via dynamic multilayer neural network with different time scales. By means of a Lyapunov‐like analysis, we determine stability conditions for the on‐line identification. Then, a sliding mode controller is designed for trajectory tracking with consideration of the modeling error and disturbance. The main contributions of the paper lie in the following aspects. First, we extend our prior identification results of single‐layer dynamic neural networks with multi‐time scales to those of multilayer case. Second, the e‐modification in standard use in adaptive control is introduced in the on‐line update laws to guarantee bounded weights and bounded identification errors. Third, the potential singularity problem in controller design is solved by using new update laws for the NN weights so that the control signal is guaranteed bounded. The stability of proposed controller is proved by using Lyapunov function. Simulation results demonstrate the effectiveness of the proposed algorithm. Copyright © 2014 John Wiley & Sons, Ltd.     

A small‐gain approach for adaptive output‐feedback NN control of switched pure‐feedback nonlinear systems

This paper investigates the problem of adaptive output‐feedback neural network (NN) control for a class of switched pure‐feedback uncertain nonlinear systems. A switched observer is first constructed to estimate the unmeasurable states. Next, with the help of an NN to approximate the unknown nonlinear terms, a switched small‐gain technique‐based adaptive output‐feedback NN control scheme is developed by exploiting the backstepping recursive design scheme, input‐to‐state stability analysis, the common Lyapunov function method, and the average dwell time (ADT) method. In the recursive design, the difficulty of constructing an overall Lyapunov function for the switched closed‐loop system is dealt with by decomposing the switched closed‐loop system into two interconnected switched systems and constructing two Lyapunov functions for two interconnected switched systems, respectively. The proposed controllers for individual subsystems guarantee that all signals in the closed‐loop system are semiglobally, uniformly, and ultimately bounded under a class of switching signals with ADT, and finally, two examples illustrate the effectiveness of theoretical results, which include a switched RLC circuit system.     

Modeling of non‐linear systems by FWNNs and their intelligent control

By utilizing some of the important properties of wavelets like denoising, compression, multiresolution along with the concepts of fuzzy logic and neural network, two fuzzy wavelet neural networks (FWNNs) are proposed for approximating any arbitrary non‐linear function, hence, identifying a non‐linear system. We have fuzzified the output of DWT block, which receives the given inputs, in the proposed two methods: one using compression property and other using multiresolution property. We present a new type of fuzzy neuron model, each non‐linear synapse of which is characterized by a set of fuzzy implication rules with singleton weights in their consequents. It is shown that noise and disturbance in the reference signal are reduced with wavelets and also the variation of somatic gain, the parameter that controls the slope of the activation function in the neural network, leads to more accurate output. Identification results are found to be accurate and speed of their convergence is fast. Next, we simulate a control system for keeping output at a desired level by using the identified models. Two self‐learning controllers are designed in this simulation. One is a self‐learning fuzzy PI controller and other is a NN controller. Simulation results show that the NN controller is more adaptive and fast. Copyright © 2005 John Wiley & Sons, Ltd.