Identification of nonlinear system using extreme learning machine based Hammerstein model

In this paper, a new method for nonlinear system identification via extreme learning machine neural network based Hammerstein model (ELM-Hammerstein) is proposed. The ELM-Hammerstein model consists of static ELM neural network followed by a linear dynamic subsystem. The identification of nonlinear system is achieved by determining the structure of ELM-Hammerstein model and estimating its parameters. Lipschitz quotient criterion is adopted to determine the structure of ELM-Hammerstein model from input–output data. A generalized ELM algorithm is proposed to estimate the parameters of ELM-Hammerstein model, where the parameters of linear dynamic part and the output weights of ELM neural network are estimated simultaneously. The proposed method can obtain more accurate identification results with less computation complexity. Three simulation examples demonstrate its effectiveness.     

Bernstein polynomial and discontinuous functions

We obtain new estimates of the rate of convergence of Bernstein operators for discontinuous functions on [0,1]$[0,1]$ which can be used to derive known results for continuous functions and functions of bounded variation.     

Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials

In this paper a numerical method for solving the nonlinear age-structured population models is presented which is based on Bernstein polynomials approximation. Operational matrices of integration, differentiation, dual and product are introduced and are utilized to reduce the age-structured population problem to the solution of algebraic equations. The method in general is easy to implement, and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities

This paper focuses on identification problems for Hammerstein systems with saturation and dead-zone nonlinearities. An appropriate switching function is introduced to derive an identification model with fewer parameters and all the unknown parameters can be estimated by using an iterative method. A numerical simulation is carried out to show the effectiveness of the proposed method.     

A new class of polynomials associated with Bernstein and beta polynomials

The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that interpolates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright © 2013 John Wiley & Sons, Ltd.     

Parameter and state estimation algorithm for single-input single-output linear systems using the canonical state space models

This paper presents a new parameter and state estimation algorithm for single-input single-output systems based on canonical state space models from the given input–output data. Difficulties of identification for state space models lie in that there exist unknown noise terms in the formation vector and unknown state variables. By means of the hierarchical identification principle, those noise terms in the information vector are replaced with the estimated residuals and a new least squares algorithm is proposed for parameter estimation and the system states are computed by using the estimated parameters. Finally, an example is provided.     

On derivatives of hypergeometric functions and classical polynomials with respect to parameters

Closed expressions are obtained for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial.     

q‐Bernstein polynomials related to q‐Frobenius–Euler polynomials,l‐functions,and q‐Stirling numbers

The aim of this paper was to derive new identities and relations associated with the q‐Bernstein polynomials, q‐Frobenius–Euler polynomials, l‐functions, and q‐Stirling numbers of the second kind. We also give some applications related to theses polynomials and numbers. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiple Meixner–Pollaczek polynomials and the six-vertex model

We study multiple orthogonal polynomials of Meixner–Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of (locally) block Toeplitz matrices, for which we provide some general results of independent interest.The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner–Pollaczek polynomials arise in an inhomogeneous version of this model.     

Properties of fractional calculus with respect to a function and Bernstein type polynomials

This paper is divided into two stages. In the first stage, we investigated a new approach for the $\psi$-Riemann–Liouville fractional integral and the Faa di Bruno formula for the $\psi$-Hilfer fractional derivative. In addition, we discussed other properties involving the $\psi$-Hilfer fractional derivative and the $\psi$-Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the $\psi (·)$ function are investigated and the $\psi$-Riemann–Liouville fractional integral and $\psi$-Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the $\psi$-Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to a function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro-differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.     

Fibre products of supersingular curves and the enumeration of irreducible polynomials with prescribed coefficients

For any positive integers $n\ge 3$, $r\ge 1$ we present formulae for the number of irreducible polynomials of degree n over the finite field ${\mathbb{F}}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ and $x^{n-3}$ are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the ${\mathbb{F}}_2$ base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period 24 in n.     

Identification of MIMO systems using radial basis function networks with hybrid learning algorithm

When a radial basis function network (RBFN) is used for identification of a nonlinear multi-input multi-output (MIMO) system, the number of hidden layer nodes, the initial parameters of the kernel, and the initial weights of the network must be determined first. For this purpose, a systematic way that integrates the support vector regression (SVR) and the least squares regression (LSR) is proposed to construct the initial structure of the RBFN. The first step of the proposed method is to determine the number of hidden layer nodes and the initial parameters of the kernel by the SVR method. Then the weights of the RBFN are determined by solving a simple minimization problem based on the concept of LSR. After initialization, an annealing robust learning algorithm (ARLA) is then applied to train the RBFN. With the proposed initialization approach, one can find that the designed RBFN has few hidden layer nodes while maintaining good performance. To show the feasibility and superiority of the annealing robust radial basis function networks (ARRBFNs) for identification of MIMO systems, several illustrative examples are included.     

ETSK模糊模型辨识及模糊控制算法

本文提出一种基于扩张原理的ＥＴＳＫ（ＥｘｔｅｎｄｅｄＴＳＫ）模型,导出了该模型的输入输出解析式,给出了辨识这种模型的方法。本文还导出了ＥＴＳＫ模型的一种等价形式——变权ＴＳＫ模型,从而将ＥＴＳＫ模型规则后件中的模糊数及其扩展运算转化为普通数的运算,使基于ＥＴＳＫ模型的模糊控制算法ＭＢＦＣ（Ｍｏｄｅｌ－ＢａｓｅｄＦｕｚｚｙＣｏｎｔｒｏｌ）易于实现。仿真辨识结果表明,ＥＴＳＫ模型的辨识效果和预报精度优于ＴＳＫ和ＬＭ模型;ＭＢＦＣ算法的控制效果优于通常模型ＰＩ控制算法     

Identification of fuzzy relation models using hierarchical fair competition-based parallel genetic algorithms and information granulation

The paper is concerned with a hybrid optimization of fuzzy inference systems based on hierarchical fair competition-based parallel genetic algorithms (HFCGA) and information granulation. The process of information granulation is realized with the aid of the C-Means clustering. HFCGA being a multi-population based parallel genetic algorithms (PGA) is exploited here to realize structure optimization and carry out parameter estimation of the fuzzy models. The HFCGA becomes helpful in the context of fuzzy models as it restricts a premature convergence encountered quite often in optimization problems. It concerns a set of parameters of the model including among others the number of input variables to be used, a specific subset of input variables, and the number of membership functions. In the hybrid optimization process, two general optimization mechanisms are explored. The structural development of the fuzzy model is realized via the HFCGA optimization and C-Means, whereas to deal with the parametric optimization we proceed with a standard least square method and the use of the HFCGA technique. A suite of comparative studies demonstrates that the proposed algorithm leads to the models whose performance is superior in comparison with some other constructs commonly used in fuzzy modeling.     

Solution of Volterra's population model via block‐pulse functions and Lagrange‐interpolating polynomials

A numerical method for solving Volterra's population model for population growth of a species in a closed system is proposed. Volterra's model is a nonlinear integro‐differential equation where the integral term represents the effects of toxin. The approach is based on hybrid function approximations. The properties of hybrid functions that consist of block‐pulse and Lagrange‐interpolating polynomials are presented. The associated operational matrices of integration and product are then utilized to reduce the solution of Volterra's model to the solution of a system of algebraic equations. The method is easy to implement and computationally very attractive. Applications are demonstrated through an illustrative example. Copyright © 2008 John Wiley & Sons, Ltd.     

农林产品质量分级的物元分析识别模型研究

论述了物元分析识别评判模型,指出该方法在农林系统识别产品质量是较好的一种方法,并用其对我国北方6省12个品种的枣果质量进行分级,为发展优质品种提供科学根据     

Improvement of random coefficient differential models of growth of anaerobic photosynthetic bacteria by combining Bayesian inference and gPC

The time evolution of microorganisms, such as bacteria, is of great interest in biology. In the article by D. Stanescu et al. [Electronic Transactions on Numerical Analysis, 34, 44–58 (2009)], a logistic model was proposed to model the growth of anaerobic photosynthetic bacteria. In the laboratory experiment, actual data for two species of bacteria were considered: Rhodobacter capsulatus and Chlorobium vibrioforme. In this paper, we suggest a new nonlinear model by assuming that the population growth rate is not proportional to the size of the bacteria population, but to the number of interactions between the microorganisms, and by taking into account the beginning of the death phase in the kinetic curve. Stanescu et al. evaluated the effect of randomness into the model coefficients by using generalized polynomial chaos (gPC) expansions, by setting arbitrary distributions without taking into account the likelihood of the data. By contrast, we utilize a Bayesian inverse approach for parameter estimation to obtain reliable posterior distributions for the random input coefficients in both the logistic and our new model. Since our new model does not possess an explicit solution, we use gPC expansions to construct the Bayesian model and to accelerate the Markov chain Monte Carlo algorithm for the Bayesian inference.     

Identification of Pipeline Inner Wall Geometry Based on the POD⁃RBF Method北大核心CSCD

Based on the proper orthogonal decomposition⁃radial basis function (POD⁃RBF), a geometric identification method for pipeline inner wall was proposed to solve the internal corrosion detection problem of natural gas and oil pipelines. In view of the static magnetic field, the simplified finite element model for the pipelines was established, and the variable⁃geometry sample library was constructed, to realize the response prediction of arbitrary geometry by the POD⁃RBF. The proposed method achieves reduced⁃order analysis and avoids repeated solution of the stiffness matrix due to the geometrical change during the identification process. Hence, it can significantly improve the computation efficiency. Finally, the grey wolf optimization (GWO) algorithm was used to optimize the objective function and avoid the calculation of the sensitivity in the process of geometry change. The numerical examples show that, the proposed method has high efficiency and accuracy in the geometric identification of the pipeline inner wall, with good stability even under introduced noises. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Dynamic analysis and system identification of an LCD glass-handling robot driven by a PMSM

In this paper, Hamilton’s principle is employed to derive Lagrange’s equations of an liquid crystal display (LCD) glass-handling robot driven by a permanent magnet synchronous motor (PMSM). The robot has three arms driven by two timing belts. The dynamic formulations can be expressed by one and four independent variables, which are named as the rigid and flexible models, respectively. In order to verify the dynamic formulation is correct, we reduce the flexible model to the rigid one under some assumptions. In this paper, we adopt the real-coded genetic algorithm (RGA) to identify all the parameters of the robot and PMSM simultaneously. It is found that the RGA can identify system parameters which are difficult to be measured in practical problems, for examples, the inductance, stator resistance, motor torque constant, damping coefficient of the motor and timing belts. In numerical simulations, vibrations due to flexibility of the timing belts are investigated for the angular displacements, speeds, accelerations of arms, and the horizontal and vertical displacements of the robot. The angular displacements of the robot arm and the translational positions of the robot end are obtained in the numerical simulations and experimental results. From their comparisons, it is demonstrated that identification results of the dynamic model with four independent variables present the better matching with experimental results of the system.