A new forecasting method for time continuous model of dynamic system

Usually a linear differential equation is used to represent continuous dynamic systems, but a linear difference equation is used to represent discrete dynamic systems. AGO is one of the most important characteristics of grey theory, and its main purpose is to reduce the randomness of data. A linear differential equation, instead of a linear difference equation, is used to replace the grey differential equation to analyze discrete systems in this paper. Approximating a k-order derivative by operating after spline curve fitting of 1-AGO data, a model is directly established by means of the least square method. ARIMA models are used to analyze the leading indicator in advance, and the Fourier series with suitably chosen values of parameters is used in the fitting of leading indicator. This model is called the GDM(2, 2, 1) model.     

Simple, robust, selftuning controller

This paper proposes a robust method for automatic tuning of parameters of a discrete PID controller. The tuning rules for SISO and MIMO systems are based on automatic determination of critical gain and critical frequency from the estimated model parameters. The plant model can be expressed by a transfer function in continuous and/or discrete form or by differential and/or difference equation. A simple control law using Takahashi discrete form is proposed. Simulations results prove that it is easy to use being able to handle minimum and nonminimum phase plant as well.     

灰色GM(1,1,k,k2)模型背景值及时间响应函数优化

本文提出了一种新的带有时间幂次项的灰色GM(1,1,k,k²)模型,给出了其灰微分方程和白化微分方程基本形式。基于最小二乘法获得了该模型参数估计值,并推导了该模型时间响应函数。鉴于GM(1,1,k,k²)模型灰微分方程与白化微分方程之间存在跳跃关系,首先对灰微分方程的背景值进行了优化,并推导了优化后的背景值计算公式。为了克服初始值的影响,根据误差平方和最小,进一步优化了GM(1,1,k,k²)模型时间响应函数。最后,该优化后的GM(1,1,k,k²)模型被应用于软土地基沉降预测,获得了较好的模拟预测效果,说明模型是可行的。     

Working with differential,functional and difference equations using functional networks

In this paper we first analyze the problem of equivalence of differential, functional and difference equations and give methods to move between them. We also introduce functional networks, a powerful alternative to neural networks, which allow neural functions to be different, multidimensional, multiargument and constrained by link connections, and use them for predicting values of magnitudes satisfying differential, functional and/or difference equations, and for obtaining the difference and differential equation associated with a set of data. The estimation of the differential or difference equation coefficients is done by simply solving systems of linear equations, in the cases of equally or unequally spaced or missing data points. Some examples of applications are given to illustrate the method.     

Stability criteria for differential equations with variable time delays

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay differential equations arising in population dynamics may serve as good teaching examples for these methods. The determination of linear stability properties for an ordinary differential equation with a varying time delay is carried out through discrete point analysis, either by seeking explicit solutions or leading to the consideration of a difference equation and the roots of a characteristic polynomial. Numerical simulations carried out using MATLAB Simulink are compared to the analytical solutions, and computation is also used to suggest extensions to some results.     

SEM Modeling with Singular Moment Matrices Part II: ML-Estimation of Sampled Stochastic Differential Equations

Linear stochastic differential equations are expressed as an exact discrete model (EDM) and estimated with structural equation models (SEMs) and the Kalman filter (KF) algorithm. The oversampling approach is introduced in order to formulate the EDM on a time grid which is finer than the sampling intervals. This leads to a simple computation of the nonlinear parameter functionals of the EDM. For small discretization intervals, the functionals can be linearized, and standard software permitting only linear parameter restrictions can be used. However, in this case the SEM approach must handle large matrices leading to degraded performance and possible numerical problems. The methods are compared using coupled linear random oscillators with time-varying parameters and irregular sampling times.     

捕食模型高精度参数的估计

研究了捕食者模型在多种观测值条件下的非线性微分方程组参数拟合问题.首先利用龙格-库塔法进行微分方程数值计算,通过首次积分项变形建立线性回归方程,进行最小二乘拟合;其次,考虑到实验数据包含随机误差的扰动,引进正规方程组对模型进行误差分析;最后针对时间变量也出现误差,采用拉依达准则筛选,然后提出了一种较为简单的参数分段动态估计算法.     

Mathieu equation with application to analysis of dynamic characteristics of resonant inertial sensors

In this paper, Mathieu equation is applied to analyze the dynamic characteristics of resonant inertial sensors. Unlike previous work, Mathieu equation is not just a differential equation and analyzes the stability of the transition curves, but become an important method in analyzing parametric resonant characteristics and approximate output of resonant inertial sensors. It is demonstrated that the mathematical model of resonant inertial sensors is described by Mathieu equation. The relevant Mathieu equation theory and dynamic characteristics analysis methods were proposed, which include both stability and dynamic linear output. Finally, theoretical and experimental analysis show that the correlation of the theoretical curve and the experimental result coincide so perfectly, which means proposed analysis methods for Mathieu equation could be used to analyze the dynamic output characteristic of resonant inertial sensors. The theoretical analyzing approach of Mathieu equation and experimental results of resonant inertial sensors are obtained, which provide an application area for Mathieu equation and a reference for the robust design for resonant inertial sensors.     

Problems with the estimation of stochastic differential equations using structural equations models

The present paper deals with the identification and maximum likelihood estimation of systems of linear stochastic differential equations using panel data. So we only have a sample of discrete observations over time of the relevant variables for each individual. A popular approach in the social sciences advocates the estimation of the “exact discrete model” after a reparameterization with LISREL or similar programs for structural equations models. The “exact discrete model” corresponds to the continuous time model in the sense that observations at equidistant points in time that are generated by the latter system also satisfy the former. In the LISREL approach the reparameterized discrete time model is estimated first without taking into account the nonlinear mapping from the continuous to the discrete time parameters. In a second step, using the inverse mapping, the fundamental system parameters of the continuous time system in which we are interested, are inferred. However, some severe problems arise with this “indirect approach”. First, an identification problem may arise in multiple equation systems, since the matrix exponential function denning some of the new parameters is in general not one‐to‐one, and hence the inverse mapping mentioned above does not exist. Second, usually some sort of approximation of the time paths of the exogenous variables is necessary before the structural parameters of the system can be estimated with discrete data. Two simple approximation methods are discussed. In both approximation methods the resulting new discrete time parameters are connected in a complicated way. So estimating the reparameterized discrete model by OLS without restrictions does not yield maximum likelihood estimates of the desired continuous time parameters as claimed by some authors. Third, a further limitation of estimating the reparameterized model with programs for structural equations models is that even simple restrictions on the original fundamental parameters of the continuous time system cannot be dealt with. This issue is also discussed in some detail. For these reasons the “indirect method” cannot be recommended. In many cases the approach leads to misleading inferences. We strongly advocate the direct estimation of the continuous time parameters. This approach is more involved, because the exact discrete model is nonlinear in the original parameters. A computer program by Hermann Singer that provides appropriate maximum likelihood estimates is described.     

ON LINEARIZED FINITE DIFFERENCE SIMULATION FOR THE MODEL OF NUCLEAR REACTOR DYNAMICS

1 hoeductIOuThe dynamics models of one--dimensional continuous medium nuclear reactor are the foelowing initial--boundary value problem of the formsubject to the innal conditionsand the boundary conditionsIn (1. 1), x denotes position along the reactor, which is regarded as a rod of length L, t denotes the time, u(t) the logarithm of the loud reactor POwer, v(x,t) the deviation of the temperature from equilibrium, a(x) the ratio of the temperature coefficient of reactivity to theynean life of…     

On Discrete Painlevé Equations Associated with the Lattice KdV Systems and the Painlevé VI Equation

A new integrable nonautonomous nonlinear ordinary difference equation is presented that can be considered to be a discrete analogue of the Painlevé V equation. Its derivation is based on the similarity reduction on the two-dimensional lattice of integrable partial differential equations of Korteweg–de Vries (KdV) type. The new equation, which is referred to as generalized discrete Painlevé equation (GDP), contains various "discrete Painlevé equations" as subcases for special values/limits of the parameters, some of which have already been given in the literature. The general solution of the GDP can be expressed in terms of Painlevé VI (PVI) transcendents. In fact, continuous PVI emerges as the equation obeyed by the solutions of the discrete equation in terms of the lattice parameters rather than the lattice variables that label the lattice sites. We show that the bilinear form of PVI is embedded naturally in the lattice systems leading to the GDP. Further results include the establishment of Bäcklund and Schlesinger transformations for the GDP, the corresponding isomonodromic deformation problem, and the self-duality of its bilinear scheme.     

Quasi-interpolation for linear functional data

Quasi-interpolation has been studied extensively in the literature. However, most studies of quasi-interpolation are usually only for discrete function values (or a finite linear combination of discrete function values). Note that in practical applications, more commonly, we can sample the linear functional data (the discrete values of the right-hand side of some differential equations) rather than the discrete function values (e.g., remote sensing, seismic data, etc). Therefore, it is more meaningful to study quasi-interpolation for the linear functional data. The main result of this paper is to propose such a quasi-interpolation scheme. Error estimate of the scheme is also given in the paper. Based on the error estimate, one can find a quasi-interpolant that provides an optimal approximation order with respect to the smoothness of the right-hand side of the differential equation. The scheme can be applied in many situations such as the numerical solution of the differential equation, construction of the Lyapunov function and so on. Respective examples are presented in the end of this paper.     

A method of adjoint operators for solving boundary value problems for second-order ordinary differential equations

In this paper, for a linear boundary value problem we propose a method that reduces the differential problem to a discrete (difference) problem. The difference equations, which are an exact analog of the differential equation, are constructed by an adjoint operator method. The adjoint equations are solved by a factorization method.     

Dynamic analysis of repetitive decision-free discreteevent processes: The algebra of timed marked graphs and algorithmic issues

A model to analyze certain classes of discrete event dynamic systems is presented. Previous research on timed marked graphs is reviewed and extended. This model is useful to analyze asynchronous and repetitive production processes. In particular, applications to certain classes of flexible manufacturing systems are provided in a companion paper. Here, an algebraic representation of timed marked graphs in terms of reccurrence equations is provided. These equations are linear in a nonconventional algebra, that is described. Also, an algorithm to properly characterize the periodic behavior of repetitive production processes is descrbed. This model extends the concepts from PERT/CPM analysis to repetitive production processes.     

关于抛物型方程离散流量的收敛性

关于用差分方法求解具有间断系数的二阶抛型方程的问题, Ａ．Ｈ． ＴＮＸＯＨＯＢ与 Ａ．Ａ．Ｃａｍａｐｃ　从１９６１年山开始曾经作过详尽的研究,他们的结果都总结在专著［２］中,有关的文献也可以在该书中找到．他们指出在系数间断点处附近的网格点上格式的截断误差为Ｏ（１）,但差分格式的解在极大意义下收敛于原微分方程的连续解．他们在构造差分格式,并论证其收敛性时,充分利用了原微分方程中流连续的性质,但是却没有讨论差分格式中的离散流量的收敛性．８０年代 Ｔ．Ａ． Ｍａｎｔｅｎｆｆｅｌ, Ａ．Ｂ． Ｗｈｉｔｅ, Ｊｒ…     

Dynamic reconstruction of disturbances in stochastic differential equations

The reconstruction of the unknown deterministic disturbance in an Ito stochastic differential equation is studied using the Osipov-Kryazhimskii dynamic inversion theory. Inexact discrete observations of the current phase state are used as input data. A finite-step solving algorithm based on the method of auxiliary controllable models is proposed. Its convergence is proved, and the compatibility conditions for the parameters are given.     

干扰解耦自校正调节器

一、引言 关于确定性系统的自校正调节器理论已经有了比较成熟的结果,然而没有一种算法考虑过确定性外干扰的影响。一般说来,一个实际受控过程,除系统的参数未知外,还往往受到各种类型的确定型干扰的影响。因此,讨论带确定性干扰自校正调节器的算法,无论从理论的完整性,还是从实际应用考虑都是有意义的。为了说明确定型干扰解耦自校     

Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

This paper considers a non-linear system of ordinary differential equations, which arises in the study of hantavirus epidemics. The system has the property that the total population obeys the logistic equation. For this system, we use linearization and the dynamical properties of the logistic equation to analyze the dynamics of the subpopulation system. In view of the usual numerical instabilities associated with standard finite difference methods used for simulating such systems, we construct non-standard finite difference (NSFD) schemes, which preserve the dynamic properties of the system, and may therefore be used for its simulation.     

On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev′e Equation: an Operator-Splitting Approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevé transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painlevé ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.     

T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise

The paper deals with the T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise. A difference equation is obtained by applying the numerical method to a linear test equation, in which the Wiener increment is approximated by a discrete random variable with two-point distribution. The conditions under which the method is T-stable are considered and the numerical experiments are given.