不同空间域下扩散反应方程的边界控制

发布时间：2018-01-01 14:11

本文关键词：不同空间域下扩散反应方程的边界控制　出处：《东华大学》2017年硕士论文　论文类型：学位论文

【摘要】：在物理学、化学、生物学等各种工程领域中存在着大量的扩散反应现象,扩散反应方程(又称热方程)是描述这些工程领域中各种扩散反应现象的数学模型。扩散反应方程的边界控制问题在物理学、生物学和工程实践中有着广泛的应用前景,比如温度控制、湿纺碳纤维成形过程和化学反应过程等,因此备受广大研究人员的关注。扩散反应方程的边界控制的主要目标是通过赋予边界控制律,使得不同空间域下的系统能够在控制律的作用下最终稳定至平衡态。本文主要系统研究了三种空间域下扩散反应方程的边界控制问题,针对具体的扩散反应系统,建立相应的PDE模型。在Backstepping控制算法的基础上,引入Volterra积分映射进行可逆变换,建立原系统和稳定的目标系统的等价关系,将原系统转化为指数稳定的目标系统,利用其等价性,获得核函数方程,通过边界控制器和核函数的关系,得到边界控制器的精确解,最后基于Lyapunov稳定性理论,证明了系统的稳定性。首先,研究了n维球对称空间域下扩散反应方程的边界控制问题,因为许多实际系统可用n维超球坐标系来描述,且系统具有球对称的性质,所以可通过研究半径方向的状态变化,得到系统的全局动态过程,通过将高维的对称系统转化为等价的径向一维方程,运用Backstepping方法设计边界控制器,利用容易测量的边界状态值,设计了状态观测器来估计系统在空间域的所有状态,从而实现输出反馈控制。同时,利用Lyapunov方法,证明了施加反馈控制的闭环系统具有H1范数下的指数稳定性。其次,本文还研究了圆柱面空间域下二维扩散反应方程的边界控制问题,圆柱的底面是一个圆形区域,而圆柱面的边界就是其表面区域。状态变量关于圆心旋转对称,采用傅里叶级数展开法,二维系统可转化为等价的一维抛物方程,实现了系统的降维。设计稳定的目标系统,引入Volterra可逆积分映射,建立原系统到目标系统的等价关系,利用原系统与目标系统的等价性,获得核函数方程,最后利用边界控制器和核函数的关系,得到边界控制器的精确解。本文还设计了边界观测器,通过边界微分的测量值,去估计整个系统在圆柱面空间域下的所有状态值,实现全状态反馈。最后,本文研究了输入时滞系统扩散反应方程的边界控制问题,特别对反应系数依赖于空间变量的时滞系统,通过引入运输方程,和扩散反应方程系统构成级联系统,把原系统中的时滞项转移到运输方程系统中,利用Volterra可逆积分映射,建立了级联系统和稳定的目标系统的等价关系。利用等价关系获得三个核函数方程,运用分离变量法分别求解出各个核函数的解,根据边界控制器和核函数的关系,最终得到边界控制器的解。本文运用有限差分法进行数值仿真,验证了边界控制器能使开环不稳定的系统迅速收敛到稳态值。
[Abstract]:There are a lot of diffusion reactions in physics, chemistry, biology and other engineering fields. Diffusion reaction equation (also called thermal equation) is a mathematical model to describe various diffusion reaction phenomena in these engineering fields. The boundary control problem of diffusion reaction equation is in physics. Biological and engineering practices have a wide range of applications, such as temperature control, wet spinning carbon fiber forming process and chemical reaction process. The main goal of the boundary control of diffusion reaction equation is to assign the boundary control law. The system in different space domain can be stabilized to equilibrium state under the action of control law. In this paper, the boundary control problem of diffusion reaction equation in three kinds of space domain is studied systematically. For the specific diffusion reaction system, the corresponding PDE model is established. Based on the Backstepping control algorithm, the Volterra integral mapping is introduced for reversible transformation. The equivalent relation between the original system and the stable target system is established, and the original system is transformed into the exponential stable target system. By using its equivalence, the kernel function equation is obtained, and the relation between the boundary controller and the kernel function is obtained. The exact solution of the boundary controller is obtained, and the stability of the system is proved based on Lyapunov stability theory. Firstly, the boundary control problem of diffusion reaction equation in n-dimensional spherical symmetric space is studied. Because many practical systems can be described in n-dimensional hypersphere coordinate system, and the system has the property of spherical symmetry, the global dynamic process of the system can be obtained by studying the state changes in the radius direction. By transforming the high-dimensional symmetric system into an equivalent radial one-dimensional equation, a boundary controller is designed by using the Backstepping method, and the boundary state value is easily measured. A state observer is designed to estimate all the states of the system in spatial domain, and the output feedback control is realized. At the same time, the Lyapunov method is used. It is proved that the closed-loop system with feedback control has exponential stability under H _ 1-norm. Secondly, the boundary control problem of two-dimensional diffusion reaction equations in cylindrical space is also studied. The bottom surface of a cylinder is a circular region and the boundary of a cylinder is its surface. The state variable is symmetric about the center of the circle and the Fourier series expansion method is used to transform the two-dimensional system into an equivalent one-dimensional parabolic equation. The dimensionality reduction of the system, the design of a stable target system, the introduction of Volterra reversible integral mapping, the establishment of the equivalent relationship between the original system and the target system, and the use of the equivalence between the original system and the target system. The kernel function equation is obtained and the exact solution of the boundary controller is obtained by using the relation between the boundary controller and the kernel function. The boundary observer is also designed and measured by the boundary differential. To estimate all the state values of the whole system in the cylindrical space domain, and realize the full state feedback. Finally, the boundary control problem of the diffusion reaction equation of the input time-delay system is studied in this paper. Especially for time-delay systems whose response coefficients are dependent on spatial variables, transport equations and diffusion reaction equations are introduced to form cascaded systems, and the time-delay terms in the original system are transferred to the transport equation system. The equivalence relation between cascade system and stable target system is established by using the Volterra reversible integral mapping, and three kernel function equations are obtained by using the equivalence relation. According to the relationship between the boundary controller and the kernel function, the solution of the boundary controller is obtained. The finite difference method is used to carry out the numerical simulation in this paper. It is proved that the boundary controller can make the open loop unstable system converge rapidly to the steady state.
【学位授予单位】：东华大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O231

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