【作者】 王维；

【导师】 丁建完；

【作者基本信息】 华中科技大学 ， 机械电子工程， 2017， 硕士

【摘要】 随着工业的发展和科技的进步,集机、电、液、控等于一体的复杂物理系统越来越常见。这些复杂的多领域系统模型所对应的数学描述往往是大规模的微分代数方程。求解微分方程是微分代数方程求解的基础。对于复杂的微分方程系统通过方程变换和符号积分求其解析解基本不可能,工程上一般求其数值解。研究微分方程的数值解意义重大。相较于时间离散的微分方程数值求解方法,量化状态系统(Quantized State System,QSS)方法具有许多优点。基于此,本文对一阶、二阶QSS方法进行了研究。具体内容如下:首先,阐述QSS方法求解的原理包括:量化过程、四种内部转换和外部转换顺序、仿真推进和输入的量化原则,总结QSS方法求解微分方程的一般过程。然后,给出QSS1和QSS2方法的时间推进函数的Modelica实现,结合具体的微分方程实例,采用文本建模的方式给出QSS1、QSS2方法求解的Modelica模型,并对建立的模型进行了仿真,分析部分结果曲线,表明算法的有效性。基于QSS方法求解微分方程的一般过程,总结建立QSS方法求解任意规模、连续常微分方程的Modelica模型一般方法。最后,基于线性时不变系统,采用扰动理论推导QSS方法求解的误差上界,分析QSS求解的快速频繁震荡特性。探讨QSS1和QSS2方法具有相同误差上界的原因以及QSS方法误差上界的保守性,同时对比时间离散的数值积分方法,分析QSS方法的稀疏利用特性。阐述QSS方法求解刚性系统的局限性,同时通过与Euler方法对比指出QSS1方法在求解刚性系统方面仍具有优势。上述研究成果展示了QSS方法的有效性,并能为相关平台的开发提供技术指导。更多还原

【Abstract】 With the development of industry,science and technology,the complex physical systems that integrate mechanical,electric,hydraulic and control are increasingly common.The mathematical descriptions of these complex models of multi-domain systems are often largescale differential algebraic equations.Solving differential equations is the basis for solving differential algebraic equations.It is almost impossible to obtain exact solution of the complex differential equation system by means of equation transformation and symbolic integration.Thus,numerical solution of the differential equation instead of its analytical solution is preferred in engineering.It is significant to study the numerical solution of differential equations.Compared with the time-discrete numerical solution of ordinary differential equations,Quantized State System methods enjoy many advantages.This paper is devoted to studying the first and second order QSS methods.Specific contents are described as follows:Firstly,the theory of QSS methods,including process of quantization,order of internal and external transformations,the principle of simulation forward,and quantization of input functions,is described.The general process of QSS methods for solving ordinary differential equations,is summarized.Secondly,the Modelica implementation of the time push function of the QSS1 and QSS2 methods is given.the Modelica model of QSS1 and QSS2 methods for solving exact equations,is given and simulated and some partial curves are analyzed.Based on the general process of solving ordinary differential equations by QSS method,this paper summarizes the general method of establishing Modelica models of QSS methods for solving arbitrary continuous differential equations.Finally,based on the linear time-invariant system,the upper bound of the error of QSS methods,is deduced by using the perturbation theory,and the property of fast frequent oscillatory of QSS is analyzed.The reasons why the QSS1 and QSS2 methods have the same upper bound of the error and the conservativeness of the upper bound of the error,are discussed.At the same time,the sparse characteristic of QSS method,is analyzed.In this paper,limitations of QSS methods for solving stiff systems,is expounded,and by comparing Euler,the advantages of QSS1 for solving stiff systems are stailed.The above-mentioned research results show the effectiveness of the QSS methods,thus providing technical guidance for the development of the relevant platform.更多还原