Parameter optimization of orthonormal basis functions for efficient rational approximations

In hands-free telephone or video conference application, there exists an acoustic feedback coupling between the loudspeaker and microphone in an enclosed environment, which creates the acoustic echo. FIR filters are commonly used in acoustic echo cancellers because of their simple structure. However, in this paper, the Kautz and Laguerre filter structures are shown to be more efficient echo cancellers than the FIR filters, because they can describe accurately the acoustic echo system with fewer parameters. These filters are built from their respective orthonormal Kautz and Laguerre basis functions. The proposal is motivated by some theoretical and numerical results that the time-varying acoustic echo path is basically due to its time-varying zeros and not its time-invariant acoustical poles. Therefore, the poles of the Kautz and the Laguerre filters are estimated, and can be kept fixed or updated occasionally if required. The poles are estimated by a batch Gauss-Newton algorithm. Then, the coefficients of the Kautz and Laguerre filters can be estimated by most recursive algorithms that are suitable for linear regression models, e.g., the normalized LMS algorithm. Generally, it is shown that the proposed Kautz and Laguerre filters, as the filter structures in an acoustic echo canceller, have better convergence and tracking properties than the FIR and IIR filters.

This paper presents a rational approximation method for fractional calculus operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mo>±</mml:mo><mml:mi mathvariant="normal">α</mml:mi></mml:mrow></mml:msup></mml:math>in the given frequency range and the error, which is based on the best rational approximation definition. The fractional integral operator is selected as an example to describe the construction of the rational approximation functions. An application case (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mn>0.3</mml:mn></mml:mrow></mml:msup></mml:math>) is used to illustrate the effectiveness of the proposed method. The obtained approximation function in the frequency domain is a best rational approximation function, which can further improve the accuracy of the approximation without increasing the orders. On the basis of the presented rational approximation method, a rational approximation equation of a fractional-order PID controller is obtained. Finally, the method for analyzing the optimization and frequency characteristics of the fractional-order controller is implemented to demonstrate the good frequency characteristic and best structure. The results from theoretical analysis and experimental verification show that the proposed method provides a new design idea for the effective application of the fractional-order PID controller in engineering.

The rational approximation is an important tool for establishing a dynamic analysis model in time domain for semi‐infinite water and soil. It accurately represents the interaction between the semi‐infinite medium and an engineering structure. Current research on rational approximation method focuses mainly on the establishment of time‐domain analysis models. However, the stability, accuracy, and calculation efficiency of the rational approximation model cannot be guaranteed simultaneously. Based on linear system control theory, this work decomposes the continuous‐time rational approximation (CRA) and the discrete‐time rational approximation (DRA) into a combination of first‐ and second‐order subsystems, and the stability boundaries for the identified parameters are established according to the stability conditions of each subsystem. A genetic algorithm and a sequential quadratic programming algorithm are used to construct a parameter identification method that can ensure stability in the time domain. Through parameter identification of complex frequency response functions, it is demonstrated that the method proposed in this work can guarantee the stability and accuracy simultaneously, and the calculation efficiency is also substantially improved over that of the existing methods.

Computer-based derivation, reformulation, and simulation of enzyme reaction models

Rational approximation is fundamental for practical implementation of infinite dimensional fractional-order system in the area of system and control. In this paper, a class of fractional-order single-input-single-output (SISO) system has been realized in delta-domain using two rational approximation techniques, namely continued fraction expansion (CFE) approximation and Oustaloup approximation, respectively. Both the techniques use indirect discretization method for obtaining the integer-order approximants of the ideal fractional-order SISO system in delta-domain. The frequency responses of two discretized integer-order approximants of the original fractional-order system are compared subsequently using MATLAB.

Über die kreiseleigenschaften des atomkerns

In this paper we address the identification problem of a continuous-time deterministic system. We consider that the continuous-time system can be approximated by using the orthonormal continuous basis function of Laguerre. We assume that only discrete-time measurements are available and we obtain the exact discrete-time model in terms of the continuous-time model parameters of Laguerre basis function, the sample time and the Euler-Fröbenius polynomials. The estimation problem is solved by using the Least Squares method. We illustrate the benefits of our proposal with numerical simulations.

A rational approximation of the 1s basis function: χ=(ζrn′+ a)–1 exp (–ζrn) is used with (non-) integer values of n and n′ to approximate the (1 σg)2 ground state of H2(at R= 1.4 a.u.) and He2+2(at R= 1.3 a.u.) within a CI framework and using numerical integration over the molecular orbitals. Results are compared with the usual 1s STO, 1s GTO and the non-integer-n 1s STO. The most significant improvement in energy is reached by using orbitals with non-integer values of n′.

Frequency dependent transmission line models of the traveling wave type are usually implemented via rational approximation of the characteristic admittance and the propagation function. For the propagation function, a delay needs first to be compensated before proceeding with the rational approximation. It has been shown that assumption of a lossless time delay can lead to significant loss of accuracy of the rational approximation itself. In this paper is shown a procedure suitable for approximating the magnitude of the propagation function with a minimum phase shift rational function. This enables a more efficient search for the optimal delay that gives best accuracy for a given order of the rational approximation, since the function does not need to be refitted in the delay optimization loop

This paper presents a versatile hyper-ellipsoidal basis function for function approximation in a given high dimensional space. This hyper-ellipsoidal basis function can be translated and rotated to cover the data based upon the distribution of data in a given high dimensional space. Based on this function, we propose a one-pass hyper-ellipsoidal learning algorithm for which any new incoming data can be fed for learning without involving the previously learned one. This learning algorithm is used to adjust the parameters of the versatile hyper-ellipsoidal basis function. In addition, we propose the hyper-ellipsoidal basis function (HEBF) neural network that uses the one-pass hyper-ellipsoidal neural learning algorithm. The structure of this neural network is similar to the radial basis function (RBF) neural networks. The hidden neurons in the HEBF neural network can be increased or decreased during learning process. The number of the hidden neurons in the network can be grown based on geometric growth criterion and can be reduced by merging the two hidden neurons into a new hidden neuron based on merging criterion during learning process. The merging process can be done independently without considering the learned data set.KeywordsRadial Basis FunctionHide NeuronRadial Basis Function Neural NetworkHigh Dimensional SpaceInitial WidthThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Robust identification with FIR models fails to be successful when the number of coefficients to be estimated becomes large, caused by lightly damped modes of the plant or poles very close to the unit circle. The paper presents a two-stage algorithm to obtain a low-order approximate model in frequency domain in a generalized orthonormal basis with guaranteed H/sup /spl infin// error bound for deterministic linear time-invariant stable systems, the first stage being an L/sup /spl infin// rational approximation and the second nonlinear step being an H/sup /spl infin// rational approximation.

In this paper, we present four projection-based model order reduction algorithms grounded in Laguerre functions, specifically tailored for linear systems with quadratic outputs. The first algorithm utilises a Galerkin projection strategy, where the projection matrix is construced from the Laguerre expansion coefficient vectors of the system's state impulse response. The resulting reduced-order model accurately matches selected Laguerre expansion coefficients of the original system's impulse response output. The remaining three algorithms implement Petrov-Galerkin projections combined with the low-rank square root balancing method. While these methods share an identical low-rank factor for the controllability Gramian, they differ in their selection of the observability Gramian, which is chosen from either the linearised system, the quadratic-bilinearized system, or the original linear system with quadratic outputs. A notable advantage of these algorithms is their ability to avoid solving large-scale Lyapunov equations. Instead, all low-rank factors are directly constructed from the Laguerre expansion coefficient vectors via recurrence formulas, substantially improving computational efficiency. The effectiveness and performance of the proposed algorithms are demonstrated through a series of numerical experiments.

The paper presents a modeling concept for signal paths in generalized high density interconnect (HDI) boards and packages. It is based on partitioning the complex HDI structure into generic substructures and on subsequent application of matrix rational approximation and model order reduction methods. The modeling procedure results in reduced order models that are easily applicable to SPICE system level simulations and allow for numerically effective analysis and simulation of high-speed signal behavior in modern multilayer high density passive interconnecting structures.

The continuity of the optimal value function of a parametric convex semi-infinite program is secured by a weak regularity condition that also implies the convergence of certain discretization methods for semi-infinite problems. Since each discretization level yields a parametric program, a sequence of optimal value functions occurs. The regularity condition implies that, with increasing refinement of the discretization, this sequence converges uniformly with respect to the parameter to the optimal value function corresponding to the original semi-infinite problem. Our result is applicable to the convergence analysis of numerical algorithms based on parametric programming, for example, rational approximation and computation of the eigenvalues of the Laplacian.

This paper considered the solution of special second order delay differential equations. The method was generated by interpolation and collocation techniques using a combination of power series and exponential function. The approximate basis function is interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve special second order delay differential equations. The basic properties of the methods such as order, error constants, consistency and convergence were also examined. Numerical results are presented to show that the proposed method is suitable for solving second order delay differential equations.